{
  "n_equations": 1388,
  "eigenvalues": [
    0.968750000000003,
    0.9696969696969726,
    0.9705882352941232,
    0.9920634920634899,
    0.9984639016897063
  ],
  "eigenvectors": [
    [
      -3.1627001031859965e-15,
      -3.164968871180872e-15,
      -3.173039394659518e-15,
      -3.145833878328221e-15,
      -3.1463441590937174e-15,
      -3.169031786686088e-15,
      -3.1748199647643256e-15,
      -3.197686818572771e-15,
      -3.193016444250793e-15,
      -3.1930016417418985e-15,
      -3.17565841067863e-15,
      -3.2030443217810973e-15,
      -3.1876629399281203e-15,
      -9.954301984081377e-16,
      -3.202457511905002e-15,
      -3.1738863026750836e-15,
      -3.1538372598490965e-15,
      -3.1789836227219854e-15,
      -3.2046166808158922e-15,
      -3.203939650606641e-15,
      -3.169543498113519e-15,
      -3.1911484761111402e-15,
      -3.174436184721048e-15,
      -8.844544553899976e-16,
      -8.823567910811356e-16,
      -9.052590723155702e-16,
      -8.986588244318905e-16,
      -9.01735433004006e-16,
      -9.038517198091645e-16,
      -8.96510566077379e-16,
      -9.138732919878113e-16,
      -8.833349953170442e-16,
      -9.052599712831478e-16,
      -9.070541208561697e-16,
      -8.807745828526303e-16,
      -0.17677669529663725,
      -0.17677669529663714,
      -0.17677669529663717,
      -0.17677669529663725,
      -0.17677669529663717,
      -0.17677669529663714,
      -0.17677669529663725,
      -0.17677669529663706,
      -0.17677669529663714,
      -0.17677669529663712,
      -0.17677669529663725,
      -0.17677669529663714,
      -0.17677669529663725,
      -0.17677669529663725,
      -0.17677669529663725,
      -0.17677669529663714,
      -0.17677669529663712,
      -0.17677669529663714,
      -0.17677669529663703,
      -0.17677669529663723,
      -0.17677669529663717,
      -0.17677669529663714,
      -0.17677669529663712,
      -0.17677669529663717,
      -0.17677669529663714,
      -0.17677669529663728,
      -0.1767766952966372,
      -0.17677669529663712,
      -0.17677669529663723,
      -0.176776695296637,
      -1.6397236523783057e-16,
      -1.3916529040751545e-16,
      -1.5916714260237383e-16,
      -1.5882766741321045e-16,
      -1.5039557854669116e-16,
      -1.6979931611909257e-16,
      -1.6104073591296516e-16,
      -1.4993479834354359e-16,
      -1.534647828094197e-16,
      -1.7203183122882325e-16,
      -1.6527812574974108e-16,
      -1.3895488812841826e-16,
      -1.338275675939306e-16,
      -1.6423968281664684e-16,
      -1.4619107967653036e-16,
      -1.6419051199016084e-16,
      -1.294858262885583e-16,
      -1.4415911152382937e-16,
      -1.3624603185097028e-16,
      -1.4530460159763835e-16,
      -1.0459006683591244e-14,
      -1.0461820313239665e-14,
      -1.046176758236464e-14,
      -1.0462032473665586e-14,
      -1.0467023765895351e-14,
      -1.0524544404069479e-14,
      -1.0516593557211677e-14,
      -1.049484970434378e-14,
      -1.0464821838018397e-14,
      -1.0475584978238574e-14,
      -1.0473231082533995e-14,
      -1.0495889130821368e-14,
      -1.0474742674680316e-14,
      -1.0534954870193056e-14,
      -1.0461820127480668e-14,
      -1.0484031167386423e-14,
      -1.046105690111306e-14,
      -1.0519571125778833e-14,
      -1.0503866679249078e-14,
      -1.0464193124994502e-14,
      -1.048854007408454e-14,
      -1.046289571106852e-14,
      -7.698564215765662e-16,
      -1.0483601969339556e-14,
      -1.0523689241522906e-14,
      -1.052845523100161e-14,
      -1.0468966357002097e-14,
      -1.0501938596895527e-14,
      -1.0460749057945454e-14,
      -1.0462626983776451e-14,
      -1.0465927407711652e-14,
      -1.0512258950320217e-14,
      -1.0463123223367227e-14,
      -1.0501684582185589e-14,
      -1.0501262353180619e-14,
      -8.75740843817778e-16,
      -8.833387889098168e-16,
      -9.01863680211921e-16,
      -8.887814415771339e-16,
      -8.866095595515147e-16,
      -8.903496664464915e-16,
      -8.89899269160482e-16,
      -9.076753117994923e-16,
      -8.859374560373898e-16,
      -9.042775734477854e-16,
      -8.931541861962914e-16,
      -8.695521124587636e-16,
      -8.806298939223968e-16,
      -8.8834485689959e-16,
      -9.032178705756295e-16,
      -8.930543015935018e-16,
      -9.03907368326219e-16,
      -8.944573261589905e-16,
      -7.59282577020754e-16,
      -7.542056090658788e-16,
      -7.556630442759393e-16,
      -7.396777703130021e-16,
      -7.600695634914718e-16,
      -7.6669873801906135e-16,
      -7.645993917258937e-16,
      -7.557997228082696e-16,
      -7.326260363187909e-16,
      -7.585735233390168e-16,
      -7.720497839145524e-16,
      -7.714752461996846e-16,
      -7.63980334167575e-16,
      -7.615100748580245e-16,
      -7.341360002884724e-16,
      -7.57616427129526e-16,
      -7.631710627246266e-16,
      -7.492333565550446e-16,
      -7.321568196062488e-16,
      -7.578329350250836e-16,
      2.341787420430452e-16,
      2.0684554660766146e-16,
      2.2167741651722883e-16,
      2.314280535451077e-16,
      2.1011941951091603e-16,
      2.3225424413330545e-16,
      2.1243711776946523e-16,
      2.1300312667147437e-16,
      2.0240074242308518e-16,
      2.0454088202485631e-16,
      2.0720785717084543e-16,
      2.000651266475798e-16,
      2.242534713946984e-16,
      2.296002683637218e-16,
      2.1488761201300965e-16,
      2.2022590767114458e-16,
      2.2991882172577795e-16,
      -8.87094386027805e-17,
      -9.751500468540426e-17,
      -9.434092022656327e-17,
      -9.053531780371445e-17,
      -1.0175293895907919e-16,
      -1.0089439213006106e-16,
      -9.40179215584779e-17,
      -8.305340709656324e-17,
      -8.821614405851469e-17,
      -8.821876967075286e-17,
      -8.676281493449052e-17,
      -9.044064111079399e-17,
      -9.431629744793937e-17,
      -9.987146956728363e-17,
      -8.738981940480936e-17,
      1.5266866279401755e-16,
      1.2063706637618023e-16,
      1.349628423325823e-16,
      1.3652284180945607e-16,
      1.288219779291371e-16,
      1.3388527534821995e-16,
      1.2884631548751132e-16,
      1.4716351801537552e-16,
      1.4528462197963557e-16,
      1.4971290791173137e-16,
      1.638606677740558e-16,
      1.4782927054837152e-16,
      1.6388602259037606e-16,
      1.3208292410499849e-16,
      1.3513878053712914e-16,
      1.4818067269804824e-16,
      1.3524506244530098e-16,
      1.4556313571527667e-16,
      1.4867732234448672e-16,
      1.6372350819494596e-16,
      -5.697863401572136e-17,
      -1.2409671328082817e-16,
      -1.4832008224659815e-16,
      -1.1925325836320998e-16,
      -1.8283450050766065e-17,
      -9.614607425629219e-17,
      -1.9175861172317875e-17,
      -1.635358624817837e-13,
      -1.635495153229392e-13,
      -1.6357775095320586e-13,
      -1.634668000350048e-13,
      -1.63549718681921e-13,
      -1.634947229733927e-13,
      -1.6352228715041764e-13,
      -1.635359896564509e-13,
      -1.634251726226106e-13,
      -1.6357753240430998e-13,
      -1.6350832471274222e-13,
      -1.634531085035576e-13,
      -1.6341144484273965e-13,
      -1.6354987802596751e-13,
      -1.6363284324520965e-13,
      -1.6352223733340093e-13,
      -1.6343895885096115e-13,
      -1.635912423332498e-13,
      -1.6346695123528985e-13,
      -1.6345311001470867e-13,
      -1.6354950780036443e-13,
      -1.634389128519816e-13,
      -1.6349402004362752e-13,
      -3.885762342402281e-16,
      -3.8431231234318205e-16,
      -3.8061433115014657e-16,
      -3.777381103628411e-16,
      -3.9613653882184886e-16,
      -3.873624752245256e-16,
      -3.772682621722084e-16,
      -3.9259514650343236e-16,
      -3.9379269201386145e-16,
      -3.939461635653004e-16,
      -7.617409213538796e-17,
      -6.301946382224753e-17,
      -8.016896521186991e-17,
      -8.249970484889574e-17,
      -1.0078495171527273e-16,
      -7.249048427361486e-17,
      -7.762240449994125e-17,
      -8.852029281341148e-17,
      -9.298155151303115e-17,
      -6.114843363535055e-17,
      -6.647024046283864e-17,
      -8.870931118082468e-17,
      -5.498533507703029e-17,
      -5.555078525963731e-17,
      -1.0094624278333788e-16,
      -6.280163653512989e-17,
      -9.722684660212734e-17,
      -1.0733778120965811e-16,
      4.2180310992383526e-17,
      3.625702632004581e-17,
      2.5774161093072443e-17,
      3.077393902601917e-17,
      2.937102764195465e-17,
      4.323523104019137e-17,
      4.137345745721523e-17,
      2.5131511284305922e-17,
      3.548042321860535e-16,
      3.710533522180301e-16,
      3.574642196783926e-16,
      3.652898408840685e-16,
      3.43752768889264e-16,
      3.535738793325805e-16,
      3.5493135911644116e-16,
      3.6761607870848207e-16,
      3.5332137140915884e-16,
      3.3690013913766497e-16,
      3.493656850428316e-16,
      3.570690882622013e-16,
      3.6550872471117034e-16,
      3.744814470852441e-16,
      3.686621879011246e-16,
      3.838659159125606e-16,
      3.64200314649243e-16,
      3.6196247590942574e-16,
      3.759730537574851e-16,
      -3.242603365072697e-16,
      -3.1817045620107615e-16,
      -3.1994455329545e-16,
      -3.132343271185314e-16,
      -3.2930167552064784e-16,
      -3.2227881679604163e-16,
      -3.204367339393015e-16,
      -3.252561871768311e-16,
      -3.217271647756551e-16,
      -3.1828686327902575e-16,
      -3.071778968778129e-16,
      -3.1787745951746317e-16,
      -3.2383629841718247e-16,
      -9.870376460667005e-16,
      -9.955041544315953e-16,
      -9.975965047240487e-16,
      -9.992896657238952e-16,
      -9.9550085169539e-16,
      -9.955002503990087e-16,
      -9.948427449904565e-16,
      -9.979965302012202e-16,
      -1.0018471786821933e-15,
      -9.921544261990965e-16,
      -9.921192169738591e-16,
      -9.96134232951495e-16,
      -9.3401328808231e-18,
      -1.2071755263907103e-17,
      4.486375204231882e-17,
      1.899723868034982e-17,
      -8.99457081356367e-18,
      3.50744103744889e-17,
      3.124795237200943e-16,
      1.1521662127199095e-16,
      4.780573665585974e-17,
      2.701516832281225e-16,
      -3.583250584673672e-18,
      -1.634668845944928e-13,
      -1.6353563665518862e-13,
      -1.9178553758743386e-16,
      -1.8615360919031136e-16,
      -1.8347883147570796e-16,
      -1.7769424837401204e-16,
      -1.8835988778785196e-16,
      -1.8056593253688328e-16,
      -1.950289856676822e-16,
      -1.9267730549157823e-16,
      -1.8849502988918788e-16,
      -1.9632465222148275e-16,
      -1.8167984123526218e-16,
      -1.8294563946462763e-16,
      -1.9174434437182776e-16,
      1.3595895242968226e-16,
      1.233822072288504e-16,
      1.1817803680091998e-16,
      1.360673726469308e-16,
      1.24683249835833e-16,
      1.2349062744609896e-16,
      1.3487475025719675e-16,
      1.1828645701816853e-16,
      1.205632815803881e-16,
      1.3097162243624894e-16,
      1.3075478200175183e-16,
      1.1893697832165984e-16,
      1.199127602768968e-16,
      1.1340754724198376e-16,
      1.1514227071796057e-16,
      1.323810852604801e-16,
      1.1546753136970622e-16,
      1.207801220148852e-16,
      1.199127602768968e-16,
      1.2045486136313954e-16,
      1.1611805267319752e-16,
      1.220811646218678e-16,
      1.3021268091550908e-16,
      1.2999584048101198e-16,
      1.2674323396355547e-16,
      1.233822072288504e-16,
      1.2999584048101198e-16,
      1.1590121223870042e-16,
      1.325979256949772e-16,
      1.354168513434395e-16,
      1.1210650463500116e-16,
      1.1557595158695477e-16,
      1.124317652867468e-16,
      1.2555061157382141e-16,
      1.1915381875615694e-16,
      1.1384122811097797e-16,
      1.3660947373317356e-16,
      1.22514845490862e-16,
      1.3487475025719675e-16,
      1.218643241873707e-16,
      -2.6202481833700576e-16,
      -2.704428006879171e-16,
      -2.6316543323885386e-16,
      -2.6620563428651893e-16,
      -2.6511370420369033e-16,
      -2.621707650518078e-16,
      -2.5726271863891815e-16,
      -2.6587064042923603e-16,
      -2.6615345702244956e-16,
      -2.552536047668796e-16,
      -2.6495328523411227e-16,
      -2.6533349527929825e-16,
      -2.544874452449713e-16,
      -2.652477927297659e-16,
      -2.6537809505547194e-16,
      -2.5900280613307056e-16,
      1.3064636178450328e-16,
      1.3010426069826053e-16,
      1.249000902703301e-16,
      1.196959198423997e-16,
      1.3444106938820255e-16,
      1.1676857397668883e-16,
      1.1785277614917433e-16,
      1.1362438767648086e-16,
      1.334652874329656e-16,
      1.3400738851920835e-16,
      1.1579279202145187e-16,
      1.3530843112619095e-16,
      1.2847795743953228e-16,
      1.1752751549742868e-16,
      1.2891163830852648e-16,
      1.323810852604801e-16,
      1.238158880978446e-16,
      1.2435798918408736e-16,
      -0.17677669529663725,
      1.1167282376600696e-16,
      1.1752751549742868e-16,
      1.2175590397012215e-16,
      1.1275702593849246e-16,
      1.24683249835833e-16,
      1.3866945786089602e-16,
      1.2609271266006417e-16,
      1.2414114874959026e-16,
      1.201296007113939e-16,
      1.3921155894713877e-16,
      1.2663481374630692e-16,
      1.20346441145891e-16,
      1.251169307048272e-16,
      2.7539528598681816e-16,
      2.6598259098145687e-16,
      2.6263738586248637e-16,
      2.7424961226660386e-16,
      2.696679167675067e-16,
      2.7958546587645883e-16,
      2.7743447475671735e-16,
      2.7942059911986976e-16,
      2.7383290901677414e-16,
      2.7345098695031083e-16,
      5.71799139014006e-16,
      5.652538231289404e-16,
      3.1790267214575196e-16,
      3.3232043430677395e-16,
      3.5971351444831273e-16,
      -9.751168654544364e-17,
      3.087604827308653e-16,
      3.114617114759652e-16,
      3.218034109487343e-16,
      3.2483844387131606e-16,
      3.072517782374006e-16,
      -9.91422897604243e-16,
      3.4268790796101217e-16,
      3.127937863685656e-16,
      3.25521426674413e-16,
      3.601805494860277e-16,
      3.1895051644139826e-16,
      3.0841890202405353e-16,
      3.3527658526611727e-16,
      1.2761059570154387e-16,
      1.2565903179106996e-16,
      1.222980050563649e-16,
      1.2620113287731272e-16,
      1.2620113287731272e-16,
      1.251169307048272e-16,
      1.1600963245594897e-16,
      1.2902005852577503e-16,
      5.732226736615093e-16,
      5.665391295039825e-16,
      5.774263250301574e-16,
      5.759602472755895e-16,
      5.794451301778488e-16,
      5.630040044990704e-16,
      1.2891163830852648e-16,
      1.3064636178450328e-16,
      1.205632815803881e-16,
      1.330316065639714e-16,
      1.3834419720915037e-16,
      1.349831704744453e-16,
      1.2435798918408736e-16,
      1.3639263329867646e-16,
      1.2218958483911635e-16,
      1.3335686721571705e-16,
      1.2847795743953228e-16,
      1.2685165418080402e-16,
      1.1861171766991419e-16,
      1.3660947373317356e-16,
      -2.6297775078466613e-16,
      -2.606287144439918e-16,
      -1.634668060115236e-13,
      -1.6343908010125374e-13,
      -1.6361899169484179e-13,
      -1.6353636949010658e-13,
      -1.6360525038598462e-13,
      -1.634524456503192e-13,
      -1.6349466002295566e-13,
      -1.6348012688997502e-13,
      1.1514227071796057e-16,
      1.1828645701816853e-16,
      1.1687699419393738e-16,
      1.3747683547116196e-16,
      1.1470858984896637e-16,
      -0.17677669529663714,
      1.1904539853890839e-16,
      1.2218958483911635e-16,
      1.341158087364569e-16,
      1.1449174941446927e-16,
      1.2912847874302358e-16,
      1.1345232466492181e-17,
      -7.356952635661388e-18,
      2.801083166035463e-17,
      -9.04395402010808e-18,
      5.788004641153626e-18,
      2.812221068232276e-17,
      1.2414114874959026e-16,
      1.2457482961858446e-16,
      5.676927285598718e-16,
      5.773491353373425e-16,
      1.1655173354219173e-16,
      1.227316859253591e-16,
      1.2956215961201778e-16,
      1.2392430831509316e-16,
      1.356336917779366e-16,
      1.3986208025063007e-16,
      -2.6060367237906635e-16,
      -2.5716351763763018e-16,
      1.2696007439805257e-16,
      1.3129688308799459e-16,
      1.1123914289701275e-16,
      1.1796119636642288e-16,
      1.1633489310769463e-16,
      1.3053794156725473e-16,
      1.227316859253591e-16,
      -2.562957366039987e-16,
      1.2522535092207576e-16,
      1.2999584048101198e-16,
      6.6123725552961e-16,
      6.54284083897873e-16,
      6.608421358129931e-16,
      6.671383356255682e-16,
      6.597401421641503e-16,
      6.518353028875845e-16,
      6.487016119823662e-16,
      6.531967020122132e-16,
      6.52997834389839e-16,
      6.601047911098629e-16,
      6.628025762293695e-16,
      6.573309203339473e-16,
      6.555897169333418e-16,
      6.598858540813084e-16,
      6.586040752625067e-16,
      4.683587974342112e-16,
      4.592704346285713e-16,
      4.576234821132964e-16,
      4.561925174832823e-16,
      4.470279371256706e-16,
      4.635305246301694e-16,
      4.374401636599851e-16,
      4.767374557739005e-16,
      4.557164457217262e-16,
      -1.7831220319253274e-16,
      -1.483025963399272e-16,
      -1.6366071608534146e-16,
      -1.5762320459536223e-16,
      -1.7994684009093005e-16,
      -1.7909426268347713e-16,
      -1.8232937765835405e-16,
      -1.7489843714163864e-16,
      -1.6422105621852814e-16,
      -5.648734776789092e-16,
      -5.246447142030408e-16,
      -3.588391329193187e-16,
      -4.91561797546201e-17,
      -4.150300230833903e-17,
      -4.248843770560046e-17,
      -4.459452078427125e-17,
      -4.13198135664003e-17,
      -3.9801138671058115e-17,
      -4.521611623553023e-17,
      -4.950811764903356e-17,
      -5.226592012626562e-17,
      -4.8065559886902043e-17,
      -4.153742594288549e-17,
      -3.8379416177398975e-17,
      -4.991517690274294e-17,
      -3.8190454592973894e-17,
      -4.7311899496677063e-17,
      -3.6852643130403034e-17,
      6.952942593820518e-17,
      5.748284170535251e-17,
      9.097460739988927e-17,
      9.223724873636043e-17,
      7.826516152231596e-17,
      7.38430267498112e-17,
      8.308631790277881e-17,
      1.0846777282595383e-16,
      1.0791206864900999e-16,
      1.0405390325324584e-16,
      1.8551429447445647e-16,
      6.569220290019399e-17,
      -2.6199910778679906e-17,
      9.263097818021485e-17,
      8.910423484010614e-17,
      -1.813058306184485e-16,
      6.911960019542549e-17,
      -1.5837853469644247e-16,
      -1.3743862736369822e-16,
      -1.6140118985473506e-16,
      -1.4051912179520255e-16,
      -1.705254196737185e-16,
      -1.9480661870267712e-16,
      -1.905864783375241e-16,
      -1.6481661344566863e-16,
      -1.6292885103854645e-16,
      -1.858123062397905e-16,
      -1.8731638258459265e-16,
      -6.356875573805184e-16,
      -6.220093747569884e-16,
      -6.251646133318962e-16,
      -6.31325754361394e-16,
      -6.301978228870375e-16,
      -6.173452899835848e-16,
      -6.340984896645236e-16,
      -6.35002526611935e-16,
      -6.414466525435136e-16,
      -6.283425449081467e-16,
      7.857882666761954e-17,
      8.024458294420651e-17,
      7.944287102584826e-17,
      8.822625143943545e-17,
      7.854493979750127e-17,
      8.143608182168183e-17,
      8.262553241849627e-17,
      7.598697905301829e-17,
      8.322450988873984e-17,
      8.746207511409605e-17,
      -2.5936643901993716e-17,
      -2.6098048212000155e-17,
      -9.162715725616005e-18,
      -3.844621670656871e-18,
      -9.942174387458197e-16,
      -8.430015882256052e-19,
      6.099324367623346e-17,
      5.099122434231734e-17,
      3.624554366679703e-17,
      5.364526630123164e-17,
      4.3829576097444104e-17,
      3.683799496905751e-17,
      5.096652459659934e-17,
      3.649310528975801e-17,
      4.497813480413386e-17,
      4.862579785765974e-17,
      3.8706398697100386e-17,
      2.7410766949290565e-17,
      3.660713118277751e-17,
      5.0600770461017446e-17,
      -2.191112552167088e-18,
      -8.313571447291697e-18,
      -1.2504544401093425e-17,
      -1.1901794318946327e-17,
      -2.901239096299242e-18,
      -2.2209695356308944e-18,
      -7.423247006743501e-18,
      -4.955079077631346e-18,
      -3.519373444768111e-17,
      -1.0693146698307737e-17,
      -1.6316163521638585e-17,
      -9.908421709298027e-18,
      -2.778568043875792e-18,
      -5.691741334474807e-17,
      2.0121638256217713e-17,
      4.653945126861299e-18,
      -3.8106904676487435e-17,
      3.1570605996884145e-17,
      4.251477518058703e-17,
      4.6708782395843116e-17,
      2.9615967164348236e-16,
      2.2243532505179984e-16,
      4.2667320584498797e-16,
      1.8331252274903112e-16,
      1.2101887460883156e-16,
      9.222337395668267e-17,
      9.520386919626262e-17,
      9.05939757240514e-17,
      9.465083066305637e-17,
      9.47655371398972e-17,
      9.064851401527911e-17,
      9.546074987634166e-17,
      -1.1068756878422397e-16,
      -1.984843524424758e-16,
      -1.136951618613346e-16,
      -1.7805721420578696e-16,
      -2.727752776405832e-16,
      -1.0931775044313017e-16,
      -1.3976311518594172e-16,
      -8.348718412495207e-18,
      5.048201291558343e-17,
      6.173679582326741e-18,
      -5.2469431855543476e-18,
      -2.6506144200953464e-17,
      -1.652516231684875e-17,
      -1.9100904279473298e-16,
      -3.018391293532896e-16,
      -4.492097195554102e-16,
      -1.9870203223555921e-16,
      -1.492516170559518e-16,
      -6.948052623771401e-17,
      -3.2433805444731864e-16,
      4.6855599816696005e-17,
      3.4276965184373745e-16,
      3.066629392245781e-16,
      3.537666969261011e-16,
      8.136509111019691e-17,
      4.443232967695477e-17,
      1.2956215961201778e-16,
      1.371515748194163e-16,
      1.3357370765021415e-16,
      1.1340754724198376e-16,
      1.1503385050071202e-16,
      1.2218958483911635e-16,
      -2.7168625891832957e-16,
      9.42897581954129e-17,
      9.383513305765238e-17,
      9.211118465994557e-17,
      9.43838056964757e-17,
      9.524591885882998e-17,
      9.284307346227586e-17,
      8.882761990261842e-17,
      9.124951522784834e-17,
      8.696719383506654e-17,
      8.70075116628973e-17,
      7.64572077069976e-17,
      -1.4930569891589603e-16,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      1.3877787807814457e-17,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      0.0,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      0.0,
      -6.938893903907228e-18,
      0.0,
      0.0,
      0.0,
      1.3877787807814457e-17,
      0.0,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      -6.938893903907228e-18,
      0.0,
      1.3877787807814457e-17,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      1.3877787807814457e-17,
      0.0,
      0.0,
      6.938893903907228e-18,
      0.0,
      0.0,
      0.0,
      0.0,
      0.0,
      0.0,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      0.0,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      -6.938893903907228e-18,
      1.3877787807814457e-17,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      0.0,
      -6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      0.0,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      1.3877787807814457e-17,
      0.0,
      6.938893903907228e-18,
      0.0,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      1.3877787807814457e-17,
      1.3877787807814457e-17,
      0.0,
      0.0,
      1.3877787807814457e-17,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      1.3877787807814457e-17,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      0.0,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      -6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      0.0,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      0.0,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      1.3877787807814457e-17,
      0.0,
      6.938893903907228e-18,
      0.0,
      -6.938893903907228e-18,
      0.0,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      -6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      0.0,
      1.3877787807814457e-17,
      1.3877787807814457e-17,
      1.3877787807814457e-17,
      0.0,
      6.938893903907228e-18,
      0.0,
      -6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      0.0,
      0.0,
      0.0,
      1.3877787807814457e-17,
      0.0,
      0.0,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      0.0,
      -6.938893903907228e-18,
      -6.938893903907228e-18,
      -6.938893903907228e-18,
      0.0,
      0.0,
      -6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      0.0,
      1.3877787807814457e-17,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      1.3877787807814457e-17,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      0.0,
      0.0,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      1.3877787807814457e-17,
      -6.938893903907228e-18,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      0.0,
      0.0,
      0.0,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      0.0,
      -6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      -6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      0.0,
      1.3877787807814457e-17,
      0.0,
      6.938893903907228e-18,
      0.0,
      0.0,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      -6.938893903907228e-18,
      -6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      1.3877787807814457e-17,
      0.0,
      0.0,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      1.3877787807814457e-17,
      0.0,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      0.0,
      0.0,
      0.0,
      0.0,
      -6.938893903907228e-18,
      0.0,
      1.3877787807814457e-17,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      0.0,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      0.0,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      0.0,
      1.3877787807814457e-17,
      0.0,
      -6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      0.0,
      0.0,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      0.0,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      0.0,
      0.0,
      0.0,
      0.0,
      1.3877787807814457e-17,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      0.0,
      0.0,
      -6.938893903907228e-18,
      0.0,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      0.0,
      -6.938893903907228e-18,
      0.0,
      1.3877787807814457e-17,
      -6.938893903907228e-18,
      1.3877787807814457e-17,
      0.0,
      0.0,
      1.3877787807814457e-17,
      1.3877787807814457e-17,
      0.0,
      6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      -6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      0.0,
      0.0,
      0.0,
      0.0,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      1.3877787807814457e-17,
      0.0,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      0.0,
      0.0,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      1.3877787807814457e-17,
      0.0,
      0.0,
      0.0,
      -6.938893903907228e-18,
      0.0,
      0.0,
      0.0,
      0.0,
      0.0,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      1.3877787807814457e-17,
      0.0,
      0.0,
      6.938893903907228e-18,
      0.0,
      1.3877787807814457e-17,
      0.0,
      0.0,
      0.0,
      6.938893903907228e-18,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      0.0,
      0.0,
      0.0,
      0.0,
      1.3877787807814457e-17,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      0.0,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      0.0,
      6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      0.0,
      0.0,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      -6.938893903907228e-18,
      0.0,
      0.0,
      1.3877787807814457e-17,
      0.0,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      -6.938893903907228e-18,
      6.938893903907228e-18,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      0.0,
      0.0,
      6.938893903907228e-18,
      1.3877787807814457e-17,
      -6.938893903907228e-18,
      -6.938893903907228e-18,
      0.0,
      6.938893903907228e-18,
      6.938893903907228e-18,
      0.0,
      -6.938893903907228e-18,
      6.938893903907228e-18
    ],
    [
      3.288660425431229e-15,
      3.3160477423363294e-15,
      3.3288163846220155e-15,
      3.309165546628788e-15,
      3.313603689962846e-15,
      3.308603961047715e-15,
      3.298763653350507e-15,
      3.310495546824989e-15,
      3.3398991480309593e-15,
      3.325365612180705e-15,
      3.3048815581469964e-15,
      3.3252787233528286e-15,
      3.321687288287196e-15,
      1.7958444629628764e-16,
      3.3365924260688884e-15,
      3.3249058105264654e-15,
      3.3082704934156608e-15,
      3.3345024751450214e-15,
      3.32508844284679e-15,
      3.328100168488844e-15,
      3.3120192028805286e-15,
      3.328830130968876e-15,
      3.3166788178956243e-15,
      2.2005608414957218e-16,
      2.2052793544604556e-16,
      2.341762845909384e-16,
      2.236964838753884e-16,
      2.3430777744010954e-16,
      2.3032258199887383e-16,
      2.132747531103546e-16,
      2.2989283955238667e-16,
      2.1353083826341165e-16,
      5.999337295387754e-16,
      6.136579901176774e-16,
      5.870713797012208e-16,
      1.6594074497641895e-13,
      1.6596233597701435e-13,
      1.6598310339023318e-13,
      1.6601197830346053e-13,
      1.6596498259259628e-13,
      1.6592504663443425e-13,
      1.6591097816418526e-13,
      1.6591474104176153e-13,
      1.6590432538050876e-13,
      1.6605353988127712e-13,
      1.6606926844763583e-13,
      1.6598341920902127e-13,
      1.6594287550327114e-13,
      1.659509361650086e-13,
      1.659306108103493e-13,
      1.6593400081235426e-13,
      1.6600339191909485e-13,
      1.6602120993033672e-13,
      1.6585614499166793e-13,
      1.6592153890719603e-13,
      1.6590370808089176e-13,
      1.6599254903448816e-13,
      1.659754350490885e-13,
      1.659322785966147e-13,
      1.6606372536716998e-13,
      1.659214650764192e-13,
      1.6607937200328136e-13,
      1.6595556174205481e-13,
      1.659621538250909e-13,
      1.6592697448141372e-13,
      7.507746541308245e-16,
      7.249979360025294e-16,
      7.442640955603793e-16,
      7.428726298026048e-16,
      7.428965314938541e-16,
      7.436487950466651e-16,
      7.475428667884462e-16,
      7.338959188511926e-16,
      7.380217468023099e-16,
      7.467664315667647e-16,
      7.43275394315359e-16,
      7.287112696941473e-16,
      7.309606646523055e-16,
      7.499498330744186e-16,
      7.317178584807008e-16,
      7.438071037873211e-16,
      7.293505657456389e-16,
      7.310416922369571e-16,
      7.249073112535147e-16,
      7.32671750225279e-16,
      2.8937518131097304e-14,
      2.894306928497384e-14,
      2.893949086359363e-14,
      2.8943847422461466e-14,
      2.8951745846289525e-14,
      2.8982668971921764e-14,
      2.898052948353626e-14,
      2.896535707990123e-14,
      2.894635391889064e-14,
      2.895079743553396e-14,
      2.894589836136919e-14,
      2.896434811414816e-14,
      2.895026097354475e-14,
      2.8993502276403733e-14,
      2.8944479509631416e-14,
      2.8954852411588294e-14,
      2.8941689185820585e-14,
      2.8981635554089396e-14,
      2.89725580331018e-14,
      2.8943120281485786e-14,
      2.8958505378692105e-14,
      2.893725070588001e-14,
      8.0936125663880565e-16,
      2.895559296591799e-14,
      2.8981317891918757e-14,
      2.8987841562868653e-14,
      2.8947554261563826e-14,
      2.896313757998944e-14,
      2.893911305277795e-14,
      2.8941908759259826e-14,
      2.894626627558922e-14,
      2.89753162819504e-14,
      2.8946205811167986e-14,
      2.897326471790093e-14,
      2.8969423038018967e-14,
      5.760275076476769e-16,
      5.782772651700484e-16,
      6.027992389587896e-16,
      5.91682332076767e-16,
      5.815298288391636e-16,
      5.786880777144468e-16,
      5.972540112777158e-16,
      5.978443629731508e-16,
      5.920279662022377e-16,
      5.966028504670971e-16,
      5.951587268781125e-16,
      5.67552156599111e-16,
      5.796994675056358e-16,
      5.845580825366273e-16,
      5.981037374719356e-16,
      5.953856338630619e-16,
      5.868835969099884e-16,
      5.810892872850809e-16,
      7.91262192358161e-16,
      7.969910600322468e-16,
      7.92617176935827e-16,
      7.816291650276524e-16,
      7.907982088668289e-16,
      7.862690259535562e-16,
      8.013224952043673e-16,
      7.974445900455577e-16,
      7.739414901608466e-16,
      7.891058739526462e-16,
      8.010136499847215e-16,
      8.026609484450178e-16,
      8.004948007947965e-16,
      7.921352570350111e-16,
      7.687084112333822e-16,
      7.961458422789767e-16,
      7.943950690250695e-16,
      7.989186995427482e-16,
      7.893030036431224e-16,
      7.964850456095575e-16,
      9.513126897341782e-17,
      9.135369805865732e-17,
      9.181582740228426e-17,
      9.004770522419392e-17,
      8.99930651553585e-17,
      9.434697137096568e-17,
      9.108686341723629e-17,
      9.105092520992616e-17,
      8.686807494874235e-17,
      9.330014591875266e-17,
      8.820940108022444e-17,
      8.968810850515725e-17,
      9.007053865042883e-17,
      9.027644948350783e-17,
      9.006202525466117e-17,
      9.446300011993718e-17,
      8.772946064887825e-17,
      -2.031173252782092e-16,
      -2.178444164716567e-16,
      -2.1494062397509284e-16,
      -2.179804734198634e-16,
      -2.2180962823470488e-16,
      -2.0856294944588624e-16,
      -2.1861008989951662e-16,
      -2.1371248092012262e-16,
      -2.1251473003124065e-16,
      -2.1492920737872286e-16,
      -2.1361246732928804e-16,
      -2.148303124354673e-16,
      -2.271221207958901e-16,
      -2.2204563694912297e-16,
      -2.086986265852801e-16,
      -4.135443684718219e-16,
      -3.8971809943090233e-16,
      -3.915855887952818e-16,
      -4.01830780661822e-16,
      -3.881072006659421e-16,
      -3.970173582478352e-16,
      -3.9502330812129605e-16,
      -4.0593372876267383e-16,
      -4.0361366364690115e-16,
      -4.0690130202340123e-16,
      -4.158631409798064e-16,
      -4.0831505215980433e-16,
      -4.1722464638000093e-16,
      -3.914276354883442e-16,
      -3.9652415504893405e-16,
      -4.0490017256373533e-16,
      -3.946042292873864e-16,
      -4.053376165917078e-16,
      -4.0702682782282285e-16,
      -4.202747785436877e-16,
      -1.880340943067745e-16,
      -2.0002238716557983e-16,
      -2.1033610467780568e-16,
      -1.971311132436182e-16,
      -1.7795951607659301e-16,
      -1.8952675758804263e-16,
      -1.7616755430016462e-16,
      -0.17407765595569807,
      -0.17407765595569794,
      -0.174077655955698,
      -0.17407765595569816,
      -0.1740776559556981,
      -0.1740776559556979,
      -0.17407765595569807,
      -0.1740776559556981,
      -0.17407765595569816,
      -0.17407765595569824,
      -0.17407765595569805,
      -0.17407765595569802,
      -0.174077655955698,
      -0.1740776559556981,
      -0.17407765595569807,
      -0.17407765595569802,
      -0.174077655955698,
      -0.17407765595569813,
      -0.17407765595569796,
      -0.17407765595569802,
      -0.17407765595569813,
      -0.17407765595569813,
      -0.17407765595569813,
      1.8971914738461e-16,
      1.8798206539688245e-16,
      1.8641842935044332e-16,
      1.8588860175331562e-16,
      1.9433187909568326e-16,
      1.8976468108780056e-16,
      1.8439708645153674e-16,
      1.9247893218219816e-16,
      1.9305416223459188e-16,
      1.9310559582245657e-16,
      -2.486878923335451e-18,
      -2.5804822850045726e-18,
      7.375062661946116e-19,
      -1.98120581041716e-18,
      2.5249866894837944e-19,
      -3.832350983627336e-18,
      1.4375860508924583e-18,
      -1.893779563646809e-18,
      -4.924433375431516e-19,
      -1.3760413222832397e-18,
      -1.43292419685258e-18,
      -2.900849237663781e-19,
      -4.987390493862121e-18,
      -2.782128889747798e-18,
      1.928330579468175e-18,
      6.2590616008905554e-21,
      -6.736453543776059e-19,
      7.354572372582356e-19,
      -1.2444450054223545e-17,
      -2.2564292439094913e-17,
      -4.395469441542559e-17,
      -3.5509227588429114e-17,
      -3.675323534421714e-17,
      -1.602863664409246e-17,
      -1.5189584383280902e-17,
      -5.584343365845552e-17,
      -1.331006429210365e-16,
      -1.3713276207761862e-16,
      -1.277143382031347e-16,
      -1.3250722423619e-16,
      -1.2384472273832547e-16,
      -1.3118925430906735e-16,
      -1.2627309962088796e-16,
      -1.3544555180453423e-16,
      -1.3170247013237668e-16,
      -1.210486854223588e-16,
      -1.272640064897653e-16,
      -1.2830979715522466e-16,
      -1.2543263971851016e-16,
      -1.3600388183006523e-16,
      -1.2933888570081782e-16,
      -1.4145796524183567e-16,
      -1.3401346324190302e-16,
      -1.2852786796298873e-16,
      -1.3841032243955379e-16,
      1.389002841056391e-17,
      2.900792323295239e-17,
      1.947748738414289e-17,
      2.34680269448294e-17,
      1.1997299924293503e-17,
      1.0731769835296646e-17,
      2.094610029784724e-17,
      1.7193793556889707e-17,
      2.453054524084508e-17,
      2.3497257298779518e-17,
      2.84766217483355e-17,
      2.463474519034001e-17,
      1.0088740116538976e-17,
      1.7470851571096918e-16,
      1.7765424430423845e-16,
      1.7614526980918626e-16,
      1.754134838693652e-16,
      1.7522785397227752e-16,
      1.734073654017331e-16,
      1.7370587272154406e-16,
      1.7136242026637816e-16,
      1.76275084908025e-16,
      1.7054454417202666e-16,
      1.7757982294357644e-16,
      1.7590209458943564e-16,
      -2.2583611025016317e-18,
      -1.4232382647177102e-18,
      3.19705249227578e-17,
      1.6593889565948988e-17,
      -9.505031959704729e-20,
      2.523058970768828e-17,
      2.3240120744769913e-16,
      1.3080370233582387e-16,
      1.0215501741315593e-16,
      2.1380426446400763e-16,
      8.302369245927722e-17,
      -0.17407765595569802,
      -0.17407765595569805,
      3.382000081491474e-17,
      4.5677744725743727e-17,
      5.341120176589639e-17,
      5.473479230841027e-17,
      4.242648517565706e-17,
      4.794145128338201e-17,
      2.818850378251482e-17,
      3.910772785589891e-17,
      4.267142481615439e-17,
      3.84229881281156e-17,
      5.3150156452820534e-17,
      5.2390136438781834e-17,
      3.869676327881467e-17,
      -1.1102230246251565e-16,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      -1.1102230246251565e-16,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      -1.1102230246251565e-16,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      -1.1102230246251565e-16,
      -1.1102230246251565e-16,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      -8.326672684688674e-17,
      -8.326672684688674e-17,
      -1.1102230246251565e-16,
      -8.326672684688674e-17,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      -8.326672684688674e-17,
      -9.71445146547012e-17,
      -1.1102230246251565e-16,
      -1.1102230246251565e-16,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      -1.1102230246251565e-16,
      -8.326672684688674e-17,
      -1.1102230246251565e-16,
      -1.1102230246251565e-16,
      -8.326672684688674e-17,
      -8.326672684688674e-17,
      -8.326672684688674e-17,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      -8.326672684688674e-17,
      -1.1102230246251565e-16,
      -9.71445146547012e-17,
      -1.1102230246251565e-16,
      -9.71445146547012e-17,
      2.8876742504461124e-16,
      3.0164291253048915e-16,
      2.933014210210681e-16,
      2.972009159205286e-16,
      2.9338530461022395e-16,
      2.886366201620227e-16,
      2.8306661263757315e-16,
      2.9937533848504543e-16,
      2.972680976395018e-16,
      2.795494914642694e-16,
      2.937126795480162e-16,
      2.942387555987857e-16,
      2.8280786029039113e-16,
      2.929551639739714e-16,
      2.9378018408961895e-16,
      2.835523970115214e-16,
      -1.1102230246251565e-16,
      -1.1102230246251565e-16,
      -9.71445146547012e-17,
      -8.326672684688674e-17,
      -1.1102230246251565e-16,
      -8.326672684688674e-17,
      -8.326672684688674e-17,
      -8.326672684688674e-17,
      -1.1102230246251565e-16,
      -1.1102230246251565e-16,
      -8.326672684688674e-17,
      -1.1102230246251565e-16,
      -1.1102230246251565e-16,
      -9.71445146547012e-17,
      -1.1102230246251565e-16,
      -1.1102230246251565e-16,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      1.6596178130624485e-13,
      -8.326672684688674e-17,
      -8.326672684688674e-17,
      -9.71445146547012e-17,
      -8.326672684688674e-17,
      -9.71445146547012e-17,
      -1.1102230246251565e-16,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      -8.326672684688674e-17,
      -1.1102230246251565e-16,
      -1.1102230246251565e-16,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      -1.3131422495721742e-16,
      -1.2667788551250652e-16,
      -1.251226058788428e-16,
      -1.3109049028462117e-16,
      -1.2899215945403502e-16,
      -1.3381276580264407e-16,
      -1.326707677136162e-16,
      -1.3389631930121837e-16,
      -1.3097722589887677e-16,
      -1.3046986686207378e-16,
      -2.808744680628825e-16,
      -2.778522865031996e-16,
      5.846739127311399e-17,
      6.396431427868134e-17,
      4.112104716745571e-17,
      -2.1977484692400457e-16,
      6.326845866120707e-17,
      5.51815336489846e-17,
      5.32130482079806e-17,
      5.606340615775859e-17,
      5.890377537071724e-17,
      1.6876143959670684e-16,
      4.543595359873064e-17,
      6.130295378491221e-17,
      4.3733658947101586e-17,
      5.0109958606494366e-17,
      5.592983105827567e-17,
      5.274390384162381e-17,
      5.694328051454418e-17,
      -1.1102230246251565e-16,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      -8.326672684688674e-17,
      -1.1102230246251565e-16,
      -2.816969776286763e-16,
      -2.7813745527284883e-16,
      -2.8423114870135206e-16,
      -2.8312951587373215e-16,
      -2.837350544516029e-16,
      -2.75152990802115e-16,
      -1.1102230246251565e-16,
      -1.1102230246251565e-16,
      -9.71445146547012e-17,
      -1.1102230246251565e-16,
      -1.249000902703301e-16,
      -1.1102230246251565e-16,
      -9.71445146547012e-17,
      -1.249000902703301e-16,
      -9.71445146547012e-17,
      -1.1102230246251565e-16,
      -1.1102230246251565e-16,
      -1.1102230246251565e-16,
      -8.326672684688674e-17,
      -1.249000902703301e-16,
      2.9096460906351325e-16,
      2.878693000841457e-16,
      -0.1740776559556981,
      -0.17407765595569813,
      -0.1740776559556982,
      -0.17407765595569794,
      -0.17407765595569805,
      -0.17407765595569805,
      -0.17407765595569816,
      -0.17407765595569805,
      -8.326672684688674e-17,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      -1.249000902703301e-16,
      -8.326672684688674e-17,
      1.6591225494747504e-13,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      -1.1102230246251565e-16,
      -8.326672684688674e-17,
      -1.1102230246251565e-16,
      -3.7783725070549293e-17,
      -4.4626029821196846e-17,
      -2.694303937495837e-17,
      -5.083829861476008e-17,
      -3.866883507942962e-17,
      -2.5757953892576153e-17,
      -9.71445146547012e-17,
      -9.71445146547012e-17,
      -2.7910356611774406e-16,
      -2.837670494037017e-16,
      -8.326672684688674e-17,
      -9.71445146547012e-17,
      -1.1102230246251565e-16,
      -9.71445146547012e-17,
      -1.1102230246251565e-16,
      -1.249000902703301e-16,
      2.8598303691270835e-16,
      2.8428592863611923e-16,
      -1.1102230246251565e-16,
      -1.1102230246251565e-16,
      -8.326672684688674e-17,
      -9.71445146547012e-17,
      -8.326672684688674e-17,
      -1.1102230246251565e-16,
      -9.71445146547012e-17,
      2.8215902864031664e-16,
      -9.71445146547012e-17,
      -1.1102230246251565e-16,
      -7.179284757353871e-17,
      -8.141017926920517e-17,
      -8.39693353146975e-17,
      -8.012431490721178e-17,
      -8.297445619262174e-17,
      -8.988643126431686e-17,
      -8.791420033211647e-17,
      -9.901823053187024e-17,
      -8.988249861428904e-17,
      -8.343760519212591e-17,
      -8.105452026941693e-17,
      -8.411744271559967e-17,
      -8.297534031103476e-17,
      -7.753190948212968e-17,
      -8.749947018718194e-17,
      -9.903490365022431e-17,
      -9.628302758706466e-17,
      -1.0085975794180532e-16,
      -1.030879056041987e-16,
      -8.999793745582584e-17,
      -9.959167214766152e-17,
      -8.867138579600058e-17,
      -1.0866676469668694e-16,
      -9.553090326323693e-17,
      2.524635076390494e-17,
      1.6078882283981718e-17,
      1.5058483245693962e-17,
      1.8776067480210807e-17,
      2.761260380543097e-17,
      2.8462400112943296e-17,
      2.4199101345852115e-17,
      2.429925346237883e-17,
      2.098876972541476e-17,
      6.626536164574365e-16,
      5.799533667720729e-16,
      2.1165843139780192e-16,
      -1.5918719128836992e-16,
      -1.5196054208161377e-16,
      -1.579977566854827e-16,
      -1.5585732289695453e-16,
      -1.5302769652376283e-16,
      -1.5531714496076608e-16,
      -1.5772803086493913e-16,
      -1.6017434191384697e-16,
      -1.5997194556996251e-16,
      -1.6015912659332374e-16,
      -1.561530692948207e-16,
      -1.5742707624760133e-16,
      -1.5330140745405478e-16,
      -1.5332588526005194e-16,
      -1.5920423115024973e-16,
      -1.5545020546006112e-16,
      -3.903970920706069e-17,
      -3.696697365150301e-17,
      -2.5589847965993664e-17,
      -2.7436715971044666e-17,
      -3.397784053350961e-17,
      -3.43455794499066e-17,
      -3.2472899919336996e-17,
      -1.687797206350768e-17,
      -2.056627754957597e-17,
      -2.101885078861918e-17,
      1.3875936472697845e-16,
      6.768500570071892e-17,
      2.1269361979412935e-17,
      7.945030200202888e-17,
      7.964915212712607e-17,
      6.671146823310648e-17,
      -3.95328840861372e-17,
      8.310561972363866e-17,
      9.343774523528687e-17,
      7.456652263845952e-17,
      8.311861954689419e-17,
      6.781433572604551e-17,
      5.0475347097907695e-17,
      5.940981841412141e-17,
      7.214952305396989e-17,
      6.940321381877549e-17,
      5.429005999683513e-17,
      5.448415323644553e-17,
      3.158449677829216e-16,
      3.08191185337595e-16,
      3.105759695176982e-16,
      3.134817578850168e-16,
      3.1314093907586846e-16,
      3.076472038025046e-16,
      3.1469401805093573e-16,
      3.145957036652706e-16,
      3.16897079894538e-16,
      3.1055091356222583e-16,
      -9.162579706674806e-18,
      -5.958992801205076e-18,
      -1.1473831221510404e-17,
      2.5556839756883464e-19,
      -1.0143160549440364e-17,
      -6.8712203393139525e-18,
      -6.724252438931183e-18,
      -1.0891433098803314e-17,
      -2.5129593442056066e-18,
      3.1540052629587765e-18,
      -3.452856793573192e-17,
      -3.217730202074582e-17,
      -2.577663464427256e-17,
      -2.418339543727917e-17,
      1.657742447548632e-16,
      -2.1505384048342464e-17,
      2.1137185080153808e-16,
      2.0067470582803096e-16,
      2.0530521016381052e-16,
      2.0281174744924125e-16,
      2.0993351601080295e-16,
      1.969691368833117e-16,
      8.559392920524075e-18,
      -2.527401450364207e-17,
      -6.682742515912142e-18,
      3.7659657071573024e-18,
      -2.3372586515838163e-17,
      -3.7771068699137996e-17,
      -2.0517660616180252e-17,
      7.9567622057825e-18,
      1.5887203336450975e-17,
      2.9846737354608395e-18,
      -6.315990435538116e-18,
      -2.015583730345452e-18,
      1.107170299384087e-17,
      1.4069087314934363e-17,
      5.47599063400668e-18,
      9.912995124187504e-18,
      7.615423611699171e-17,
      9.19847321521182e-17,
      8.801089990342768e-17,
      9.137881996245421e-17,
      9.51508374536082e-17,
      7.035350340114815e-17,
      -5.117384800060964e-17,
      -5.867749721724105e-17,
      -6.926915061759742e-17,
      -4.2441610514865624e-17,
      -4.161894800579714e-17,
      -4.119133383840085e-17,
      2.493901884073675e-16,
      2.0671368584986644e-16,
      3.1481860661545195e-16,
      1.9299303582891566e-16,
      1.6164168951363712e-16,
      1.5749459788832322e-16,
      1.5712208373347637e-16,
      1.582265480549032e-16,
      1.590720765778748e-16,
      1.6281826798066278e-16,
      1.5926607409468697e-16,
      1.6062685464895673e-16,
      -4.788922669596053e-16,
      -4.792045791249246e-16,
      -4.8522658228493e-16,
      -4.941480045075473e-16,
      -5.492441232988112e-16,
      -4.711578526286529e-16,
      -4.943781943711052e-16,
      -2.7570881400239172e-17,
      -2.0183021480655373e-18,
      -2.104944958703182e-17,
      -2.583403199881218e-17,
      -3.9089735699403606e-17,
      -3.2647113769137094e-17,
      -1.8243618323979973e-16,
      -2.0893434248252223e-16,
      -2.9697414070099404e-16,
      -1.7735637106451543e-16,
      2.865496784104323e-16,
      2.7170454639871244e-16,
      3.1502089670265226e-16,
      3.0878110331810986e-16,
      5.177155540037483e-17,
      5.922114484130884e-17,
      4.731572459406202e-17,
      -9.012172043011227e-18,
      -1.3144507783733184e-18,
      -1.1102230246251565e-16,
      -1.249000902703301e-16,
      -1.249000902703301e-16,
      -8.326672684688674e-17,
      -8.326672684688674e-17,
      -9.71445146547012e-17,
      3.0370104286815014e-16,
      1.6791504792752633e-16,
      1.6798309669883795e-16,
      1.6395672310892233e-16,
      1.6738234540032713e-16,
      1.685522861677926e-16,
      1.6855830640931236e-16,
      1.6134166140060614e-16,
      1.6920480393346439e-16,
      1.6357248899236308e-16,
      8.529979122694639e-19,
      -1.7005895780498047e-17,
      -1.6258839476813876e-16,
      -9.595189226496714e-18,
      2.168404344971009e-19,
      -1.2793585635328952e-17,
      -9.595189226496714e-18,
      -3.2526065174565133e-18,
      -6.2341624917916505e-18,
      -2.9815559743351372e-18,
      -2.9815559743351372e-18,
      -6.2341624917916505e-18,
      -9.432558900623889e-18,
      -1.2902005852577503e-17,
      -2.927345865710862e-18,
      -9.75781955236954e-18,
      -9.540979117872439e-18,
      -6.2341624917916505e-18,
      -2.9815559743351372e-18,
      -9.540979117872439e-18,
      -3.0357660829594124e-18,
      -2.927345865710862e-18,
      -6.342582709040201e-18,
      -6.342582709040201e-18,
      -6.179952383167375e-18,
      -6.451002926288751e-18,
      -2.9815559743351372e-18,
      -6.179952383167375e-18,
      -9.486769009248164e-18,
      -2.9815559743351372e-18,
      -3.0357660829594124e-18,
      -3.0899761915836876e-18,
      -3.144186300207963e-18,
      -9.703609443745265e-18,
      -6.288372600415926e-18,
      -6.451002926288751e-18,
      -3.144186300207963e-18,
      -9.595189226496714e-18,
      -9.540979117872439e-18,
      -9.432558900623889e-18,
      -6.5052130349130266e-18,
      1.6263032587282567e-19,
      -6.342582709040201e-18,
      2.710505431213761e-19,
      -2.9815559743351372e-18,
      -9.432558900623889e-18,
      -6.396792817664476e-18,
      -9.64939933512099e-18,
      -2.9815559743351372e-18,
      -9.486769009248164e-18,
      -6.342582709040201e-18,
      -3.0357660829594124e-18,
      -3.144186300207963e-18,
      -6.342582709040201e-18,
      -9.595189226496714e-18,
      -3.0899761915836876e-18,
      -9.540979117872439e-18,
      -6.288372600415926e-18,
      -6.396792817664476e-18,
      -3.3068166260807885e-18,
      2.710505431213761e-19,
      -1.2956215961201778e-17,
      1.6263032587282567e-19,
      -9.540979117872439e-18,
      -1.2956215961201778e-17,
      -9.378348791999613e-18,
      -3.0899761915836876e-18,
      -3.0357660829594124e-18,
      -9.486769009248164e-18,
      -6.451002926288751e-18,
      -6.396792817664476e-18,
      -3.144186300207963e-18,
      -3.198396408832238e-18,
      -6.396792817664476e-18,
      -6.451002926288751e-18,
      -1.2793585635328952e-17,
      -6.342582709040201e-18,
      -6.342582709040201e-18,
      -9.486769009248164e-18,
      -3.0899761915836876e-18,
      -3.3068166260807885e-18,
      -9.540979117872439e-18,
      -3.2526065174565133e-18,
      -2.8731357570865868e-18,
      -6.288372600415926e-18,
      -9.540979117872439e-18,
      -9.75781955236954e-18,
      -3.0899761915836876e-18,
      -9.64939933512099e-18,
      -6.2341624917916505e-18,
      -3.198396408832238e-18,
      -9.432558900623889e-18,
      -6.179952383167375e-18,
      -9.432558900623889e-18,
      1.6263032587282567e-19,
      -9.378348791999613e-18,
      -3.2526065174565133e-18,
      -9.595189226496714e-18,
      -9.432558900623889e-18,
      -6.179952383167375e-18,
      -3.3068166260807885e-18,
      -9.540979117872439e-18,
      -9.324138683375338e-18,
      -9.703609443745265e-18,
      -3.144186300207963e-18,
      -3.3068166260807885e-18,
      -3.0357660829594124e-18,
      -6.396792817664476e-18,
      -9.595189226496714e-18,
      -9.432558900623889e-18,
      -3.0899761915836876e-18,
      -1.2956215961201778e-17,
      -9.75781955236954e-18,
      -3.2526065174565133e-18,
      -1.2902005852577503e-17,
      -3.198396408832238e-18,
      -9.595189226496714e-18,
      -2.927345865710862e-18,
      -3.3068166260807885e-18,
      -6.288372600415926e-18,
      1.6263032587282567e-19,
      -6.396792817664476e-18,
      -9.64939933512099e-18,
      1.6263032587282567e-19,
      -9.540979117872439e-18,
      -9.595189226496714e-18,
      -2.927345865710862e-18,
      -1.2847795743953228e-17,
      -1.2793585635328952e-17,
      -3.0899761915836876e-18,
      -2.927345865710862e-18,
      -1.2847795743953228e-17,
      -9.486769009248164e-18,
      -9.540979117872439e-18,
      -6.2341624917916505e-18,
      -9.595189226496714e-18,
      -9.595189226496714e-18,
      -9.703609443745265e-18,
      1.0842021724855044e-19,
      -1.2793585635328952e-17,
      -3.0899761915836876e-18,
      -9.595189226496714e-18,
      -9.486769009248164e-18,
      -1.2956215961201778e-17,
      -9.75781955236954e-18,
      -2.9815559743351372e-18,
      -2.8731357570865868e-18,
      -3.0357660829594124e-18,
      -6.396792817664476e-18,
      -9.703609443745265e-18,
      -3.0899761915836876e-18,
      -1.2847795743953228e-17,
      -9.64939933512099e-18,
      -2.8731357570865868e-18,
      -2.9815559743351372e-18,
      -6.288372600415926e-18,
      -6.342582709040201e-18,
      -9.486769009248164e-18,
      -3.0357660829594124e-18,
      -9.703609443745265e-18,
      -3.2526065174565133e-18,
      -9.595189226496714e-18,
      -3.144186300207963e-18,
      -3.3068166260807885e-18,
      -3.0899761915836876e-18,
      2.710505431213761e-19,
      -9.540979117872439e-18,
      -3.3068166260807885e-18,
      -9.595189226496714e-18,
      -6.2341624917916505e-18,
      -6.288372600415926e-18,
      -3.0899761915836876e-18,
      -3.198396408832238e-18,
      -9.595189226496714e-18,
      -3.0357660829594124e-18,
      -3.0899761915836876e-18,
      -9.486769009248164e-18,
      -9.486769009248164e-18,
      -3.144186300207963e-18,
      -1.2847795743953228e-17,
      -9.703609443745265e-18,
      -6.396792817664476e-18,
      -1.2902005852577503e-17,
      -9.486769009248164e-18,
      -6.2341624917916505e-18,
      -9.540979117872439e-18,
      -9.486769009248164e-18,
      -6.342582709040201e-18,
      -6.179952383167375e-18,
      -9.595189226496714e-18,
      -1.2956215961201778e-17,
      2.168404344971009e-19,
      1.6263032587282567e-19,
      -6.2341624917916505e-18,
      -9.64939933512099e-18,
      -9.595189226496714e-18,
      -9.540979117872439e-18,
      -9.595189226496714e-18,
      -9.378348791999613e-18,
      -9.64939933512099e-18,
      -9.432558900623889e-18,
      -9.75781955236954e-18,
      -9.378348791999613e-18,
      -9.378348791999613e-18,
      2.710505431213761e-19,
      -9.64939933512099e-18,
      -6.451002926288751e-18,
      -3.2526065174565133e-18,
      -3.0357660829594124e-18,
      -1.2956215961201778e-17,
      -6.179952383167375e-18,
      -9.64939933512099e-18,
      2.710505431213761e-19,
      2.168404344971009e-19,
      -2.927345865710862e-18,
      -6.342582709040201e-18,
      -9.812029660993815e-18,
      -9.378348791999613e-18,
      -6.2341624917916505e-18,
      -3.0357660829594124e-18,
      -9.378348791999613e-18,
      -1.2847795743953228e-17,
      -1.2847795743953228e-17,
      -3.0357660829594124e-18,
      -6.179952383167375e-18,
      -6.396792817664476e-18,
      1.6263032587282567e-19,
      -9.486769009248164e-18,
      -9.595189226496714e-18,
      -3.144186300207963e-18,
      -9.486769009248164e-18,
      -1.2902005852577503e-17,
      -9.595189226496714e-18,
      -3.0899761915836876e-18,
      -9.595189226496714e-18,
      -6.451002926288751e-18,
      -9.595189226496714e-18,
      -6.342582709040201e-18,
      2.710505431213761e-19,
      -9.64939933512099e-18,
      -9.486769009248164e-18,
      -6.288372600415926e-18,
      -3.0899761915836876e-18,
      -9.75781955236954e-18,
      -9.595189226496714e-18,
      -2.8731357570865868e-18,
      -3.0357660829594124e-18,
      -2.927345865710862e-18,
      -3.198396408832238e-18,
      -1.2956215961201778e-17,
      -2.927345865710862e-18,
      -6.451002926288751e-18,
      -9.540979117872439e-18,
      -1.2847795743953228e-17,
      -9.595189226496714e-18,
      -9.64939933512099e-18,
      -3.198396408832238e-18,
      -6.2341624917916505e-18,
      -1.2847795743953228e-17,
      -3.0899761915836876e-18,
      -3.0899761915836876e-18,
      -9.595189226496714e-18,
      -9.486769009248164e-18,
      1.6263032587282567e-19,
      -2.927345865710862e-18,
      -6.342582709040201e-18,
      -3.0899761915836876e-18,
      -6.2341624917916505e-18,
      -2.8731357570865868e-18,
      -3.198396408832238e-18,
      -3.0357660829594124e-18,
      -3.3068166260807885e-18,
      -3.3068166260807885e-18,
      -3.2526065174565133e-18,
      4.336808689942018e-19,
      -2.927345865710862e-18,
      -3.2526065174565133e-18,
      -3.0899761915836876e-18,
      -9.64939933512099e-18,
      -3.144186300207963e-18,
      -9.486769009248164e-18,
      -3.198396408832238e-18,
      -6.396792817664476e-18,
      -9.540979117872439e-18,
      -6.342582709040201e-18,
      -3.2526065174565133e-18,
      -6.342582709040201e-18,
      -9.432558900623889e-18,
      -9.378348791999613e-18,
      -3.0357660829594124e-18,
      -3.0357660829594124e-18,
      -9.75781955236954e-18,
      -9.64939933512099e-18,
      -2.8731357570865868e-18,
      -9.812029660993815e-18,
      -6.451002926288751e-18,
      -9.540979117872439e-18,
      -6.179952383167375e-18,
      -2.8731357570865868e-18,
      -3.0357660829594124e-18,
      -6.396792817664476e-18,
      -6.179952383167375e-18,
      -9.540979117872439e-18,
      -6.342582709040201e-18,
      -6.288372600415926e-18,
      -6.288372600415926e-18,
      -2.8731357570865868e-18,
      -3.0357660829594124e-18,
      -3.0899761915836876e-18,
      -6.396792817664476e-18,
      -3.3068166260807885e-18,
      -6.288372600415926e-18,
      -6.396792817664476e-18,
      -9.432558900623889e-18,
      1.0842021724855044e-19,
      -3.3068166260807885e-18,
      -2.8731357570865868e-18,
      -3.0899761915836876e-18,
      -3.0899761915836876e-18,
      -3.0357660829594124e-18,
      2.710505431213761e-19,
      2.168404344971009e-19,
      -2.9815559743351372e-18,
      -6.288372600415926e-18,
      -1.2956215961201778e-17,
      -6.342582709040201e-18,
      -9.486769009248164e-18,
      -9.486769009248164e-18,
      -3.3068166260807885e-18,
      -9.540979117872439e-18,
      -6.288372600415926e-18,
      -9.378348791999613e-18,
      -9.703609443745265e-18,
      3.2526065174565133e-19,
      -3.198396408832238e-18,
      -2.9815559743351372e-18,
      -9.486769009248164e-18,
      -9.703609443745265e-18,
      -9.540979117872439e-18,
      -9.486769009248164e-18,
      3.63207727782644e-18,
      -9.703609443745265e-18,
      -2.9815559743351372e-18,
      -3.3068166260807885e-18,
      -1.2847795743953228e-17,
      -9.703609443745265e-18,
      -6.451002926288751e-18,
      -1.2902005852577503e-17,
      -6.559423143537302e-18,
      -9.595189226496714e-18,
      -6.396792817664476e-18,
      -3.2526065174565133e-18,
      -9.378348791999613e-18,
      -6.179952383167375e-18,
      -9.64939933512099e-18,
      -6.288372600415926e-18,
      -9.64939933512099e-18,
      -9.703609443745265e-18,
      -9.378348791999613e-18,
      -6.2341624917916505e-18,
      3.2526065174565133e-19,
      -9.540979117872439e-18,
      -9.540979117872439e-18,
      -3.2526065174565133e-18,
      1.6263032587282567e-19,
      -3.2526065174565133e-18,
      -3.0357660829594124e-18,
      -6.2341624917916505e-18,
      -9.432558900623889e-18,
      -9.595189226496714e-18,
      -1.2902005852577503e-17,
      -1.2956215961201778e-17,
      -3.0899761915836876e-18,
      -2.9815559743351372e-18,
      -9.378348791999613e-18,
      -9.595189226496714e-18,
      -9.703609443745265e-18,
      -6.288372600415926e-18,
      -1.2956215961201778e-17,
      -9.432558900623889e-18,
      -3.0357660829594124e-18,
      -9.540979117872439e-18,
      -3.2526065174565133e-18,
      -3.198396408832238e-18,
      -2.9815559743351372e-18,
      -3.144186300207963e-18,
      2.710505431213761e-19,
      -3.3068166260807885e-18,
      -3.0899761915836876e-18,
      -1.2793585635328952e-17,
      -1.2902005852577503e-17,
      -6.288372600415926e-18,
      -2.9815559743351372e-18,
      -6.288372600415926e-18,
      2.168404344971009e-19,
      -9.378348791999613e-18,
      -6.288372600415926e-18,
      1.6263032587282567e-19,
      -9.540979117872439e-18,
      -6.342582709040201e-18,
      -6.396792817664476e-18,
      -2.9815559743351372e-18,
      -3.0357660829594124e-18,
      -2.927345865710862e-18,
      -9.540979117872439e-18,
      -6.2341624917916505e-18,
      -9.75781955236954e-18,
      -6.342582709040201e-18,
      -6.179952383167375e-18,
      -3.0357660829594124e-18,
      -6.613633252161577e-18,
      -9.75781955236954e-18,
      -1.2847795743953228e-17,
      -2.927345865710862e-18,
      -6.2341624917916505e-18,
      -3.3610267347050637e-18,
      -3.2526065174565133e-18,
      -1.2847795743953228e-17,
      -6.2341624917916505e-18,
      -1.2847795743953228e-17,
      -9.486769009248164e-18,
      -6.396792817664476e-18,
      -3.198396408832238e-18,
      -9.486769009248164e-18,
      -6.342582709040201e-18,
      -9.432558900623889e-18,
      -3.0357660829594124e-18,
      -9.703609443745265e-18,
      -9.75781955236954e-18,
      -3.198396408832238e-18,
      -2.927345865710862e-18,
      -1.2902005852577503e-17,
      3.7947076036992655e-19,
      -3.2526065174565133e-18,
      -3.144186300207963e-18,
      -2.9815559743351372e-18,
      -6.2341624917916505e-18,
      -6.342582709040201e-18,
      -6.179952383167375e-18,
      -1.2847795743953228e-17,
      -9.64939933512099e-18,
      -9.703609443745265e-18,
      -9.703609443745265e-18,
      -9.540979117872439e-18,
      -6.451002926288751e-18,
      -1.2793585635328952e-17,
      -9.64939933512099e-18,
      -3.0357660829594124e-18,
      -6.342582709040201e-18,
      -1.2847795743953228e-17,
      -6.451002926288751e-18,
      -2.9815559743351372e-18,
      -6.396792817664476e-18,
      -3.198396408832238e-18,
      -6.288372600415926e-18,
      -9.540979117872439e-18,
      -6.179952383167375e-18,
      -9.595189226496714e-18,
      -9.595189226496714e-18,
      3.2526065174565133e-19,
      -9.486769009248164e-18,
      -9.595189226496714e-18,
      -6.396792817664476e-18,
      -2.9815559743351372e-18,
      -1.2902005852577503e-17,
      -6.288372600415926e-18,
      -9.75781955236954e-18,
      -9.64939933512099e-18,
      -3.2526065174565133e-18,
      -3.0899761915836876e-18,
      -6.451002926288751e-18,
      -3.0357660829594124e-18,
      -3.144186300207963e-18,
      -9.378348791999613e-18,
      -1.2902005852577503e-17,
      -9.486769009248164e-18,
      -3.2526065174565133e-18,
      -9.486769009248164e-18,
      -6.288372600415926e-18,
      -1.2902005852577503e-17,
      -3.0357660829594124e-18,
      -6.396792817664476e-18,
      -6.342582709040201e-18,
      -6.288372600415926e-18,
      -9.75781955236954e-18,
      -2.9815559743351372e-18,
      -9.595189226496714e-18,
      -6.288372600415926e-18,
      -3.2526065174565133e-18,
      -9.540979117872439e-18,
      1.6263032587282567e-19,
      -2.9815559743351372e-18,
      -3.198396408832238e-18,
      -9.486769009248164e-18,
      -3.144186300207963e-18,
      -6.613633252161577e-18,
      1.6263032587282567e-19,
      -6.396792817664476e-18,
      2.168404344971009e-19,
      -9.540979117872439e-18,
      -6.179952383167375e-18,
      -3.0899761915836876e-18,
      -3.0899761915836876e-18,
      1.6263032587282567e-19,
      -2.9815559743351372e-18,
      -1.2902005852577503e-17,
      1.6263032587282567e-19,
      -1.2902005852577503e-17,
      -2.927345865710862e-18,
      -3.0899761915836876e-18,
      -9.432558900623889e-18,
      -1.2956215961201778e-17,
      -3.0899761915836876e-18,
      -9.540979117872439e-18,
      -3.144186300207963e-18,
      -3.0899761915836876e-18,
      -6.342582709040201e-18,
      -9.540979117872439e-18,
      -6.2341624917916505e-18,
      -3.0357660829594124e-18,
      1.6263032587282567e-19,
      -3.0357660829594124e-18,
      -6.288372600415926e-18,
      -3.198396408832238e-18,
      -2.9815559743351372e-18,
      -2.9815559743351372e-18,
      -3.144186300207963e-18,
      -6.288372600415926e-18,
      -9.432558900623889e-18,
      -1.2956215961201778e-17,
      -3.0899761915836876e-18,
      2.710505431213761e-19,
      -6.396792817664476e-18,
      -6.2341624917916505e-18,
      -9.540979117872439e-18,
      -6.288372600415926e-18,
      -1.2793585635328952e-17,
      -6.179952383167375e-18,
      -3.0357660829594124e-18,
      -9.486769009248164e-18,
      -3.198396408832238e-18,
      -2.9815559743351372e-18,
      -9.540979117872439e-18,
      -2.927345865710862e-18,
      -3.2526065174565133e-18,
      -9.540979117872439e-18,
      -6.613633252161577e-18,
      -3.0899761915836876e-18,
      -9.540979117872439e-18,
      -6.342582709040201e-18,
      -3.3068166260807885e-18,
      -9.540979117872439e-18,
      -9.486769009248164e-18,
      -3.2526065174565133e-18,
      -3.0899761915836876e-18,
      -1.2847795743953228e-17,
      -9.703609443745265e-18,
      -9.64939933512099e-18,
      -1.2956215961201778e-17,
      -2.8731357570865868e-18,
      -6.396792817664476e-18,
      -6.342582709040201e-18,
      -2.927345865710862e-18,
      -9.540979117872439e-18,
      -9.378348791999613e-18,
      -1.2956215961201778e-17,
      -6.451002926288751e-18,
      -3.0899761915836876e-18,
      -3.0899761915836876e-18,
      -3.2526065174565133e-18,
      -3.0357660829594124e-18,
      -6.342582709040201e-18,
      2.168404344971009e-19,
      -3.198396408832238e-18,
      -6.396792817664476e-18,
      -9.595189226496714e-18,
      -3.3068166260807885e-18,
      -1.2956215961201778e-17,
      -3.144186300207963e-18,
      -3.0357660829594124e-18,
      -9.486769009248164e-18,
      -3.0357660829594124e-18,
      -1.2793585635328952e-17,
      -3.198396408832238e-18,
      -3.0899761915836876e-18,
      3.7947076036992655e-19,
      -9.64939933512099e-18,
      2.168404344971009e-19,
      -3.0899761915836876e-18,
      -9.75781955236954e-18,
      -9.703609443745265e-18,
      -6.288372600415926e-18,
      -1.2902005852577503e-17,
      -9.75781955236954e-18,
      -3.0357660829594124e-18,
      -6.179952383167375e-18,
      -3.0357660829594124e-18,
      -3.0357660829594124e-18,
      -3.0899761915836876e-18,
      -3.198396408832238e-18,
      -1.2847795743953228e-17,
      -6.451002926288751e-18,
      -9.540979117872439e-18,
      -9.595189226496714e-18,
      -1.2956215961201778e-17,
      -9.432558900623889e-18,
      -9.540979117872439e-18,
      -9.540979117872439e-18,
      2.710505431213761e-19,
      -3.144186300207963e-18,
      -9.432558900623889e-18,
      -6.288372600415926e-18,
      -9.378348791999613e-18,
      -3.144186300207963e-18,
      -9.595189226496714e-18,
      -3.198396408832238e-18,
      -2.8731357570865868e-18,
      -9.540979117872439e-18,
      -3.198396408832238e-18,
      -3.0899761915836876e-18,
      -9.595189226496714e-18,
      -6.2341624917916505e-18,
      -6.451002926288751e-18,
      -3.0899761915836876e-18,
      2.710505431213761e-19,
      -9.486769009248164e-18,
      -9.378348791999613e-18,
      -9.540979117872439e-18,
      -3.2526065174565133e-18,
      1.6263032587282567e-19,
      -3.0357660829594124e-18,
      -3.0357660829594124e-18,
      -1.2902005852577503e-17,
      -2.927345865710862e-18,
      -6.396792817664476e-18,
      -9.432558900623889e-18,
      -9.595189226496714e-18,
      -6.288372600415926e-18,
      1.6263032587282567e-19,
      -9.595189226496714e-18,
      -9.595189226496714e-18,
      -9.324138683375338e-18,
      -2.927345865710862e-18,
      -3.0899761915836876e-18,
      -9.595189226496714e-18,
      -9.432558900623889e-18,
      -3.2526065174565133e-18,
      1.6263032587282567e-19,
      -2.9815559743351372e-18,
      -9.540979117872439e-18,
      -6.2341624917916505e-18,
      -3.0357660829594124e-18,
      1.6263032587282567e-19,
      -9.595189226496714e-18
    ],
    [
      9.413482840832077e-16,
      9.287179941382296e-16,
      9.349263543352453e-16,
      9.354626666234307e-16,
      9.367704143213197e-16,
      9.315445491590397e-16,
      9.480906831382353e-16,
      9.358265521789812e-16,
      9.365640188257747e-16,
      9.436300809805508e-16,
      9.358215722868447e-16,
      9.334600514425229e-16,
      9.297907274316018e-16,
      -1.1884309332763572e-15,
      9.406082402142558e-16,
      9.322890382060169e-16,
      9.210659714100634e-16,
      9.26313544967353e-16,
      9.39391502909271e-16,
      9.358640095116054e-16,
      9.427991292795393e-16,
      9.426729659159127e-16,
      9.326523024676274e-16,
      2.0533646069407438e-16,
      2.257873657688928e-16,
      2.8960992489908e-16,
      2.496103046476419e-16,
      2.6966506989600003e-16,
      2.22316199708757e-16,
      2.5103092633951795e-16,
      2.7845100198575187e-16,
      2.812261913153051e-16,
      7.158580214798178e-16,
      7.27365733807576e-16,
      7.277279627476403e-16,
      -1.0807298829175614e-14,
      -1.0808951978546861e-14,
      -1.0812786226994975e-14,
      -1.0805026707127793e-14,
      -1.0822782439539934e-14,
      -1.0821371000698247e-14,
      -1.083111847103199e-14,
      -1.0816958977008879e-14,
      -1.0820355005565886e-14,
      -1.0819633778762209e-14,
      -1.0796020036326375e-14,
      -1.0812130840107858e-14,
      -1.0818101420541505e-14,
      -1.080845996245314e-14,
      -1.0809389468719614e-14,
      -1.081670319772707e-14,
      -1.0825952102846535e-14,
      -1.0828103772627603e-14,
      -1.081146403827563e-14,
      -1.0828153897917856e-14,
      -1.0833808784736313e-14,
      -1.082089781211467e-14,
      -1.0819169001814698e-14,
      -1.0837546642037209e-14,
      -1.0813422326913922e-14,
      -1.0827070816786966e-14,
      -1.0805856250090851e-14,
      -1.0813586256543476e-14,
      -1.0809605309773133e-14,
      -1.081331814982323e-14,
      -1.3140075953550216e-15,
      -1.3144574421577457e-15,
      -1.3374537344783296e-15,
      -1.3172720861950517e-15,
      -1.3217502274758035e-15,
      -1.335977281045805e-15,
      -1.3287106698777632e-15,
      -1.3195941954829715e-15,
      -1.3288327469493805e-15,
      -1.3292814233611002e-15,
      -1.3507454923231077e-15,
      -1.3228808921151496e-15,
      -1.3498690298243654e-15,
      -1.3583810580516692e-15,
      -1.3351780674403232e-15,
      -1.3397510139103109e-15,
      -1.3381222924608064e-15,
      -1.3111680592494193e-15,
      -1.3064629372180193e-15,
      -1.3285690916832288e-15,
      0.17149858514250912,
      0.17149858514250907,
      0.17149858514250918,
      0.17149858514250912,
      0.171498585142509,
      0.17149858514250924,
      0.17149858514250912,
      0.17149858514250904,
      0.17149858514250904,
      0.17149858514250915,
      0.1714985851425091,
      0.17149858514250915,
      0.1714985851425091,
      0.17149858514250904,
      0.1714985851425091,
      0.1714985851425091,
      0.1714985851425092,
      0.1714985851425092,
      0.1714985851425091,
      0.17149858514250907,
      0.17149858514250918,
      0.1714985851425092,
      7.586663912546588e-16,
      0.17149858514250904,
      0.17149858514250932,
      0.17149858514250904,
      0.17149858514250904,
      0.17149858514250924,
      0.17149858514250907,
      0.17149858514250918,
      0.17149858514250915,
      0.17149858514250912,
      0.17149858514250904,
      0.1714985851425091,
      0.1714985851425092,
      7.300502283476857e-16,
      7.287633516942415e-16,
      7.270529327601722e-16,
      7.248337159271107e-16,
      7.293911710468669e-16,
      7.256760397767399e-16,
      7.271991661627252e-16,
      7.227453390704095e-16,
      7.234001483706288e-16,
      7.165427337806858e-16,
      7.173014316493898e-16,
      7.215723526999567e-16,
      7.246334936002891e-16,
      7.193125825700836e-16,
      7.232923640826959e-16,
      7.220140975977631e-16,
      7.24733764585635e-16,
      7.203822442094088e-16,
      7.607946830827655e-16,
      7.578961560026172e-16,
      7.64110405175849e-16,
      7.655412894509865e-16,
      7.593339083646432e-16,
      7.5879029094497e-16,
      7.521118744768629e-16,
      7.5580032607310555e-16,
      7.700300794603892e-16,
      7.646196650500803e-16,
      7.565898059694171e-16,
      7.585130070147096e-16,
      7.607251901185675e-16,
      7.472730368316442e-16,
      7.604077356762574e-16,
      7.540834243065883e-16,
      7.562751129734227e-16,
      7.61405126459535e-16,
      7.67890416812098e-16,
      7.60734994954404e-16,
      -3.9749179067396997e-16,
      -4.13466043283003e-16,
      -3.9675094168674014e-16,
      -3.962533236378699e-16,
      -4.140461723712622e-16,
      -3.962567396577607e-16,
      -4.108349654641553e-16,
      -4.0997116800742607e-16,
      -4.190129592082424e-16,
      -4.113943426642403e-16,
      -4.0886015074726715e-16,
      -4.22716209674327e-16,
      -3.990435535961989e-16,
      -3.9443139454381375e-16,
      -4.0793109879777983e-16,
      -4.066704700145e-16,
      -3.949753855963768e-16,
      8.401975340992192e-16,
      8.350321765777314e-16,
      8.13876911777333e-16,
      8.419080551983403e-16,
      8.259840516237489e-16,
      8.18773975336961e-16,
      8.365798347937947e-16,
      8.492922603265299e-16,
      8.465293334889945e-16,
      8.317022691201616e-16,
      8.446885479360129e-16,
      8.50425396887859e-16,
      8.292596651580162e-16,
      8.265569860407913e-16,
      8.336452379843456e-16,
      7.882293774285977e-16,
      7.911622394659634e-16,
      7.904205169011628e-16,
      7.898915690228642e-16,
      7.8058967999565435e-16,
      7.866442407365826e-16,
      7.769229652845697e-16,
      7.88484926825031e-16,
      7.902357843555632e-16,
      7.859566077761057e-16,
      7.9775190368885465e-16,
      7.896693108143456e-16,
      7.940152909209152e-16,
      7.815666533167174e-16,
      7.821306832394967e-16,
      7.884769300267717e-16,
      7.83664093074534e-16,
      7.986608895188825e-16,
      7.85096018293872e-16,
      7.914828980533823e-16,
      -2.5551620991844854e-17,
      -7.419590077583393e-17,
      -8.630531606577116e-17,
      -7.112393119518847e-17,
      -9.049304716438409e-19,
      -4.372385966010217e-17,
      -5.350720019561778e-18,
      2.9407166357973154e-14,
      2.942566762525881e-14,
      2.9405418538265635e-14,
      2.939482277298452e-14,
      2.9403934023881184e-14,
      2.939734432955716e-14,
      2.939843292007036e-14,
      2.940479303814658e-14,
      2.939149503389688e-14,
      2.939612758596356e-14,
      2.9406123468214413e-14,
      2.938963392813649e-14,
      2.938530039286876e-14,
      2.9399566867424063e-14,
      2.940565235696088e-14,
      2.940527795479676e-14,
      2.940281820393938e-14,
      2.9410876207862455e-14,
      2.9400813335665306e-14,
      2.938082279306668e-14,
      2.941969271528632e-14,
      2.9393165994375446e-14,
      2.941090379858359e-14,
      4.3666464581112824e-17,
      4.962939977910221e-17,
      7.601558896574857e-17,
      6.679767902262104e-17,
      3.075473647746584e-17,
      4.555238786464024e-17,
      6.236309600406782e-17,
      4.529396226289092e-17,
      4.5998659884203825e-17,
      3.328505304969278e-17,
      -3.3089030215862136e-16,
      -3.199891514542649e-16,
      -3.3282526834115665e-16,
      -3.3222735690559563e-16,
      -3.4619323083572296e-16,
      -3.257187922313164e-16,
      -3.331234092097499e-16,
      -3.3530219546713138e-16,
      -3.450597939111922e-16,
      -3.1791373225518855e-16,
      -3.2837892483107566e-16,
      -3.3546835734904314e-16,
      -3.16066877069839e-16,
      -3.170626394893118e-16,
      -3.4195553391640785e-16,
      -3.2541510957923314e-16,
      -3.3579346698015866e-16,
      -3.499276895434879e-16,
      4.789790933482e-17,
      4.091266095669818e-17,
      1.4574855289331377e-17,
      2.0300984963989063e-17,
      2.2891497181155687e-17,
      5.3074245783434e-17,
      5.181496422111884e-17,
      -3.5352959233402456e-18,
      5.516348920417359e-16,
      5.590619759225477e-16,
      5.542200394858948e-16,
      5.68202193993637e-16,
      5.455272736532676e-16,
      5.508636399958447e-16,
      5.573710933343327e-16,
      5.666457978472688e-16,
      5.49465922975137e-16,
      5.480621997114799e-16,
      5.497025101259474e-16,
      5.593940967263392e-16,
      5.614842013605134e-16,
      5.645687709611373e-16,
      5.614359054396874e-16,
      5.662054148669815e-16,
      5.610639022673434e-16,
      5.613918167657789e-16,
      5.654695534372833e-16,
      -8.608886920848045e-16,
      -8.68455090264734e-16,
      -8.532496766926665e-16,
      -8.43293788428992e-16,
      -8.78881476445254e-16,
      -8.735921850850024e-16,
      -8.402902024406502e-16,
      -8.405963787499112e-16,
      -8.562333560659349e-16,
      -8.568146788638504e-16,
      -8.583001014130352e-16,
      -8.597863904549888e-16,
      -8.622067044792264e-16,
      -1.1731160151797082e-15,
      -1.1529538043928625e-15,
      -1.160358185896918e-15,
      -1.1477765911106384e-15,
      -1.18169835233838e-15,
      -1.1385295252724377e-15,
      -1.164914100847107e-15,
      -1.1556939776791475e-15,
      -1.1682501944799221e-15,
      -1.16711115056897e-15,
      -1.1779456814972946e-15,
      -1.1758585237194397e-15,
      -4.042330286828018e-17,
      -4.864675346758277e-17,
      4.332281214499395e-17,
      5.77123039113977e-18,
      -3.6569917581335625e-17,
      2.9847943843459804e-17,
      1.719239225684257e-16,
      2.060224408746108e-16,
      2.2214600503578953e-16,
      1.62179990115045e-16,
      2.3482191411735047e-16,
      2.939657568003502e-14,
      2.940647561266036e-14,
      -6.976425170290658e-16,
      -6.619573077545523e-16,
      -6.722277836879902e-16,
      -6.714887569962876e-16,
      -7.00374282732571e-16,
      -6.697377156959295e-16,
      -7.060141772063225e-16,
      -6.891103198478452e-16,
      -6.926375679650338e-16,
      -6.948120747598709e-16,
      -6.809009216171699e-16,
      -6.918879939194047e-16,
      -6.965304229957701e-16,
      1.8969201209806386e-15,
      1.92597673920325e-15,
      1.9337829948451457e-15,
      1.906461100098511e-15,
      1.9077621427054936e-15,
      1.906461100098511e-15,
      1.9016906105395748e-15,
      1.921206249644314e-15,
      1.932048271369169e-15,
      1.8951853975046618e-15,
      1.9160020792163834e-15,
      1.9290125052862095e-15,
      1.9160020792163834e-15,
      1.921639930513308e-15,
      1.9307472287621863e-15,
      1.902124291408569e-15,
      1.9068947809675052e-15,
      1.9437576548320123e-15,
      1.9068947809675052e-15,
      1.908629504443482e-15,
      1.9359513991901167e-15,
      1.9199052070373313e-15,
      1.9077621427054936e-15,
      1.9094968661814704e-15,
      1.9264104200722443e-15,
      1.8925833122906965e-15,
      1.9060274192295168e-15,
      1.9203388879063255e-15,
      1.897787482718627e-15,
      1.8925833122906965e-15,
      1.9008232488015864e-15,
      1.916869440954372e-15,
      1.9311809096311805e-15,
      1.9155683983473892e-15,
      1.9142673557404066e-15,
      1.906461100098511e-15,
      1.9094968661814704e-15,
      1.919471526168337e-15,
      1.9090631853124762e-15,
      1.9142673557404066e-15,
      9.400917385164184e-17,
      9.249769904556571e-17,
      9.247602602986758e-17,
      9.01905662982906e-17,
      9.26492687820295e-17,
      9.518891532368568e-17,
      9.349641416660444e-17,
      8.980604660617296e-17,
      9.249845554142315e-17,
      9.966819049103005e-17,
      9.21310837731637e-17,
      9.358159651365659e-17,
      9.915312599159212e-17,
      9.320380914021707e-17,
      9.338665935276816e-17,
      9.614940945083261e-17,
      1.914701036609401e-15,
      1.9060274192295168e-15,
      1.9199052070373313e-15,
      1.9394208461420703e-15,
      1.880440247958859e-15,
      1.928145143548221e-15,
      1.9220736113823023e-15,
      1.92597673920325e-15,
      1.9051600574915284e-15,
      1.904726376622534e-15,
      1.912966313133424e-15,
      1.910797908788453e-15,
      1.9068947809675052e-15,
      1.9225072922512965e-15,
      1.9090631853124762e-15,
      1.8947517166356675e-15,
      1.8986548444566154e-15,
      1.9285788244172153e-15,
      -1.0810261568683273e-14,
      1.921206249644314e-15,
      1.930313547893192e-15,
      1.923374653989285e-15,
      1.9294461861552037e-15,
      1.91253263226443e-15,
      1.8947517166356675e-15,
      1.9077621427054936e-15,
      1.8973538018496328e-15,
      1.9290125052862095e-15,
      1.891715950552708e-15,
      1.910364227919459e-15,
      1.902124291408569e-15,
      1.9142673557404066e-15,
      -5.0304937019926626e-17,
      -8.001397834512749e-17,
      -8.320399269011926e-17,
      -4.679490317242876e-17,
      -6.74924290698383e-17,
      -3.099834724296905e-17,
      -3.722616897394996e-17,
      -3.8150041993832416e-17,
      -7.202829036837392e-17,
      -6.026848136820148e-17,
      -1.0436192598881519e-16,
      -1.0776672733214903e-16,
      2.2217801402716284e-15,
      2.2155277951164567e-15,
      2.2352666981459405e-15,
      8.359644433435131e-16,
      2.220398642688822e-15,
      2.1876672997947258e-15,
      2.1908024194914505e-15,
      2.2164110487925457e-15,
      2.2012319841961548e-15,
      -1.151512894918116e-15,
      2.221783522363215e-15,
      2.203388201469138e-15,
      2.215648951141392e-15,
      2.239969639250682e-15,
      2.2214461133590464e-15,
      2.1964438758159653e-15,
      2.2064601894475543e-15,
      1.900389567932592e-15,
      1.893450674028685e-15,
      1.9181704835613544e-15,
      1.9012569296705806e-15,
      1.9094968661814704e-15,
      1.9160020792163834e-15,
      1.938553484404082e-15,
      1.910364227919459e-15,
      -7.519335056685186e-17,
      -1.0581396843288074e-16,
      -9.444067705529441e-17,
      -6.957366628080511e-17,
      -9.842603114667997e-17,
      -1.3617792865421582e-16,
      1.8882465036007545e-15,
      1.8860780992557835e-15,
      1.9285788244172153e-15,
      1.9060274192295168e-15,
      1.908195823574488e-15,
      1.8865117801247777e-15,
      1.910364227919459e-15,
      1.912966313133424e-15,
      1.906461100098511e-15,
      1.919037845299343e-15,
      1.91253263226443e-15,
      1.9220736113823023e-15,
      1.9311809096311805e-15,
      1.8973538018496328e-15,
      9.334923543233322e-17,
      9.358547888688416e-17,
      2.939161235101824e-14,
      2.9397445350779755e-14,
      2.940665869281024e-14,
      2.9395680075478415e-14,
      2.940557306289036e-14,
      2.940640133651362e-14,
      2.9390388055373524e-14,
      2.9411677740181e-14,
      1.9268441009412385e-15,
      1.919471526168337e-15,
      1.897787482718627e-15,
      1.8995222061946038e-15,
      1.9337829948451457e-15,
      -1.081310084009887e-14,
      1.9467934209149718e-15,
      1.915134717478395e-15,
      1.895619078373656e-15,
      1.9073284618364994e-15,
      1.9060274192295168e-15,
      6.979475284292159e-17,
      5.0935655311559217e-17,
      9.263194651326367e-17,
      3.973333092223188e-17,
      6.32963812905131e-17,
      9.604233683146184e-17,
      1.9207725687753197e-15,
      1.919471526168337e-15,
      -1.2317839258897163e-16,
      -7.226657400099677e-17,
      1.919471526168337e-15,
      1.936385080059111e-15,
      1.90429269575354e-15,
      1.9251093774652617e-15,
      1.9116652705264414e-15,
      1.9142673557404066e-15,
      9.505551763243242e-17,
      9.556801192885983e-17,
      1.9099305470504646e-15,
      1.9251093774652617e-15,
      1.921206249644314e-15,
      1.921206249644314e-15,
      1.923808334858279e-15,
      1.9160020792163834e-15,
      1.9186041644303486e-15,
      9.891160130502125e-17,
      1.9307472287621863e-15,
      1.9090631853124762e-15,
      7.555183905025573e-16,
      7.419201794182502e-16,
      7.636915209736196e-16,
      7.548555075601855e-16,
      7.504544303776529e-16,
      7.492980914530474e-16,
      7.381580144861395e-16,
      7.663271264521506e-16,
      7.736451063220771e-16,
      7.5978837930709e-16,
      7.610871590975243e-16,
      7.595731658606217e-16,
      7.678310295504398e-16,
      7.491544240119676e-16,
      7.514077605139851e-16,
      -1.3587288526254087e-16,
      -1.499372253260613e-16,
      -1.476127091814475e-16,
      -1.2597745587293456e-16,
      -1.1957122785351193e-16,
      -1.4959667119283028e-16,
      -7.959640167059328e-17,
      -1.5582239024642839e-16,
      -1.1117607765635463e-16,
      8.78140527474905e-17,
      1.5868097793914403e-17,
      4.501908617348083e-17,
      4.5456598034448486e-17,
      8.98979687380174e-17,
      8.957252954314721e-17,
      9.477257289812066e-17,
      8.642719594238011e-17,
      6.18157139295669e-17,
      1.0690003878520522e-16,
      9.243825329052308e-17,
      9.915989819834652e-17,
      4.2789418465095925e-16,
      4.477646483808489e-16,
      4.301018418020059e-16,
      4.321955420126554e-16,
      4.317729814760636e-16,
      4.408480628550578e-16,
      4.3863411604802635e-16,
      4.2017683836032286e-16,
      4.246873808659583e-16,
      4.2879541352540506e-16,
      4.3736956479853875e-16,
      4.48102484563032e-16,
      4.287071416214604e-16,
      4.444227994414226e-16,
      4.2882521618438086e-16,
      4.441163945487815e-16,
      -2.352446741742229e-17,
      -2.6789014426992588e-17,
      -1.410483672333525e-17,
      -1.4983314783607112e-17,
      -1.9594354359995363e-17,
      -2.1103481312737552e-17,
      -1.7420954686509868e-17,
      -4.7867740164510195e-18,
      -8.638133001192058e-18,
      -9.932277756461334e-18,
      8.377446110304442e-17,
      1.1912292466479882e-16,
      1.4011009322400316e-16,
      1.145296140160685e-16,
      1.1538303458794638e-16,
      6.884179190127274e-17,
      -2.2298929117357894e-17,
      8.080196597990353e-17,
      8.603903733006582e-17,
      8.06177280904872e-17,
      8.684536965616391e-17,
      7.723066261568235e-17,
      6.757637185832525e-17,
      6.75363652981642e-17,
      7.562249328106127e-17,
      7.647001854485646e-17,
      6.878610853422954e-17,
      6.733224793611415e-17,
      4.4619602512082676e-17,
      7.405093100377409e-17,
      6.657135108639517e-17,
      5.2905983851529235e-17,
      4.2081432523878806e-17,
      9.461745057323073e-17,
      8.828861849458075e-17,
      5.236892142374396e-17,
      6.028544640375924e-17,
      8.253820899417023e-17,
      3.633991559956787e-16,
      3.596106192532003e-16,
      3.617513093979037e-16,
      3.744891050771679e-16,
      3.5862987613546444e-16,
      3.5969620006256747e-16,
      3.675365541962445e-16,
      3.51442299604334e-16,
      3.6231622994320737e-16,
      3.6563490271468144e-16,
      -2.58181703737391e-18,
      2.9424307521190813e-18,
      2.428696964668368e-17,
      3.336799967247891e-17,
      -1.1684287794537345e-15,
      3.675476127979867e-17,
      -3.5648930092035655e-16,
      -3.7105321445824666e-16,
      -3.8427450226491507e-16,
      -3.7606653032941845e-16,
      -4.047127198349092e-16,
      -3.9655115957426976e-16,
      8.633173474205832e-17,
      3.9192862806630374e-17,
      6.971780479688326e-17,
      8.92849538239558e-17,
      5.0619832244680826e-17,
      3.2145439867392485e-17,
      5.009330465470233e-17,
      7.631193866072837e-17,
      7.116897162867254e-18,
      -1.0321617904855119e-17,
      -3.119719309718283e-17,
      -1.8532202807976077e-17,
      1.6980067795102063e-18,
      2.3656581680682778e-18,
      -7.187221818934895e-18,
      -1.2929972897931078e-19,
      -1.8388479354250312e-16,
      -1.476218203564166e-16,
      -1.6426542729343903e-16,
      -1.5376301697557704e-16,
      -1.4298035378537432e-16,
      -2.2047225140631535e-16,
      1.387683896294508e-16,
      1.0852506321328462e-16,
      5.501971776177885e-17,
      1.5693023810173245e-16,
      1.669884295338123e-16,
      1.8004108004322977e-16,
      3.395279642819641e-16,
      3.468458471789851e-16,
      2.918582893005619e-16,
      3.7836632729531953e-16,
      3.773699532663341e-16,
      -5.605395155480186e-16,
      -5.533782542037922e-16,
      -5.594453779418683e-16,
      -5.6094948521507e-16,
      -5.419180495149327e-16,
      -5.371641806550571e-16,
      -5.44090567493296e-16,
      -1.4990211452888893e-16,
      -2.09324154301431e-16,
      -1.486369784899211e-16,
      -1.542465302768761e-16,
      -2.2649247123639576e-16,
      -1.4927214350415225e-16,
      -1.45552923661512e-16,
      8.594284523550738e-18,
      8.932662780366987e-17,
      2.8587837089676904e-17,
      1.2897480293031842e-17,
      -1.7087584587722837e-17,
      -6.126340011594221e-18,
      -3.2975406353396915e-16,
      -3.152675023896444e-16,
      -2.777708438814152e-16,
      -3.362940934379434e-16,
      -4.729071360034815e-16,
      -4.4054281786200205e-16,
      -6.241613538743881e-16,
      -4.0916502941625867e-16,
      2.2372645994796977e-15,
      2.2086831458992423e-15,
      2.225909533094298e-15,
      3.600332052863711e-16,
      1.1113659890270004e-16,
      1.904726376622534e-15,
      1.899955887063598e-15,
      1.90429269575354e-15,
      1.9133999940024182e-15,
      1.9290125052862095e-15,
      1.910364227919459e-15,
      8.917417408044522e-17,
      -5.559510531597692e-16,
      -5.541648985481957e-16,
      -5.454171887432563e-16,
      -5.509014102635395e-16,
      -5.535339634122464e-16,
      -5.440204137402629e-16,
      -5.61892255268552e-16,
      -5.659428917448006e-16,
      -5.514469505066086e-16,
      3.726927565153693e-16,
      3.622996223593077e-16,
      -3.419012902911926e-16,
      2.5348646792711094e-16,
      2.4974597043203595e-16,
      2.4909544912854464e-16,
      2.507759624958972e-16,
      2.453007415248454e-16,
      2.495833401061631e-16,
      2.625395560673649e-16,
      2.443249595696084e-16,
      2.5083017260452145e-16,
      2.4931228956304174e-16,
      2.5809432716017433e-16,
      2.5565487227208195e-16,
      2.458970527197124e-16,
      2.601543112878968e-16,
      2.5451645999097217e-16,
      2.518059545597584e-16,
      2.585822181377928e-16,
      2.5538382172896057e-16,
      2.5278173651499536e-16,
      2.566306542273189e-16,
      2.456802122852153e-16,
      2.489328188026718e-16,
      2.42861286636753e-16,
      2.5061333217002435e-16,
      2.53106997166741e-16,
      2.512096433648914e-16,
      2.502338614096544e-16,
      2.5831116759467143e-16,
      2.5083017260452145e-16,
      2.4497548087309973e-16,
      2.4752335597844066e-16,
      2.4952912999753885e-16,
      2.507217523872729e-16,
      2.463849436973309e-16,
      2.587990585722899e-16,
      2.564138137928218e-16,
      2.5890747878953846e-16,
      2.554922419462091e-16,
      2.5093859282177e-16,
      2.485533480423019e-16,
      2.494207097802903e-16,
      2.5473330042546927e-16,
      2.582027473774229e-16,
      2.5278173651499536e-16,
      2.5961221020165404e-16,
      2.4275286641950444e-16,
      2.6020852139652106e-16,
      2.504507018441515e-16,
      2.458970527197124e-16,
      2.4974597043203595e-16,
      2.5191437477700696e-16,
      2.5560066216345767e-16,
      2.5744380585668303e-16,
      2.5126385347351565e-16,
      2.480654570646834e-16,
      2.535406780357352e-16,
      2.4795703684743486e-16,
      2.4779440652156204e-16,
      2.569017047704403e-16,
      2.45029690981724e-16,
      2.4974597043203595e-16,
      2.6020852139652106e-16,
      2.5587171270657905e-16,
      2.5261910618912253e-16,
      2.569017047704403e-16,
      2.636237582398504e-16,
      2.5126385347351565e-16,
      2.5104701303901855e-16,
      2.4817387728193196e-16,
      2.4535495163346965e-16,
      2.5240226575462543e-16,
      2.502880715182787e-16,
      2.598832607447754e-16,
      2.485533480423019e-16,
      2.498001805406602e-16,
      2.54679090316845e-16,
      2.482822974991805e-16,
      2.4649336391457943e-16,
      2.5083017260452145e-16,
      2.544080397737236e-16,
      2.4562600217659103e-16,
      2.520770051028798e-16,
      2.5782327661705295e-16,
      2.4779440652156204e-16,
      2.5305278705811673e-16,
      2.515891141252613e-16,
      2.553296116203363e-16,
      2.5565487227208195e-16,
      2.618890347638736e-16,
      2.5131806358213993e-16,
      2.5435382966509934e-16,
      2.453007415248454e-16,
      2.535406780357352e-16,
      2.4204813500738886e-16,
      2.498543906492845e-16,
      2.585822181377928e-16,
      2.535406780357352e-16,
      2.491496592371689e-16,
      2.514264837993885e-16,
      2.5804011705155006e-16,
      2.522938455373769e-16,
      2.5343225781848666e-16,
      2.561427632497004e-16,
      2.5538382172896057e-16,
      2.496375502147874e-16,
      2.465475740232037e-16,
      2.555464520548334e-16,
      2.470354650008222e-16,
      2.590701091154113e-16,
      2.5034228162690297e-16,
      2.4822808739055624e-16,
      2.5917852933265983e-16,
      2.432407573971229e-16,
      2.4996281086653305e-16,
      2.4362022815749285e-16,
      2.4031341153141206e-16,
      2.494207097802903e-16,
      2.540285690133537e-16,
      2.513722736907642e-16,
      2.575522260739316e-16,
      2.45029690981724e-16,
      2.586364282464171e-16,
      2.467644144577008e-16,
      2.4779440652156204e-16,
      2.5413698923060224e-16,
      2.553296116203363e-16,
      2.504507018441515e-16,
      2.5722696542218593e-16,
      2.599374708533997e-16,
      2.520770051028798e-16,
      2.573353856394345e-16,
      2.4801124695605914e-16,
      2.5326962749261384e-16,
      2.563596036841975e-16,
      2.5218542532012833e-16,
      2.4887860869404754e-16,
      2.55275401511712e-16,
      2.535948881443595e-16,
      2.5256489608049826e-16,
      2.5766064629118013e-16,
      2.5722696542218593e-16,
      2.470354650008222e-16,
      2.448128505472269e-16,
      2.539743589047294e-16,
      2.507217523872729e-16,
      2.4757756608706494e-16,
      2.515891141252613e-16,
      2.4795703684743486e-16,
      2.574980159653073e-16,
      2.483365076078048e-16,
      2.5364909825298376e-16,
      2.5543803183758484e-16,
      2.513722736907642e-16,
      2.544622498823479e-16,
      2.5435382966509934e-16,
      2.5370330836160804e-16,
      2.538117285788566e-16,
      2.4730651554394356e-16,
      2.520770051028798e-16,
      2.5104701303901855e-16,
      2.5104701303901855e-16,
      2.4399969891786277e-16,
      2.507759624958972e-16,
      2.491496592371689e-16,
      2.4844492782505334e-16,
      2.574980159653073e-16,
      2.5738959574805875e-16,
      2.5169753434250985e-16,
      2.480654570646834e-16,
      2.5283594662361963e-16,
      2.544080397737236e-16,
      2.527275264063711e-16,
      2.467644144577008e-16,
      2.544622498823479e-16,
      2.512096433648914e-16,
      2.5348646792711094e-16,
      2.6129272356900657e-16,
      2.504507018441515e-16,
      2.5261910618912253e-16,
      2.583653777032957e-16,
      2.516433242338856e-16,
      2.540285690133537e-16,
      2.5408277912197796e-16,
      2.575522260739316e-16,
      2.498543906492845e-16,
      2.569017047704403e-16,
      2.5809432716017433e-16,
      2.559259228152033e-16,
      2.474691458698164e-16,
      2.4801124695605914e-16,
      2.528901567322439e-16,
      2.5055912206140007e-16,
      2.526733162977468e-16,
      2.5587171270657905e-16,
      2.5695591487906455e-16,
      2.6221429541561925e-16,
      2.5110122314764283e-16,
      2.594495798757812e-16,
      2.5169753434250985e-16,
      2.579316968343015e-16,
      2.565764441186946e-16,
      2.4822808739055624e-16,
      2.507759624958972e-16,
      2.540285690133537e-16,
      2.4557179206796675e-16,
      2.5234805564600116e-16,
      2.6107588313450947e-16,
      2.5256489608049826e-16,
      2.540285690133537e-16,
      2.4692704478357363e-16,
      2.4334917761437147e-16,
      2.507759624958972e-16,
      2.461681032628338e-16,
      2.550585610772149e-16,
      2.5722696542218593e-16,
      2.4801124695605914e-16,
      2.455175819593425e-16,
      2.601543112878968e-16,
      2.5386593868748086e-16,
      2.574980159653073e-16,
      2.474149357611921e-16,
      2.4969176032341167e-16,
      2.533780477098624e-16,
      2.443791696782327e-16,
      2.625395560673649e-16,
      2.52510685971874e-16,
      2.5093859282177e-16,
      2.557090823807062e-16,
      2.559801329238276e-16,
      2.5218542532012833e-16,
      2.4736072565256784e-16,
      2.5191437477700696e-16,
      2.5625118346694897e-16,
      2.5435382966509934e-16,
      2.5093859282177e-16,
      2.489870289112961e-16,
      2.5522119140308774e-16,
      2.53106997166741e-16,
      2.518059545597584e-16,
      2.484991379336776e-16,
      2.520227949942555e-16,
      2.5148069390801275e-16,
      2.5017965130103015e-16,
      2.5034228162690297e-16,
      2.4627652348008233e-16,
      2.5261910618912253e-16,
      2.588532686809142e-16,
      2.533780477098624e-16,
      2.55275401511712e-16,
      2.603169416137696e-16,
      2.5500435096859064e-16,
      2.498543906492845e-16,
      2.445960101127298e-16,
      2.4383706859198995e-16,
      2.4990860075790877e-16,
      2.56847494661816e-16,
      2.465475740232037e-16,
      2.3884973859855663e-16,
      2.5874484846366563e-16,
      2.5630539357557325e-16,
      2.4931228956304174e-16,
      2.535406780357352e-16,
      2.5283594662361963e-16,
      2.4795703684743486e-16,
      2.4996281086653305e-16,
      2.4909544912854464e-16,
      2.515891141252613e-16,
      2.423733956591345e-16,
      2.4405390902648705e-16,
      2.4730651554394356e-16,
      2.465475740232037e-16,
      2.575522260739316e-16,
      2.522396354287526e-16,
      2.428070765281287e-16,
      2.4421653935235987e-16,
      2.4774019641293776e-16,
      2.373318555570769e-16,
      2.5543803183758484e-16,
      2.516433242338856e-16,
      2.446502202213541e-16,
      2.559801329238276e-16,
      2.564138137928218e-16,
      2.4291549674537727e-16,
      2.498001805406602e-16,
      2.5581750259795477e-16,
      2.546248802082207e-16,
      2.514264837993885e-16,
      2.5478751053409354e-16,
      2.5299857694949246e-16,
      2.542454094478508e-16,
      2.5413698923060224e-16,
      2.565764441186946e-16,
      2.5847379792054426e-16,
      2.563596036841975e-16,
      2.5175174445113413e-16,
      2.533780477098624e-16,
      2.498001805406602e-16,
      2.4904123901992037e-16,
      2.5538382172896057e-16,
      2.5457067009959644e-16,
      2.559801329238276e-16,
      2.5457067009959644e-16,
      2.5104701303901855e-16,
      2.5148069390801275e-16,
      2.5126385347351565e-16,
      2.491496592371689e-16,
      2.544622498823479e-16,
      2.476317761956892e-16,
      2.520227949942555e-16,
      2.4817387728193196e-16,
      2.6443690986921453e-16,
      2.4969176032341167e-16,
      2.476859863043135e-16,
      2.5668486433594317e-16,
      2.574980159653073e-16,
      2.520770051028798e-16,
      2.4817387728193196e-16,
      2.5370330836160804e-16,
      2.46601784131828e-16,
      2.4736072565256784e-16,
      2.4627652348008233e-16,
      2.539743589047294e-16,
      2.559801329238276e-16,
      2.5782327661705295e-16,
      2.5321541738398956e-16,
      2.474691458698164e-16,
      2.5364909825298376e-16,
      2.509928029303943e-16,
      2.5560066216345767e-16,
      2.539743589047294e-16,
      2.4708967510944646e-16,
      2.4443337978685697e-16,
      2.4714388521807074e-16,
      2.509928029303943e-16,
      2.5261910618912253e-16,
      2.5587171270657905e-16,
      2.546248802082207e-16,
      2.456802122852153e-16,
      2.537575184702323e-16,
      2.4952912999753885e-16,
      2.500170209751573e-16,
      2.515891141252613e-16,
      2.478486166301863e-16,
      2.4774019641293776e-16,
      2.563596036841975e-16,
      2.4952912999753885e-16,
      2.458970527197124e-16,
      2.5131806358213993e-16,
      2.4671020434907653e-16,
      2.610216730258852e-16,
      2.4752335597844066e-16,
      2.5055912206140007e-16,
      2.4795703684743486e-16,
      2.5896168889816273e-16,
      2.516433242338856e-16,
      2.526733162977468e-16,
      2.4774019641293776e-16,
      2.396086801192965e-16,
      2.533780477098624e-16,
      2.4887860869404754e-16,
      2.511554332562671e-16,
      2.4074709240040626e-16,
      2.426444462022559e-16,
      2.4421653935235987e-16,
      2.579316968343015e-16,
      2.625395560673649e-16,
      2.505049119527758e-16,
      2.618890347638736e-16,
      2.582027473774229e-16,
      2.522396354287526e-16,
      2.4708967510944646e-16,
      2.601001011792725e-16,
      2.5673907444456745e-16,
      2.577690665084287e-16,
      2.529443668408682e-16,
      2.608048325913881e-16,
      2.574980159653073e-16,
      2.5408277912197796e-16,
      2.507759624958972e-16,
      2.4860755815092617e-16,
      2.512096433648914e-16,
      2.5055912206140007e-16,
      2.5055912206140007e-16,
      2.4210234511601314e-16,
      2.4752335597844066e-16,
      2.487159783681747e-16,
      2.6107588313450947e-16,
      2.5804011705155006e-16,
      2.5169753434250985e-16,
      2.5175174445113413e-16,
      2.518601646683827e-16,
      2.454633718507182e-16,
      2.538117285788566e-16,
      2.564138137928218e-16,
      2.4600547293696096e-16,
      2.458970527197124e-16,
      2.4839071771642907e-16,
      2.529443668408682e-16,
      2.5392014879610514e-16,
      2.467644144577008e-16,
      2.5473330042546927e-16,
      2.4947491988891457e-16,
      2.5261910618912253e-16,
      2.5088438271314573e-16,
      2.6129272356900657e-16,
      2.5603434303245187e-16,
      2.430781270712501e-16,
      2.455175819593425e-16,
      2.5500435096859064e-16,
      2.5234805564600116e-16,
      2.564138137928218e-16,
      2.4887860869404754e-16,
      2.4015078120553923e-16,
      2.500170209751573e-16,
      2.588532686809142e-16,
      2.502338614096544e-16,
      2.5841958781192e-16,
      2.559801329238276e-16,
      2.498001805406602e-16,
      2.3771132631744685e-16,
      2.607506224827638e-16,
      2.4952912999753885e-16,
      2.546248802082207e-16,
      2.4839071771642907e-16,
      2.572811755308102e-16,
      2.5305278705811673e-16,
      2.5370330836160804e-16,
      2.516433242338856e-16,
      2.5852800802916853e-16,
      2.533780477098624e-16,
      2.4492127076447545e-16,
      2.4427074946098415e-16,
      2.468186245663251e-16,
      2.5473330042546927e-16,
      2.4801124695605914e-16,
      2.5522119140308774e-16,
      2.5478751053409354e-16,
      2.481196671733077e-16,
      2.5191437477700696e-16,
      2.474149357611921e-16,
      2.520770051028798e-16,
      2.4584284261108813e-16,
      2.539743589047294e-16,
      2.5061333217002435e-16,
      2.480654570646834e-16,
      2.4540916174209393e-16,
      2.5088438271314573e-16,
      2.4882439858542327e-16,
      2.417770844642675e-16,
      2.500170209751573e-16,
      2.496375502147874e-16,
      2.500712310837816e-16,
      2.461681032628338e-16,
      2.4671020434907653e-16,
      2.540285690133537e-16,
      2.524564758632497e-16,
      2.472523054353193e-16,
      2.622685055242435e-16,
      2.5581750259795477e-16,
      2.5131806358213993e-16,
      2.5175174445113413e-16,
      2.557090823807062e-16,
      2.5516698129446347e-16,
      2.4188550468151604e-16,
      2.504507018441515e-16,
      2.4161445413839466e-16,
      2.513722736907642e-16,
      2.4925807945441747e-16,
      2.4822808739055624e-16,
      2.5066754227864863e-16,
      2.559259228152033e-16,
      2.5581750259795477e-16,
      2.621058751983707e-16,
      2.489328188026718e-16,
      2.592327394412841e-16,
      2.4774019641293776e-16,
      2.535948881443595e-16,
      2.5153490401663703e-16,
      2.474149357611921e-16,
      2.5191437477700696e-16,
      2.500170209751573e-16,
      2.504507018441515e-16,
      2.489870289112961e-16,
      2.5896168889816273e-16,
      2.5343225781848666e-16,
      2.485533480423019e-16,
      2.5831116759467143e-16,
      2.410723530521519e-16,
      2.4882439858542327e-16,
      2.479028267388106e-16,
      2.505049119527758e-16,
      2.478486166301863e-16,
      2.4904123901992037e-16,
      2.5451645999097217e-16,
      2.4736072565256784e-16,
      2.461681032628338e-16,
      2.514264837993885e-16,
      2.5326962749261384e-16,
      2.5652223401007035e-16,
      2.5495014085996637e-16,
      2.4513811119897255e-16,
      2.443791696782327e-16,
      2.592869495499084e-16,
      2.358681826242215e-16,
      2.520227949942555e-16,
      2.4584284261108813e-16,
      2.520227949942555e-16,
      2.5429961955647506e-16,
      2.5847379792054426e-16,
      2.4475864043860263e-16,
      2.524564758632497e-16,
      2.5538382172896057e-16,
      2.4687283467494936e-16,
      2.5495014085996637e-16,
      2.511554332562671e-16,
      2.555464520548334e-16,
      2.4969176032341167e-16,
      2.4557179206796675e-16,
      2.4822808739055624e-16,
      2.5538382172896057e-16,
      2.4492127076447545e-16,
      2.439454888092385e-16,
      2.4974597043203595e-16,
      2.4866176825955044e-16,
      2.557090823807062e-16,
      2.5543803183758484e-16,
      2.572811755308102e-16,
      2.4822808739055624e-16,
      2.4399969891786277e-16,
      2.489870289112961e-16,
      2.520770051028798e-16,
      2.5110122314764283e-16,
      2.4513811119897255e-16,
      2.5516698129446347e-16,
      2.4866176825955044e-16,
      2.54679090316845e-16,
      2.5543803183758484e-16,
      2.4470443032997835e-16,
      2.48770188476799e-16,
      2.528901567322439e-16,
      2.570643350963131e-16,
      2.5169753434250985e-16,
      2.482822974991805e-16,
      2.5126385347351565e-16,
      2.472523054353193e-16,
      2.5516698129446347e-16,
      2.548959307513421e-16,
      2.415602440297704e-16,
      2.597206304189026e-16,
      2.539743589047294e-16,
      2.4671020434907653e-16,
      2.5061333217002435e-16,
      2.588532686809142e-16,
      2.4882439858542327e-16,
      2.548417206427178e-16,
      2.5478751053409354e-16,
      2.528901567322439e-16,
      2.479028267388106e-16,
      2.564138137928218e-16,
      2.4990860075790877e-16,
      2.5543803183758484e-16,
      2.590701091154113e-16,
      2.554922419462091e-16,
      2.480654570646834e-16,
      2.463849436973309e-16,
      2.4969176032341167e-16,
      2.48770188476799e-16,
      2.4752335597844066e-16,
      2.463849436973309e-16,
      2.5169753434250985e-16,
      2.4692704478357363e-16,
      2.557090823807062e-16,
      2.483365076078048e-16,
      2.53106997166741e-16,
      2.4882439858542327e-16,
      2.522396354287526e-16,
      2.5088438271314573e-16,
      2.535948881443595e-16,
      2.4779440652156204e-16,
      2.608048325913881e-16,
      2.481196671733077e-16,
      2.4605968304558523e-16,
      2.571185452049374e-16,
      2.5321541738398956e-16,
      2.458970527197124e-16,
      2.5196858488563123e-16,
      2.4752335597844066e-16,
      2.520227949942555e-16,
      2.5934115965853266e-16,
      2.6264797628461345e-16,
      2.498543906492845e-16,
      2.482822974991805e-16,
      2.463849436973309e-16,
      2.4952912999753885e-16,
      2.502880715182787e-16,
      2.4795703684743486e-16,
      2.47198095326695e-16,
      2.6026273150514534e-16,
      2.5131806358213993e-16,
      2.512096433648914e-16,
      2.527275264063711e-16,
      2.5234805564600116e-16,
      2.621058751983707e-16,
      2.504507018441515e-16,
      2.5738959574805875e-16,
      2.49366499671666e-16,
      2.5451645999097217e-16,
      2.575522260739316e-16,
      2.4839071771642907e-16,
      2.638405986743475e-16,
      2.4296970685400154e-16,
      2.4708967510944646e-16,
      2.4540916174209393e-16,
      2.524564758632497e-16,
      2.551127711858392e-16,
      2.463307335887066e-16,
      2.472523054353193e-16,
      2.5646802390144607e-16,
      2.5131806358213993e-16,
      2.4708967510944646e-16,
      2.4492127076447545e-16,
      2.4974597043203595e-16,
      2.6178061454662505e-16,
      2.618890347638736e-16,
      2.559259228152033e-16,
      2.4535495163346965e-16,
      2.439454888092385e-16,
      2.4378285848336567e-16,
      2.520770051028798e-16,
      2.5169753434250985e-16,
      2.404760418572849e-16,
      2.5012544119240587e-16,
      2.528901567322439e-16,
      2.5538382172896057e-16,
      2.518601646683827e-16,
      2.3489240066898454e-16,
      2.597206304189026e-16,
      2.577690665084287e-16,
      2.673642557349254e-16,
      2.548417206427178e-16,
      2.5451645999097217e-16,
      2.570643350963131e-16,
      2.614553538948794e-16,
      2.482822974991805e-16,
      2.417770844642675e-16,
      2.438912787006142e-16,
      2.5896168889816273e-16,
      2.586364282464171e-16,
      2.459512628283367e-16,
      2.4291549674537727e-16,
      2.529443668408682e-16
    ],
    [
      3.493780770914175e-16,
      3.392164031485812e-16,
      3.305702890151546e-16,
      3.4330861463802004e-16,
      3.2418797311998336e-16,
      3.476788053131056e-16,
      3.5370471777030846e-16,
      3.3171791019785753e-16,
      3.2416690429017553e-16,
      3.507490359683755e-16,
      3.5048146674087783e-16,
      3.4735041072928173e-16,
      3.4117522786467957e-16,
      7.724003668080848e-16,
      3.3233523753823595e-16,
      3.344757652325576e-16,
      3.332231018863626e-16,
      3.3939136652574527e-16,
      3.364537497910129e-16,
      3.4323643506133264e-16,
      3.4324903309314765e-16,
      3.1777605030457617e-16,
      3.3016491229409007e-16,
      4.608382801101979e-16,
      4.450183187981881e-16,
      4.276366380784855e-16,
      4.379460974127009e-16,
      4.3378911228934437e-16,
      4.3171267463860613e-16,
      4.3025891745404936e-16,
      4.2761690627879564e-16,
      4.438625714931616e-16,
      8.241215533323072e-17,
      8.365589788818722e-17,
      1.0175630291221905e-16,
      2.402707489627988e-16,
      2.3722755628112875e-16,
      2.455030515001649e-16,
      2.3415201595403157e-16,
      2.6020532877449418e-16,
      2.726901792071435e-16,
      2.8088993281196626e-16,
      2.645505604627061e-16,
      2.61342220007151e-16,
      2.619823220302862e-16,
      2.273751094367517e-16,
      2.3909254509816126e-16,
      2.7172799224433383e-16,
      2.50906795416964e-16,
      2.3335808114357725e-16,
      2.359840393434174e-16,
      2.705832975342654e-16,
      2.7554348253522584e-16,
      2.2381780191578283e-16,
      2.821612627238484e-16,
      2.802663256645704e-16,
      2.7372255642804913e-16,
      2.60675954777851e-16,
      2.8594355286113423e-16,
      2.6499465449432314e-16,
      2.7959956863090614e-16,
      2.4332042923090893e-16,
      2.289991508672572e-16,
      2.429012804504664e-16,
      2.303789100351209e-16,
      -9.991712530838543e-17,
      -8.47879180079518e-17,
      -1.015702167608083e-16,
      -1.060602249582558e-16,
      -1.0334704505321195e-16,
      -1.0221823957490557e-16,
      -1.1025302368820953e-16,
      -1.0224829828779155e-16,
      -9.908878222538216e-17,
      -1.0232823012818334e-16,
      -9.133594622503902e-17,
      -8.782922312828534e-17,
      -8.20201837937288e-17,
      -9.423895819686584e-17,
      -9.343324803597126e-17,
      -1.1592635732583742e-16,
      -8.923078425088499e-17,
      -1.0435692361990653e-16,
      -8.768299580145835e-17,
      -8.664221739662782e-17,
      -3.709917861665239e-15,
      -3.706786164307919e-15,
      -3.715094776918995e-15,
      -3.688723071229559e-15,
      -3.669436285734983e-15,
      -3.757198020199313e-15,
      -3.804006916837638e-15,
      -3.7322230243042325e-15,
      -3.713609097773471e-15,
      -3.734956606596273e-15,
      -3.721284537727528e-15,
      -3.745286085622024e-15,
      -3.729392844851503e-15,
      -3.780979385996912e-15,
      -3.716304663683357e-15,
      -3.692819266592771e-15,
      -3.730726465704585e-15,
      -3.758048527655904e-15,
      -3.7483985571413694e-15,
      -3.693023100097489e-15,
      -3.687989690894149e-15,
      -3.70193994856611e-15,
      1.115102435553441e-16,
      -3.772806896207796e-15,
      -3.759816355765629e-15,
      -3.78239992527116e-15,
      -3.692115001001267e-15,
      -3.728889643663023e-15,
      -3.676707359130716e-15,
      -3.7160949704062065e-15,
      -3.7030818064463645e-15,
      -3.759416560840379e-15,
      -3.695851539133455e-15,
      -3.7230333495094266e-15,
      -3.7436422104698335e-15,
      1.0032935395174498e-16,
      9.843694972630297e-17,
      9.488236390289332e-17,
      8.845728416109884e-17,
      8.75433787872242e-17,
      9.744560538119262e-17,
      8.91834592081554e-17,
      9.339850487745102e-17,
      8.95246827444091e-17,
      8.486926760051258e-17,
      8.63847113809282e-17,
      1.0360561822715672e-16,
      1.0524172175103514e-16,
      9.330556251213144e-17,
      8.297434666018445e-17,
      8.954323830287904e-17,
      8.86691945468405e-17,
      8.948962299053054e-17,
      1.2003276669870782e-16,
      1.1604719026639882e-16,
      1.114744475951956e-16,
      1.275565568075934e-16,
      1.186575334928338e-16,
      1.1866953266055183e-16,
      1.1106925075592496e-16,
      1.1885500825909741e-16,
      1.270377985411676e-16,
      1.2698055557626629e-16,
      1.1346618453840173e-16,
      1.168790616826283e-16,
      1.1653115220757215e-16,
      1.104397263503953e-16,
      1.3171990782922806e-16,
      1.13167379319011e-16,
      1.0978352537452842e-16,
      1.1856524605962867e-16,
      1.2382410980277034e-16,
      1.205583736247867e-16,
      1.0440686451595219e-16,
      8.725490806732499e-17,
      9.226517848176599e-17,
      1.0348944680702534e-16,
      8.974016335423837e-17,
      1.0565625149261327e-16,
      9.452044171276052e-17,
      9.148304795070042e-17,
      8.02511411848568e-17,
      8.536998831125459e-17,
      8.579392118538595e-17,
      8.01567060476525e-17,
      9.929537688598871e-17,
      1.041751844031311e-16,
      9.635169015394812e-17,
      9.725613279512976e-17,
      1.0330299617486371e-16,
      -2.7627344791888585e-16,
      -2.9383953899569476e-16,
      -2.99154929852771e-16,
      -2.9385446601662513e-16,
      -2.887817367906543e-16,
      -2.995528190094972e-16,
      -2.7541757472365146e-16,
      -2.7042196515348877e-16,
      -2.7182272280859686e-16,
      -2.6369993190620353e-16,
      -2.6319122364341653e-16,
      -2.6634898137128265e-16,
      -2.9001651967151537e-16,
      -2.887185468023506e-16,
      -2.676753797100887e-16,
      9.220950461604174e-17,
      7.537270563502098e-17,
      7.176289800060253e-17,
      8.758792959675628e-17,
      8.417743690016416e-17,
      8.605511450792016e-17,
      8.611109046000918e-17,
      9.842456853115805e-17,
      9.176962059783151e-17,
      9.418279359838959e-17,
      1.0375376083078203e-16,
      8.918473214583531e-17,
      9.928897275220265e-17,
      7.739297107518558e-17,
      8.431513900507901e-17,
      8.836808284759997e-17,
      7.907990096313341e-17,
      9.270834562534363e-17,
      1.0319272349983851e-16,
      1.0119098464691236e-16,
      2.117848309025906e-17,
      2.0634917030435567e-17,
      2.0533804595090982e-17,
      2.5928240094982747e-17,
      1.8694961541140893e-17,
      3.217040894358973e-17,
      3.3995211206988453e-17,
      -2.4353301151141587e-16,
      -2.859591653397808e-16,
      -2.4436855251541654e-16,
      -1.6251782376605458e-16,
      -2.256441226999056e-16,
      -2.3248712199800835e-16,
      -1.838919531403997e-16,
      -2.26179332708226e-16,
      -1.8381819561740936e-16,
      -2.2564533793841125e-16,
      -1.9266225525730463e-16,
      -1.9325867478402542e-16,
      -1.9304344636174348e-16,
      -1.9585158397825294e-16,
      -1.9496675303754632e-16,
      -2.229916020135248e-16,
      -2.442530532374897e-16,
      -2.254429178012718e-16,
      -2.3251996881494365e-16,
      -1.7167397683911024e-16,
      -2.4668390668431357e-16,
      -2.527137114121933e-16,
      -2.5540604157617965e-16,
      -1.8212325455361218e-16,
      -1.662891505423292e-16,
      -1.570392566320307e-16,
      -1.649680076253794e-16,
      -1.7518238954440697e-16,
      -1.7274840450696002e-16,
      -1.69730892301484e-16,
      -1.7452197141255388e-16,
      -1.6513112332175458e-16,
      -1.7676249021037373e-16,
      3.8425420730966565e-17,
      4.7672802098090684e-17,
      3.9692165070873244e-17,
      3.589157184565846e-17,
      2.973003505793258e-17,
      4.04566061523163e-17,
      4.036429146334844e-17,
      2.7320308588142887e-17,
      2.937901723357873e-17,
      4.5093906274069067e-17,
      3.962075569405573e-17,
      3.376231920188859e-17,
      4.55475257108699e-17,
      5.0371060951000996e-17,
      3.1085303746220214e-17,
      4.739884853271731e-17,
      2.450620611980401e-17,
      2.4086133980455206e-17,
      -2.516487561013001e-17,
      -7.343173545296987e-19,
      5.2322183733602337e-17,
      3.4634465730948394e-17,
      3.2176746245864896e-17,
      -2.183437650609721e-17,
      -1.9985946284273016e-17,
      8.466969737246212e-17,
      4.981981444853793e-18,
      1.2330173380443923e-17,
      4.999807236577174e-18,
      9.04727565352429e-18,
      -4.978611812065848e-18,
      1.8203337500325277e-18,
      3.0074243252182363e-18,
      1.3619938188478033e-17,
      -4.537755205136455e-19,
      -5.122774189926542e-18,
      8.11007641989464e-19,
      3.464207588127145e-18,
      3.013190753402782e-18,
      1.0026175431847048e-17,
      9.298796307715226e-18,
      1.9030533784566148e-17,
      5.314302341665209e-18,
      -2.4729527801780313e-19,
      1.4618355455190964e-17,
      8.874516049681588e-16,
      9.31144173992982e-16,
      9.478258917329446e-16,
      9.258759448761584e-16,
      9.196543303067816e-16,
      9.195926972752187e-16,
      9.214359962689547e-16,
      9.129601678211379e-16,
      9.53340180536167e-16,
      9.504364871497011e-16,
      9.597408277389548e-16,
      9.475692794205874e-16,
      9.08686028602675e-16,
      7.365384688362215e-16,
      7.72693098204502e-16,
      7.760959482888918e-16,
      7.755024784110053e-16,
      7.488556463146003e-16,
      7.695390241461764e-16,
      7.412922143591596e-16,
      7.551283964692322e-16,
      7.862133167573794e-16,
      7.385986052084995e-16,
      7.72576571173885e-16,
      7.678578916324076e-16,
      1.4749682419940172e-18,
      -1.4093267326467098e-18,
      4.233114538585026e-18,
      3.500480568588379e-18,
      3.700377513685558e-18,
      -1.871343405255977e-18,
      -2.816962960680112e-17,
      3.118887003507261e-17,
      3.045403157757326e-17,
      8.150053071214536e-18,
      5.786210735405469e-17,
      -1.9308511340008797e-16,
      -2.9433181312049004e-16,
      6.814748637411863e-16,
      7.243165236118273e-16,
      7.292173941629049e-16,
      7.629192392452995e-16,
      7.278964254953383e-16,
      7.551326458016703e-16,
      6.80911682922054e-16,
      7.021147343572398e-16,
      6.830072747205217e-16,
      6.851933932770392e-16,
      7.5378806538905e-16,
      7.64368772613395e-16,
      6.847097641953418e-16,
      0.08908708063747475,
      0.0890870806374748,
      0.08908708063747474,
      0.08908708063747481,
      0.08908708063747478,
      0.08908708063747477,
      0.08908708063747477,
      0.08908708063747475,
      0.08908708063747478,
      0.08908708063747475,
      0.08908708063747484,
      0.08908708063747484,
      0.08908708063747478,
      0.08908708063747484,
      0.08908708063747482,
      0.0890870806374748,
      0.08908708063747484,
      0.08908708063747474,
      0.08908708063747478,
      0.08908708063747484,
      0.08908708063747484,
      0.08908708063747482,
      0.08908708063747488,
      0.0890870806374748,
      0.08908708063747481,
      0.08908708063747475,
      0.0890870806374748,
      0.08908708063747481,
      0.0890870806374748,
      0.08908708063747474,
      0.08908708063747481,
      0.08908708063747478,
      0.08908708063747478,
      0.08908708063747478,
      0.08908708063747475,
      0.08908708063747485,
      0.08908708063747472,
      0.08908708063747486,
      0.08908708063747478,
      0.08908708063747478,
      2.6708729566245054e-17,
      1.800461340336116e-17,
      2.2189668427201995e-17,
      1.8199526516854355e-17,
      2.2712728790649638e-17,
      2.9296152368321174e-17,
      3.255433739102095e-17,
      1.792370697255954e-17,
      1.913486970099873e-17,
      3.5657717343771115e-17,
      2.222913514909444e-17,
      2.2622411624872737e-17,
      3.423810792228857e-17,
      2.1872405587899437e-17,
      2.195596174881391e-17,
      3.0030078856715134e-17,
      0.08908708063747481,
      0.0890870806374748,
      0.08908708063747478,
      0.08908708063747478,
      0.08908708063747475,
      0.08908708063747467,
      0.0890870806374748,
      0.08908708063747481,
      0.08908708063747475,
      0.08908708063747482,
      0.08908708063747478,
      0.08908708063747475,
      0.0890870806374748,
      0.08908708063747478,
      0.08908708063747477,
      0.0890870806374748,
      0.08908708063747484,
      0.08908708063747484,
      2.4634707445896654e-16,
      0.08908708063747485,
      0.08908708063747478,
      0.08908708063747482,
      0.08908708063747481,
      0.0890870806374748,
      0.0890870806374748,
      0.08908708063747484,
      0.08908708063747478,
      0.08908708063747475,
      0.0890870806374748,
      0.08908708063747481,
      0.08908708063747481,
      0.0890870806374748,
      1.2502974855272835e-16,
      1.0815340662682026e-16,
      1.0980722035564158e-16,
      1.2268068436376413e-16,
      1.1814329234661488e-16,
      1.3810347652555242e-16,
      1.375392860294683e-16,
      1.2680713995638877e-16,
      1.2051763771360225e-16,
      1.2534934243762112e-16,
      2.486607281784642e-16,
      2.5488832252950546e-16,
      -2.5914465389255067e-16,
      -2.35917266317489e-16,
      -2.5696391108455726e-16,
      -2.818198017281286e-16,
      -2.626058365506192e-16,
      -2.616898336753787e-16,
      -2.700992736684768e-16,
      -2.6782364298106105e-16,
      -2.6473586194320372e-16,
      7.636634614273073e-16,
      -2.426545183523496e-16,
      -2.462737788024689e-16,
      -2.672322314513872e-16,
      -2.5393325927921165e-16,
      -2.634287325031203e-16,
      -2.618418801286871e-16,
      -2.601726480835963e-16,
      0.0890870806374748,
      0.08908708063747471,
      0.08908708063747484,
      0.08908708063747477,
      0.08908708063747475,
      0.08908708063747474,
      0.0890870806374748,
      0.08908708063747478,
      2.711185942180257e-16,
      2.5884797845277765e-16,
      2.54726321968709e-16,
      2.6529824225579165e-16,
      2.5577123929511283e-16,
      2.4743094368277617e-16,
      0.0890870806374748,
      0.0890870806374748,
      0.08908708063747481,
      0.08908708063747481,
      0.08908708063747475,
      0.08908708063747472,
      0.0890870806374748,
      0.08908708063747478,
      0.08908708063747478,
      0.08908708063747481,
      0.0890870806374748,
      0.08908708063747475,
      0.08908708063747481,
      0.08908708063747475,
      2.5662524596073285e-17,
      2.917931749877168e-17,
      -1.7495536030976712e-16,
      -1.92938549928851e-16,
      -2.1644782669152036e-16,
      -1.6316254840609225e-16,
      -2.64853584585848e-16,
      -2.0389888902467897e-16,
      -2.018614754890612e-16,
      -2.944694771364731e-16,
      0.0890870806374748,
      0.08908708063747478,
      0.08908708063747478,
      0.0890870806374748,
      0.0890870806374748,
      2.423869202825506e-16,
      0.0890870806374748,
      0.0890870806374748,
      0.08908708063747481,
      0.08908708063747482,
      0.08908708063747481,
      -2.2162017689473154e-17,
      -2.270333405627815e-17,
      -1.8626504091578443e-17,
      -1.9371013944830842e-17,
      -2.562299486678294e-17,
      -2.498194722187785e-17,
      0.08908708063747482,
      0.08908708063747477,
      2.458193846798485e-16,
      2.726243452709892e-16,
      0.08908708063747478,
      0.08908708063747472,
      0.08908708063747481,
      0.08908708063747482,
      0.0890870806374748,
      0.08908708063747477,
      2.9243772307339034e-17,
      3.0549066563499283e-17,
      0.08908708063747472,
      0.0890870806374748,
      0.08908708063747475,
      0.0890870806374748,
      0.08908708063747478,
      0.08908708063747481,
      0.08908708063747477,
      3.575213031121376e-17,
      0.08908708063747481,
      0.08908708063747478,
      -4.766599584224482e-16,
      -4.723439984281324e-16,
      -4.740640937014684e-16,
      -4.86416455807104e-16,
      -4.789722060053611e-16,
      -4.658300028259863e-16,
      -4.647471123369209e-16,
      -4.4320162855648333e-16,
      -4.581819301306258e-16,
      -4.631782911045191e-16,
      -4.650508895321628e-16,
      -4.788961260207823e-16,
      -4.465421339166233e-16,
      -4.809617983999151e-16,
      -4.683087276198653e-16,
      -2.5264134927777043e-16,
      -2.4461553166165063e-16,
      -2.624104345054479e-16,
      -2.651365565269233e-16,
      -2.7704719316396406e-16,
      -2.6402022457638663e-16,
      -2.9779959139985513e-16,
      -2.4343317251070604e-16,
      -2.7332740581482845e-16,
      1.0764853697644301e-16,
      1.5510605819059515e-16,
      1.310174636582694e-16,
      1.3923721109055083e-16,
      1.0042916067843319e-16,
      1.0148958172381573e-16,
      9.796372820549213e-17,
      1.1012490282249484e-16,
      1.2933050278835167e-16,
      -4.936563513431599e-16,
      -5.244941742166793e-16,
      -6.451826114823837e-16,
      -1.7664731699252636e-16,
      -1.527893139511468e-16,
      -1.649535788842172e-16,
      -1.5787472479694198e-16,
      -1.6300379584669979e-16,
      -1.5134121363371812e-16,
      -1.5688673716705733e-16,
      -1.8226470698571196e-16,
      -1.754771853151998e-16,
      -1.6243418151020517e-16,
      -1.5266820887992022e-16,
      -1.4445160355595162e-16,
      -1.7783713362408949e-16,
      -1.4845348492670125e-16,
      -1.765820185756501e-16,
      -1.4353392243188869e-16,
      1.8477541093796186e-18,
      -1.1668044060003432e-17,
      2.3226645096424598e-17,
      2.1746423930515403e-17,
      1.4060615460299202e-17,
      4.8258377229722855e-18,
      1.5107006806658114e-17,
      3.3980271956499254e-17,
      3.6711884962660467e-17,
      3.2618168553596866e-17,
      -1.9090804623012934e-17,
      8.881579777922885e-18,
      3.160265846851604e-17,
      -2.8158537728572228e-18,
      -6.973057153928752e-18,
      -9.925997661503267e-17,
      -2.496650118701438e-19,
      -6.416904158600618e-17,
      -4.087869440886285e-17,
      -6.729331080007292e-17,
      -3.374394568787696e-17,
      -8.423332649592998e-17,
      -1.0808294933674578e-16,
      -1.0815724753553781e-16,
      -7.298750993095532e-17,
      -6.522870775802482e-17,
      -1.0535589532240786e-16,
      -1.0647497456395988e-16,
      -3.0389789123565777e-16,
      -2.6311826920619495e-16,
      -2.8469398292442686e-16,
      -2.913780672596986e-16,
      -3.0412088122095255e-16,
      -2.9250330090643536e-16,
      -2.76944146399904e-16,
      -3.020716099928067e-16,
      -2.848670255318257e-16,
      -2.81092353440607e-16,
      -4.149373409472136e-16,
      -3.8293913962829547e-16,
      -4.157972848908411e-16,
      -3.677912289071919e-16,
      -4.1108879105555043e-16,
      -3.9376268455653447e-16,
      -3.9485125068311515e-16,
      -4.0606424429225047e-16,
      -3.695151842709748e-16,
      -3.5728499074346413e-16,
      -1.3978594405330982e-17,
      -1.1498147036638028e-17,
      -9.056373722678621e-18,
      -8.294923678350977e-18,
      7.533202679609518e-16,
      -9.855219354027843e-18,
      2.077769168858969e-17,
      2.506221155394673e-17,
      4.538925746182226e-17,
      1.53952153608756e-17,
      1.558456830819881e-17,
      3.4955170840064794e-17,
      -5.910597212429284e-18,
      7.479617487884102e-17,
      2.972649541349034e-17,
      -8.349993016001663e-18,
      7.45810025934804e-17,
      9.734463011565746e-17,
      6.47620872924185e-17,
      1.323776921380282e-17,
      -3.321514030326035e-17,
      -2.4357561013595672e-18,
      3.0332706810829405e-17,
      8.537212207381312e-18,
      -2.7835056411134235e-17,
      -3.06286709751251e-17,
      -8.314966721090172e-18,
      -2.1620055920141513e-17,
      3.4826758004774056e-17,
      2.943386676247297e-17,
      2.7409842001373337e-17,
      2.2054707768664108e-17,
      3.3219949011277e-17,
      3.147837334861144e-17,
      -3.2788047641381764e-17,
      -3.234343198762855e-17,
      -2.8412629633648136e-17,
      -3.157510931694095e-17,
      -2.6647258805801777e-17,
      -3.1318541311472726e-17,
      -9.387043013343405e-17,
      -5.1857243486762364e-17,
      -1.354169530863748e-16,
      -5.331197812132302e-17,
      -2.6137612344140686e-17,
      2.3930494203978975e-16,
      2.5531040453953945e-16,
      2.426678530141899e-16,
      2.340155927277882e-16,
      2.530748475506646e-16,
      2.6610041656640387e-16,
      2.6216834666047394e-16,
      -2.122170546581082e-17,
      -2.1234510062277096e-17,
      -1.7202422006653732e-17,
      -2.5386054969730223e-17,
      -2.4989098864879474e-17,
      -2.7916893420499e-17,
      -4.454422528178073e-17,
      -7.137064829072638e-18,
      -9.24519015552021e-18,
      -5.801193614669865e-18,
      -6.891223146839039e-18,
      -6.216225960899902e-18,
      -6.406379600393483e-18,
      1.5727146119210887e-17,
      2.8126004764740556e-17,
      8.259175026377293e-17,
      2.6194881624292408e-17,
      1.8371943547366451e-16,
      1.2480702915210079e-16,
      3.53635523456605e-16,
      -3.333688138314516e-19,
      -2.4874153134649617e-16,
      -2.630996831452955e-16,
      -2.4011075958159166e-16,
      -3.9563796036892074e-16,
      -1.3742890468981851e-17,
      0.08908708063747482,
      0.0890870806374748,
      0.08908708063747477,
      0.08908708063747484,
      0.08908708063747478,
      0.08908708063747484,
      1.448544654026748e-17,
      2.3946995429382166e-16,
      2.484566100829631e-16,
      2.419586753262938e-16,
      2.4737163801274156e-16,
      2.5053157561606986e-16,
      2.6004329791811825e-16,
      2.326260926541082e-16,
      2.3164070704789156e-16,
      2.246835342706681e-16,
      -3.479538567930917e-16,
      -4.075873142642172e-16,
      -2.177108417094333e-17,
      6.102757188486407e-15,
      6.0958182945825e-15,
      6.106226635438361e-15,
      6.088879400678593e-15,
      6.0923488476305465e-15,
      6.1201044232461754e-15,
      6.1096960823903146e-15,
      6.0958182945825e-15,
      6.113165529342268e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.0958182945825e-15,
      6.088879400678593e-15,
      6.1096960823903146e-15,
      6.113165529342268e-15,
      6.1096960823903146e-15,
      6.0958182945825e-15,
      6.106226635438361e-15,
      6.081940506774686e-15,
      6.1096960823903146e-15,
      6.099287741534454e-15,
      6.102757188486407e-15,
      6.106226635438361e-15,
      6.102757188486407e-15,
      6.106226635438361e-15,
      6.1096960823903146e-15,
      6.106226635438361e-15,
      6.102757188486407e-15,
      6.1096960823903146e-15,
      6.099287741534454e-15,
      6.099287741534454e-15,
      6.102757188486407e-15,
      6.106226635438361e-15,
      6.116634976294222e-15,
      6.102757188486407e-15,
      6.099287741534454e-15,
      6.1096960823903146e-15,
      6.123573870198129e-15,
      6.106226635438361e-15,
      6.1096960823903146e-15,
      6.1096960823903146e-15,
      6.099287741534454e-15,
      6.1096960823903146e-15,
      6.106226635438361e-15,
      6.1096960823903146e-15,
      6.116634976294222e-15,
      6.1096960823903146e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.106226635438361e-15,
      6.106226635438361e-15,
      6.106226635438361e-15,
      6.106226635438361e-15,
      6.088879400678593e-15,
      6.0958182945825e-15,
      6.106226635438361e-15,
      6.113165529342268e-15,
      6.085409953726639e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.102757188486407e-15,
      6.106226635438361e-15,
      6.0958182945825e-15,
      6.116634976294222e-15,
      6.1096960823903146e-15,
      6.0958182945825e-15,
      6.102757188486407e-15,
      6.085409953726639e-15,
      6.113165529342268e-15,
      6.099287741534454e-15,
      6.1096960823903146e-15,
      6.106226635438361e-15,
      6.1096960823903146e-15,
      6.113165529342268e-15,
      6.1096960823903146e-15,
      6.106226635438361e-15,
      6.099287741534454e-15,
      6.106226635438361e-15,
      6.0958182945825e-15,
      6.1096960823903146e-15,
      6.088879400678593e-15,
      6.106226635438361e-15,
      6.1201044232461754e-15,
      6.1096960823903146e-15,
      6.106226635438361e-15,
      6.102757188486407e-15,
      6.099287741534454e-15,
      6.1096960823903146e-15,
      6.116634976294222e-15,
      6.099287741534454e-15,
      6.113165529342268e-15,
      6.106226635438361e-15,
      6.0958182945825e-15,
      6.099287741534454e-15,
      6.0958182945825e-15,
      6.0958182945825e-15,
      6.102757188486407e-15,
      6.0958182945825e-15,
      6.102757188486407e-15,
      6.102757188486407e-15,
      6.116634976294222e-15,
      6.1096960823903146e-15,
      6.0958182945825e-15,
      6.106226635438361e-15,
      6.099287741534454e-15,
      6.0958182945825e-15,
      6.123573870198129e-15,
      6.102757188486407e-15,
      6.1096960823903146e-15,
      6.113165529342268e-15,
      6.113165529342268e-15,
      6.106226635438361e-15,
      6.102757188486407e-15,
      6.1096960823903146e-15,
      6.088879400678593e-15,
      6.102757188486407e-15,
      6.099287741534454e-15,
      6.106226635438361e-15,
      6.0958182945825e-15,
      6.102757188486407e-15,
      6.099287741534454e-15,
      6.0958182945825e-15,
      6.088879400678593e-15,
      6.1096960823903146e-15,
      6.0958182945825e-15,
      6.0923488476305465e-15,
      6.0923488476305465e-15,
      6.102757188486407e-15,
      6.0958182945825e-15,
      6.113165529342268e-15,
      6.0958182945825e-15,
      6.102757188486407e-15,
      6.0923488476305465e-15,
      6.113165529342268e-15,
      6.1096960823903146e-15,
      6.0923488476305465e-15,
      6.0958182945825e-15,
      6.116634976294222e-15,
      6.088879400678593e-15,
      6.099287741534454e-15,
      6.116634976294222e-15,
      6.113165529342268e-15,
      6.0923488476305465e-15,
      6.106226635438361e-15,
      6.116634976294222e-15,
      6.106226635438361e-15,
      6.113165529342268e-15,
      6.1201044232461754e-15,
      6.106226635438361e-15,
      6.099287741534454e-15,
      6.106226635438361e-15,
      6.116634976294222e-15,
      6.1096960823903146e-15,
      6.106226635438361e-15,
      6.0923488476305465e-15,
      6.113165529342268e-15,
      6.123573870198129e-15,
      6.102757188486407e-15,
      6.102757188486407e-15,
      6.1096960823903146e-15,
      6.0958182945825e-15,
      6.099287741534454e-15,
      6.099287741534454e-15,
      6.113165529342268e-15,
      6.0958182945825e-15,
      6.106226635438361e-15,
      6.106226635438361e-15,
      6.0923488476305465e-15,
      6.102757188486407e-15,
      6.106226635438361e-15,
      6.099287741534454e-15,
      6.102757188486407e-15,
      6.123573870198129e-15,
      6.088879400678593e-15,
      6.1096960823903146e-15,
      6.113165529342268e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.099287741534454e-15,
      6.0958182945825e-15,
      6.0958182945825e-15,
      6.116634976294222e-15,
      6.099287741534454e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.102757188486407e-15,
      6.1096960823903146e-15,
      6.113165529342268e-15,
      6.088879400678593e-15,
      6.0958182945825e-15,
      6.099287741534454e-15,
      6.0923488476305465e-15,
      6.099287741534454e-15,
      6.0923488476305465e-15,
      6.102757188486407e-15,
      6.127043317150083e-15,
      6.102757188486407e-15,
      6.1096960823903146e-15,
      6.1096960823903146e-15,
      6.1096960823903146e-15,
      6.106226635438361e-15,
      6.0958182945825e-15,
      6.0958182945825e-15,
      6.0958182945825e-15,
      6.099287741534454e-15,
      6.1096960823903146e-15,
      6.106226635438361e-15,
      6.102757188486407e-15,
      6.102757188486407e-15,
      6.088879400678593e-15,
      6.113165529342268e-15,
      6.099287741534454e-15,
      6.0958182945825e-15,
      6.113165529342268e-15,
      6.116634976294222e-15,
      6.106226635438361e-15,
      6.102757188486407e-15,
      6.113165529342268e-15,
      6.102757188486407e-15,
      6.1096960823903146e-15,
      6.116634976294222e-15,
      6.099287741534454e-15,
      6.113165529342268e-15,
      6.088879400678593e-15,
      6.1096960823903146e-15,
      6.1096960823903146e-15,
      6.106226635438361e-15,
      6.116634976294222e-15,
      6.099287741534454e-15,
      6.0958182945825e-15,
      6.099287741534454e-15,
      6.0958182945825e-15,
      6.1201044232461754e-15,
      6.0958182945825e-15,
      6.1201044232461754e-15,
      6.099287741534454e-15,
      6.106226635438361e-15,
      6.1096960823903146e-15,
      6.099287741534454e-15,
      6.113165529342268e-15,
      6.106226635438361e-15,
      6.099287741534454e-15,
      6.113165529342268e-15,
      6.1201044232461754e-15,
      6.085409953726639e-15,
      6.0958182945825e-15,
      6.1096960823903146e-15,
      6.0923488476305465e-15,
      6.099287741534454e-15,
      6.113165529342268e-15,
      6.0958182945825e-15,
      6.1096960823903146e-15,
      6.106226635438361e-15,
      6.099287741534454e-15,
      6.1096960823903146e-15,
      6.113165529342268e-15,
      6.099287741534454e-15,
      6.116634976294222e-15,
      6.102757188486407e-15,
      6.1096960823903146e-15,
      6.106226635438361e-15,
      6.0958182945825e-15,
      6.1096960823903146e-15,
      6.113165529342268e-15,
      6.0923488476305465e-15,
      6.0958182945825e-15,
      6.1096960823903146e-15,
      6.099287741534454e-15,
      6.0958182945825e-15,
      6.1096960823903146e-15,
      6.0958182945825e-15,
      6.0958182945825e-15,
      6.0958182945825e-15,
      6.0958182945825e-15,
      6.106226635438361e-15,
      6.102757188486407e-15,
      6.099287741534454e-15,
      6.106226635438361e-15,
      6.113165529342268e-15,
      6.102757188486407e-15,
      6.0958182945825e-15,
      6.0958182945825e-15,
      6.0958182945825e-15,
      6.099287741534454e-15,
      6.116634976294222e-15,
      6.0923488476305465e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.106226635438361e-15,
      6.099287741534454e-15,
      6.099287741534454e-15,
      6.113165529342268e-15,
      6.113165529342268e-15,
      6.088879400678593e-15,
      6.099287741534454e-15,
      6.106226635438361e-15,
      6.116634976294222e-15,
      6.099287741534454e-15,
      6.1096960823903146e-15,
      6.0923488476305465e-15,
      6.113165529342268e-15,
      6.1096960823903146e-15,
      6.113165529342268e-15,
      6.1096960823903146e-15,
      6.116634976294222e-15,
      6.113165529342268e-15,
      6.106226635438361e-15,
      6.099287741534454e-15,
      6.099287741534454e-15,
      6.106226635438361e-15,
      6.1201044232461754e-15,
      6.088879400678593e-15,
      6.106226635438361e-15,
      6.099287741534454e-15,
      6.106226635438361e-15,
      6.116634976294222e-15,
      6.102757188486407e-15,
      6.106226635438361e-15,
      6.088879400678593e-15,
      6.0958182945825e-15,
      6.102757188486407e-15,
      6.0958182945825e-15,
      6.1096960823903146e-15,
      6.113165529342268e-15,
      6.1201044232461754e-15,
      6.102757188486407e-15,
      6.113165529342268e-15,
      6.106226635438361e-15,
      6.102757188486407e-15,
      6.1096960823903146e-15,
      6.0923488476305465e-15,
      6.106226635438361e-15,
      6.102757188486407e-15,
      6.088879400678593e-15,
      6.102757188486407e-15,
      6.106226635438361e-15,
      6.102757188486407e-15,
      6.0923488476305465e-15,
      6.127043317150083e-15,
      6.106226635438361e-15,
      6.102757188486407e-15,
      6.0923488476305465e-15,
      6.106226635438361e-15,
      6.1096960823903146e-15,
      6.1096960823903146e-15,
      6.0958182945825e-15,
      6.085409953726639e-15,
      6.099287741534454e-15,
      6.099287741534454e-15,
      6.1096960823903146e-15,
      6.099287741534454e-15,
      6.106226635438361e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.085409953726639e-15,
      6.1201044232461754e-15,
      6.106226635438361e-15,
      6.0958182945825e-15,
      6.106226635438361e-15,
      6.0958182945825e-15,
      6.0923488476305465e-15,
      6.0923488476305465e-15,
      6.099287741534454e-15,
      6.106226635438361e-15,
      6.099287741534454e-15,
      6.1096960823903146e-15,
      6.0958182945825e-15,
      6.106226635438361e-15,
      6.113165529342268e-15,
      6.1096960823903146e-15,
      6.0958182945825e-15,
      6.116634976294222e-15,
      6.1096960823903146e-15,
      6.113165529342268e-15,
      6.1096960823903146e-15,
      6.106226635438361e-15,
      6.102757188486407e-15,
      6.116634976294222e-15,
      6.099287741534454e-15,
      6.0923488476305465e-15,
      6.106226635438361e-15,
      6.1201044232461754e-15,
      6.123573870198129e-15,
      6.0958182945825e-15,
      6.106226635438361e-15,
      6.1096960823903146e-15,
      6.116634976294222e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.102757188486407e-15,
      6.123573870198129e-15,
      6.085409953726639e-15,
      6.113165529342268e-15,
      6.102757188486407e-15,
      6.113165529342268e-15,
      6.102757188486407e-15,
      6.088879400678593e-15,
      6.116634976294222e-15,
      6.106226635438361e-15,
      6.116634976294222e-15,
      6.113165529342268e-15,
      6.102757188486407e-15,
      6.0923488476305465e-15,
      6.113165529342268e-15,
      6.102757188486407e-15,
      6.102757188486407e-15,
      6.102757188486407e-15,
      6.116634976294222e-15,
      6.1201044232461754e-15,
      6.099287741534454e-15,
      6.116634976294222e-15,
      6.0923488476305465e-15,
      6.0958182945825e-15,
      6.099287741534454e-15,
      6.116634976294222e-15,
      6.0923488476305465e-15,
      6.113165529342268e-15,
      6.088879400678593e-15,
      6.085409953726639e-15,
      6.099287741534454e-15,
      6.088879400678593e-15,
      6.102757188486407e-15,
      6.106226635438361e-15,
      6.113165529342268e-15,
      6.123573870198129e-15,
      6.116634976294222e-15,
      6.0958182945825e-15,
      6.102757188486407e-15,
      6.113165529342268e-15,
      6.0923488476305465e-15,
      6.0923488476305465e-15,
      6.106226635438361e-15,
      6.1096960823903146e-15,
      6.0923488476305465e-15,
      6.106226635438361e-15,
      6.102757188486407e-15,
      6.099287741534454e-15,
      6.102757188486407e-15,
      6.1096960823903146e-15,
      6.113165529342268e-15,
      6.099287741534454e-15,
      6.106226635438361e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.1201044232461754e-15,
      6.099287741534454e-15,
      6.099287741534454e-15,
      6.116634976294222e-15,
      6.1096960823903146e-15,
      6.106226635438361e-15,
      6.106226635438361e-15,
      6.081940506774686e-15,
      6.0923488476305465e-15,
      6.085409953726639e-15,
      6.102757188486407e-15,
      6.113165529342268e-15,
      6.0958182945825e-15,
      6.106226635438361e-15,
      6.102757188486407e-15,
      6.106226635438361e-15,
      6.102757188486407e-15,
      6.116634976294222e-15,
      6.106226635438361e-15,
      6.102757188486407e-15,
      6.1096960823903146e-15,
      6.0958182945825e-15,
      6.1096960823903146e-15,
      6.1096960823903146e-15,
      6.116634976294222e-15,
      6.116634976294222e-15,
      6.1201044232461754e-15,
      6.0923488476305465e-15,
      6.113165529342268e-15,
      6.099287741534454e-15,
      6.102757188486407e-15,
      6.088879400678593e-15,
      6.0958182945825e-15,
      6.106226635438361e-15,
      6.1096960823903146e-15,
      6.113165529342268e-15,
      6.099287741534454e-15,
      6.099287741534454e-15,
      6.102757188486407e-15,
      6.102757188486407e-15,
      6.106226635438361e-15,
      6.088879400678593e-15,
      6.116634976294222e-15,
      6.0958182945825e-15,
      6.102757188486407e-15,
      6.0923488476305465e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.102757188486407e-15,
      6.0923488476305465e-15,
      6.0958182945825e-15,
      6.0958182945825e-15,
      6.116634976294222e-15,
      6.099287741534454e-15,
      6.099287741534454e-15,
      6.099287741534454e-15,
      6.102757188486407e-15,
      6.102757188486407e-15,
      6.102757188486407e-15,
      6.099287741534454e-15,
      6.106226635438361e-15,
      6.116634976294222e-15,
      6.113165529342268e-15,
      6.102757188486407e-15,
      6.1096960823903146e-15,
      6.130512764102036e-15,
      6.102757188486407e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.123573870198129e-15,
      6.099287741534454e-15,
      6.099287741534454e-15,
      6.102757188486407e-15,
      6.106226635438361e-15,
      6.099287741534454e-15,
      6.099287741534454e-15,
      6.1096960823903146e-15,
      6.1096960823903146e-15,
      6.116634976294222e-15,
      6.106226635438361e-15,
      6.113165529342268e-15,
      6.1096960823903146e-15,
      6.1096960823903146e-15,
      6.088879400678593e-15,
      6.102757188486407e-15,
      6.1096960823903146e-15,
      6.099287741534454e-15,
      6.116634976294222e-15,
      6.106226635438361e-15,
      6.1096960823903146e-15,
      6.0958182945825e-15,
      6.0923488476305465e-15,
      6.1096960823903146e-15,
      6.0958182945825e-15,
      6.113165529342268e-15,
      6.0958182945825e-15,
      6.106226635438361e-15,
      6.116634976294222e-15,
      6.102757188486407e-15,
      6.099287741534454e-15,
      6.106226635438361e-15,
      6.0958182945825e-15,
      6.099287741534454e-15,
      6.106226635438361e-15,
      6.116634976294222e-15,
      6.1096960823903146e-15,
      6.106226635438361e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.102757188486407e-15,
      6.088879400678593e-15,
      6.1096960823903146e-15,
      6.106226635438361e-15,
      6.1096960823903146e-15,
      6.099287741534454e-15,
      6.106226635438361e-15,
      6.099287741534454e-15,
      6.0923488476305465e-15,
      6.0958182945825e-15,
      6.113165529342268e-15,
      6.102757188486407e-15,
      6.113165529342268e-15,
      6.1096960823903146e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.099287741534454e-15,
      6.0958182945825e-15,
      6.0958182945825e-15,
      6.116634976294222e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.102757188486407e-15,
      6.113165529342268e-15,
      6.106226635438361e-15,
      6.099287741534454e-15,
      6.106226635438361e-15,
      6.0923488476305465e-15,
      6.102757188486407e-15,
      6.088879400678593e-15,
      6.1096960823903146e-15,
      6.1096960823903146e-15,
      6.106226635438361e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.113165529342268e-15,
      6.1096960823903146e-15,
      6.106226635438361e-15,
      6.099287741534454e-15,
      6.113165529342268e-15,
      6.106226635438361e-15,
      6.0958182945825e-15,
      6.113165529342268e-15,
      6.102757188486407e-15,
      6.102757188486407e-15,
      6.099287741534454e-15,
      6.102757188486407e-15,
      6.1096960823903146e-15,
      6.116634976294222e-15,
      6.0958182945825e-15,
      6.099287741534454e-15,
      6.1201044232461754e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.113165529342268e-15,
      6.102757188486407e-15,
      6.102757188486407e-15,
      6.099287741534454e-15,
      6.099287741534454e-15,
      6.1201044232461754e-15,
      6.102757188486407e-15,
      6.106226635438361e-15,
      6.113165529342268e-15,
      6.1096960823903146e-15,
      6.099287741534454e-15,
      6.102757188486407e-15,
      6.099287741534454e-15,
      6.116634976294222e-15,
      6.116634976294222e-15,
      6.102757188486407e-15,
      6.106226635438361e-15,
      6.099287741534454e-15,
      6.113165529342268e-15,
      6.102757188486407e-15,
      6.078471059822732e-15,
      6.0958182945825e-15,
      6.1096960823903146e-15,
      6.102757188486407e-15,
      6.088879400678593e-15,
      6.106226635438361e-15,
      6.099287741534454e-15,
      6.099287741534454e-15,
      6.1096960823903146e-15,
      6.1096960823903146e-15,
      6.113165529342268e-15,
      6.113165529342268e-15,
      6.099287741534454e-15,
      6.127043317150083e-15,
      6.102757188486407e-15,
      6.102757188486407e-15,
      6.1096960823903146e-15,
      6.099287741534454e-15,
      6.106226635438361e-15,
      6.1201044232461754e-15,
      6.0958182945825e-15,
      6.106226635438361e-15
    ],
    [
      2.7608827555597776e-16,
      3.318486105006714e-16,
      3.4408026959842623e-16,
      3.2899645731719555e-16,
      3.5146386918962813e-16,
      2.8762165204507006e-16,
      2.652343170978957e-16,
      3.274542397594679e-16,
      3.5978829030936895e-16,
      2.968199067122131e-16,
      2.7847625282320186e-16,
      3.0826674160106615e-16,
      3.087434465118635e-16,
      -2.9130088481032536e-16,
      3.475904842502671e-16,
      3.4026097125928426e-16,
      3.3365878515282465e-16,
      3.2726864705488695e-16,
      3.2853475549815597e-16,
      3.299255890259203e-16,
      2.9522599222735725e-16,
      3.4461913035971567e-16,
      3.226652530978665e-16,
      -1.3061852035432995e-16,
      -1.5052561096110193e-16,
      -1.465277443897469e-16,
      -1.5611137334143078e-16,
      -1.5945258925359666e-16,
      -1.6202398030220893e-16,
      -1.5933474519796075e-16,
      -1.4917265374771305e-16,
      -1.398604205427861e-16,
      2.1890199960446524e-16,
      2.0235220619134742e-16,
      1.5975549661691565e-16,
      3.167585964182277e-17,
      1.3849889420008947e-17,
      -4.208607582511963e-18,
      8.252436675206271e-18,
      -2.5586772413234726e-17,
      -2.9068668172369744e-17,
      1.8830789356091863e-17,
      -1.9995932047371732e-17,
      -1.1071041634496451e-17,
      -2.7902662810454887e-17,
      3.301630106441494e-17,
      -4.1383431015803796e-18,
      -7.81293401048726e-18,
      2.2645023183581992e-17,
      4.069205566328617e-17,
      1.4756292819170588e-18,
      -2.247856681602764e-17,
      -1.2848482563121763e-18,
      -5.044774014435255e-18,
      9.931840990036799e-18,
      -1.1391071557590194e-17,
      -1.3489308619856445e-17,
      -3.4515687659245124e-17,
      8.776128247037651e-19,
      -3.6852799881695897e-17,
      9.908472131177841e-18,
      2.3831541703014256e-17,
      -1.629955678952291e-17,
      1.3818924261233594e-17,
      -2.858679592983338e-17,
      -4.589681998318858e-16,
      -5.221331769642407e-16,
      -4.492588362884678e-16,
      -4.693150021140214e-16,
      -4.48105024531095e-16,
      -4.485792111742755e-16,
      -4.603979202602449e-16,
      -4.756040086620439e-16,
      -4.59883228265859e-16,
      -4.4327272379041996e-16,
      -4.682029164733108e-16,
      -5.009840273522662e-16,
      -4.886565169227248e-16,
      -4.369667494772324e-16,
      -4.793884767736957e-16,
      -4.711327699459344e-16,
      -4.994921003440295e-16,
      -4.790429943130231e-16,
      -5.022876200704415e-16,
      -4.910866274212907e-16,
      9.796446388468e-16,
      9.942595669819193e-16,
      9.949087783935061e-16,
      9.806939058897628e-16,
      9.767834317609835e-16,
      1.0434144163043165e-15,
      9.886698502136097e-16,
      1.0078557214990721e-15,
      9.91338900486895e-16,
      1.0291382338903341e-15,
      1.028076685330794e-15,
      1.0095984648964927e-15,
      1.012853514659795e-15,
      1.002555315617278e-15,
      9.922985306416561e-16,
      1.0119587676041145e-15,
      9.934974056608995e-16,
      1.0071391690799758e-15,
      1.0070294524760173e-15,
      1.0107895708300936e-15,
      1.0067384533952062e-15,
      9.756204662251662e-16,
      1.9592148416817454e-16,
      9.87594953664512e-16,
      1.0063603813016497e-15,
      1.0076554693615117e-15,
      9.942413436836403e-16,
      1.0197576042149823e-15,
      9.898047829775255e-16,
      1.027620597136019e-15,
      9.864503080127805e-16,
      1.0125165167728193e-15,
      9.775643742728052e-16,
      1.0130877591795638e-15,
      9.923814331663763e-16,
      1.503650279261133e-16,
      1.598320290587088e-16,
      1.821110503592243e-16,
      1.8202996132517226e-16,
      1.8100755369004105e-16,
      1.7356848465943105e-16,
      1.884252029400813e-16,
      1.999536673625088e-16,
      1.8281475534682845e-16,
      2.0447636398619342e-16,
      2.111152911613289e-16,
      1.4436735010620365e-16,
      1.4220116365540592e-16,
      1.7015292856800612e-16,
      2.1210204843166096e-16,
      1.7949511695111724e-16,
      1.915351364116753e-16,
      1.7709853550302655e-16,
      1.6810880433444407e-16,
      1.8935859550299333e-16,
      1.8871075621954713e-16,
      1.516877219710892e-16,
      1.8463502352177237e-16,
      1.7647252474854174e-16,
      1.9839061782650947e-16,
      2.0188003891712514e-16,
      1.433787086816105e-16,
      1.6138703230031267e-16,
      2.0541187676164944e-16,
      1.9918977361645475e-16,
      1.8389556565551866e-16,
      2.1268373199560456e-16,
      1.3739721113325227e-16,
      2.0183180270042238e-16,
      1.8872161818020216e-16,
      1.640907467515652e-16,
      1.6348995520722728e-16,
      1.7560305464834097e-16,
      -1.1249942610924316e-16,
      -7.922851168274208e-17,
      -9.089214200463655e-17,
      -1.1655907180254464e-16,
      -8.720905186096063e-17,
      -1.253251780885228e-16,
      -8.994790427321623e-17,
      -8.606244757649753e-17,
      -6.973854572739451e-17,
      -7.314948139805407e-17,
      -7.580026157270278e-17,
      -6.910460311215376e-17,
      -1.1625697671137742e-16,
      -1.154399678869079e-16,
      -9.885285723782158e-17,
      -1.001722060323002e-16,
      -1.1310480255498555e-16,
      2.0957196475371488e-16,
      2.4573969685341596e-16,
      2.403571719201743e-16,
      2.4626434001905946e-16,
      2.3111338960766284e-16,
      2.399861414884276e-16,
      2.1520829393534643e-16,
      2.0908115628934683e-16,
      2.0782485642007398e-16,
      1.8918284889638204e-16,
      2.0477780923328582e-16,
      1.8978000058967806e-16,
      2.3873897770643287e-16,
      2.263313478269726e-16,
      2.0152987301149077e-16,
      2.718548424128626e-16,
      3.0672973140262724e-16,
      3.104765590522847e-16,
      2.997588444451446e-16,
      3.212223790822105e-16,
      3.0074535119057023e-16,
      3.179323713265331e-16,
      2.7424025452040674e-16,
      2.8076472758715157e-16,
      2.7363403177405246e-16,
      2.5505830379925525e-16,
      2.7251543221799127e-16,
      2.568249301363838e-16,
      2.986487486730236e-16,
      3.169821795914863e-16,
      2.7038051963715807e-16,
      3.0999026941676445e-16,
      2.750017279443188e-16,
      2.562892928891567e-16,
      2.366578829735226e-16,
      8.34968109732896e-17,
      7.496657596129423e-17,
      7.425660829251375e-17,
      7.908183489447251e-17,
      8.087972178032264e-17,
      9.13781121321846e-17,
      9.771505743361668e-17,
      4.198428768121843e-17,
      7.786900214291174e-17,
      4.170488304646981e-17,
      9.680172846148453e-18,
      6.332776957872391e-17,
      1.2967896437714202e-17,
      2.747424426387511e-17,
      6.318402411898004e-17,
      2.737263324117636e-17,
      6.318841557108191e-17,
      1.696240428626049e-17,
      1.6491580756490272e-17,
      1.6478378692155487e-17,
      5.606442258334156e-17,
      5.643570575355338e-17,
      2.3973425194596643e-17,
      4.169290548212038e-17,
      6.351116347496646e-17,
      1.30199011846175e-17,
      -1.3325223393564052e-18,
      8.143749696009145e-17,
      3.0964291463528345e-17,
      7.064655477417698e-17,
      1.5245818626946366e-17,
      1.5752848884207435e-17,
      9.961244680414375e-18,
      9.064904392668079e-18,
      1.7324216253234535e-17,
      2.2976747545965593e-17,
      1.1868007131306992e-17,
      1.8445377415752382e-17,
      1.672236181868204e-17,
      1.3229626247046472e-17,
      -1.2351103414770494e-16,
      -1.460797743833587e-16,
      -1.2442140334040459e-16,
      -1.0985589036754975e-16,
      -8.424035703125402e-17,
      -1.2714942832724787e-16,
      -1.261101066476622e-16,
      -9.393906799838721e-17,
      -9.566266080165975e-17,
      -1.4678541383638382e-16,
      -1.394154588167988e-16,
      -1.0659724805785354e-16,
      -1.5176561874374266e-16,
      -1.5291835859831884e-16,
      -9.381700795538984e-17,
      -1.4946606059908026e-16,
      -8.644080538590655e-17,
      -8.827369575105305e-17,
      -8.648615100118124e-17,
      -7.078488893942801e-17,
      -2.860906383959791e-17,
      -4.412087511296855e-17,
      -4.894449786073555e-17,
      -8.947823276842618e-17,
      -8.434120724736542e-17,
      -1.4490994731456287e-17,
      2.060096515063966e-16,
      1.7331246280599165e-16,
      2.024296224588318e-16,
      1.9002798656279106e-16,
      2.2916322722604653e-16,
      2.0386017723485092e-16,
      2.094113923827829e-16,
      1.8398870364532338e-16,
      2.0897325824683092e-16,
      2.3959600891549544e-16,
      2.227292401909443e-16,
      1.9966339833606748e-16,
      1.9673210677758678e-16,
      1.8187147097185895e-16,
      1.8584874582530853e-16,
      1.5137926458685288e-16,
      2.009079036471019e-16,
      1.9656803675241221e-16,
      1.7329528208828278e-16,
      -1.0390664184598879e-16,
      -1.0923305139608981e-16,
      -8.769941168070078e-17,
      -1.0753267485294679e-16,
      -1.055665500086338e-16,
      -9.313249417185034e-17,
      -1.0617647497155548e-16,
      -7.859234222356564e-17,
      -1.0294838001637838e-16,
      -8.404664185964292e-17,
      -9.555614970903215e-17,
      -1.0484478358175601e-16,
      -1.0433748716621702e-16,
      -2.8081083420707505e-16,
      -2.8915278123284076e-16,
      -2.8871709405729703e-16,
      -2.840661868198818e-16,
      -2.8590652041572573e-16,
      -2.7079937783048114e-16,
      -2.723243525997894e-16,
      -2.7537703106983136e-16,
      -3.054886043191756e-16,
      -2.7514989148977897e-16,
      -2.846763986442662e-16,
      -2.884106869793271e-16,
      -2.0444671643017423e-18,
      -3.7937109135941985e-18,
      -1.0954580092955775e-17,
      -2.8164871535680275e-18,
      -1.3947377187134934e-18,
      -4.042646632697957e-18,
      1.6098718188820247e-16,
      1.0218477904165264e-16,
      7.417974003825574e-17,
      1.7670837845525195e-16,
      7.356201939024978e-17,
      1.666675256592258e-17,
      6.699668552424812e-17,
      -6.065075260973051e-18,
      -1.9945674456823906e-18,
      -3.4607920392880004e-18,
      9.889635211891167e-19,
      -9.346217721224732e-18,
      4.397694641600307e-19,
      9.410526430003174e-18,
      -4.2027185791049465e-18,
      3.4641735354385123e-19,
      -2.1219666727589514e-18,
      7.401795059696945e-18,
      -6.852905687317356e-18,
      3.367235844251202e-18,
      1.3884727344909066e-14,
      1.3891665476425622e-14,
      1.3898604023594825e-14,
      1.3898605175437848e-14,
      1.389860451437594e-14,
      1.3884726703108659e-14,
      1.3905543979860465e-14,
      1.388472633054667e-14,
      1.3877787492690283e-14,
      1.3884727185178592e-14,
      1.3919421547985972e-14,
      1.3926359664234987e-14,
      1.3870848626516307e-14,
      1.3898603811969473e-14,
      1.3891665003383407e-14,
      1.3912482662500372e-14,
      1.3898603967752071e-14,
      1.3870848610789914e-14,
      1.3898604251985768e-14,
      1.3905543203235269e-14,
      1.3884726112455386e-14,
      1.3884726527644568e-14,
      1.389860483099338e-14,
      1.3912482571168448e-14,
      1.3884726763744424e-14,
      1.3912482292538078e-14,
      1.3905543726587432e-14,
      1.3891665035288337e-14,
      1.3926360571978448e-14,
      1.3891666261461789e-14,
      1.3898603920204329e-14,
      1.3891664916234995e-14,
      1.389166490278422e-14,
      1.3898604542365704e-14,
      1.3912481953527051e-14,
      1.390554277500124e-14,
      1.3877788360209096e-14,
      1.3891665420381397e-14,
      1.3870849516804643e-14,
      1.3877787661309633e-14,
      1.9223130730739777e-17,
      5.441633279983979e-17,
      3.723525533806068e-17,
      4.644744876754472e-17,
      3.402688812643647e-17,
      1.452375978595778e-17,
      4.700661756896712e-18,
      5.349541488796895e-17,
      4.533229971977912e-17,
      -1.092530654377044e-17,
      3.5866937459774886e-17,
      3.589285147417135e-17,
      -8.29392866388505e-18,
      3.475016631082807e-17,
      3.5652042957975293e-17,
      1.0737311648529516e-17,
      1.3898604758481611e-14,
      1.3898604804214264e-14,
      1.3870848881575166e-14,
      1.3891665076167087e-14,
      1.3884727400500445e-14,
      1.3884726176076216e-14,
      1.3912481755513507e-14,
      1.3891665024288675e-14,
      1.3912482833696864e-14,
      1.3912482850379824e-14,
      1.3884726174119636e-14,
      1.389166625346486e-14,
      1.3905543595506164e-14,
      1.3870848553173883e-14,
      1.388472694754814e-14,
      1.3898604933446921e-14,
      1.3898604540817905e-14,
      1.3905543301672724e-14,
      1.3707491343935918e-17,
      1.3898603757971077e-14,
      1.3884726198180569e-14,
      1.3884726448318596e-14,
      1.3856970369901965e-14,
      1.3905543406274844e-14,
      1.3912483049951088e-14,
      1.3933299068142284e-14,
      1.3891665623050028e-14,
      1.3877787288911507e-14,
      1.3905544129571951e-14,
      1.3912482429595387e-14,
      1.3919421008392305e-14,
      1.3870848954878062e-14,
      -2.4576617120628097e-17,
      -1.7326340696012922e-17,
      -1.8784723703169686e-17,
      -2.6462650243633657e-17,
      -1.6625663039930938e-17,
      -2.647330118797154e-17,
      -2.1675406170165424e-17,
      -2.72203760138467e-17,
      -2.0903234901695428e-17,
      -2.3000344097418192e-17,
      -3.8424726107570304e-17,
      -3.261697452135129e-17,
      6.478296407115701e-16,
      5.869852117339026e-16,
      5.980994439094433e-16,
      2.1557655907137269e-16,
      6.342639089551571e-16,
      6.299419010502963e-16,
      6.180278741290635e-16,
      6.240563338023143e-16,
      6.229502784706704e-16,
      -2.7510589496600857e-16,
      6.068948890945539e-16,
      6.398179654892876e-16,
      6.476735182701494e-16,
      5.932634278264909e-16,
      6.545246766090863e-16,
      6.31409097160869e-16,
      6.257109745073495e-16,
      1.3898604675459097e-14,
      1.3891665720736453e-14,
      1.3905543237496971e-14,
      1.3877787922211743e-14,
      1.3905543476090473e-14,
      1.3863910029857325e-14,
      1.3877787172605806e-14,
      1.388472688092641e-14,
      -2.884893678243135e-17,
      -3.544373496675062e-17,
      -2.4896412044965777e-17,
      -2.298432085773084e-17,
      -3.490875746834169e-17,
      -2.1142366355329487e-17,
      1.3926360486077709e-14,
      1.3912482712603052e-14,
      1.3877787385176493e-14,
      1.3891666088719757e-14,
      1.3912483084529562e-14,
      1.3933299623990127e-14,
      1.3891665589306007e-14,
      1.3856971808554366e-14,
      1.3912482201687353e-14,
      1.3898604933689385e-14,
      1.3891665786396789e-14,
      1.3905543527273706e-14,
      1.390554288363584e-14,
      1.3891666417402878e-14,
      2.4141228951001374e-17,
      1.5913004807059756e-17,
      3.8050988190856306e-17,
      1.6725788983526102e-17,
      7.415163756988071e-17,
      9.50385413054455e-18,
      5.97016268030315e-17,
      4.575218212095499e-17,
      5.906012275512452e-18,
      6.704840141888382e-17,
      1.3884725969626856e-14,
      1.3884726299994959e-14,
      1.3898604148173216e-14,
      1.390554407956088e-14,
      1.3877787174441636e-14,
      1.9311754157932304e-17,
      1.3898603957270353e-14,
      1.3905543198930694e-14,
      1.3891666142161123e-14,
      1.3884726086039716e-14,
      1.3898604787939684e-14,
      1.1017426248534165e-16,
      1.0859362231437789e-16,
      1.0938932388451578e-16,
      1.087691796254047e-16,
      1.0862800656562848e-16,
      1.1444161808232383e-16,
      1.3884726619828484e-14,
      1.3884726650479725e-14,
      -2.037980365869427e-17,
      -1.8157862272945e-17,
      1.3877787270417534e-14,
      1.3898604175099657e-14,
      1.3891665880916487e-14,
      1.3884726595758794e-14,
      1.3912482956820517e-14,
      1.3884727509172998e-14,
      1.3999717063844606e-17,
      6.949581106501982e-18,
      1.3877787960333468e-14,
      1.3870849146709676e-14,
      1.3870848177948675e-14,
      1.3877787393383076e-14,
      1.3884726197070762e-14,
      1.387778815946154e-14,
      1.3884726616534522e-14,
      -5.790644089788152e-18,
      1.391248222426046e-14,
      1.3898604769861924e-14,
      1.9938784691735322e-16,
      1.7944113033780247e-16,
      2.0287816206497979e-16,
      2.1864928706708363e-16,
      2.0466145950522752e-16,
      1.9487017356558268e-16,
      1.7477422400146086e-16,
      1.7395682840029673e-16,
      1.7707660049768725e-16,
      1.9261261339440094e-16,
      2.005484457492491e-16,
      2.05061664003441e-16,
      1.8166521442724756e-16,
      1.9537737348348503e-16,
      1.8942328616206955e-16,
      7.987757680629481e-17,
      8.41159318648764e-17,
      7.683508693914108e-17,
      7.586748818194406e-17,
      6.706368500641623e-17,
      7.461160534963801e-17,
      2.6940840473995706e-17,
      1.0276063835500932e-16,
      6.849847213198242e-17,
      -3.897722988402738e-17,
      8.820544192269438e-19,
      -1.9586017299897632e-17,
      -3.3612871874113556e-18,
      -4.169180511249249e-17,
      -4.5270416906392836e-17,
      -4.090228317340441e-17,
      -3.251568151271426e-17,
      -1.9495749506955294e-17,
      5.897018915860949e-16,
      5.996548213459436e-16,
      6.516078103580461e-16,
      1.3169183421670238e-16,
      1.0578402160400205e-16,
      1.2111728123207358e-16,
      1.0692731814106033e-16,
      1.1170831008998408e-16,
      9.417224936071676e-17,
      1.1007739544595e-16,
      1.3830955048832828e-16,
      1.2398910025482005e-16,
      1.1085624069218386e-16,
      1.0079848249693965e-16,
      8.522337726309293e-17,
      1.313165298468368e-16,
      9.444968992391431e-17,
      1.3476698180847812e-16,
      8.762117489225478e-17,
      4.350779881429738e-18,
      3.0550247814276986e-18,
      5.670008497048509e-18,
      6.39351478903812e-18,
      5.430209006580136e-18,
      4.30197517898171e-18,
      5.556336660892998e-18,
      5.545533419694795e-18,
      9.889840882329019e-18,
      9.357274581258398e-18,
      8.07136396002031e-17,
      4.311617319298076e-17,
      9.553189996692067e-18,
      4.9059747605278195e-17,
      4.4247527740522783e-17,
      1.0494081308653858e-17,
      3.7086664501858015e-18,
      1.0226358135172125e-17,
      1.1937167631195803e-17,
      1.0259390826510815e-17,
      1.1721267339540385e-17,
      8.6838327504217e-18,
      1.0926737687132482e-17,
      7.310202156609966e-18,
      9.90825109005788e-18,
      1.1352142069574239e-17,
      6.288646564739684e-18,
      9.05587158126571e-18,
      8.703275939384268e-18,
      2.533760429325617e-17,
      1.700674230154093e-17,
      1.4557451666586584e-17,
      1.1550998824759846e-17,
      -5.825025874377581e-20,
      4.267301491575244e-18,
      9.519925300033683e-18,
      1.5261147179920055e-17,
      8.917286778885043e-18,
      5.859978607094124e-17,
      5.2932043443329824e-17,
      5.624016799856283e-17,
      5.195238697526682e-17,
      5.581804487138495e-17,
      5.077630446635793e-17,
      5.919866817483939e-17,
      4.933993444146724e-17,
      5.795372413577759e-17,
      4.9719513625422246e-17,
      4.416254871467881e-17,
      4.4464766783924287e-17,
      4.275020771542007e-17,
      4.1865860767023093e-17,
      -2.845513211848967e-16,
      4.161803559799829e-17,
      -4.603005226383559e-16,
      -4.3874872284077944e-16,
      -4.49489974432393e-16,
      -4.516494719004551e-16,
      -4.3940670543236453e-16,
      -4.62009485576574e-16,
      -8.830065379823593e-17,
      -2.4857245734819578e-17,
      -6.317006743104985e-17,
      -9.483256213178199e-17,
      -2.9084214287057776e-17,
      -1.2168123025155482e-17,
      -4.291166697760523e-17,
      -7.941208538768765e-17,
      -1.0018645654523543e-17,
      1.2489676264268864e-17,
      3.5178814394972556e-17,
      2.1164047368686454e-17,
      -6.5545377569862336e-18,
      -9.056671435418426e-18,
      7.651080192181987e-18,
      -9.602595454607843e-19,
      -2.31505535819969e-16,
      -2.3644306593434775e-16,
      -2.3701450216980147e-16,
      -2.3568816487597157e-16,
      -2.4861174603965817e-16,
      -2.3330760277062566e-16,
      1.9612118007988045e-16,
      1.973739146224125e-16,
      2.089026054179589e-16,
      1.8762787794384642e-16,
      1.9751561999265157e-16,
      1.9807278425568585e-16,
      1.1829409527392746e-16,
      1.19840314801901e-16,
      1.7185053266154898e-16,
      9.594700675432897e-17,
      8.288256381737598e-17,
      -1.5930869831168384e-16,
      -1.6347709729710682e-16,
      -1.5718511916816256e-16,
      -1.4636338829559295e-16,
      -1.725794228551603e-16,
      -1.8366364466805881e-16,
      -1.9191001989673711e-16,
      2.709924474466992e-16,
      2.456716727492271e-16,
      2.6327215097208967e-16,
      2.6201830184072535e-16,
      2.584950841502554e-16,
      2.6494918353961735e-16,
      2.584348408816367e-16,
      2.0859880695432834e-17,
      1.946711680569084e-17,
      2.1448587033186448e-17,
      2.0292042193054087e-17,
      2.1019010292159542e-17,
      2.1268118392809543e-17,
      -1.1786117266671224e-16,
      -1.7558739234210745e-16,
      -2.0819750969422096e-16,
      -1.0593951671339865e-16,
      1.1245758725301406e-16,
      1.456644805055114e-16,
      -8.771173406722077e-17,
      2.494523649634785e-16,
      5.991519037698242e-16,
      6.413851543580361e-16,
      5.71801486626408e-16,
      6.214384578343185e-17,
      3.940710375581146e-17,
      1.3912482652737903e-14,
      1.3898605308440087e-14,
      1.3891666250290805e-14,
      1.3863909455579492e-14,
      1.3891664904598994e-14,
      1.3912482157755507e-14,
      6.392678888525437e-17,
      -1.5226002533234237e-16,
      -1.6402510635477615e-16,
      -1.5280362894035704e-16,
      -1.598982405556894e-16,
      -1.7256972322791865e-16,
      -1.8625361577462493e-16,
      -1.4246855660388526e-16,
      -1.4369480452386833e-16,
      -1.3434904380290198e-16,
      5.5472859501787593e-17,
      6.158275859950321e-17,
      -1.1471768013917505e-16,
      -0.03919309008348101,
      -0.03919309008348098,
      -0.03919309008348104,
      -0.039193090083480986,
      -0.039193090083481,
      -0.03919309008348105,
      -0.039193090083480986,
      -0.03919309008348103,
      -0.03919309008348103,
      -0.039193090083481034,
      -0.039193090083481014,
      -0.039193090083480986,
      -0.039193090083481,
      -0.03919309008348101,
      -0.039193090083481034,
      -0.03919309008348103,
      -0.03919309008348101,
      -0.03919309008348102,
      -0.03919309008348097,
      -0.03919309008348104,
      -0.03919309008348099,
      -0.039193090083481014,
      -0.03919309008348102,
      -0.03919309008348101,
      -0.03919309008348103,
      -0.03919309008348103,
      -0.03919309008348103,
      -0.039193090083481014,
      -0.039193090083481034,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348103,
      -0.039193090083481,
      -0.03919309008348102,
      -0.03919309008348102,
      -0.039193090083481,
      -0.039193090083481014,
      -0.03919309008348102,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348102,
      -0.039193090083481,
      -0.03919309008348102,
      -0.03919309008348104,
      -0.039193090083481,
      -0.039193090083481055,
      -0.039193090083481014,
      -0.039193090083481034,
      -0.039193090083481,
      -0.039193090083481034,
      -0.03919309008348102,
      -0.039193090083481014,
      -0.03919309008348099,
      -0.03919309008348099,
      -0.03919309008348103,
      -0.03919309008348099,
      -0.03919309008348101,
      -0.03919309008348097,
      -0.039193090083481014,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083480986,
      -0.039193090083480986,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348096,
      -0.039193090083481,
      -0.03919309008348097,
      -0.039193090083481014,
      -0.03919309008348101,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348105,
      -0.039193090083481014,
      -0.03919309008348104,
      -0.039193090083481,
      -0.03919309008348099,
      -0.03919309008348101,
      -0.039193090083481014,
      -0.03919309008348102,
      -0.03919309008348101,
      -0.03919309008348103,
      -0.03919309008348103,
      -0.039193090083481014,
      -0.03919309008348099,
      -0.039193090083481034,
      -0.039193090083481014,
      -0.03919309008348101,
      -0.039193090083481014,
      -0.03919309008348104,
      -0.039193090083481014,
      -0.03919309008348103,
      -0.03919309008348102,
      -0.03919309008348101,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348103,
      -0.03919309008348097,
      -0.03919309008348101,
      -0.03919309008348104,
      -0.03919309008348104,
      -0.03919309008348103,
      -0.03919309008348102,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.03919309008348102,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.03919309008348104,
      -0.03919309008348104,
      -0.039193090083480986,
      -0.039193090083481,
      -0.03919309008348102,
      -0.039193090083480986,
      -0.03919309008348101,
      -0.039193090083481,
      -0.03919309008348102,
      -0.03919309008348099,
      -0.03919309008348103,
      -0.039193090083481,
      -0.039193090083480986,
      -0.039193090083481,
      -0.03919309008348103,
      -0.039193090083480986,
      -0.03919309008348099,
      -0.03919309008348099,
      -0.03919309008348101,
      -0.03919309008348098,
      -0.039193090083481014,
      -0.03919309008348099,
      -0.039193090083481014,
      -0.039193090083480986,
      -0.039193090083481,
      -0.03919309008348104,
      -0.03919309008348099,
      -0.03919309008348102,
      -0.039193090083481034,
      -0.03919309008348101,
      -0.039193090083480986,
      -0.03919309008348102,
      -0.039193090083481014,
      -0.03919309008348099,
      -0.03919309008348103,
      -0.039193090083481014,
      -0.03919309008348099,
      -0.039193090083481014,
      -0.03919309008348104,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348103,
      -0.03919309008348103,
      -0.03919309008348103,
      -0.039193090083481034,
      -0.039193090083480986,
      -0.039193090083481014,
      -0.039193090083481055,
      -0.039193090083481,
      -0.039193090083481,
      -0.039193090083481014,
      -0.03919309008348101,
      -0.039193090083480986,
      -0.039193090083481014,
      -0.03919309008348103,
      -0.03919309008348099,
      -0.03919309008348103,
      -0.03919309008348101,
      -0.039193090083481014,
      -0.03919309008348101,
      -0.03919309008348103,
      -0.039193090083481,
      -0.039193090083481,
      -0.039193090083481034,
      -0.039193090083480986,
      -0.03919309008348103,
      -0.039193090083481,
      -0.03919309008348101,
      -0.03919309008348102,
      -0.03919309008348099,
      -0.039193090083480986,
      -0.039193090083480986,
      -0.039193090083481,
      -0.03919309008348101,
      -0.03919309008348099,
      -0.039193090083481,
      -0.03919309008348103,
      -0.039193090083481,
      -0.03919309008348101,
      -0.039193090083480965,
      -0.039193090083481034,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348101,
      -0.03919309008348101,
      -0.039193090083481034,
      -0.03919309008348103,
      -0.03919309008348099,
      -0.039193090083481,
      -0.039193090083481014,
      -0.03919309008348102,
      -0.03919309008348102,
      -0.03919309008348103,
      -0.039193090083480965,
      -0.039193090083481,
      -0.039193090083481,
      -0.039193090083481,
      -0.03919309008348099,
      -0.03919309008348103,
      -0.039193090083481034,
      -0.03919309008348099,
      -0.03919309008348103,
      -0.039193090083480986,
      -0.039193090083481,
      -0.039193090083481014,
      -0.03919309008348102,
      -0.039193090083481014,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083481,
      -0.039193090083481014,
      -0.03919309008348105,
      -0.03919309008348103,
      -0.039193090083481034,
      -0.03919309008348097,
      -0.03919309008348102,
      -0.03919309008348103,
      -0.03919309008348102,
      -0.03919309008348103,
      -0.03919309008348099,
      -0.039193090083481014,
      -0.03919309008348102,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.03919309008348097,
      -0.03919309008348104,
      -0.03919309008348101,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.03919309008348101,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348101,
      -0.039193090083481014,
      -0.03919309008348098,
      -0.03919309008348097,
      -0.03919309008348103,
      -0.03919309008348098,
      -0.03919309008348099,
      -0.039193090083481,
      -0.03919309008348097,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348103,
      -0.039193090083481,
      -0.03919309008348099,
      -0.03919309008348102,
      -0.03919309008348101,
      -0.03919309008348103,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348099,
      -0.039193090083481014,
      -0.039193090083480986,
      -0.03919309008348098,
      -0.03919309008348102,
      -0.03919309008348102,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083481,
      -0.039193090083480986,
      -0.039193090083481,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083481,
      -0.039193090083481,
      -0.039193090083481014,
      -0.03919309008348105,
      -0.039193090083481,
      -0.03919309008348101,
      -0.039193090083481,
      -0.039193090083481014,
      -0.03919309008348103,
      -0.03919309008348104,
      -0.039193090083480986,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348101,
      -0.03919309008348103,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.039193090083480986,
      -0.03919309008348103,
      -0.03919309008348101,
      -0.039193090083481034,
      -0.039193090083481014,
      -0.03919309008348104,
      -0.03919309008348102,
      -0.039193090083481034,
      -0.03919309008348103,
      -0.03919309008348103,
      -0.03919309008348103,
      -0.03919309008348102,
      -0.039193090083481014,
      -0.039193090083481034,
      -0.03919309008348101,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083481034,
      -0.039193090083480986,
      -0.039193090083480986,
      -0.03919309008348101,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.03919309008348104,
      -0.03919309008348099,
      -0.03919309008348101,
      -0.039193090083481,
      -0.03919309008348098,
      -0.039193090083481014,
      -0.03919309008348103,
      -0.03919309008348103,
      -0.039193090083481,
      -0.03919309008348103,
      -0.03919309008348103,
      -0.03919309008348101,
      -0.039193090083481,
      -0.03919309008348103,
      -0.03919309008348101,
      -0.039193090083481014,
      -0.03919309008348097,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.03919309008348098,
      -0.039193090083481014,
      -0.039193090083481,
      -0.039193090083481,
      -0.03919309008348101,
      -0.039193090083481,
      -0.03919309008348102,
      -0.039193090083481014,
      -0.03919309008348099,
      -0.039193090083480965,
      -0.039193090083481,
      -0.03919309008348101,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348099,
      -0.03919309008348103,
      -0.039193090083481,
      -0.03919309008348099,
      -0.03919309008348104,
      -0.039193090083481014,
      -0.03919309008348101,
      -0.039193090083481,
      -0.039193090083481014,
      -0.03919309008348099,
      -0.039193090083480986,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.039193090083480986,
      -0.039193090083481034,
      -0.03919309008348099,
      -0.03919309008348099,
      -0.039193090083481014,
      -0.03919309008348101,
      -0.03919309008348097,
      -0.03919309008348101,
      -0.03919309008348103,
      -0.039193090083481014,
      -0.03919309008348103,
      -0.039193090083481,
      -0.039193090083481,
      -0.03919309008348102,
      -0.039193090083481,
      -0.039193090083481,
      -0.03919309008348102,
      -0.03919309008348105,
      -0.039193090083481034,
      -0.03919309008348103,
      -0.039193090083481,
      -0.039193090083481014,
      -0.03919309008348101,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348103,
      -0.03919309008348097,
      -0.03919309008348103,
      -0.039193090083481,
      -0.03919309008348102,
      -0.039193090083481,
      -0.03919309008348099,
      -0.03919309008348102,
      -0.03919309008348103,
      -0.03919309008348102,
      -0.03919309008348103,
      -0.03919309008348103,
      -0.039193090083480986,
      -0.03919309008348102,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.03919309008348102,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.03919309008348101,
      -0.039193090083480986,
      -0.039193090083481,
      -0.03919309008348099,
      -0.039193090083481014,
      -0.039193090083480986,
      -0.03919309008348103,
      -0.03919309008348099,
      -0.039193090083481,
      -0.03919309008348101,
      -0.03919309008348096,
      -0.03919309008348104,
      -0.03919309008348102,
      -0.039193090083481014,
      -0.03919309008348104,
      -0.039193090083481014,
      -0.03919309008348097,
      -0.03919309008348103,
      -0.039193090083481,
      -0.039193090083480986,
      -0.03919309008348099,
      -0.03919309008348103,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.03919309008348102,
      -0.039193090083480986,
      -0.03919309008348101,
      -0.039193090083481,
      -0.039193090083481014,
      -0.03919309008348106,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.039193090083481034,
      -0.039193090083481,
      -0.039193090083481,
      -0.039193090083481,
      -0.039193090083481014,
      -0.03919309008348104,
      -0.039193090083481034,
      -0.039193090083481,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083480986,
      -0.039193090083481014,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083480986,
      -0.03919309008348103,
      -0.039193090083481,
      -0.039193090083481014,
      -0.03919309008348101,
      -0.03919309008348099,
      -0.03919309008348103,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.03919309008348102,
      -0.03919309008348102,
      -0.03919309008348102,
      -0.039193090083481034,
      -0.039193090083481014,
      -0.03919309008348099,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.03919309008348099,
      -0.039193090083480986,
      -0.03919309008348099,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.03919309008348102,
      -0.039193090083481,
      -0.039193090083481,
      -0.03919309008348101,
      -0.03919309008348101,
      -0.03919309008348101,
      -0.03919309008348099,
      -0.03919309008348102,
      -0.039193090083481,
      -0.03919309008348099,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083481034,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.03919309008348103,
      -0.039193090083481,
      -0.039193090083481014,
      -0.03919309008348102,
      -0.03919309008348099,
      -0.03919309008348103,
      -0.039193090083481014,
      -0.039193090083481,
      -0.039193090083481014,
      -0.03919309008348099,
      -0.03919309008348101,
      -0.03919309008348103,
      -0.03919309008348103,
      -0.03919309008348101,
      -0.03919309008348101,
      -0.039193090083481014,
      -0.039193090083481055,
      -0.03919309008348101,
      -0.039193090083481014,
      -0.03919309008348101,
      -0.039193090083481034,
      -0.039193090083481014,
      -0.03919309008348099,
      -0.039193090083481014,
      -0.03919309008348103,
      -0.03919309008348101,
      -0.039193090083481014,
      -0.03919309008348103,
      -0.039193090083481,
      -0.039193090083481034,
      -0.039193090083481014,
      -0.03919309008348102,
      -0.039193090083481014,
      -0.03919309008348102,
      -0.03919309008348099,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348101,
      -0.039193090083480986,
      -0.03919309008348101,
      -0.039193090083481,
      -0.03919309008348102,
      -0.03919309008348103,
      -0.03919309008348101,
      -0.03919309008348101,
      -0.039193090083481,
      -0.03919309008348103,
      -0.039193090083481014,
      -0.03919309008348099,
      -0.03919309008348101,
      -0.03919309008348102,
      -0.03919309008348097,
      -0.039193090083481,
      -0.039193090083481014,
      -0.03919309008348101,
      -0.03919309008348103,
      -0.03919309008348101,
      -0.03919309008348104,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348101,
      -0.039193090083481014,
      -0.03919309008348101,
      -0.03919309008348103,
      -0.039193090083481014,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348099,
      -0.039193090083481014,
      -0.03919309008348099,
      -0.039193090083481,
      -0.039193090083481014,
      -0.03919309008348102,
      -0.03919309008348101,
      -0.039193090083481,
      -0.039193090083480986,
      -0.039193090083480986,
      -0.03919309008348103,
      -0.03919309008348102,
      -0.039193090083481,
      -0.039193090083480986,
      -0.03919309008348102,
      -0.03919309008348103,
      -0.03919309008348099,
      -0.039193090083481014,
      -0.03919309008348098,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348102,
      -0.039193090083481014,
      -0.039193090083481,
      -0.039193090083481,
      -0.03919309008348103,
      -0.039193090083481014,
      -0.03919309008348101,
      -0.039193090083481,
      -0.03919309008348097,
      -0.039193090083481014,
      -0.03919309008348103,
      -0.039193090083481,
      -0.03919309008348103,
      -0.039193090083481014,
      -0.03919309008348103,
      -0.039193090083481,
      -0.03919309008348103,
      -0.03919309008348103,
      -0.03919309008348103,
      -0.039193090083481,
      -0.03919309008348102,
      -0.03919309008348102,
      -0.03919309008348102,
      -0.03919309008348101,
      -0.03919309008348102,
      -0.03919309008348099,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348101,
      -0.039193090083481014,
      -0.039193090083481014,
      -0.03919309008348103,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083481,
      -0.039193090083481,
      -0.039193090083481,
      -0.03919309008348101,
      -0.039193090083481,
      -0.039193090083481014,
      -0.03919309008348102,
      -0.039193090083481,
      -0.039193090083481,
      -0.03919309008348099,
      -0.039193090083480986,
      -0.039193090083481,
      -0.039193090083481014,
      -0.039193090083481,
      -0.03919309008348098,
      -0.039193090083481014,
      -0.03919309008348103,
      -0.03919309008348102,
      -0.03919309008348103,
      -0.03919309008348102,
      -0.03919309008348101,
      -0.03919309008348104,
      -0.039193090083481014,
      -0.03919309008348103,
      -0.03919309008348101,
      -0.039193090083481,
      -0.03919309008348103,
      -0.039193090083481014,
      -0.039193090083481034,
      -0.03919309008348102,
      -0.03919309008348097,
      -0.03919309008348099
    ]
  ],
  "equations": [
    {
      "eq_number": 1,
      "title": "Newton's Three Laws of Motion",
      "domain_id": 1,
      "significance": "Foundation of all classical mechanics; inertial frames; F=dp/dt; action=reaction"
    },
    {
      "eq_number": 2,
      "title": "Lagrangian Mechanics (Principle of Least Action)",
      "domain_id": 1,
      "significance": "Action S=\u222bL dt; \u03b4S=0 \u2192 Euler-Lagrange equations"
    },
    {
      "eq_number": 3,
      "title": "Hamiltonian Mechanics",
      "domain_id": 1,
      "significance": "Canonical eqs: q\u0307=\u2202H/\u2202p, \u1e57=\u2212\u2202H/\u2202q; symplectic structure"
    },
    {
      "eq_number": 4,
      "title": "Hamilton-Jacobi Equation",
      "domain_id": 1,
      "significance": "\u2202S/\u2202t + H(q,\u2202S/\u2202q,t)=0; bridges classical\u2192quantum"
    },
    {
      "eq_number": 5,
      "title": "Euler-Lagrange Equation",
      "domain_id": 1,
      "significance": "d/dt(\u2202L/\u2202q\u0307) \u2212 \u2202L/\u2202q = 0; from \u03b4S=0"
    },
    {
      "eq_number": 6,
      "title": "D'Alembert's Principle",
      "domain_id": 1,
      "significance": "Virtual work for dynamics: \u03a3(F_i\u2212\u1e57_i)\u00b7\u03b4r_i=0"
    },
    {
      "eq_number": 7,
      "title": "Euler's Rigid Body Rotation Equations",
      "domain_id": 1,
      "significance": "I\u00b7\u03c9\u0307 + \u03c9\u00d7(I\u00b7\u03c9) = \u03c4; angular momentum dynamics"
    },
    {
      "eq_number": 8,
      "title": "Conservation of Momentum",
      "domain_id": 1,
      "significance": "dP/dt = \u03a3F_ext; P constant when \u03a3F_ext=0"
    },
    {
      "eq_number": 9,
      "title": "Conservation of Angular Momentum",
      "domain_id": 1,
      "significance": "dL/dt = \u03c4_ext; L=I\u03c9 constant when \u03c4=0"
    },
    {
      "eq_number": 10,
      "title": "Conservation of Energy",
      "domain_id": 1,
      "significance": "dE/dt=0 for isolated system; time translation symmetry"
    },
    {
      "eq_number": 11,
      "title": "Work-Energy Theorem",
      "domain_id": 1,
      "significance": "W=\u0394KE; \u222bF\u00b7dr = \u00bdmv\u00b2_f \u2212 \u00bdmv\u00b2_i"
    },
    {
      "eq_number": 12,
      "title": "Impulse-Momentum Theorem",
      "domain_id": 1,
      "significance": "J=\u222bF dt=\u0394p"
    },
    {
      "eq_number": 13,
      "title": "Center of Mass Equation",
      "domain_id": 1,
      "significance": "MR\u0308_cm=\u03a3F_ext; COM moves like point particle"
    },
    {
      "eq_number": 14,
      "title": "Hooke's Law",
      "domain_id": 18,
      "significance": "F=\u2212kx; \u03c3=E\u03b5; linear elastic response"
    },
    {
      "eq_number": 15,
      "title": "Parallel Axis Theorem",
      "domain_id": 1,
      "significance": "I=I_cm+Md\u00b2"
    },
    {
      "eq_number": 16,
      "title": "Coriolis Force",
      "domain_id": 1,
      "significance": "F_cor=\u22122m \u03c9\u00d7v' (rotating frame)"
    },
    {
      "eq_number": 17,
      "title": "Centrifugal Force",
      "domain_id": 1,
      "significance": "F_cf=\u2212m \u03c9\u00d7(\u03c9\u00d7r) (rotating frame)"
    },
    {
      "eq_number": 18,
      "title": "Simple Harmonic Motion",
      "domain_id": 1,
      "significance": "x\u0308+\u03c9\u00b2x=0; x=A cos(\u03c9t+\u03c6); T=2\u03c0/\u03c9"
    },
    {
      "eq_number": 19,
      "title": "Damped Harmonic Oscillator",
      "domain_id": 1,
      "significance": "x\u0308+2\u03b2x\u0307+\u03c9\u2080\u00b2x=0; under/over/critically damped"
    },
    {
      "eq_number": 20,
      "title": "Forced Oscillator + Resonance",
      "domain_id": 1,
      "significance": "x\u0308+2\u03b2x\u0307+\u03c9\u2080\u00b2x=(F\u2080/m)cos \u03c9t; A=F\u2080/m/\u221a((\u03c9\u2080\u00b2\u2212\u03c9\u00b2)\u00b2+4\u03b2\u00b2\u03c9\u00b2)"
    },
    {
      "eq_number": 21,
      "title": "Coupled Oscillators (Normal Modes)",
      "domain_id": 1,
      "significance": "m\u1e8d\u2081=\u2212k x\u2081\u2212k'(x\u2081\u2212x\u2082); symmetric/antisymmetric modes"
    },
    {
      "eq_number": 22,
      "title": "Pendulum Equation",
      "domain_id": 1,
      "significance": "\u03b8\u0308+(g/L)sin \u03b8=0; small angle: \u03c9=\u221a(g/L)"
    },
    {
      "eq_number": 23,
      "title": "Kinematics (Constant Acceleration)",
      "domain_id": 1,
      "significance": "v=v\u2080+at, x=x\u2080+v\u2080t+\u00bdat\u00b2, v\u00b2=v\u2080\u00b2+2a\u0394x"
    },
    {
      "eq_number": 24,
      "title": "Universal Gravitation Law",
      "domain_id": 2,
      "significance": "F=\u2212G m\u2081m\u2082/r\u00b2 r\u0302"
    },
    {
      "eq_number": 25,
      "title": "Gravitational Potential Energy",
      "domain_id": 2,
      "significance": "U=\u2212GMm/r; F=\u2212\u2207U"
    },
    {
      "eq_number": 26,
      "title": "Kepler's First Law",
      "domain_id": 2,
      "significance": "Planetary orbits are ellipses with Sun at one focus"
    },
    {
      "eq_number": 27,
      "title": "Kepler's Second Law",
      "domain_id": 2,
      "significance": "Equal areas swept in equal times (areal velocity constant)"
    },
    {
      "eq_number": 28,
      "title": "Kepler's Third Law",
      "domain_id": 2,
      "significance": "T\u00b2\u221da\u00b3; T\u00b2=(4\u03c0\u00b2/GM)a\u00b3"
    },
    {
      "eq_number": 29,
      "title": "Escape Velocity",
      "domain_id": 2,
      "significance": "v_esc=\u221a(2GM/r)"
    },
    {
      "eq_number": 30,
      "title": "Orbital Velocity (Circular)",
      "domain_id": 2,
      "significance": "v_orb=\u221a(GM/r)"
    },
    {
      "eq_number": 31,
      "title": "Poisson Equation (Gravity)",
      "domain_id": 2,
      "significance": "\u2207\u00b2\u03a6=4\u03c0G\u03c1"
    },
    {
      "eq_number": 32,
      "title": "Tidal Force",
      "domain_id": 2,
      "significance": "F_tide\u22482GMm\u0394r/r\u00b3"
    },
    {
      "eq_number": 33,
      "title": "Gravitational Time Dilation (GR)",
      "domain_id": 6,
      "significance": "\u0394t'=\u0394t\u221a(1\u22122GM/rc\u00b2)"
    },
    {
      "eq_number": 34,
      "title": "Precession of Perihelion (GR)",
      "domain_id": 6,
      "significance": "\u0394\u03c6=6\u03c0GM/(a(1\u2212e\u00b2)c\u00b2) per orbit"
    },
    {
      "eq_number": 35,
      "title": "Lense-Thirring Precession (Frame Dragging)",
      "domain_id": 6,
      "significance": "\u03a9_LT=GJ/(2c\u00b2r\u00b3)(3(r\u0302\u00b7J\u0302)r\u0302\u2212J\u0302)"
    },
    {
      "eq_number": 36,
      "title": "Coulomb's Law",
      "domain_id": 3,
      "significance": "F=(1/4\u03c0\u03b5\u2080)q\u2081q\u2082/r\u00b2 r\u0302"
    },
    {
      "eq_number": 37,
      "title": "Lorentz Force Law",
      "domain_id": 3,
      "significance": "F=q(E+v\u00d7B)"
    },
    {
      "eq_number": 38,
      "title": "Maxwell's Equations (Differential)",
      "domain_id": 3,
      "significance": "\u2207\u00b7E=\u03c1/\u03b5\u2080, \u2207\u00b7B=0, \u2207\u00d7E=\u2212\u2202B/\u2202t, \u2207\u00d7B=\u03bc\u2080J+\u03bc\u2080\u03b5\u2080\u2202E/\u2202t"
    },
    {
      "eq_number": 39,
      "title": "Maxwell's Equations (Integral)",
      "domain_id": 3,
      "significance": "\u222eE\u00b7dA=Q/\u03b5\u2080, \u222eB\u00b7dA=0, \u222eE\u00b7dl=\u2212d\u03a6_B/dt, \u222eB\u00b7dl=\u03bc\u2080I+\u03bc\u2080\u03b5\u2080d\u03a6_E/dt"
    },
    {
      "eq_number": 40,
      "title": "Maxwell's Equations (Covariant / Tensor)",
      "domain_id": 3,
      "significance": "\u2202_\u03bcF^{\u03bc\u03bd}=\u03bc\u2080J^\u03bd; \u2202_\u03bcF\u0303^{\u03bc\u03bd}=0"
    },
    {
      "eq_number": 41,
      "title": "Scalar and Vector Potentials",
      "domain_id": 3,
      "significance": "B=\u2207\u00d7A; E=\u2212\u2207\u03c6\u2212\u2202A/\u2202t"
    },
    {
      "eq_number": 42,
      "title": "Gauge Invariance (U(1) in E&M)",
      "domain_id": 3,
      "significance": "A_\u03bc\u2192A_\u03bc+\u2202_\u03bc\u039b; E,B unchanged"
    },
    {
      "eq_number": 43,
      "title": "Biot-Savart Law",
      "domain_id": 3,
      "significance": "dB=(\u03bc\u2080/4\u03c0) I dl\u00d7r\u0302/r\u00b2"
    },
    {
      "eq_number": 44,
      "title": "Amp\u00e8re's Force Law (Wire)",
      "domain_id": 3,
      "significance": "dF=I dl\u00d7B"
    },
    {
      "eq_number": 45,
      "title": "Ohm's Law",
      "domain_id": 3,
      "significance": "V=IR; J=\u03c3E"
    },
    {
      "eq_number": 46,
      "title": "Kirchhoff's Current Law (KCL)",
      "domain_id": 3,
      "significance": "\u03a3I_in=\u03a3I_out at junction"
    },
    {
      "eq_number": 47,
      "title": "Kirchhoff's Voltage Law (KVL)",
      "domain_id": 3,
      "significance": "\u03a3V around closed loop=0"
    },
    {
      "eq_number": 48,
      "title": "Faraday's Law of Induction",
      "domain_id": 3,
      "significance": "\u03b5=\u2212d\u03a6_B/dt; induced EMF=\u2212flux change"
    },
    {
      "eq_number": 49,
      "title": "Lenz's Law",
      "domain_id": 3,
      "significance": "Induced current opposes flux change"
    },
    {
      "eq_number": 50,
      "title": "Poynting's Theorem",
      "domain_id": 3,
      "significance": "\u2202u/\u2202t+\u2207\u00b7S=\u2212J\u00b7E; S=(1/\u03bc\u2080)E\u00d7B"
    },
    {
      "eq_number": 51,
      "title": "Electromagnetic Wave Equation",
      "domain_id": 3,
      "significance": "\u25a1E=0; \u25a1B=0; c=1/\u221a(\u03bc\u2080\u03b5\u2080)"
    },
    {
      "eq_number": 52,
      "title": "EM Stress-Energy Tensor",
      "domain_id": 3,
      "significance": "T^{\u03bc\u03bd}=(1/\u03bc\u2080)[F^\u03bc_\u03b1 F^{\u03bd\u03b1}+\u00bcg^{\u03bc\u03bd}F\u00b2]"
    },
    {
      "eq_number": 53,
      "title": "Lienard-Wiechert Potentials",
      "domain_id": 3,
      "significance": "Retarded potentials for arbitrarily moving point charge"
    },
    {
      "eq_number": 54,
      "title": "Larmor Formula (Non-rel. Radiation)",
      "domain_id": 3,
      "significance": "P=q\u00b2 a\u00b2/(6\u03c0\u03b5\u2080 c\u00b3)"
    },
    {
      "eq_number": 55,
      "title": "Li\u00e9nard Formula (Relativistic Radiation)",
      "domain_id": 3,
      "significance": "P=(q\u00b2\u03b3\u2076/6\u03c0\u03b5\u2080c\u00b3)[a\u00b2\u2212(v\u00d7a)\u00b2/c\u00b2]"
    },
    {
      "eq_number": 56,
      "title": "Abraham-Lorentz Force (Radiation Reaction)",
      "domain_id": 3,
      "significance": "F_rad=(q\u00b2/6\u03c0\u03b5\u2080c\u00b3)d\u00b3r/dt\u00b3"
    },
    {
      "eq_number": 57,
      "title": "Coulomb Gauge",
      "domain_id": 3,
      "significance": "\u2207\u00b7A=0"
    },
    {
      "eq_number": 58,
      "title": "Lorenz Gauge",
      "domain_id": 3,
      "significance": "\u2202_\u03bc A^\u03bc=0"
    },
    {
      "eq_number": 59,
      "title": "RC Circuit Charging",
      "domain_id": 3,
      "significance": "q(t)=C \u03b5(1\u2212e^{\u2212t/RC}); \u03c4=RC"
    },
    {
      "eq_number": 60,
      "title": "RL Circuit Time Constant",
      "domain_id": 3,
      "significance": "I(t)=I\u2080 e^{\u2212tR/L}; \u03c4=L/R"
    },
    {
      "eq_number": 61,
      "title": "LC Oscillation",
      "domain_id": 3,
      "significance": "\u03c9\u2080=1/\u221a(LC); q\u0308+\u03c9\u2080\u00b2 q=0"
    },
    {
      "eq_number": 62,
      "title": "RLC Damped Oscillation",
      "domain_id": 3,
      "significance": "q\u0308+(R/L)q\u0307+(1/LC)q=0; \u03b3=R/(2L)"
    },
    {
      "eq_number": 63,
      "title": "Skin Effect",
      "domain_id": 3,
      "significance": "\u03b4=\u221a(2/\u03c9\u03bc\u03c3); penetration depth"
    },
    {
      "eq_number": 64,
      "title": "Dielectric Polarization (Linear)",
      "domain_id": 3,
      "significance": "D=\u03b5\u2080E+P=\u03b5_r \u03b5\u2080 E"
    },
    {
      "eq_number": 65,
      "title": "Magnetic Susceptibility",
      "domain_id": 3,
      "significance": "M=\u03c7_m H; B=\u03bc\u2080(H+M)=\u03bc_r \u03bc\u2080 H"
    },
    {
      "eq_number": 66,
      "title": "Zeroth Law of Thermodynamics",
      "domain_id": 4,
      "significance": "Thermal equilibrium is transitive; defines temperature"
    },
    {
      "eq_number": 67,
      "title": "First Law of Thermodynamics",
      "domain_id": 4,
      "significance": "dU=\u03b4Q\u2212\u03b4W; \u0394U=Q\u2212W"
    },
    {
      "eq_number": 68,
      "title": "Second Law of Thermodynamics",
      "domain_id": 4,
      "significance": "dS_total\u22650; entropy never decreases"
    },
    {
      "eq_number": 69,
      "title": "Third Law of Thermodynamics",
      "domain_id": 4,
      "significance": "S\u21920 as T\u21920 (perfect crystal)"
    },
    {
      "eq_number": 70,
      "title": "Ideal Gas Law",
      "domain_id": 4,
      "significance": "pV=nRT=N k_B T"
    },
    {
      "eq_number": 71,
      "title": "Van der Waals Equation of State",
      "domain_id": 4,
      "significance": "(p+an\u00b2/V\u00b2)(V\u2212nb)=nRT"
    },
    {
      "eq_number": 72,
      "title": "Kinetic Theory: Pressure",
      "domain_id": 4,
      "significance": "p=(1/3) N m \u27e8v\u00b2\u27e9/V"
    },
    {
      "eq_number": 73,
      "title": "Equipartition Theorem",
      "domain_id": 4,
      "significance": "\u27e8E\u27e9=f k_B T/2; C_V=(f/2)R"
    },
    {
      "eq_number": 74,
      "title": "Maxwell-Boltzmann Speed Distribution",
      "domain_id": 4,
      "significance": "f(v)=4\u03c0(m/2\u03c0k_B T)^{3/2} v\u00b2 e^{\u2212mv\u00b2/2kBT}"
    },
    {
      "eq_number": 75,
      "title": "Carnot Efficiency",
      "domain_id": 4,
      "significance": "\u03b7_max=1\u2212T_c/T_h"
    },
    {
      "eq_number": 76,
      "title": "Clausius-Clapeyron Relation",
      "domain_id": 4,
      "significance": "dP/dT=L/(T \u0394V) for phase coexistence"
    },
    {
      "eq_number": 77,
      "title": "Gibbs Phase Rule",
      "domain_id": 4,
      "significance": "F=C\u2212P+2"
    },
    {
      "eq_number": 78,
      "title": "Helmholtz Free Energy",
      "domain_id": 4,
      "significance": "F=U\u2212TS; \u0394F\u22640 at const T,V (spontaneous)"
    },
    {
      "eq_number": 79,
      "title": "Gibbs Free Energy",
      "domain_id": 4,
      "significance": "G=H\u2212TS; \u0394G\u22640 at const T,P (spontaneous)"
    },
    {
      "eq_number": 80,
      "title": "Enthalpy",
      "domain_id": 4,
      "significance": "H=U+pV; \u0394H=Q_p"
    },
    {
      "eq_number": 81,
      "title": "Maxwell Relations (Thermodynamics)",
      "domain_id": 4,
      "significance": "(\u2202T/\u2202V)_S=\u2212(\u2202p/\u2202S)_V; (\u2202T/\u2202p)_S=(\u2202V/\u2202S)_p; (\u2202S/\u2202V)_T=(\u2202p/\u2202T)_V; (\u2202S/\u2202p)_T=\u2212(\u2202V/\u2202T)_p"
    },
    {
      "eq_number": 82,
      "title": "TdS Equations",
      "domain_id": 4,
      "significance": "T dS=C_V dT+T(\u2202p/\u2202T)_V dV; T dS=C_p dT\u2212T(\u2202V/\u2202T)_p dp"
    },
    {
      "eq_number": 83,
      "title": "Specific Heat Relations (C_p\u2212C_V)",
      "domain_id": 4,
      "significance": "C_p\u2212C_V=\u2212T(\u2202V/\u2202T)_p\u00b2/(\u2202V/\u2202p)_T=TV\u03b1\u00b2/\u03ba_T"
    },
    {
      "eq_number": 84,
      "title": "Joule-Thomson Coefficient",
      "domain_id": 4,
      "significance": "\u03bc_JT=(\u2202T/\u2202p)_H=(V/C_p)(T\u03b1\u22121)"
    },
    {
      "eq_number": 85,
      "title": "Entropy of Mixing",
      "domain_id": 4,
      "significance": "\u0394S_mix=\u2212k_B(N\u2081 ln x\u2081+N\u2082 ln x\u2082)"
    },
    {
      "eq_number": 86,
      "title": "Planck's Blackbody Radiation Law",
      "domain_id": 5,
      "significance": "B_\u03bd=(2h\u03bd\u00b3/c\u00b2)/(e^{h\u03bd/kT}\u22121)"
    },
    {
      "eq_number": 87,
      "title": "Wien's Displacement Law",
      "domain_id": 5,
      "significance": "\u03bb_max T=2.898\u00d710\u207b\u00b3 m\u00b7K"
    },
    {
      "eq_number": 88,
      "title": "Stefan-Boltzmann Law",
      "domain_id": 5,
      "significance": "j*=\u03c3 T\u2074; \u03c3=2\u03c0\u2075k_B\u2074/(15h\u00b3c\u00b2)"
    },
    {
      "eq_number": 89,
      "title": "Photoelectric Effect Equation (Einstein)",
      "domain_id": 5,
      "significance": "K_max=h\u03bd\u2212\u03c6; photon quanta"
    },
    {
      "eq_number": 90,
      "title": "Einstein A and B Coefficients",
      "domain_id": 5,
      "significance": "A_21/B_21=8\u03c0h\u03bd\u00b3/c\u00b3; B_12/B_21=g\u2082/g\u2081"
    },
    {
      "eq_number": 91,
      "title": "Compton Scattering Formula",
      "domain_id": 5,
      "significance": "\u0394\u03bb=(h/m_e c)(1\u2212cos \u03b8); \u0394\u03bb_max\u22480.00486 nm"
    },
    {
      "eq_number": 92,
      "title": "de Broglie Wavelength",
      "domain_id": 5,
      "significance": "\u03bb=h/p=h/(\u03b3mv)"
    },
    {
      "eq_number": 93,
      "title": "Schr\u00f6dinger Equation (Time-Dependent)",
      "domain_id": 5,
      "significance": "i\u210f\u2202\u03c8/\u2202t=\u0124\u03c8"
    },
    {
      "eq_number": 94,
      "title": "Time-Independent Schr\u00f6dinger Equation",
      "domain_id": 5,
      "significance": "\u0124\u03c8=E\u03c8"
    },
    {
      "eq_number": 95,
      "title": "Born Rule (Probability Interpretation)",
      "domain_id": 5,
      "significance": "\u03c1(r,t)=|\u03c8(r,t)|\u00b2"
    },
    {
      "eq_number": 96,
      "title": "Probability Current (QM)",
      "domain_id": 5,
      "significance": "j=(\u210f/2mi)(\u03c8*\u2207\u03c8\u2212\u03c8\u2207\u03c8*); \u2202\u03c1/\u2202t+\u2207\u00b7j=0"
    },
    {
      "eq_number": 97,
      "title": "Canonical Commutation Relations",
      "domain_id": 5,
      "significance": "[x\u0302_i,p\u0302_j]=i\u210f\u03b4_{ij}"
    },
    {
      "eq_number": 98,
      "title": "Heisenberg Uncertainty Principle",
      "domain_id": 5,
      "significance": "\u0394x\u00b7\u0394p\u2265\u210f/2; \u0394E\u00b7\u0394t\u2265\u210f/2"
    },
    {
      "eq_number": 99,
      "title": "Harmonic Oscillator Energy Levels (QM)",
      "domain_id": 5,
      "significance": "E_n=\u210f\u03c9(n+\u00bd); \u00e2|n\u27e9=\u221an|n\u22121\u27e9, \u00e2\u2020|n\u27e9=\u221a(n+1)|n+1\u27e9"
    },
    {
      "eq_number": 100,
      "title": "Hydrogen Atom Energy Levels",
      "domain_id": 5,
      "significance": "E_n=\u2212R_y/n\u00b2; R_y=13.605693123 eV"
    },
    {
      "eq_number": 101,
      "title": "Angular Momentum Quantization",
      "domain_id": 5,
      "significance": "L\u00b2|l,m\u27e9=\u210f\u00b2 l(l+1); L_z|l,m\u27e9=\u210f m"
    },
    {
      "eq_number": 102,
      "title": "Spin-\u00bd Algebra (Pauli Matrices)",
      "domain_id": 5,
      "significance": "S=(\u210f/2)\u03c3; [\u03c3_i,\u03c3_j]=2i\u03b5_{ijk}\u03c3_k; {\u03c3_i,\u03c3_j}=2\u03b4_{ij}"
    },
    {
      "eq_number": 103,
      "title": "Spin-Orbit Coupling",
      "domain_id": 5,
      "significance": "H_SO=(1/2m\u00b2c\u00b2)(1/r)(dV/dr) L\u00b7S"
    },
    {
      "eq_number": 104,
      "title": "Dirac Equation",
      "domain_id": 5,
      "significance": "(i\u210f\u03b3^\u03bc\u2202_\u03bc\u2212mc)\u03c8=0"
    },
    {
      "eq_number": 105,
      "title": "Klein-Gordon Equation",
      "domain_id": 5,
      "significance": "(\u25a1+m\u00b2c\u00b2/\u210f\u00b2)\u03c6=0"
    },
    {
      "eq_number": 106,
      "title": "Fine Structure Formula (Hydrogen)",
      "domain_id": 5,
      "significance": "\u0394E_FS=(R_y \u03b1\u00b2/n\u00b3)[1/(j+\u00bd)\u22123/(4n)]"
    },
    {
      "eq_number": 107,
      "title": "Lamb Shift",
      "domain_id": 5,
      "significance": "\u0394E(2S\u22122P)\u22481057.8 MHz; QED vacuum effects"
    },
    {
      "eq_number": 108,
      "title": "Anomalous Magnetic Moment (Electron)",
      "domain_id": 7,
      "significance": "a_e=(g\u22122)/2\u22480.00115965218091; QED+EW+hadronic"
    },
    {
      "eq_number": 109,
      "title": "Pauli Exclusion Principle",
      "domain_id": 5,
      "significance": "No two identical fermions in same quantum state; \u03c8 antisymmetric"
    },
    {
      "eq_number": 110,
      "title": "Spin-Statistics Theorem",
      "domain_id": 5,
      "significance": "Half-int spin\u2192fermion (anticommutators); int\u2192boson (commutators)"
    },
    {
      "eq_number": 111,
      "title": "Fermi's Golden Rule",
      "domain_id": 5,
      "significance": "\u0393_{i\u2192f}=(2\u03c0/\u210f)|\u27e8f|V|i\u27e9|\u00b2 \u03c1(E_f)"
    },
    {
      "eq_number": 112,
      "title": "Time-Dependent Perturbation Theory (1st Order)",
      "domain_id": 5,
      "significance": "c_f(t)=\u2212(i/\u210f)\u222b\u2080\u1d57 \u27e8f|V(t')|i\u27e9 e^{i\u03c9_fi t'} dt'"
    },
    {
      "eq_number": 113,
      "title": "WKB Approximation",
      "domain_id": 5,
      "significance": "\u03c8\u223c(1/\u221ap)exp(\u00b1i\u222b p dx/\u210f); Bohr-Sommerfeld quantization"
    },
    {
      "eq_number": 114,
      "title": "Born Approximation (Scattering)",
      "domain_id": 5,
      "significance": "f(\u03b8,\u03c6)=\u2212(2m/\u210f\u00b2)(1/4\u03c0)\u222b e^{\u2212iq\u00b7r} V(r) d\u00b3r"
    },
    {
      "eq_number": 115,
      "title": "Partial Wave Expansion (Scattering)",
      "domain_id": 5,
      "significance": "f(\u03b8)=(1/k)\u03a3(2l+1)e^{i\u03b4_l} sin \u03b4_l P_l(cos \u03b8)"
    },
    {
      "eq_number": 116,
      "title": "Optical Theorem",
      "domain_id": 5,
      "significance": "Im f(0)=(k/4\u03c0)\u03c3_total"
    },
    {
      "eq_number": 117,
      "title": "Feynman Path Integral",
      "domain_id": 5,
      "significance": "\u27e8x_f,t_f|x_i,t_i\u27e9=\u222b D[x(t)] exp(iS[x]/\u210f)"
    },
    {
      "eq_number": 118,
      "title": "Von Neumann Equation",
      "domain_id": 5,
      "significance": "i\u210f \u2202\u03c1\u0302/\u2202t=[\u0124,\u03c1\u0302]"
    },
    {
      "eq_number": 119,
      "title": "Ehrenfest Theorem",
      "domain_id": 5,
      "significance": "d\u27e8A\u27e9/dt=(1/i\u210f)\u27e8[A,\u0124]\u27e9+\u27e8\u2202A/\u2202t\u27e9"
    },
    {
      "eq_number": 120,
      "title": "Bell's Inequality",
      "domain_id": 5,
      "significance": "|E(a,b)\u2212E(a,c)|\u22641+E(b,c)"
    },
    {
      "eq_number": 121,
      "title": "Lorentz Transformations (Boost)",
      "domain_id": 6,
      "significance": "x'=\u03b3(x\u2212vt); t'=\u03b3(t\u2212vx/c\u00b2); \u03b3=1/\u221a(1\u2212v\u00b2/c\u00b2)"
    },
    {
      "eq_number": 122,
      "title": "Minkowski Spacetime Interval",
      "domain_id": 6,
      "significance": "ds\u00b2=\u2212c\u00b2dt\u00b2+dx\u00b2+dy\u00b2+dz\u00b2=\u03b7_\u03bc\u03bd dx^\u03bc dx^\u03bd"
    },
    {
      "eq_number": 123,
      "title": "Time Dilation",
      "domain_id": 6,
      "significance": "\u0394t'=\u03b3\u0394t (moving clock runs slow)"
    },
    {
      "eq_number": 124,
      "title": "Length Contraction",
      "domain_id": 6,
      "significance": "L'=L/\u03b3 (moving object contracts)"
    },
    {
      "eq_number": 125,
      "title": "Relativistic Energy-Momentum Relation",
      "domain_id": 6,
      "significance": "E\u00b2=(pc)\u00b2+(mc\u00b2)\u00b2; E=\u03b3mc\u00b2; p=\u03b3mv"
    },
    {
      "eq_number": 126,
      "title": "Mass-Energy Equivalence",
      "domain_id": 6,
      "significance": "E=mc\u00b2; \u0394E=\u0394m c\u00b2"
    },
    {
      "eq_number": 127,
      "title": "Relativistic Doppler Effect",
      "domain_id": 6,
      "significance": "f_obs=f_s\u221a[(1+\u03b2)/(1\u2212\u03b2)] (longitudinal); transverse: f_obs=\u03b3f_s"
    },
    {
      "eq_number": 128,
      "title": "Relativistic Velocity Addition",
      "domain_id": 6,
      "significance": "u=(u'+v)/(1+u'v/c\u00b2)"
    },
    {
      "eq_number": 129,
      "title": "Einstein Field Equations (GR)",
      "domain_id": 6,
      "significance": "G_\u03bc\u03bd+\u039bg_\u03bc\u03bd=(8\u03c0G/c\u2074)T_\u03bc\u03bd"
    },
    {
      "eq_number": 130,
      "title": "Einstein-Hilbert Action",
      "domain_id": 6,
      "significance": "S=(c\u2074/16\u03c0G)\u222b d\u2074x\u221a(\u2212g)(R\u22122\u039b)+S_matter"
    },
    {
      "eq_number": 131,
      "title": "Schwarzschild Metric",
      "domain_id": 6,
      "significance": "ds\u00b2=\u2212(1\u2212r_s/r)c\u00b2dt\u00b2+dr\u00b2/(1\u2212r_s/r)+r\u00b2d\u03a9\u00b2; r_s=2GM/c\u00b2"
    },
    {
      "eq_number": 132,
      "title": "Kerr Metric (Rotating Black Hole)",
      "domain_id": 6,
      "significance": "Rotating axisymmetric vacuum solution; a=J/Mc"
    },
    {
      "eq_number": 133,
      "title": "FLRW Metric",
      "domain_id": 6,
      "significance": "ds\u00b2=\u2212c\u00b2dt\u00b2+a\u00b2(t)[dr\u00b2/(1\u2212kr\u00b2)+r\u00b2d\u03a9\u00b2]"
    },
    {
      "eq_number": 134,
      "title": "Geodesic Equation",
      "domain_id": 6,
      "significance": "d\u00b2x^\u03bc/d\u03c4\u00b2+\u0393^\u03bc_\u03b1\u03b2(dx^\u03b1/d\u03c4)(dx^\u03b2/d\u03c4)=0"
    },
    {
      "eq_number": 135,
      "title": "Gravitational Wave (TT Gauge)",
      "domain_id": 6,
      "significance": "h_\u03bc\u03bd^{TT} has only h_+,h_\u00d7 spatial transverse components"
    },
    {
      "eq_number": 136,
      "title": "Bekenstein-Hawking Black Hole Entropy",
      "domain_id": 6,
      "significance": "S_BH=k_B A/4\u2113_P\u00b2=k_B c\u00b3A/(4G\u210f)"
    },
    {
      "eq_number": 137,
      "title": "Hawking Temperature",
      "domain_id": 6,
      "significance": "T_H=\u210fc\u00b3/(8\u03c0GMk_B); T_H(M\u2299)\u22486.2\u00d710\u207b\u2078 K"
    },
    {
      "eq_number": 138,
      "title": "Black Hole Area Theorem (Hawking 1971)",
      "domain_id": 6,
      "significance": "dA/dt\u22650; horizon area never decreases"
    },
    {
      "eq_number": 139,
      "title": "Standard Model Lagrangian (Full)",
      "domain_id": 7,
      "significance": "\u2112_SM=\u2112_gauge+\u2112_fermion+\u2112_Higgs+\u2112_Yukawa; SU(3)\u00d7SU(2)\u00d7U(1)"
    },
    {
      "eq_number": 140,
      "title": "Yang-Mills Field Strength",
      "domain_id": 7,
      "significance": "F_\u03bc\u03bd^a=\u2202_\u03bcA_\u03bd^a\u2212\u2202_\u03bdA_\u03bc^a+gf^{abc}A_\u03bc^bA_\u03bd^c"
    },
    {
      "eq_number": 141,
      "title": "QED Lagrangian",
      "domain_id": 7,
      "significance": "\u2112_QED=\u03c8\u0304(i\u2202\u0338\u2212m)\u03c8\u2212e\u03c8\u0304\u03b3^\u03bc\u03c8A_\u03bc\u2212\u00bcF_\u03bc\u03bdF^{\u03bc\u03bd}"
    },
    {
      "eq_number": 142,
      "title": "QCD Lagrangian",
      "domain_id": 7,
      "significance": "\u2112_QCD=\u03a3\u03c8\u0304_f(iD\u0338\u2212m_f)\u03c8_f\u2212\u00bcG_a^{\u03bc\u03bd}G^a_{\u03bc\u03bd}"
    },
    {
      "eq_number": 143,
      "title": "Electroweak Symmetry Breaking",
      "domain_id": 7,
      "significance": "SU(2)_L\u00d7U(1)_Y\u2192U(1)_EM via Higgs VEV"
    },
    {
      "eq_number": 144,
      "title": "QCD Beta Function (1-loop)",
      "domain_id": 7,
      "significance": "\u03b2(\u03b1_s)=\u2212(b\u2080/2\u03c0)\u03b1_s\u00b2; b\u2080=11\u22122n_f/3"
    },
    {
      "eq_number": 145,
      "title": "DGLAP Evolution Equations",
      "domain_id": 7,
      "significance": "\u2202q/\u2202lnQ\u00b2=(\u03b1_s/2\u03c0)\u222b(dz/z)[P_qq q+P_qg g]; gluon evolution similarly"
    },
    {
      "eq_number": 146,
      "title": "CKM Matrix (Quark Mixing)",
      "domain_id": 7,
      "significance": "3\u00d73 unitary; 4 parameters (3 angles+1 CP phase)"
    },
    {
      "eq_number": 147,
      "title": "PMNS Matrix (Neutrino Mixing)",
      "domain_id": 7,
      "significance": "3\u00d73 leptonic mixing; \u03b8\u2081\u2082\u224833\u00b0,\u03b8\u2082\u2083\u224845\u00b0,\u03b8\u2081\u2083\u22488.5\u00b0"
    },
    {
      "eq_number": 148,
      "title": "Gell-Mann\u2013Oakes\u2013Renner Relation",
      "domain_id": 7,
      "significance": "m_\u03c0\u00b2=\u2212(m_u+m_d)\u27e8\u03c8\u0304\u03c8\u27e9/f_\u03c0\u00b2"
    },
    {
      "eq_number": 149,
      "title": "Higgs Mechanism (Mass Generation)",
      "domain_id": 7,
      "significance": "Scalar VEV v=246 GeV\u2192W,Z masses; fermion masses via Yukawa"
    },
    {
      "eq_number": 150,
      "title": "Weinberg Angle",
      "domain_id": 7,
      "significance": "sin\u00b2\u03b8_W=1\u2212M_W\u00b2/M_Z\u00b2; 0.23121\u00b10.00004"
    },
    {
      "eq_number": 151,
      "title": "Faddeev-Popov Gauge Fixing + Ghosts",
      "domain_id": 7,
      "significance": "Anticommuting scalar ghosts cancel unphysical gluon d.o.f."
    },
    {
      "eq_number": 152,
      "title": "BRST Symmetry",
      "domain_id": 7,
      "significance": "Residual global symmetry after gauge fixing"
    },
    {
      "eq_number": 153,
      "title": "Running Coupling (RGE, General)",
      "domain_id": 7,
      "significance": "\u03bc dg/d\u03bc=\u03b2(g); \u03bc d m/d\u03bc=\u03b3_m m"
    },
    {
      "eq_number": 154,
      "title": "Fermi's Theory (4-Fermion, Low-Energy EW)",
      "domain_id": 7,
      "significance": "\u2112_eff=\u2212(G_F/\u221a2) J_\u03bc^{CC} J^{CC\u2020\u03bc}"
    },
    {
      "eq_number": 155,
      "title": "Pati-Salam Model (SU(4)\u00d7SU(2)\u00d7SU(2))",
      "domain_id": 7,
      "significance": "Partial unification with lepton as 4th color"
    },
    {
      "eq_number": 156,
      "title": "Grand Unified Theories (GUTs)",
      "domain_id": 7,
      "significance": "SU(5), SO(10), E\u2086 unification at ~10\u00b9\u2076 GeV"
    },
    {
      "eq_number": 157,
      "title": "Axion (Peccei-Quinn Solution to Strong CP)",
      "domain_id": 7,
      "significance": "a\u2192\u03b3\u03b3; m_a~\u03bceV\u2212meV"
    },
    {
      "eq_number": 158,
      "title": "Muon g\u22122 Anomaly",
      "domain_id": 7,
      "significance": "a_\u03bc(exp) = 0.001165920705(148) (Fermilab final, June 2025); a_\u03bc(theory) = 0.00116592033(62) (lattice QCD white paper, May 2025). Now consistent; long-standing 4.2\u03c3 tension resolved."
    },
    {
      "eq_number": 159,
      "title": "First Friedmann Equation",
      "domain_id": 8,
      "significance": "H\u00b2=(\u0227/a)\u00b2=8\u03c0G\u03c1/3\u2212kc\u00b2/a\u00b2+\u039bc\u00b2/3; H\u2080=67.4 km/s/Mpc"
    },
    {
      "eq_number": 160,
      "title": "Second Friedmann Equation",
      "domain_id": 8,
      "significance": "\u00e4/a=\u22124\u03c0G(\u03c1+3p/c\u00b2)/3+\u039bc\u00b2/3"
    },
    {
      "eq_number": 161,
      "title": "Cosmological Fluid Equation",
      "domain_id": 8,
      "significance": "\u03c1\u0307+3H(\u03c1+p/c\u00b2)=0"
    },
    {
      "eq_number": 162,
      "title": "Redshift Relation",
      "domain_id": 8,
      "significance": "1+z=a\u2080/a(t); \u03bb_obs=\u03bb_emit(1+z)"
    },
    {
      "eq_number": 163,
      "title": "Hubble-Lema\u00eetre Law",
      "domain_id": 8,
      "significance": "v=H\u2080 d (low z)"
    },
    {
      "eq_number": 164,
      "title": "CMB Blackbody Spectrum",
      "domain_id": 8,
      "significance": "T\u2080=2.72548\u00b10.00057 K; \u0394T/T\u2080<50 ppm"
    },
    {
      "eq_number": 165,
      "title": "BBN Primordial Element Abundances",
      "domain_id": 8,
      "significance": "Y_p=0.24709\u00b10.00025; D/H=(2.527\u00b10.030)\u00d710\u207b\u2075"
    },
    {
      "eq_number": 166,
      "title": "Sound Horizon at Recombination",
      "domain_id": 8,
      "significance": "r_s\u2248147 Mpc (comoving); BAO standard ruler"
    },
    {
      "eq_number": 167,
      "title": "Sachs-Wolfe Effect (CMB)",
      "domain_id": 8,
      "significance": "\u0394T/T=\u2212\u03a6/(3c\u00b2) at large angular scales"
    },
    {
      "eq_number": 168,
      "title": "Dark Energy Equation of State",
      "domain_id": 8,
      "significance": "w=p/\u03c1c\u00b2=\u22121.03\u00b10.03"
    },
    {
      "eq_number": 169,
      "title": "Deceleration Parameter",
      "domain_id": 8,
      "significance": "q\u2080=\u2212\u00e4a/\u0227\u00b2=\u22120.53\u00b10.02"
    },
    {
      "eq_number": 170,
      "title": "Matter Power Spectrum",
      "domain_id": 8,
      "significance": "P(k)~k^{n_s}; n_s=0.9649\u00b10.0042"
    },
    {
      "eq_number": 171,
      "title": "Cosmic Distance Ladder Relations",
      "domain_id": 8,
      "significance": "d_L=(1+z)\u03c7; \u03bc=5log\u2081\u2080(d_L/10pc)"
    },
    {
      "eq_number": 172,
      "title": "Inflation (Slow-Roll)",
      "domain_id": 8,
      "significance": "\u03b5=\u2212H\u0307/H\u00b2\u226a1; \u03b7\u226a1 scalar spectral index"
    },
    {
      "eq_number": 173,
      "title": "Hubble Tension",
      "domain_id": 8,
      "significance": "H\u2080(CMB)=67.4\u00b10.5 vs H\u2080(local)=73.0\u00b11.0 (5\u03c3)"
    },
    {
      "eq_number": 174,
      "title": "S\u2088 Tension",
      "domain_id": 8,
      "significance": "\u03c3\u2088(\u03a9_m/0.3)^{0.5}=0.832\u00b10.013 (CMB) vs ~0.76 (WL)"
    },
    {
      "eq_number": 175,
      "title": "Age of the Universe",
      "domain_id": 8,
      "significance": "t\u2080=13.797\u00b10.023 Gyr (Planck 2018)"
    },
    {
      "eq_number": 176,
      "title": "Navier-Stokes Equation (Incompressible)",
      "domain_id": 9,
      "significance": "\u2202v/\u2202t+(v\u00b7\u2207)v=\u2212(1/\u03c1)\u2207p+\u03bd\u2207\u00b2v+g; \u2207\u00b7v=0"
    },
    {
      "eq_number": 177,
      "title": "Continuity Equation (Fluid)",
      "domain_id": 9,
      "significance": "\u2202\u03c1/\u2202t+\u2207\u00b7(\u03c1v)=0"
    },
    {
      "eq_number": 178,
      "title": "Euler Equation (Inviscid)",
      "domain_id": 9,
      "significance": "\u2202v/\u2202t+(v\u00b7\u2207)v=\u2212(1/\u03c1)\u2207p+g (\u03bc=0 limit)"
    },
    {
      "eq_number": 179,
      "title": "Bernoulli's Equation",
      "domain_id": 9,
      "significance": "p+\u00bd\u03c1v\u00b2+\u03c1gz=constant (steady, incompressible, inviscid)"
    },
    {
      "eq_number": 180,
      "title": "Stokes Law (Drag on Sphere)",
      "domain_id": 9,
      "significance": "F_d=6\u03c0\u03bcRv (Re\u226a1)"
    },
    {
      "eq_number": 181,
      "title": "Poiseuille Flow (Hagen-Poiseuille)",
      "domain_id": 9,
      "significance": "Q=\u03c0GR\u2074/(8\u03bc); v_z(r)=(G/4\u03bc)(R\u00b2\u2212r\u00b2)"
    },
    {
      "eq_number": 182,
      "title": "Reynolds Number",
      "domain_id": 9,
      "significance": "Re=\u03c1UL/\u03bc; transition at Re~2300 (pipe)"
    },
    {
      "eq_number": 183,
      "title": "Kolmogorov Energy Spectrum",
      "domain_id": 9,
      "significance": "E(k)=C_K \u03b5^{2/3}k^{\u22125/3}; C_K\u22481.5"
    },
    {
      "eq_number": 184,
      "title": "Froude Number",
      "domain_id": 9,
      "significance": "Fr=v/\u221a(gL); wave/gravity scaling"
    },
    {
      "eq_number": 185,
      "title": "Mach Number",
      "domain_id": 9,
      "significance": "Ma=v/c_s; compressibility measure"
    },
    {
      "eq_number": 186,
      "title": "Kutta-Joukowski Theorem (Lift)",
      "domain_id": 9,
      "significance": "L'=\u03c1v\u0393 (lift per unit span)"
    },
    {
      "eq_number": 187,
      "title": "Torricelli's Law (Efflux Speed)",
      "domain_id": 9,
      "significance": "v=\u221a(2gh); speed of fluid from orifice"
    },
    {
      "eq_number": 188,
      "title": "Archimedes' Principle",
      "domain_id": 9,
      "significance": "F_b=\u03c1_fluid V_displaced g"
    },
    {
      "eq_number": 189,
      "title": "Surface Tension (Young-Laplace)",
      "domain_id": 9,
      "significance": "\u0394p=2\u03b3/R (spherical); \u0394p=\u03b3(1/R\u2081+1/R\u2082)"
    },
    {
      "eq_number": 190,
      "title": "Kelvin-Helmholtz Instability Condition",
      "domain_id": 9,
      "significance": "Instability when (\u03c1\u2081\u03c1\u2082/\u03c1\u2081+\u03c1\u2082)(v\u2081\u2212v\u2082)\u00b2>2\u221a(\u03c1\u2081\u03c1\u2082)g\u03b3"
    },
    {
      "eq_number": 191,
      "title": "Snell's Law of Refraction",
      "domain_id": 10,
      "significance": "n\u2081 sin\u03b8\u2081=n\u2082 sin\u03b8\u2082"
    },
    {
      "eq_number": 192,
      "title": "Thin Lens Equation",
      "domain_id": 10,
      "significance": "1/f=1/d_o+1/d_i"
    },
    {
      "eq_number": 193,
      "title": "Lens Maker's Formula",
      "domain_id": 10,
      "significance": "1/f=(n\u22121)(1/R\u2081\u22121/R\u2082)"
    },
    {
      "eq_number": 194,
      "title": "Magnification (Geometric Optics)",
      "domain_id": 10,
      "significance": "M=\u2212d_i/d_o=h_i/h_o"
    },
    {
      "eq_number": 195,
      "title": "Scalar Wave Equation (d'Alembert)",
      "domain_id": 10,
      "significance": "\u2202\u00b2u/\u2202t\u00b2=c\u00b2\u2207\u00b2u"
    },
    {
      "eq_number": 196,
      "title": "Young's Double-Slit Interference",
      "domain_id": 10,
      "significance": "\u0394y=\u03bbL/d (fringe spacing); d sin\u03b8=m\u03bb (maxima)"
    },
    {
      "eq_number": 197,
      "title": "Single-Slit Diffraction",
      "domain_id": 10,
      "significance": "I(\u03b8)=I\u2080[sin(\u03b2/2)/(\u03b2/2)]\u00b2; \u03b2=2\u03c0a sin\u03b8/\u03bb"
    },
    {
      "eq_number": 198,
      "title": "Grating Equation",
      "domain_id": 10,
      "significance": "d(sin\u03b8_i+sin\u03b8_m)=m\u03bb"
    },
    {
      "eq_number": 199,
      "title": "Bragg's Law (X-ray Diffraction)",
      "domain_id": 10,
      "significance": "n\u03bb=2d sin\u03b8"
    },
    {
      "eq_number": 200,
      "title": "Fresnel Equations (Amplitude Reflection/Transmission)",
      "domain_id": 10,
      "significance": "r_s=(n\u2081cos\u03b8_i\u2212n\u2082cos\u03b8_t)/(n\u2081cos\u03b8_i+n\u2082cos\u03b8_t); etc."
    },
    {
      "eq_number": 201,
      "title": "Brewster's Angle",
      "domain_id": 10,
      "significance": "\u03b8_B=arctan(n\u2082/n\u2081); reflected p-pol vanishes"
    },
    {
      "eq_number": 202,
      "title": "Malus's Law",
      "domain_id": 10,
      "significance": "I=I\u2080 cos\u00b2\u03b8"
    },
    {
      "eq_number": 203,
      "title": "Rayleigh Criterion (Resolution Limit)",
      "domain_id": 10,
      "significance": "\u03b8_min=1.22\u03bb/D"
    },
    {
      "eq_number": 204,
      "title": "Abbe Sine Condition",
      "domain_id": 10,
      "significance": "n y sin\u03b8=n' y' sin\u03b8'"
    },
    {
      "eq_number": 205,
      "title": "Numerical Aperture",
      "domain_id": 10,
      "significance": "NA=n sin\u03b8; resolution d=\u03bb/(2 NA)"
    },
    {
      "eq_number": 206,
      "title": "Fermat's Principle of Least Time",
      "domain_id": 10,
      "significance": "\u03b4\u222b n ds=0; light path minimizes optical path length"
    },
    {
      "eq_number": 207,
      "title": "Huygens-Fresnel Principle",
      "domain_id": 10,
      "significance": "Every point on wavefront = source of spherical wavelets"
    },
    {
      "eq_number": 208,
      "title": "Fabry-P\u00e9rot Etalon Transmission",
      "domain_id": 10,
      "significance": "T=T_max/[1+(2F/\u03c0)\u00b2 sin\u00b2(\u03b4/2)]"
    },
    {
      "eq_number": 209,
      "title": "Critical Angle (Total Internal Reflection)",
      "domain_id": 10,
      "significance": "\u03b8_c=arcsin(n\u2082/n\u2081)"
    },
    {
      "eq_number": 210,
      "title": "Fraunhofer vs Fresnel Diffraction Condition",
      "domain_id": 10,
      "significance": "Fresnel number F=a\u00b2/\u03bbz; F\u226a1\u2192Fraunhofer; F\u226b1\u2192Fresnel"
    },
    {
      "eq_number": 211,
      "title": "Speed of Sound",
      "domain_id": 11,
      "significance": "c=\u221a(K/\u03c1); c_air=\u221a(\u03b3RT/M)\u2248331.3+0.606\u00b7T_\u00b0C m/s"
    },
    {
      "eq_number": 212,
      "title": "Doppler Effect (Sound)",
      "domain_id": 11,
      "significance": "f'=f(c\u00b1v_o)/(c\u2213v_s)"
    },
    {
      "eq_number": 213,
      "title": "Standing Waves (String/Column)",
      "domain_id": 11,
      "significance": "\u03bb_n=2L/n (fixed-fixed/open-open); \u03bb_n=4L/n (fixed-free)"
    },
    {
      "eq_number": 214,
      "title": "Decibel Scale (SPL)",
      "domain_id": 11,
      "significance": "L_p=10 log\u2081\u2080(p\u00b2/p\u2080\u00b2) dB; p\u2080=20 \u03bcPa"
    },
    {
      "eq_number": 215,
      "title": "Shock Wave Rankine-Hugoniot Relations",
      "domain_id": 11,
      "significance": "Conservation eqs across shock: \u03c1\u2081v\u2081=\u03c1\u2082v\u2082; p\u2081+\u03c1\u2081v\u2081\u00b2=p\u2082+\u03c1\u2082v\u2082\u00b2; etc."
    },
    {
      "eq_number": 216,
      "title": "Beat Frequency",
      "domain_id": 11,
      "significance": "f_beat=|f\u2081\u2212f\u2082|"
    },
    {
      "eq_number": 217,
      "title": "Helmholtz Resonator Frequency",
      "domain_id": 11,
      "significance": "f=(c/2\u03c0)\u221a(A/VL_eff)"
    },
    {
      "eq_number": 218,
      "title": "Drude Model (Electrical Conductivity)",
      "domain_id": 12,
      "significance": "\u03c3=n e\u00b2\u03c4/m; J=\u03c3E"
    },
    {
      "eq_number": 219,
      "title": "Bloch's Theorem",
      "domain_id": 12,
      "significance": "\u03c8_k(r)=e^{ik\u00b7r} u_k(r); u_k periodic"
    },
    {
      "eq_number": 220,
      "title": "Kronig-Penney Model (1D Band Structure)",
      "domain_id": 12,
      "significance": "cos ka=cos \u03b1a+(P/\u03b1a)sin \u03b1a"
    },
    {
      "eq_number": 221,
      "title": "Fermi-Dirac Distribution",
      "domain_id": 12,
      "significance": "f(E)=1/[e^{(E\u2212\u03bc)/k_B T}+1]"
    },
    {
      "eq_number": 222,
      "title": "Free Electron Density of States",
      "domain_id": 12,
      "significance": "g(E)=(1/2\u03c0\u00b2)(2m/\u210f\u00b2)^{3/2}\u221aE"
    },
    {
      "eq_number": 223,
      "title": "Sommerfeld Model (Electron Heat Capacity)",
      "domain_id": 12,
      "significance": "C_V=(\u03c0\u00b2/2)n k_B(k_B T/E_F)"
    },
    {
      "eq_number": 224,
      "title": "BCS Theory (Superconductivity)",
      "domain_id": 12,
      "significance": "T_c=1.13\u0398_D e^{\u22121/N(0)V}; \u0394(T); Cooper pairs"
    },
    {
      "eq_number": 225,
      "title": "BCS Gap Equation at T=0",
      "domain_id": 12,
      "significance": "\u0394(0)=1.76 k_B T_c"
    },
    {
      "eq_number": 226,
      "title": "London Equations (Perfect Diamagnetism)",
      "domain_id": 12,
      "significance": "\u2202J_s/\u2202t=(n_s e\u00b2/m)E; \u2207\u00d7J_s=\u2212(n_s e\u00b2/m)B"
    },
    {
      "eq_number": 227,
      "title": "Josephson Effects (DC + AC)",
      "domain_id": 12,
      "significance": "I=I_c sin \u03c6 (DC); d\u03c6/dt=(2e/\u210f)V=(2\u03c0/\u03a6\u2080)V (AC)"
    },
    {
      "eq_number": 228,
      "title": "Curie's Law (Paramagnetism)",
      "domain_id": 12,
      "significance": "\u03c7=C/T; C=N\u03bc\u00b2/(3k_B)"
    },
    {
      "eq_number": 229,
      "title": "Curie-Weiss Law (Ferromagnetism)",
      "domain_id": 12,
      "significance": "\u03c7=C/(T\u2212T_c) above Curie point"
    },
    {
      "eq_number": 230,
      "title": "Heisenberg Exchange Interaction",
      "domain_id": 12,
      "significance": "H=\u2212J \u03a3_{\u27e8ij\u27e9} S_i\u00b7S_j"
    },
    {
      "eq_number": 231,
      "title": "Magnetic Hysteresis Loop",
      "domain_id": 12,
      "significance": "B\u2212H loop; remanence B_r, coercivity H_c"
    },
    {
      "eq_number": 232,
      "title": "Einstein Relation (Diffusion)",
      "domain_id": 12,
      "significance": "D=\u03bc k_B T/e"
    },
    {
      "eq_number": 233,
      "title": "Seebeck Effect (Thermoelectricity)",
      "domain_id": 12,
      "significance": "\u0394V=S \u0394T; S=\u2212(\u03c0\u00b2k_B\u00b2T/3e)(d ln \u03c3/dE)_{E=\u03bc}"
    },
    {
      "eq_number": 234,
      "title": "Hall Effect",
      "domain_id": 12,
      "significance": "V_H=(I B)/(n e d); R_H=1/(n e)"
    },
    {
      "eq_number": 235,
      "title": "Quantum Hall Effect (Integer)",
      "domain_id": 12,
      "significance": "R_H=h/(\u03bd e\u00b2); \u03bd integer; exact quantization"
    },
    {
      "eq_number": 236,
      "title": "Wiedemann-Franz Law",
      "domain_id": 12,
      "significance": "\u03ba/(\u03c3T)=L; L=(\u03c0\u00b2/3)(k_B/e)\u00b2\u22482.44\u00d710\u207b\u2078 W\u03a9/K\u00b2"
    },
    {
      "eq_number": 237,
      "title": "Debye Model (Lattice Heat Capacity)",
      "domain_id": 12,
      "significance": "C_V\u2248(12\u03c0\u2074/5) N k_B (T/\u0398_D)\u00b3 for T\u226a\u0398_D"
    },
    {
      "eq_number": 238,
      "title": "Mott Insulator Transition",
      "domain_id": 12,
      "significance": "U/t\u226bW\u2192Mott insulating gap; metal-insulator transition"
    },
    {
      "eq_number": 239,
      "title": "Density Functional Theory (Kohn-Sham Equations)",
      "domain_id": 12,
      "significance": "(\u2212\u00bd\u2207\u00b2+v_eff(r))\u03c6_i(r)=\u03b5_i \u03c6_i(r)"
    },
    {
      "eq_number": 240,
      "title": "Landau Fermi Liquid Theory",
      "domain_id": 12,
      "significance": "Quasiparticles with renormalized mass m*/m; same quantum numbers"
    },
    {
      "eq_number": 241,
      "title": "Radioactive Decay Law",
      "domain_id": 13,
      "significance": "N(t)=N\u2080 e^{\u2212\u03bbt}; T_{1/2}=ln 2/\u03bb; \u03c4=1/\u03bb"
    },
    {
      "eq_number": 242,
      "title": "Bethe-Weizs\u00e4cker (Semi-Empirical) Mass Formula",
      "domain_id": 13,
      "significance": "B=a_vA\u2212a_sA^{2/3}\u2212a_cZ\u00b2/A^{1/3}\u2212a_a(N\u2212Z)\u00b2/A+\u03b4(A,Z)"
    },
    {
      "eq_number": 243,
      "title": "Geiger-Nuttall Law (\u03b1-Decay)",
      "domain_id": 13,
      "significance": "log T_{1/2}=A+B/\u221aE_\u03b1"
    },
    {
      "eq_number": 244,
      "title": "Nuclear Shell Model (Magic Numbers)",
      "domain_id": 13,
      "significance": "Magic no: 2,8,20,28,50,82,126; spin-orbit coupling"
    },
    {
      "eq_number": 245,
      "title": "Q-Value of Nuclear Reaction",
      "domain_id": 13,
      "significance": "Q=(m_initial\u2212m_final)c\u00b2"
    },
    {
      "eq_number": 246,
      "title": "Neutrino Oscillation Probability",
      "domain_id": 13,
      "significance": "P(\u03bd_\u03b1\u2192\u03bd_\u03b2)=sin\u00b2(2\u03b8) sin\u00b2(\u0394m\u00b2 L/4E)"
    },
    {
      "eq_number": 247,
      "title": "Four-Factor Formula (Nuclear Reactor)",
      "domain_id": 13,
      "significance": "k_eff=\u03b7 \u03b5 p f; criticality when k_eff=1"
    },
    {
      "eq_number": 248,
      "title": "Rutherford Scattering Cross-Section",
      "domain_id": 13,
      "significance": "d\u03c3/d\u03a9=(Z\u2081Z\u2082e\u00b2/16\u03c0\u03b5\u2080E)\u00b2 csc\u2074(\u03b8/2)"
    },
    {
      "eq_number": 249,
      "title": "M\u00f6ssbauer Effect (Recoilless \u03b3 Emission)",
      "domain_id": 13,
      "significance": "Fraction f=exp(\u2212k\u00b2\u27e8x\u00b2\u27e9)"
    },
    {
      "eq_number": 250,
      "title": "Breit-Wigner Resonance (Nuclear Reactions)",
      "domain_id": 13,
      "significance": "\u03c3(E)=\u03c0\u019b\u00b2 g (\u0393_a \u0393_b)/[(E\u2212E_R)\u00b2+\u0393\u00b2/4]"
    },
    {
      "eq_number": 251,
      "title": "Lane-Emden Equation (Polytropic Stars)",
      "domain_id": 14,
      "significance": "(1/\u03be\u00b2)d(\u03be\u00b2 d\u03b8/d\u03be)/d\u03be=\u2212\u03b8^n"
    },
    {
      "eq_number": 252,
      "title": "Eddington Luminosity Limit",
      "domain_id": 14,
      "significance": "L_Edd=4\u03c0GM m_p c/\u03c3_T\u22481.3\u00d710\u00b3\u00b9(M/M\u2299) W"
    },
    {
      "eq_number": 253,
      "title": "Chandrasekhar Limit (White Dwarf)",
      "domain_id": 14,
      "significance": "M_Ch\u22481.44 M\u2299 (electron degeneracy pressure)"
    },
    {
      "eq_number": 254,
      "title": "TOV Limit (Neutron Star Maximum Mass)",
      "domain_id": 14,
      "significance": "M_max\u22482\u22123 M\u2299 (equation of state dependent)"
    },
    {
      "eq_number": 255,
      "title": "Hertzsprung-Russell Diagram + Main Sequence",
      "domain_id": 14,
      "significance": "L\u221dM^{3.5} (MS, M>0.5M\u2299); stellar radii, T_eff"
    },
    {
      "eq_number": 256,
      "title": "Mass-Luminosity Relation",
      "domain_id": 14,
      "significance": "L/L\u2299\u2248(M/M\u2299)^{3.5} (MS, intermediate mass)"
    },
    {
      "eq_number": 257,
      "title": "Virial Theorem (Astrophysics)",
      "domain_id": 14,
      "significance": "2\u27e8T\u27e9+\u27e8U\u27e9=0 for gravitational systems"
    },
    {
      "eq_number": 258,
      "title": "Jeans Instability Criterion (Star Formation)",
      "domain_id": 14,
      "significance": "\u03bb_J=c_s\u221a(\u03c0/G\u03c1); M_J\u221dc_s\u00b3/\u221a(G\u00b3\u03c1)"
    },
    {
      "eq_number": 259,
      "title": "Schwarzschild Criterion (Convection)",
      "domain_id": 14,
      "significance": "|dT/dr|_rad>|dT/dr|_ad\u2192convective instability"
    },
    {
      "eq_number": 260,
      "title": "pp Chain Energy Release",
      "domain_id": 14,
      "significance": "4p\u2192\u2074He+2e\u207a+2\u03bd_e+26.73 MeV"
    },
    {
      "eq_number": 261,
      "title": "CNO Cycle (Massive Stars)",
      "domain_id": 14,
      "significance": "C, N, O catalytic H fusion; dominant above ~1.3 M\u2299"
    },
    {
      "eq_number": 262,
      "title": "Triple-Alpha Process (Helium Burning)",
      "domain_id": 14,
      "significance": "3 \u2074He\u2192\u00b9\u00b2C+7.65 MeV (Hoyle resonance at 7.65 MeV)"
    },
    {
      "eq_number": 263,
      "title": "Core-Collapse Supernova Mechanism",
      "domain_id": 14,
      "significance": "Fe core infall\u2192neutrino burst\u2192explosion (delayed neutrino mechanism)"
    },
    {
      "eq_number": 264,
      "title": "Type Ia Supernova (Standardizable Candle)",
      "domain_id": 14,
      "significance": "Chandrasekhar mass WD thermonuclear detonation; Phillips rel."
    },
    {
      "eq_number": 265,
      "title": "Neutron Star Equation of State (Various)",
      "domain_id": 14,
      "significance": "p(\u03c1) from nuclear matter theory; constraints from NS masses"
    },
    {
      "eq_number": 266,
      "title": "Oppenheimer-Snyder Collapse (BH Formation)",
      "domain_id": 14,
      "significance": "Dust ball collapse\u2192BH; event horizon forms"
    },
    {
      "eq_number": 267,
      "title": "Pulsar Spin-Down",
      "domain_id": 14,
      "significance": "\u0116=\u2212I \u03c9 \u03c9\u0307; B_dipole\u22483.2\u00d710\u00b9\u2079\u221a(P \u1e56) G"
    },
    {
      "eq_number": 268,
      "title": "Olbers' Paradox Resolution",
      "domain_id": 14,
      "significance": "Dark night sky\u2192finite age+expanding universe"
    },
    {
      "eq_number": 269,
      "title": "Debye Length (Plasma Screening)",
      "domain_id": 15,
      "significance": "\u03bb_D=\u221a(\u03b5\u2080 k_B T/(n e\u00b2))"
    },
    {
      "eq_number": 270,
      "title": "Plasma Frequency",
      "domain_id": 15,
      "significance": "\u03c9_p=\u221a(n e\u00b2/(\u03b5\u2080 m_e))\u224856.4\u221an (rad/s)"
    },
    {
      "eq_number": 271,
      "title": "Alfv\u00e9n Wave Speed",
      "domain_id": 15,
      "significance": "v_A=B\u2080/\u221a(\u03bc\u2080\u03c1)"
    },
    {
      "eq_number": 272,
      "title": "MHD Induction Equation",
      "domain_id": 15,
      "significance": "\u2202B/\u2202t=\u2207\u00d7(v\u00d7B)+\u03b7\u2207\u00b2B"
    },
    {
      "eq_number": 273,
      "title": "Saha Ionization Equation",
      "domain_id": 15,
      "significance": "n_{i+1}n_e/n_i=(2/\u03bb\u00b3_deB)(U_{i+1}/U_i)e^{\u2212\u03c7/(k_B T)}"
    },
    {
      "eq_number": 274,
      "title": "Gyro-frequency (Larmor Frequency)",
      "domain_id": 15,
      "significance": "\u03c9_c=qB/m; r_L=v_\u22a5/\u03c9_c"
    },
    {
      "eq_number": 275,
      "title": "Beta Parameter (Plasma Confinement)",
      "domain_id": 15,
      "significance": "\u03b2=2\u03bc\u2080 p/B\u00b2"
    },
    {
      "eq_number": 276,
      "title": "Lawson Criterion (Fusion Ignition)",
      "domain_id": 15,
      "significance": "n T \u03c4_E>3\u00d710\u00b2\u00b9 keV\u00b7s/m\u00b3 (D-T)"
    },
    {
      "eq_number": 277,
      "title": "Noether's Theorem",
      "domain_id": 16,
      "significance": "Continuous symmetry \u21d4 conserved current/charge"
    },
    {
      "eq_number": 278,
      "title": "Stokes' Theorem",
      "domain_id": 16,
      "significance": "\u222b_S (\u2207\u00d7F)\u00b7dS=\u222e_C F\u00b7dl"
    },
    {
      "eq_number": 279,
      "title": "Gauss's Divergence Theorem",
      "domain_id": 16,
      "significance": "\u222b_V \u2207\u00b7F dV=\u222e_S F\u00b7dS"
    },
    {
      "eq_number": 280,
      "title": "Green's Theorem (2D)",
      "domain_id": 16,
      "significance": "\u222c(\u2202Q/\u2202x\u2212\u2202P/\u2202y)dxdy=\u222e Pdx+Qdy"
    },
    {
      "eq_number": 281,
      "title": "Fourier Transform",
      "domain_id": 16,
      "significance": "F(k)=\u222b f(x)e^{\u2212ikx}dx; f(x)=(1/2\u03c0)\u222b F(k)e^{ikx}dk"
    },
    {
      "eq_number": 282,
      "title": "Laplace's Equation",
      "domain_id": 16,
      "significance": "\u2207\u00b2\u03c6=0; harmonic functions"
    },
    {
      "eq_number": 283,
      "title": "Poisson's Equation",
      "domain_id": 16,
      "significance": "\u2207\u00b2\u03c6=\u2212f(x); fundamental PDE of physics"
    },
    {
      "eq_number": 284,
      "title": "Bessel's Equation",
      "domain_id": 16,
      "significance": "x\u00b2 y''+x y'+(x\u00b2\u2212n\u00b2)y=0"
    },
    {
      "eq_number": 285,
      "title": "Legendre's Equation",
      "domain_id": 16,
      "significance": "(1\u2212x\u00b2)y''\u22122xy'+n(n+1)y=0"
    },
    {
      "eq_number": 286,
      "title": "Hermite's Equation",
      "domain_id": 16,
      "significance": "y''\u22122xy'+2ny=0"
    },
    {
      "eq_number": 287,
      "title": "Associated Legendre Equation",
      "domain_id": 16,
      "significance": "(1\u2212x\u00b2)y''\u22122xy'+[n(n+1)\u2212m\u00b2/(1\u2212x\u00b2)]y=0"
    },
    {
      "eq_number": 288,
      "title": "Chebyshev Polynomials",
      "domain_id": 16,
      "significance": "T_n(cos\u03b8)=cos(n\u03b8); orthogonality"
    },
    {
      "eq_number": 289,
      "title": "Laguerre Polynomials",
      "domain_id": 16,
      "significance": "x y''+(1\u2212x)y'+n y=0"
    },
    {
      "eq_number": 290,
      "title": "Spherical Harmonics (Y_l^m)",
      "domain_id": 16,
      "significance": "Y_l^m(\u03b8,\u03c6)=\u221a((2l+1)(l\u2212m)!/4\u03c0(l+m)!) P_l^m(cos\u03b8) e^{im\u03c6}"
    },
    {
      "eq_number": 291,
      "title": "Gamma Function",
      "domain_id": 16,
      "significance": "\u0393(z)=\u222b\u2080^\u221e t^{z\u22121}e^{\u2212t}dt; \u0393(n+1)=n!"
    },
    {
      "eq_number": 292,
      "title": "Error Function",
      "domain_id": 16,
      "significance": "erf(x)=(2/\u221a\u03c0)\u222b\u2080^x e^{\u2212t\u00b2}dt"
    },
    {
      "eq_number": 293,
      "title": "Delta Function (Dirac)",
      "domain_id": 16,
      "significance": "\u222b \u03b4(x\u2212a)f(x)dx=f(a); \u222b \u03b4(x)dx=1"
    },
    {
      "eq_number": 294,
      "title": "Eigenvalue Equation",
      "domain_id": 16,
      "significance": "\u00c2v=\u03bbv"
    },
    {
      "eq_number": 295,
      "title": "Separation of Variables Method",
      "domain_id": 16,
      "significance": "\u03c8(x,y,z)=X(x)Y(y)Z(z); decouples PDEs"
    },
    {
      "eq_number": 296,
      "title": "Boltzmann Distribution",
      "domain_id": 17,
      "significance": "p_i=g_i e^{\u2212\u03b2E_i}/Z; \u03b2=1/k_B T"
    },
    {
      "eq_number": 297,
      "title": "Canonical Partition Function",
      "domain_id": 17,
      "significance": "Z=\u03a3 g_i e^{\u2212\u03b2E_i}; F=\u2212k_B T ln Z"
    },
    {
      "eq_number": 298,
      "title": "Grand Canonical Partition Function",
      "domain_id": 17,
      "significance": "\u039e=\u03a3_{N} \u03a3_{E} e^{\u2212\u03b2(E\u2212\u03bcN)}; \u03a9=\u2212k_B T ln \u039e"
    },
    {
      "eq_number": 299,
      "title": "Boltzmann Entropy Formula",
      "domain_id": 17,
      "significance": "S=k_B ln \u03a9"
    },
    {
      "eq_number": 300,
      "title": "Gibbs Entropy Formula",
      "domain_id": 17,
      "significance": "S=\u2212k_B \u03a3 p_i ln p_i"
    },
    {
      "eq_number": 301,
      "title": "Fluctuation-Dissipation Theorem",
      "domain_id": 17,
      "significance": "\u27e8x\u00b2\u27e9_\u03c9=(2k_B T/\u03c9) Im \u03c7(\u03c9)"
    },
    {
      "eq_number": 302,
      "title": "Einstein-Smoluchowski Relation (Diffusion)",
      "domain_id": 17,
      "significance": "\u27e8x\u00b2\u27e9=2Dt; D=\u03bc k_B T"
    },
    {
      "eq_number": 303,
      "title": "Jarzynski Equality",
      "domain_id": 17,
      "significance": "\u27e8e^{\u2212W/k_B T}\u27e9=e^{\u2212\u0394F/k_B T}"
    },
    {
      "eq_number": 304,
      "title": "Crooks Fluctuation Theorem",
      "domain_id": 17,
      "significance": "P_F(W)/P_R(\u2212W)=e^{(W\u2212\u0394F)/k_B T}"
    },
    {
      "eq_number": 305,
      "title": "Ising Model (1D/2D Exact Solution)",
      "domain_id": 17,
      "significance": "2D Onsager solution: T_c=2.269 J/k_B"
    },
    {
      "eq_number": 306,
      "title": "Central Limit Theorem (Statistical)",
      "domain_id": 17,
      "significance": "(1/n)\u03a3 X_i \u2192 N(\u03bc,\u03c3\u00b2/n)"
    },
    {
      "eq_number": 307,
      "title": "Bose-Einstein Condensation (T_c)",
      "domain_id": 17,
      "significance": "T_c=(2\u03c0\u210f\u00b2/m k_B)(n/\u03b6(3/2))^{2/3}"
    },
    {
      "eq_number": 308,
      "title": "Kramers-Kronig Relations (Dispersion)",
      "domain_id": 17,
      "significance": "Re \u03c7(\u03c9)=(1/\u03c0) P\u222b Im \u03c7(\u03c9')/(\u03c9'\u2212\u03c9)d\u03c9'"
    },
    {
      "eq_number": 309,
      "title": "Cauchy Stress Principle",
      "domain_id": 18,
      "significance": "t=\u03c3\u00b7n; traction vector=stress tensor\u00b7normal"
    },
    {
      "eq_number": 310,
      "title": "Generalized Hooke's Law (Linear Elasticity)",
      "domain_id": 18,
      "significance": "\u03c3_{ij}=C_{ijkl} \u03b5_{kl}; 21 independent elastic constants"
    },
    {
      "eq_number": 311,
      "title": "Infinitesimal Strain Tensor",
      "domain_id": 18,
      "significance": "\u03b5_{ij}=(1/2)(\u2202_j u_i+\u2202_i u_j)"
    },
    {
      "eq_number": 312,
      "title": "Young's Modulus / Elastic Modulus",
      "domain_id": 18,
      "significance": "E=\u03c3/\u03b5 (uniaxial); stress-strain ratio"
    },
    {
      "eq_number": 313,
      "title": "Shear Modulus",
      "domain_id": 18,
      "significance": "G=\u03c4/\u03b3; G=E/[2(1+\u03bd)] (isotropic)"
    },
    {
      "eq_number": 314,
      "title": "Bulk Modulus",
      "domain_id": 18,
      "significance": "K=\u2212V dp/dV; K=E/[3(1\u22122\u03bd)] (isotropic)"
    },
    {
      "eq_number": 315,
      "title": "Poisson's Ratio",
      "domain_id": 18,
      "significance": "\u03bd=\u2212\u03b5_transvers/\u03b5_axial; \u22121<\u03bd<0.5"
    },
    {
      "eq_number": 316,
      "title": "Euler-Bernoulli Beam Equation",
      "domain_id": 18,
      "significance": "EI d\u2074w/dx\u2074=q(x); deflection"
    },
    {
      "eq_number": 317,
      "title": "Timoshenko Beam Theory",
      "domain_id": 18,
      "significance": "Shear deformation included; more accurate for short beams"
    },
    {
      "eq_number": 318,
      "title": "Elastic Wave Speeds (P and S waves)",
      "domain_id": 18,
      "significance": "v_P=\u221a((K+4G/3)/\u03c1); v_S=\u221a(G/\u03c1)"
    },
    {
      "eq_number": 319,
      "title": "Creep / Viscoelastic Maxwell Model",
      "domain_id": 18,
      "significance": "d\u03b5/dt=(1/E) d\u03c3/dt + \u03c3/\u03b7"
    },
    {
      "eq_number": 320,
      "title": "Plastic Yield (Von Mises Criterion)",
      "domain_id": 18,
      "significance": "\u03c3_v=\u221a(\u00bd[(\u03c3\u2081\u2212\u03c3\u2082)\u00b2+(\u03c3\u2082\u2212\u03c3\u2083)\u00b2+(\u03c3\u2083\u2212\u03c3\u2081)\u00b2])\u2265\u03c3_y"
    },
    {
      "eq_number": 321,
      "title": "Shannon Entropy",
      "domain_id": 19,
      "significance": "H=\u2212\u03a3 p_i log\u2082 p_i (bits)"
    },
    {
      "eq_number": 322,
      "title": "Shannon-Hartley Channel Capacity",
      "domain_id": 19,
      "significance": "C=B log\u2082(1+S/N)"
    },
    {
      "eq_number": 323,
      "title": "Nyquist-Shannon Sampling Theorem",
      "domain_id": 19,
      "significance": "f_s\u22652 f_max to perfectly reconstruct"
    },
    {
      "eq_number": 324,
      "title": "Landauer's Principle",
      "domain_id": 19,
      "significance": "Erasure of 1 bit dissipates \u2265k_B T ln 2 heat"
    },
    {
      "eq_number": 325,
      "title": "Kolmogorov Complexity (Algorithmic Info)",
      "domain_id": 19,
      "significance": "K_U(x)=min{|p|:U(p)=x}"
    },
    {
      "eq_number": 326,
      "title": "Maximum Entropy Principle (Jaynes)",
      "domain_id": 19,
      "significance": "Maximize S subject to constraints\u2192least biased distribution"
    },
    {
      "eq_number": 327,
      "title": "Speed of Light Defines Meter",
      "domain_id": 20,
      "significance": "c=299792458 m/s EXACT"
    },
    {
      "eq_number": 328,
      "title": "Planck Constant Defines Kilogram",
      "domain_id": 20,
      "significance": "h=6.62607015e-34 J\u00b7s EXACT"
    },
    {
      "eq_number": 329,
      "title": "Elementary Charge Defines Ampere",
      "domain_id": 20,
      "significance": "e=1.602176634e-19 C EXACT"
    },
    {
      "eq_number": 330,
      "title": "Boltzmann Constant Defines Kelvin",
      "domain_id": 20,
      "significance": "k_B=1.380649e-23 J/K EXACT"
    },
    {
      "eq_number": 331,
      "title": "Avogadro Number Defines Mole",
      "domain_id": 20,
      "significance": "N_A=6.02214076e23 EXACT"
    },
    {
      "eq_number": 332,
      "title": "Josephson Voltage Standard",
      "domain_id": 12,
      "significance": "V=n f/K_J; K_J=2e/h=483597.9 GHz/V EXACT"
    },
    {
      "eq_number": 333,
      "title": "Quantum Hall Resistance Standard",
      "domain_id": 12,
      "significance": "R_H=h/(i e\u00b2); R_K=h/e\u00b2=25812.80745... \u03a9"
    },
    {
      "eq_number": 334,
      "title": "Bragg's Law (Generalized, Powder Diffraction)",
      "domain_id": 22,
      "significance": "n\u03bb = 2d sin \u03b8; foundation of all crystal structure determination"
    },
    {
      "eq_number": 335,
      "title": "Laue Equations (3D Diffraction Condition)",
      "domain_id": 22,
      "significance": "a\u00b7\u0394k=2\u03c0h, b\u00b7\u0394k=2\u03c0k, c\u00b7\u0394k=2\u03c0l; constructive interference in 3D lattice"
    },
    {
      "eq_number": 336,
      "title": "Structure Factor Equation",
      "domain_id": 22,
      "significance": "F_{hkl} = \u03a3_j f_j exp[2\u03c0i(hx_j+ky_j+lz_j)]; determines diffraction intensities"
    },
    {
      "eq_number": 337,
      "title": "Atomic Scattering Factor (X-ray Form Factor)",
      "domain_id": 22,
      "significance": "f(q) = \u222b \u03c1(r) exp(iq\u00b7r) d\u00b3r; Fourier transform of electron density"
    },
    {
      "eq_number": 338,
      "title": "Reciprocal Lattice Vector Definition",
      "domain_id": 22,
      "significance": "G = h a* + k b* + l c*; a*=(b\u00d7c)/V_cell, etc."
    },
    {
      "eq_number": 339,
      "title": "Brillouin Zone Boundaries",
      "domain_id": 22,
      "significance": "2 k\u00b7G = |G|\u00b2; electron wave diffraction condition at BZ boundaries"
    },
    {
      "eq_number": 340,
      "title": "Ewald Sphere Construction",
      "domain_id": 22,
      "significance": "|k| = |k'| = 2\u03c0/\u03bb; \u0394k = G falls on sphere \u2192 diffraction"
    },
    {
      "eq_number": 341,
      "title": "Patterson Function (Interatomic Vectors)",
      "domain_id": 22,
      "significance": "P(u,v,w) = \u222b |F_{hkl}|\u00b2 exp[\u22122\u03c0i(hu+kv+lw)] d*h d*k d*l"
    },
    {
      "eq_number": 342,
      "title": "Debye-Waller Factor (Thermal Motion)",
      "domain_id": 22,
      "significance": "f_T(q) = f\u2080(q) exp(\u2212\u00bd\u27e8(u\u00b7q)\u00b2\u27e9); B = 8\u03c0\u00b2\u27e8u\u00b2\u27e9"
    },
    {
      "eq_number": 343,
      "title": "Space Group Symmetry Operations",
      "domain_id": 22,
      "significance": "230 space groups in 3D; {R|t} r = R r + t"
    },
    {
      "eq_number": 344,
      "title": "Interplanar Spacing (Cubic Systems)",
      "domain_id": 22,
      "significance": "1/d\u00b2 = (h\u00b2+k\u00b2+l\u00b2)/a\u00b2 (cubic); general: depends on lattice parameters"
    },
    {
      "eq_number": 345,
      "title": "Scherrer Equation (Crystallite Size)",
      "domain_id": 22,
      "significance": "D = K \u03bb / (\u03b2 cos \u03b8); K\u22480.9; \u03b2=FWHM in radians"
    },
    {
      "eq_number": 346,
      "title": "Williamson-Hall Analysis (Size + Strain)",
      "domain_id": 22,
      "significance": "\u03b2 cos \u03b8 = K\u03bb/D + 4\u03b5 sin \u03b8; separates size and microstrain broadening"
    },
    {
      "eq_number": 347,
      "title": "True Stress \u2014 True Strain Definition",
      "domain_id": 21,
      "significance": "\u03c3_true = F/A_inst; \u03b5_true = ln(L/L\u2080) = ln(1+\u03b5_eng)"
    },
    {
      "eq_number": 348,
      "title": "Hollomon Equation (Work Hardening)",
      "domain_id": 21,
      "significance": "\u03c3 = K \u03b5^n; n = strain hardening exponent; K = strength coefficient"
    },
    {
      "eq_number": 349,
      "title": "Hall-Petch Relationship (Grain Size Strengthening)",
      "domain_id": 21,
      "significance": "\u03c3_y = \u03c3\u2080 + k_y / \u221ad; d = grain diameter"
    },
    {
      "eq_number": 350,
      "title": "Orowan Equation (Precipitation Strengthening)",
      "domain_id": 21,
      "significance": "\u0394\u03c4 = G b / L; L = interparticle spacing; b = Burgers vector"
    },
    {
      "eq_number": 351,
      "title": "Schmid's Law (Critical Resolved Shear Stress)",
      "domain_id": 21,
      "significance": "\u03c4_CRSS = \u03c3_y cos \u03c6 cos \u03bb; m = cos \u03c6 cos \u03bb (Schmid factor)"
    },
    {
      "eq_number": 352,
      "title": "Taylor Equation (Dislocation Strengthening)",
      "domain_id": 21,
      "significance": "\u03c4 = \u03b1 G b \u221a\u03c1; \u03c1 = dislocation density; \u03b1\u22480.2\u20130.5"
    },
    {
      "eq_number": 353,
      "title": "Petch-Forwood Hardness-Yield Strength Relation",
      "domain_id": 21,
      "significance": "H \u2248 3 \u03c3_y (metals); Vickers/Brinell \u2248 3 \u00d7 yield"
    },
    {
      "eq_number": 354,
      "title": "Griffith Criterion (Brittle Fracture)",
      "domain_id": 21,
      "significance": "\u03c3_f = \u221a(2E\u03b3_s / \u03c0a); critical stress for crack propagation"
    },
    {
      "eq_number": 355,
      "title": "Stress Intensity Factor (LEFM, Mode I)",
      "domain_id": 21,
      "significance": "K_I = Y \u03c3 \u221a(\u03c0a); fracture when K_I \u2265 K_Ic"
    },
    {
      "eq_number": 356,
      "title": "J-Integral (Elastic-Plastic Fracture)",
      "domain_id": 21,
      "significance": "J = \u222b_\u0393 (W dy \u2212 T_i \u2202u_i/\u2202x ds); path-independent energy release rate"
    },
    {
      "eq_number": 357,
      "title": "Paris' Law (Fatigue Crack Growth)",
      "domain_id": 21,
      "significance": "da/dN = C (\u0394K)^m; C, m material constants; m\u22482\u20134 for metals"
    },
    {
      "eq_number": 358,
      "title": "Basquin Equation (High-Cycle Fatigue)",
      "domain_id": 21,
      "significance": "\u03c3_a = \u03c3_f' (2N_f)^b; b\u2248\u22120.05 to \u22120.12 for metals"
    },
    {
      "eq_number": 359,
      "title": "Coffin-Manson Relation (Low-Cycle Fatigue)",
      "domain_id": 21,
      "significance": "\u0394\u03b5_p/2 = \u03b5_f' (2N_f)^c; c\u2248\u22120.5 to \u22120.7"
    },
    {
      "eq_number": 360,
      "title": "Norton-Bailey Creep Law",
      "domain_id": 21,
      "significance": "\u03b5_cr = A \u03c3^n t^m (primary creep); d\u03b5_cr/dt = B \u03c3^n (secondary)"
    },
    {
      "eq_number": 361,
      "title": "Larson-Miller Parameter (Creep Rupture)",
      "domain_id": 21,
      "significance": "P = T (C + log t_r); C\u224820; T in K, t_r in hours"
    },
    {
      "eq_number": 362,
      "title": "Mohr-Coulomb Failure Criterion",
      "domain_id": 21,
      "significance": "\u03c4 = c + \u03c3_n tan \u03c6; c=cohesion, \u03c6=internal friction angle"
    },
    {
      "eq_number": 363,
      "title": "Drucker-Prager Yield Criterion",
      "domain_id": 21,
      "significance": "\u221aJ\u2082 + \u03b1 I\u2081 = k; pressure-dependent yielding"
    },
    {
      "eq_number": 364,
      "title": "Weibull Distribution (Brittle Failure Statistics)",
      "domain_id": 21,
      "significance": "P_f = 1 \u2212 exp[\u2212(\u03c3/\u03c3\u2080)^m]; m = Weibull modulus"
    },
    {
      "eq_number": 365,
      "title": "Stoney Equation (Thin Film Stress)",
      "domain_id": 21,
      "significance": "\u03c3_f = E_s h_s\u00b2 \u03ba / [6(1\u2212\u03bd_s) h_f]; substrate curvature \u2192 film stress"
    },
    {
      "eq_number": 366,
      "title": "Debye Specific Heat Model (Full)",
      "domain_id": 21,
      "significance": "C_V = 9 N k_B (T/\u0398_D)\u00b3 \u222b\u2080^{\u0398_D/T} x\u2074 e^x / (e^x\u22121)\u00b2 dx"
    },
    {
      "eq_number": 367,
      "title": "Dulong-Petit Law",
      "domain_id": 21,
      "significance": "C_V = 3R \u2248 24.94 J/(mol\u00b7K) at high T (classical limit of Debye)"
    },
    {
      "eq_number": 368,
      "title": "Einstein Heat Capacity Model",
      "domain_id": 21,
      "significance": "C_V = 3 N k_B (\u0398_E/T)\u00b2 e^{\u0398_E/T} / (e^{\u0398_E/T}\u22121)\u00b2"
    },
    {
      "eq_number": 369,
      "title": "Wiedemann-Franz Law (Electronic Thermal Conductivity)",
      "domain_id": 21,
      "significance": "\u03ba_e / (\u03c3 T) = L; L = (\u03c0\u00b2/3)(k_B/e)\u00b2 \u2248 2.44\u00d710\u207b\u2078 W \u03a9/K\u00b2"
    },
    {
      "eq_number": 370,
      "title": "Debye-Callaway Model (Lattice Thermal Conductivity)",
      "domain_id": 21,
      "significance": "\u03ba_l = (k_B/2\u03c0\u00b2v)(k_B T/\u210f)\u00b3 \u222b\u2080^{\u0398_D/T} \u03c4_c x\u2074 e^x / (e^x\u22121)\u00b2 dx"
    },
    {
      "eq_number": 371,
      "title": "Thermal Expansion Coefficient (Gr\u00fcneisen Relation)",
      "domain_id": 21,
      "significance": "\u03b1 = \u03b3 C_V / (3 B V); \u03b3 = Gr\u00fcneisen parameter; B = bulk modulus"
    },
    {
      "eq_number": 372,
      "title": "Gr\u00fcneisen Equation of State (Solids)",
      "domain_id": 21,
      "significance": "P(V) = \u2212dU\u2080/dV + \u03b3 U_th/V; \u03b3 = Gr\u00fcneisen parameter"
    },
    {
      "eq_number": 373,
      "title": "Lindemann Melting Criterion",
      "domain_id": 21,
      "significance": "T_m \u2248 C \u03b8_D\u00b2 M V^{2/3}; C depends on crystal structure"
    },
    {
      "eq_number": 374,
      "title": "Stefan-Boltzmann Radiative Heat Transfer (Between Surfaces)",
      "domain_id": 21,
      "significance": "q = \u03b5_eff \u03c3 (T\u2081\u2074\u2212T\u2082\u2074); view factor + emissivity correction"
    },
    {
      "eq_number": 375,
      "title": "Complex Dielectric Constant",
      "domain_id": 21,
      "significance": "\u03b5* = \u03b5' \u2212 i \u03b5''; tan \u03b4 = \u03b5''/\u03b5'; loss tangent"
    },
    {
      "eq_number": 376,
      "title": "Clausius-Mossotti Relation (Polarizability)",
      "domain_id": 21,
      "significance": "(\u03b5_r\u22121)/(\u03b5_r+2) = N \u03b1 / (3 \u03b5\u2080); links macro/micro dielectric properties"
    },
    {
      "eq_number": 377,
      "title": "Debye Relaxation (Dipole Response)",
      "domain_id": 21,
      "significance": "\u03b5*(\u03c9) = \u03b5_\u221e + (\u03b5_s\u2212\u03b5_\u221e) / (1 + i \u03c9 \u03c4)"
    },
    {
      "eq_number": 378,
      "title": "Cole-Cole Relaxation (Distributed)",
      "domain_id": 21,
      "significance": "\u03b5*(\u03c9) = \u03b5_\u221e + (\u03b5_s\u2212\u03b5_\u221e) / [1 + (i \u03c9 \u03c4)^{1\u2212\u03b1}]"
    },
    {
      "eq_number": 379,
      "title": "Havriliak-Negami Relaxation",
      "domain_id": 21,
      "significance": "\u03b5*(\u03c9) = \u03b5_\u221e + (\u03b5_s\u2212\u03b5_\u221e) / [1 + (i \u03c9 \u03c4)^\u03b1]^\u03b2"
    },
    {
      "eq_number": 380,
      "title": "Curie-Weiss Law for Ferroelectrics (Above T_c)",
      "domain_id": 21,
      "significance": "\u03b5_r = C / (T \u2212 T_c); C = Curie constant"
    },
    {
      "eq_number": 381,
      "title": "Piezoelectric Constitutive Equations",
      "domain_id": 21,
      "significance": "S = s^E T + d^t E; D = d T + \u03b5^T E (strain-charge form)"
    },
    {
      "eq_number": 382,
      "title": "Pyroelectric Coefficient",
      "domain_id": 21,
      "significance": "p = dP_s/dT; \u0394Q = p A \u0394T"
    },
    {
      "eq_number": 383,
      "title": "Fowler-Nordheim Tunneling (Field Emission)",
      "domain_id": 21,
      "significance": "J = (A/\u03c6)(\u03b2E)\u00b2 exp(\u2212B \u03c6^{3/2} / \u03b2E); A,B constants"
    },
    {
      "eq_number": 384,
      "title": "Poole-Frenkel Conduction (Insulators)",
      "domain_id": 21,
      "significance": "\u03c3 = \u03c3\u2080 exp[\u2212q(\u03c6_B\u2212\u221a(qE/\u03c0\u03b5))/k_B T]"
    },
    {
      "eq_number": 385,
      "title": "Varistor I-V Characteristic (Nonlinear)",
      "domain_id": 21,
      "significance": "I = k V^\u03b1; \u03b1 >> 1 (ZnO varistors \u03b1\u224820\u2013100)"
    },
    {
      "eq_number": 386,
      "title": "Percolation Threshold (Conductivity)",
      "domain_id": 21,
      "significance": "\u03c3 = \u03c3\u2080 (p \u2212 p_c)^t; p = volume fraction; p_c = percolation threshold"
    },
    {
      "eq_number": 387,
      "title": "Intrinsic Carrier Concentration (Semiconductors)",
      "domain_id": 23,
      "significance": "n_i = \u221a(N_c N_v) exp(\u2212E_g / 2 k_B T); N_c = 2(2\u03c0 m_e* k_B T/h\u00b2)^{3/2}"
    },
    {
      "eq_number": 388,
      "title": "Fermi Level in Doped Semiconductors",
      "domain_id": 23,
      "significance": "n-type: E_F = E_c \u2212 k_B T ln(N_c/N_d); p-type: E_F = E_v + k_B T ln(N_v/N_a)"
    },
    {
      "eq_number": 389,
      "title": "Mass Action Law (Semiconductors)",
      "domain_id": 23,
      "significance": "n p = n_i\u00b2; product constant at fixed T"
    },
    {
      "eq_number": 390,
      "title": "Shockley Diode Equation (Ideal)",
      "domain_id": 23,
      "significance": "I = I_s [exp(q V / n k_B T) \u2212 1]; I_s = reverse saturation current"
    },
    {
      "eq_number": 391,
      "title": "Built-in Potential (p-n Junction)",
      "domain_id": 23,
      "significance": "V_bi = (k_B T / q) ln(N_a N_d / n_i\u00b2)"
    },
    {
      "eq_number": 392,
      "title": "Depletion Width (p-n Junction)",
      "domain_id": 23,
      "significance": "W = \u221a[2\u03b5_s (V_bi\u2212V)(1/N_a+1/N_d)/q]"
    },
    {
      "eq_number": 393,
      "title": "MOS Capacitor Threshold Voltage",
      "domain_id": 23,
      "significance": "V_th = V_FB + 2\u03c6_F + \u221a(4\u03b5_s q N_a \u03c6_F)/C_ox"
    },
    {
      "eq_number": 394,
      "title": "MOSFET Drain Current (Saturation, Long Channel)",
      "domain_id": 23,
      "significance": "I_D = (\u03bc_n C_ox W / 2L) (V_GS \u2212 V_th)\u00b2"
    },
    {
      "eq_number": 395,
      "title": "Subthreshold Swing (MOSFET)",
      "domain_id": 23,
      "significance": "SS = (k_B T/q) ln(10) (1 + C_dep/C_ox); ideal: 60 mV/decade at 300K"
    },
    {
      "eq_number": 396,
      "title": "Avalanche Breakdown (Impact Ionization)",
      "domain_id": 23,
      "significance": "M = 1 / [1 \u2212 (V/V_BR)^n]; n\u22483\u20136"
    },
    {
      "eq_number": 397,
      "title": "Quantum Confinement Energy (Particle in a Box)",
      "domain_id": 23,
      "significance": "E_n = n\u00b2 \u03c0\u00b2 \u210f\u00b2 / (2 m* L\u00b2); blue shift with decreasing size"
    },
    {
      "eq_number": 398,
      "title": "Brus Equation (Semiconductor Nanocrystal Band Gap)",
      "domain_id": 23,
      "significance": "E_g(R) = E_g(bulk) + \u210f\u00b2\u03c0\u00b2/(2\u03bc R\u00b2) \u2212 1.8e\u00b2/(\u03b5_r R); \u03bc = reduced exciton mass"
    },
    {
      "eq_number": 399,
      "title": "Kane's k\u00b7p Band Model (Non-Parabolicity)",
      "domain_id": 23,
      "significance": "E(1+\u03b1E) = \u210f\u00b2 k\u00b2 / (2 m*); \u03b1 = 1/E_g; non-parabolic correction"
    },
    {
      "eq_number": 400,
      "title": "Mott Transition (Doped Semiconductor)",
      "domain_id": 23,
      "significance": "n_c^{1/3} a_B* \u2248 0.25; insulator-metal transition at critical doping"
    },
    {
      "eq_number": 401,
      "title": "Anderson Localization (Disordered Materials)",
      "domain_id": 23,
      "significance": "W/V > W_c \u2192 localized states; mobility edge at E_c"
    },
    {
      "eq_number": 402,
      "title": "Tauc Plot (Band Gap from Absorption)",
      "domain_id": 23,
      "significance": "(\u03b1 h \u03bd)^{1/r} = A (h\u03bd \u2212 E_g); r=\u00bd for direct, r=2 for indirect"
    },
    {
      "eq_number": 403,
      "title": "Stoner Criterion (Itinerant Ferromagnetism)",
      "domain_id": 21,
      "significance": "N(E_F) I > 1; spontaneous magnetization when DOS \u00d7 exchange exceeds unity"
    },
    {
      "eq_number": 404,
      "title": "Stoner-Wohlfarth Model (Single-Domain Particle)",
      "domain_id": 21,
      "significance": "E = K V sin\u00b2\u03b8 \u2212 \u03bc\u2080 M_s H V cos(\u03c6\u2212\u03b8); hysteresis from anisotropy+Zeeman"
    },
    {
      "eq_number": 405,
      "title": "N\u00e9el Temperature (Antiferromagnetism)",
      "domain_id": 21,
      "significance": "T_N = (2J S(S+1)/3k_B) z (from mean-field); sublattice ordering temperature"
    },
    {
      "eq_number": 406,
      "title": "Curie Temperature (Mean-Field Ferromagnetism)",
      "domain_id": 21,
      "significance": "T_c = (2J S(S+1)/3k_B) z; z = coordination number"
    },
    {
      "eq_number": 407,
      "title": "Bloch T^{3/2} Law (Magnetization at Low T)",
      "domain_id": 21,
      "significance": "M_s(T) = M_s(0) [1 \u2212 (T/T_c)^{3/2}] (3D Heisenberg)"
    },
    {
      "eq_number": 408,
      "title": "Landau-Lifshitz-Gilbert Equation (Magnetization Dynamics)",
      "domain_id": 21,
      "significance": "dM/dt = \u2212\u03b3 M \u00d7 H_eff + (\u03b1/M_s) M \u00d7 dM/dt"
    },
    {
      "eq_number": 409,
      "title": "Brown's Paradox (Domain Wall Motion)",
      "domain_id": 21,
      "significance": "v = (\u03b3 \u0394 / \u03b1)(H \u2212 H_c); soft magnetic materials"
    },
    {
      "eq_number": 410,
      "title": "Magnetostriction (Joule Magnetostriction)",
      "domain_id": 21,
      "significance": "\u0394L/L = (3/2) \u03bb_s (cos\u00b2\u03b8 \u2212 1/3); \u03bb_s = saturation magnetostriction"
    },
    {
      "eq_number": 411,
      "title": "Giant Magnetoresistance (GMR, CIP)",
      "domain_id": 21,
      "significance": "\u0394R/R = (R_AP\u2212R_P)/R_P; spin-dependent scattering at interfaces"
    },
    {
      "eq_number": 412,
      "title": "Tunneling Magnetoresistance (TMR, Julliere Model)",
      "domain_id": 21,
      "significance": "TMR = (R_AP\u2212R_P)/R_P = 2P\u2081P\u2082/(1\u2212P\u2081P\u2082); P = spin polarization"
    },
    {
      "eq_number": 413,
      "title": "RKKY Interaction (Indirect Exchange)",
      "domain_id": 21,
      "significance": "J(R) \u221d cos(2k_F R) / R\u00b3; oscillatory coupling through conduction electrons"
    },
    {
      "eq_number": 414,
      "title": "Superexchange (Anderson-Goodenough-Kanamori Rules)",
      "domain_id": 21,
      "significance": "J_ij \u221d \u2212b\u00b2/U (for 180\u00b0 cation-anion-cation); sign depends on orbital filling"
    },
    {
      "eq_number": 415,
      "title": "Complex Refractive Index (General)",
      "domain_id": 21,
      "significance": "\u00f1 = n + i \u03ba; I(z) = I\u2080 exp(\u2212\u03b1 z); \u03b1 = 4\u03c0\u03ba/\u03bb"
    },
    {
      "eq_number": 416,
      "title": "Kramers-Kronig Relations (Optical Constants)",
      "domain_id": 21,
      "significance": "n(\u03c9)\u22121 = (2/\u03c0) P \u222b\u2080^\u221e \u03c9' \u03ba(\u03c9')/(\u03c9'\u00b2\u2212\u03c9\u00b2) d\u03c9'; causality \u2192 dispersion relations"
    },
    {
      "eq_number": 417,
      "title": "Tauc-Lorentz Model (Amorphous Semiconductor Optics)",
      "domain_id": 21,
      "significance": "\u03b5_2(E) = [A E\u2080 C (E\u2212E_g)\u00b2] / [(E\u00b2\u2212E\u2080\u00b2)\u00b2 + C\u00b2 E\u00b2] E for E>E_g; 0 otherwise"
    },
    {
      "eq_number": 418,
      "title": "Sellmeier Equation (Refractive Index Dispersion)",
      "domain_id": 21,
      "significance": "n\u00b2(\u03bb) = 1 + \u03a3_i A_i \u03bb\u00b2 / (\u03bb\u00b2 \u2212 \u03bb_i\u00b2); empirical fit for transparent regions"
    },
    {
      "eq_number": 419,
      "title": "Cauchy Equation (Refractive Index Fit)",
      "domain_id": 21,
      "significance": "n(\u03bb) = A + B/\u03bb\u00b2 + C/\u03bb\u2074; empirical for transparent region"
    },
    {
      "eq_number": 420,
      "title": "Urbach Tail (Absorption Edge)",
      "domain_id": 21,
      "significance": "\u03b1(E) = \u03b1\u2080 exp[\u03c3 (E\u2212E\u2080) / k_B T]; exponential absorption below band edge"
    },
    {
      "eq_number": 421,
      "title": "Beer-Lambert Law (Absorption)",
      "domain_id": 3,
      "significance": "A = log\u2081\u2080(I\u2080/I) = \u03b5 c L; absorbance proportional to concentration and path"
    },
    {
      "eq_number": 422,
      "title": "Kubelka-Munk Theory (Diffuse Reflectance)",
      "domain_id": 21,
      "significance": "F(R_\u221e) = (1\u2212R_\u221e)\u00b2/(2R_\u221e) = K/S \u221d \u03b1; for thick opaque scattering media"
    },
    {
      "eq_number": 423,
      "title": "Fresnel Loss at Normal Incidence",
      "domain_id": 21,
      "significance": "R = [(n\u2081\u2212n\u2082)/(n\u2081+n\u2082)]\u00b2; reflection coefficient at normal incidence"
    },
    {
      "eq_number": 424,
      "title": "Drude Model for Free-Carrier Absorption",
      "domain_id": 21,
      "significance": "\u03b5(\u03c9) = \u03b5_\u221e \u2212 \u03c9_p\u00b2/(\u03c9\u00b2 + i \u03c9/\u03c4); \u03c9_p = \u221a(n e\u00b2/\u03b5\u2080 m*)"
    },
    {
      "eq_number": 425,
      "title": "Forster Resonance Energy Transfer (FRET) Efficiency",
      "domain_id": 21,
      "significance": "E = 1 / [1 + (r/R\u2080)\u2076]; R\u2080 = F\u00f6rster radius (~1\u201310 nm)"
    },
    {
      "eq_number": 426,
      "title": "Stokes Shift (Luminescence)",
      "domain_id": 21,
      "significance": "\u0394E = E_abs \u2212 E_em > 0; from vibrational relaxation"
    },
    {
      "eq_number": 427,
      "title": "Dexter Energy Transfer (Exchange)",
      "domain_id": 21,
      "significance": "k_ET \u221d exp(\u22122r/L); short-range (\u22721 nm) electron exchange"
    },
    {
      "eq_number": 428,
      "title": "Vickers Hardness Definition",
      "domain_id": 21,
      "significance": "HV = 1.854 F / d\u00b2; F in kgf, d = average diagonal (mm)"
    },
    {
      "eq_number": 429,
      "title": "Brinell Hardness",
      "domain_id": 21,
      "significance": "HB = 2F / [\u03c0 D (D \u2212 \u221a(D\u00b2\u2212d\u00b2))]; D = ball diameter"
    },
    {
      "eq_number": 430,
      "title": "Rockwell Hardness (Indirect)",
      "domain_id": 21,
      "significance": "HR = N \u2212 h/s; h = penetration depth; N,s depend on scale"
    },
    {
      "eq_number": 431,
      "title": "Knoop Hardness (Thin Films / Brittle)",
      "domain_id": 21,
      "significance": "HK = 14.229 F / d\u2081\u00b2; long diagonal; shallow penetration"
    },
    {
      "eq_number": 432,
      "title": "Nanoindentation (Oliver-Pharr Method)",
      "domain_id": 21,
      "significance": "H = P_max/A; E_r = \u221a\u03c0 S/(2\u03b2\u221aA); S = dP/dh at unload"
    },
    {
      "eq_number": 433,
      "title": "Charpy Impact Toughness",
      "domain_id": 21,
      "significance": "KV = m g (h_initial \u2212 h_final); energy absorbed in fracture (J)"
    },
    {
      "eq_number": 434,
      "title": "Izod Impact Test",
      "domain_id": 21,
      "significance": "Similar to Charpy; energy absorbed per unit width (J/m)"
    },
    {
      "eq_number": 435,
      "title": "Rubber Elasticity (Gaussian Chain, Affine)",
      "domain_id": 24,
      "significance": "\u03c3_true = n k_B T (\u03bb \u2212 1/\u03bb\u00b2); n = crosslink density; \u03bb = extension ratio"
    },
    {
      "eq_number": 436,
      "title": "Mooney-Rivlin Equation (Hyperelastic)",
      "domain_id": 24,
      "significance": "W = C\u2081\u2080(I\u2081\u22123) + C\u2080\u2081(I\u2082\u22123); I\u2081,I\u2082 = invariants of Cauchy-Green tensor"
    },
    {
      "eq_number": 437,
      "title": "Flory-Huggins Theory (Polymer Solution Free Energy)",
      "domain_id": 24,
      "significance": "\u0394G_mix/k_B T = n\u2081 ln \u03c6\u2081 + n\u2082 ln \u03c6\u2082 + \u03c7 n\u2081 \u03c6\u2082; \u03c7 = Flory interaction parameter"
    },
    {
      "eq_number": 438,
      "title": "Williams-Landel-Ferry (WLF) Equation",
      "domain_id": 24,
      "significance": "log a_T = \u2212C\u2081 (T\u2212T_ref) / (C\u2082 + T\u2212T_ref); time-temperature superposition"
    },
    {
      "eq_number": 439,
      "title": "Arrhenius Viscosity (Above Glass Transition)",
      "domain_id": 24,
      "significance": "\u03b7(T) = \u03b7\u2080 exp(E_a / R T) (simple) or Vogel-Fulcher-Tammann: \u03b7 = \u03b7\u2080 exp[B/(T\u2212T\u2080)]"
    },
    {
      "eq_number": 440,
      "title": "Rouse Model (Unentangled Polymer Dynamics)",
      "domain_id": 24,
      "significance": "\u03c4_R = \u03b6 N\u00b2 b\u00b2 / (3\u03c0\u00b2 k_B T); longest relaxation time of unentangled chain"
    },
    {
      "eq_number": 441,
      "title": "Reptation Model (de Gennes, Entangled Dynamics)",
      "domain_id": 24,
      "significance": "\u03c4_rep \u221d N\u00b3; D_rep \u221d N\u207b\u00b2; disentanglement time; Nobel 1991"
    },
    {
      "eq_number": 442,
      "title": "Entanglement Molecular Weight",
      "domain_id": 24,
      "significance": "M_e = \u03c1 R T / G_N\u2070; from plateau modulus G_N\u2070"
    },
    {
      "eq_number": 443,
      "title": "Flory-Fox Equation (T_g vs Molecular Weight)",
      "domain_id": 24,
      "significance": "T_g = T_g\u221e \u2212 K_F / M_n; T_g increases with MW to asymptotic limit"
    },
    {
      "eq_number": 444,
      "title": "Cahn-Hilliard Equation (Spinodal Decomposition)",
      "domain_id": 24,
      "significance": "\u2202c/\u2202t = M \u2207\u00b2[\u2202f/\u2202c \u2212 2\u03ba \u2207\u00b2c]; diffusion modulated by gradient energy"
    },
    {
      "eq_number": 445,
      "title": "Avrami Equation (Crystallization Kinetics)",
      "domain_id": 27,
      "significance": "X(t) = 1 \u2212 exp(\u2212k t^n); n = Avrami exponent (dimensionality + nucleation mode)"
    },
    {
      "eq_number": 446,
      "title": "Lauritzen-Hoffman Theory (Polymer Crystal Growth)",
      "domain_id": 27,
      "significance": "G = G\u2080 exp[\u2212U*/R(T\u2212T_\u221e)] exp[\u2212K_g / (T \u0394T f)]; secondary nucleation"
    },
    {
      "eq_number": 447,
      "title": "Young's Equation (Contact Angle)",
      "domain_id": 25,
      "significance": "\u03b3_sv = \u03b3_sl + \u03b3_lv cos \u03b8; balance of interfacial tensions"
    },
    {
      "eq_number": 448,
      "title": "Wenzel Equation (Rough Surface Wetting)",
      "domain_id": 25,
      "significance": "cos \u03b8* = r cos \u03b8; r = actual/projected area > 1; roughness amplifies wetting"
    },
    {
      "eq_number": 449,
      "title": "Cassie-Baxter Equation (Composite/Heterogeneous Wetting)",
      "domain_id": 25,
      "significance": "cos \u03b8* = f\u2081 cos \u03b8\u2081 + f\u2082 cos \u03b8\u2082; f\u2081+f\u2082=1; trapped air \u2192 superhydrophobic"
    },
    {
      "eq_number": 450,
      "title": "Laplace Pressure (Curved Interface)",
      "domain_id": 9,
      "significance": "\u0394P = \u03b3 (1/R\u2081 + 1/R\u2082); pressure inside curved surface"
    },
    {
      "eq_number": 451,
      "title": "Kelvin Equation (Capillary Condensation)",
      "domain_id": 25,
      "significance": "ln(P/P\u2080) = \u22122\u03b3 V_m / (r R T); condensation in pores below saturation"
    },
    {
      "eq_number": 452,
      "title": "Langmuir Adsorption Isotherm (Monolayer)",
      "domain_id": 25,
      "significance": "\u03b8 = K P / (1 + K P); \u03b8 = fractional coverage; K = adsorption equilibrium constant"
    },
    {
      "eq_number": 453,
      "title": "BET Isotherm (Brunauer-Emmett-Teller, Multilayer)",
      "domain_id": 25,
      "significance": "P/[V(P\u2080\u2212P)] = 1/(V_m C) + (C\u22121)P/(V_m C P\u2080); surface area from multilayer adsorption"
    },
    {
      "eq_number": 454,
      "title": "Freundlich Isotherm (Heterogeneous Surfaces)",
      "domain_id": 25,
      "significance": "q = K_F P^{1/n}; empirical; heterogeneous adsorption"
    },
    {
      "eq_number": 455,
      "title": "Gibbs Adsorption Equation",
      "domain_id": 25,
      "significance": "d\u03b3 = \u2212\u03a3 \u0393_i d\u03bc_i; \u0393_i = surface excess concentration"
    },
    {
      "eq_number": 456,
      "title": "Amontons-Coulomb Friction Law (Dry Friction)",
      "domain_id": 18,
      "significance": "F_f \u2264 \u03bc_s N (static); F_f = \u03bc_k N (kinetic); \u03bc_k < \u03bc_s"
    },
    {
      "eq_number": 457,
      "title": "Archard's Law (Adhesive Wear)",
      "domain_id": 25,
      "significance": "V = k F s / H; k = wear coefficient; H = hardness"
    },
    {
      "eq_number": 458,
      "title": "Hamaker Constant (Van der Waals Between Surfaces)",
      "domain_id": 25,
      "significance": "A = \u03c0\u00b2 C \u03c1\u2081 \u03c1\u2082; F_vdW/A = \u2212A / (6\u03c0 d\u00b3) (flat surfaces)"
    },
    {
      "eq_number": 459,
      "title": "DLVO Theory (Colloid Stability)",
      "domain_id": 25,
      "significance": "V_total(d) = V_vdW + V_edl; van der Waals + electric double-layer"
    },
    {
      "eq_number": 460,
      "title": "Zeta Potential (Smoluchowski Equation)",
      "domain_id": 25,
      "significance": "\u03b6 = \u03b7 \u03bc_e / \u03b5; \u03bc_e = electrophoretic mobility; \u03b7 = viscosity"
    },
    {
      "eq_number": 461,
      "title": "Derjaguin Approximation (Force Between Curved Surfaces)",
      "domain_id": 25,
      "significance": "F_sphere(d) = 2\u03c0R W_flat(d); relates sphere-sphere to flat-plate energy"
    },
    {
      "eq_number": 462,
      "title": "Johnson-Kendall-Roberts (JKR) Adhesion Model",
      "domain_id": 25,
      "significance": "a\u00b3 = (R/K)[F + 3\u03c0W_ad R + \u221a(6\u03c0W_adRF + (3\u03c0W_adR)\u00b2)]; elastic + adhesion contact"
    },
    {
      "eq_number": 463,
      "title": "Derjaguin-Muller-Toporov (DMT) Model",
      "domain_id": 25,
      "significance": "a\u00b3 = (R/K)[F + 2\u03c0W_ad R]; adhesion without distortion of contact profile"
    },
    {
      "eq_number": 464,
      "title": "Fick's First Law (Steady-State Diffusion)",
      "domain_id": 21,
      "significance": "J = \u2212D \u2202c/\u2202x; flux proportional to concentration gradient"
    },
    {
      "eq_number": 465,
      "title": "Fick's Second Law (Time-Dependent Diffusion)",
      "domain_id": 21,
      "significance": "\u2202c/\u2202t = D \u2202\u00b2c/\u2202x\u00b2; for constant D; general: \u2202c/\u2202t = \u2202/\u2202x(D \u2202c/\u2202x)"
    },
    {
      "eq_number": 466,
      "title": "Diffusion Solutions (Common)",
      "domain_id": 21,
      "significance": "Thin film: c(x,t) = (M/\u221a(4\u03c0Dt)) exp(\u2212x\u00b2/4Dt); Error function: c = C\u2080 erfc(x/\u221a(4Dt))"
    },
    {
      "eq_number": 467,
      "title": "Arrhenius Diffusion Coefficient",
      "domain_id": 21,
      "significance": "D = D\u2080 exp(\u2212E_a / k_B T); thermally activated diffusion"
    },
    {
      "eq_number": 468,
      "title": "Darken Equations (Interdiffusion / Kirkendall Effect)",
      "domain_id": 21,
      "significance": "D\u0303 = (X_B D_A + X_A D_B) \u03a6; \u03a6 = thermodynamic factor including non-ideality"
    },
    {
      "eq_number": 469,
      "title": "Nernst-Planck Equation (Ion Transport)",
      "domain_id": 21,
      "significance": "J_i = \u2212D_i \u2207c_i \u2212 (z_i F/RT)D_i c_i \u2207\u03c6 + c_i v; diffusion + migration + convection"
    },
    {
      "eq_number": 470,
      "title": "Stokes-Einstein Relation (Diffusion of Spheres)",
      "domain_id": 21,
      "significance": "D = k_B T / (6\u03c0 \u03b7 r); hydrodynamic radius from diffusion"
    },
    {
      "eq_number": 471,
      "title": "Tracer Diffusion Correlation Factor",
      "domain_id": 21,
      "significance": "D* = f D_rand; f = correlation factor; f<1 for vacancy mechanism"
    },
    {
      "eq_number": 472,
      "title": "Gibbs-Thomson Effect (Curvature Depression of Melting/Equilibrium Point)",
      "domain_id": 27,
      "significance": "T_m(r) = T_m(\u221e)(1 \u2212 2\u03b3_sl / (\u03c1_s \u0394H_f r)); small particles melt at lower T"
    },
    {
      "eq_number": 473,
      "title": "Classical Nucleation Theory (Homogeneous)",
      "domain_id": 27,
      "significance": "\u0394G = (4\u03c0/3)r\u00b3 \u0394G_v + 4\u03c0r\u00b2 \u03b3; r* = \u22122\u03b3/\u0394G_v; \u0394G* = 16\u03c0\u03b3\u00b3/(3\u0394G_v\u00b2)"
    },
    {
      "eq_number": 474,
      "title": "Johnson-Mehl-Avrami-Kolmogorov (JMAK) Equation",
      "domain_id": 27,
      "significance": "f = 1 \u2212 exp[\u2212(kt)^n]; n depends on nucleation+growth dimensionality"
    },
    {
      "eq_number": 475,
      "title": "Turnbull's Nucleation Rate (Steady-State)",
      "domain_id": 27,
      "significance": "I = N_v (k_B T/h) exp[\u2212(\u0394G*+\u0394G_a)/k_B T]; includes kinetic barrier"
    },
    {
      "eq_number": 476,
      "title": "Lever Rule (Phase Diagram Tie Line)",
      "domain_id": 27,
      "significance": "f_\u03b1 = (C\u2080\u2212C_\u03b2)/(C_\u03b1\u2212C_\u03b2); f_\u03b2 = (C_\u03b1\u2212C\u2080)/(C_\u03b1\u2212C_\u03b2)"
    },
    {
      "eq_number": 477,
      "title": "Gibbs-Thomson-Freundlich (Ostwald Ripening / LSW Theory)",
      "domain_id": 27,
      "significance": "\u27e8r\u27e9\u00b3 \u2212 \u27e8r\u2080\u27e9\u00b3 = k t; k \u221d \u03b3 D c_\u221e V_m\u00b2/(R T); coarsening of precipitates"
    },
    {
      "eq_number": 478,
      "title": "Darken-Gurry Plot (Solubility Limits)",
      "domain_id": 21,
      "significance": "Extensive solubility when |\u0394R_atom|<15% and |\u0394\u03c7|<0.4 (electronegativity difference)"
    },
    {
      "eq_number": 479,
      "title": "Hume-Rothery Rules (Alloy Formation)",
      "domain_id": 21,
      "significance": "(1) Size <15% (2) Similar electronegativity (3) Same valence (4) Same crystal structure"
    },
    {
      "eq_number": 480,
      "title": "Vegard's Law (Lattice Parameter in Solid Solutions)",
      "domain_id": 21,
      "significance": "a_AB = x_A a_A + x_B a_B; linear interpolation; deviations = non-ideal mixing"
    },
    {
      "eq_number": 481,
      "title": "Rule of Mixtures (Composite Modulus, Isostrain)",
      "domain_id": 21,
      "significance": "E_c = E_f V_f + E_m V_m (Voigt bound, upper); 1/E_c = V_f/E_f + V_m/E_m (Reuss bound, lower)"
    },
    {
      "eq_number": 482,
      "title": "Hashin-Shtrikman Bounds (Composite Moduli)",
      "domain_id": 21,
      "significance": "Tighter bounds than Voigt-Reuss; K_lower = K_m + V_f/[1/(K_f\u2212K_m)+3V_m/(3K_m+4G_m)]; etc."
    },
    {
      "eq_number": 483,
      "title": "Halpin-Tsai Equations (Short Fiber Composites)",
      "domain_id": 21,
      "significance": "E/E_m = (1 + \u03be \u03b7 V_f)/(1 \u2212 \u03b7 V_f); \u03b7 = (E_f/E_m\u22121)/(E_f/E_m+\u03be); \u03be = shape factor"
    },
    {
      "eq_number": 484,
      "title": "Porosity-Young's Modulus Relation (Empirical)",
      "domain_id": 21,
      "significance": "E = E\u2080 (1 \u2212 P)^n or exp(\u2212bP); P = porosity fraction; n\u22482\u20134"
    },
    {
      "eq_number": 485,
      "title": "Gibson-Ashby Model (Cellular Solids / Foams)",
      "domain_id": 21,
      "significance": "E*/E_s = C (\u03c1*/\u03c1_s)^n; n=2 open cell; n=3 closed cell; \u03c3*/\u03c3_ys \u221d (\u03c1*/\u03c1_s)^{3/2}"
    },
    {
      "eq_number": 486,
      "title": "Eshelby Inclusion Problem (Stress in Ellipsoidal Inclusion)",
      "domain_id": 21,
      "significance": "\u03b5^T = S \u03b5*; S = Eshelby tensor (depends on inclusion shape + matrix Poisson ratio)"
    },
    {
      "eq_number": 487,
      "title": "Moss-Burstein Shift (Doped Semiconductor Absorption Edge)",
      "domain_id": 21,
      "significance": "\u0394E_g = (\u210f\u00b2/2m*)(3\u03c0\u00b2n)^{2/3}; Fermi filling blocks lowest transitions"
    },
    {
      "eq_number": 488,
      "title": "Franz-Keldysh Effect (Electro-Absorption)",
      "domain_id": 21,
      "significance": "\u03b1(E,F) \u221d exp[\u2212(E_g\u2212E)^{3/2} / e\u210fF]; band edge shift in electric field"
    },
    {
      "eq_number": 489,
      "title": "Pockels Effect (Linear Electro-Optic)",
      "domain_id": 21,
      "significance": "\u0394(1/n\u00b2)_i = r_ij E_j; r_ij = linear electro-optic coefficients"
    },
    {
      "eq_number": 490,
      "title": "Kerr Effect (Quadratic Electro-Optic)",
      "domain_id": 21,
      "significance": "\u0394n = K \u03bb E\u00b2; quadratic field dependence"
    },
    {
      "eq_number": 491,
      "title": "Photoconductivity (Rose Model)",
      "domain_id": 21,
      "significance": "\u0394\u03c3 = e \u03bc \u03c4 G L / d; G=generation rate, \u03c4=lifetime; gain = \u03c4/t_transit"
    },
    {
      "eq_number": 492,
      "title": "Shockley-Read-Hall Recombination Rate",
      "domain_id": 23,
      "significance": "U = (np\u2212n_i\u00b2) / [\u03c4_p (n+n\u2081) + \u03c4_n (p+p\u2081)]; trap-assisted recombination"
    },
    {
      "eq_number": 493,
      "title": "Auger Recombination Rate",
      "domain_id": 23,
      "significance": "U_Auger = C_n n\u00b2p + C_p n p\u00b2; three-particle non-radiative recombination"
    },
    {
      "eq_number": 494,
      "title": "BCS Energy Gap at T=0",
      "domain_id": 12,
      "significance": "\u0394(0) = 1.764 k_B T_c; universal BCS ratio"
    },
    {
      "eq_number": 495,
      "title": "Ginzburg-Landau Coherence Length",
      "domain_id": 12,
      "significance": "\u03be(T) = \u03be(0) / \u221a(1\u2212T/T_c); \u03be(0) = \u221a(\u210f\u00b2/2m*|\u03b1|); spatial variation of order parameter"
    },
    {
      "eq_number": 496,
      "title": "Ginzburg-Landau Penetration Depth",
      "domain_id": 12,
      "significance": "\u03bb(T) = \u03bb(0)/\u221a(1\u2212T/T_c); magnetic field penetration into superconductor"
    },
    {
      "eq_number": 497,
      "title": "Ginzburg-Landau Parameter (\u03ba)",
      "domain_id": 12,
      "significance": "\u03ba = \u03bb/\u03be; \u03ba < 1/\u221a2 \u2192 Type I; \u03ba > 1/\u221a2 \u2192 Type II"
    },
    {
      "eq_number": 498,
      "title": "Abrikosov Vortex Lattice (Lower/Upper Critical Fields)",
      "domain_id": 12,
      "significance": "H_c1 = H_c ln \u03ba/(\u221a2 \u03ba); H_c2 = \u221a2 \u03ba H_c; vortex state between"
    },
    {
      "eq_number": 499,
      "title": "Flux Pinning (Bean Critical State Model)",
      "domain_id": 12,
      "significance": "J_c = constant; \u2207\u00d7B = \u03bc\u2080 J_c; critical state penetration profile"
    },
    {
      "eq_number": 500,
      "title": "Little-Parks Effect (Fluxoid Quantization)",
      "domain_id": 12,
      "significance": "T_c oscillates with flux through cylinder; period = \u03a6\u2080 = h/2e"
    },
    {
      "eq_number": 501,
      "title": "Andreev Reflection",
      "domain_id": 12,
      "significance": "e\u207b \u2192 NS interface reflects as h\u207a; retroreflection; sub-gap conductance enhancement"
    },
    {
      "eq_number": 502,
      "title": "Nernst Equation (Electrode Potential)",
      "domain_id": 21,
      "significance": "E = E\u2070 \u2212 (RT/nF) ln Q; E\u2070 = standard reduction potential"
    },
    {
      "eq_number": 503,
      "title": "Butler-Volmer Equation (Electrode Kinetics)",
      "domain_id": 21,
      "significance": "j = j\u2080 [exp(\u03b1_a F \u03b7/RT) \u2212 exp(\u2212\u03b1_c F \u03b7/RT)]; \u03b7 = overpotential"
    },
    {
      "eq_number": 504,
      "title": "Tafel Equation (High Overpotential Limit)",
      "domain_id": 21,
      "significance": "\u03b7 = a + b log |j|; b = 2.303 RT/(\u03b1 nF) \u2248 120 mV/decade (\u03b1=0.5 at 298K)"
    },
    {
      "eq_number": 505,
      "title": "Randles-Sevcik Equation (Cyclic Voltammetry Peak Current)",
      "domain_id": 21,
      "significance": "i_p = 0.4463 n F A C \u221a(n F v D/RT); reversible: i_p \u221d \u221av"
    },
    {
      "eq_number": 506,
      "title": "Cottrell Equation (Chronoamperometry)",
      "domain_id": 21,
      "significance": "i(t) = n F A C \u221a(D) / \u221a(\u03c0 t); diffusion-limited current decay"
    },
    {
      "eq_number": 507,
      "title": "Faraday's Laws of Electrolysis",
      "domain_id": 3,
      "significance": "m = (Q M)/(n F); mass deposited proportional to charge; Q=It"
    },
    {
      "eq_number": 508,
      "title": "Wagner Number (Current Distribution Uniformity)",
      "domain_id": 21,
      "significance": "Wa = \u03ba (d\u03b7/dj) / L; Wa \u226b 1 \u2192 uniform; Wa \u226a 1 \u2192 non-uniform"
    },
    {
      "eq_number": 509,
      "title": "Zener Anelasticity (Standard Linear Solid)",
      "domain_id": 21,
      "significance": "\u03b5 = \u03c3/E_R + (\u03c3/E_U\u2212\u03c3/E_R) (1\u2212e^{\u2212t/\u03c4}); relaxation strength \u0394 = (E_U\u2212E_R)/\u221a(E_U E_R)"
    },
    {
      "eq_number": 510,
      "title": "Debye Peak (Internal Friction, Point Defect Relaxation)",
      "domain_id": 21,
      "significance": "tan \u03b4 = \u0394 \u03c9 \u03c4 / (1 + \u03c9\u00b2 \u03c4\u00b2); \u03c4 = \u03c4\u2080 exp(E_a/k_B T); peak at \u03c9\u03c4=1"
    },
    {
      "eq_number": 511,
      "title": "Bordoni Peak (Dislocation Relaxation)",
      "domain_id": 21,
      "significance": "kink-pair formation on dislocations; tan \u03b4 peak with E_a~0.1\u20130.2 eV"
    },
    {
      "eq_number": 512,
      "title": "Granato-L\u00fccke Theory (Dislocation Damping)",
      "domain_id": 21,
      "significance": "\u03b5_d = (\u039b L\u00b2 \u03c3)/(6 G) (amplitude-independent); breakaway at high amplitude"
    },
    {
      "eq_number": 513,
      "title": "Einstein Viscosity Equation (Rigid Sphere Suspension, Dilute)",
      "domain_id": 26,
      "significance": "\u03b7 = \u03b7_s (1 + 2.5 \u03c6); \u03c6 = volume fraction; dilute limit \u03c6\u226a1"
    },
    {
      "eq_number": 514,
      "title": "Krieger-Dougherty Equation (Concentrated Suspension)",
      "domain_id": 26,
      "significance": "\u03b7 = \u03b7_s (1 \u2212 \u03c6/\u03c6_m)^{\u2212[\u03b7]\u03c6_m}; \u03c6_m = maximum packing; [\u03b7]\u22482.5"
    },
    {
      "eq_number": 515,
      "title": "Frank-Oseen Free Energy (Liquid Crystal Elastic)",
      "domain_id": 26,
      "significance": "F = \u00bd[K\u2081(\u2207\u00b7n)\u00b2 + K\u2082(n\u00b7\u2207\u00d7n)\u00b2 + K\u2083(n\u00d7\u2207\u00d7n)\u00b2]; splay, twist, bend"
    },
    {
      "eq_number": 516,
      "title": "Frederiks Transition Threshold (Liquid Crystal)",
      "domain_id": 26,
      "significance": "E_c = (\u03c0/d) \u221a(K/\u03b5\u2080\u0394\u03b5); voltage for director reorientation"
    },
    {
      "eq_number": 517,
      "title": "Rayleigh Instability (Liquid Jet Breakup)",
      "domain_id": 26,
      "significance": "\u03bb_max = 9.016 r\u2080; fastest growing wavelength \u2192 uniform droplet formation"
    },
    {
      "eq_number": 518,
      "title": "Plateau-Rayleigh Instability for Liquid Threads",
      "domain_id": 26,
      "significance": "Cylindrical liquid thread unstable for \u03bb > 2\u03c0r; surface-tension-driven breakup"
    },
    {
      "eq_number": 519,
      "title": "Kissinger Equation (DSC/DTA Peak Kinetics)",
      "domain_id": 21,
      "significance": "ln(\u03b2/T_p\u00b2) = \u2212E_a/(R T_p) + ln(A R/E_a); \u03b2 = heating rate; T_p = peak temperature"
    },
    {
      "eq_number": 520,
      "title": "Ozawa-Flynn-Wall Equation (Isoconversional Kinetics)",
      "domain_id": 21,
      "significance": "log \u03b2 = const \u2212 0.4567 E_a/(R T); model-free kinetic analysis"
    },
    {
      "eq_number": 521,
      "title": "Tammann Nucleation Diagram (Nucleation vs Growth Rate)",
      "domain_id": 27,
      "significance": "Nucleation rate I(T) and growth rate U(T) bell-shaped; overlap \u2192 crystallization window"
    },
    {
      "eq_number": 522,
      "title": "Time-Temperature-Transformation (TTT) Diagram Equation",
      "domain_id": 27,
      "significance": "\u03c4(T) \u221d exp(\u0394G*/k_B T + E_a/k_B T); C-curve shape; nose at intermediate T"
    },
    {
      "eq_number": 523,
      "title": "Thornton Structure Zone Model (Thin Film Growth)",
      "domain_id": 21,
      "significance": "T/T_m vs Ar pressure \u2192 Zone 1 (porous), Zone T (dense fibrous), Zone 2 (columnar), Zone 3 (recrystallized)"
    },
    {
      "eq_number": 524,
      "title": "Herring Scaling Laws (Sintering Kinetics)",
      "domain_id": 21,
      "significance": "(\u0394L/L\u2080)^n \u221d t; n=1 viscous flow; n=2 volume diffusion; n=3 grain boundary diffusion; n=5 surface diffusion"
    },
    {
      "eq_number": 525,
      "title": "Pilling-Bedworth Ratio (Oxide Protectiveness)",
      "domain_id": 21,
      "significance": "PBR = V_oxide / V_metal consumed; 1 < PBR < 2 \u2192 protective; PBR > 2 \u2192 spallation; PBR < 1 \u2192 porous"
    },
    {
      "eq_number": 526,
      "title": "Ellingham Diagram (Oxide Thermodynamic Stability)",
      "domain_id": 21,
      "significance": "\u0394G\u2070 = RT ln p_O\u2082; line slope = \u2212\u0394S\u2070; lower line \u2192 more stable oxide"
    },
    {
      "eq_number": 527,
      "title": "Mott-Gurney Law (Space-Charge-Limited Current)",
      "domain_id": 21,
      "significance": "J = (9/8) \u03b5 \u03bc V\u00b2 / L\u00b3; trap-free SCLC; Child's law for solids"
    },
    {
      "eq_number": 528,
      "title": "Richardson-Dushman Equation (Thermionic Emission)",
      "domain_id": 21,
      "significance": "J = A_R T\u00b2 exp(\u2212\u03c6/k_B T); A_R = 4\u03c0 m e k_B\u00b2/h\u00b3 \u2248 1.20\u00d710\u2076 A/(m\u00b2K\u00b2)"
    },
    {
      "eq_number": 529,
      "title": "Schottky Barrier Height (Metal-Semiconductor)",
      "domain_id": 23,
      "significance": "\u03c6_Bn = \u03c6_m \u2212 \u03c7_s; \u03c6_Bp = E_g/q + \u03c7_s \u2212 \u03c6_m (ideal, no interface states)"
    },
    {
      "eq_number": 530,
      "title": "Spicer's Unified Defect Model (Fermi Level Pinning at Interfaces)",
      "domain_id": 23,
      "significance": "E_F pinned by deep native defects at interface; independent of metal work function"
    },
    {
      "eq_number": 531,
      "title": "Wolff's Law (Bone Remodeling, Mechanical Adaptation)",
      "domain_id": 21,
      "significance": "Bone density distribution adapts to principal stress trajectories; \u03c3_ij \u2192 \u03c1_ij"
    },
    {
      "eq_number": 532,
      "title": "Fung's Quasi-Linear Viscoelasticity (Soft Tissue)",
      "domain_id": 21,
      "significance": "\u03c3(t) = \u222b\u2080\u1d57 G(t\u2212\u03c4) \u2202\u03c3_e(\u03b5)/\u2202\u03b5 \u00b7 \u2202\u03b5/\u2202\u03c4 d\u03c4; separable elastic + relaxation"
    },
    {
      "eq_number": 533,
      "title": "Ogden Hyperelastic Model (Biological Tissue)",
      "domain_id": 21,
      "significance": "W = \u03a3 (\u03bc_k/\u03b1_k) (\u03bb\u2081^{\u03b1_k} + \u03bb\u2082^{\u03b1_k} + \u03bb\u2083^{\u03b1_k} \u2212 3); principal stretches; fits large deformations"
    },
    {
      "eq_number": 534,
      "title": "Matthiessen's Rule (Electrical Resistivity Additivity)",
      "domain_id": 21,
      "significance": "\u03c1_total = \u03c1_thermal + \u03c1_impurity + \u03c1_deformation; independent contributions sum"
    },
    {
      "eq_number": 535,
      "title": "Nordheim's Rule (Alloy Resistivity)",
      "domain_id": 21,
      "significance": "\u03c1_alloy = \u03c1_pure + C x(1\u2212x); x = atomic fraction; max at x=0.5 for disordered binary"
    },
    {
      "eq_number": 536,
      "title": "Miedema's Rules (Alloy Formation Enthalpy)",
      "domain_id": 21,
      "significance": "\u0394H_form = f(\u0394\u03c6*, \u0394n_ws^{1/3}); work function + electron density mismatch \u2192 semi-empirical model"
    },
    {
      "eq_number": 537,
      "title": "K\u00f6hler's Rule (Magnetoresistance Scaling)",
      "domain_id": 21,
      "significance": "\u0394\u03c1(B)/\u03c1(0) = F[B/\u03c1(0)]; Kohler plot universal for given material"
    },
    {
      "eq_number": 538,
      "title": "Zener Breakdown (Band-to-Band Tunneling)",
      "domain_id": 23,
      "significance": "D = exp[\u22124\u221a(2m*) E_g^{3/2}/(3 e \u210f E)]; tunneling probability through forbidden gap"
    },
    {
      "eq_number": 539,
      "title": "Klemens Model (Thermal Boundary Resistance / Kapitza)",
      "domain_id": 21,
      "significance": "R_K = 4 / (\u03c1 c v \u03b6); acoustic mismatch model; acoustic impedance mismatch \u2192 resistance"
    },
    {
      "eq_number": 540,
      "title": "Diffuse Mismatch Model (Thermal Boundary Resistance)",
      "domain_id": 21,
      "significance": "R_K from transmission probability of phonons regardless of mode; rough interfaces"
    },
    {
      "eq_number": 541,
      "title": "Seismic Wave Equation (Elastic)",
      "domain_id": 28,
      "significance": "\u03c1 \u2202\u00b2u/\u2202t\u00b2 = (\u03bb+\u03bc)\u2207(\u2207\u00b7u) + \u03bc\u2207\u00b2u; vector elastic wave equation; P and S waves"
    },
    {
      "eq_number": 542,
      "title": "Snell's Law (Seismic Refraction at Interfaces)",
      "domain_id": 28,
      "significance": "sin i\u2081/v\u2081 = sin i\u2082/v\u2082 = p (ray parameter); seismic ray tracing through layered Earth"
    },
    {
      "eq_number": 543,
      "title": "Gutenberg-Richter Magnitude-Energy Relation",
      "domain_id": 28,
      "significance": "log\u2081\u2080 E = 4.8 + 1.5 M (E in Joules); log\u2081\u2080 E = 5.24 + 1.44 M (later refinement)"
    },
    {
      "eq_number": 544,
      "title": "Omori's Law (Aftershock Decay)",
      "domain_id": 28,
      "significance": "n(t) = K / (c + t)^p; p\u22481 (often 0.9\u20131.4); K,c constants; t=time after mainshock"
    },
    {
      "eq_number": 545,
      "title": "Frequency-Magnitude Distribution (Gutenberg-Richter Law)",
      "domain_id": 28,
      "significance": "log\u2081\u2080 N(\u2265M) = a \u2212 b M; b\u22481 (global average); N=cumulative number of earthquakes"
    },
    {
      "eq_number": 546,
      "title": "Bullen's Compressibility-Pressure Hypothesis (Earth Interior)",
      "domain_id": 28,
      "significance": "K = a + bP; Earth core compressibility varies linearly with pressure"
    },
    {
      "eq_number": 547,
      "title": "Love Wave Dispersion Equation (Surface Waves)",
      "domain_id": 28,
      "significance": "tan(\u03ba\u03b2\u2081H) = (\u03bc\u2082\u03b2\u2082)/(\u03bc\u2081\u03b2\u2081); Love wave existence condition in layer over half-space"
    },
    {
      "eq_number": 548,
      "title": "Airy Isostasy (Crustal Compensation Model)",
      "domain_id": 28,
      "significance": "Mountain root thickness = h \u03c1_c/(\u03c1_m\u2212\u03c1_c); h=elevation; \u03c1_c,\u03c1_m=crust/mantle density"
    },
    {
      "eq_number": 549,
      "title": "Free-Air Gravity Anomaly",
      "domain_id": 28,
      "significance": "\u0394g_FA = g_obs \u2212 g_theoretical(\u03bb) + \u03b4g_FAC; \u03b4g_FAC=0.3086h mGal (free-air correction, h in m)"
    },
    {
      "eq_number": 550,
      "title": "Bouguer Gravity Anomaly (Complete)",
      "domain_id": 28,
      "significance": "\u0394g_B = g_obs \u2212 g_theo(\u03bb) + 0.3086h \u2212 0.0419\u03c1h + \u03b4g_terrain; \u03c1=2.67 g/cm\u00b3 typical"
    },
    {
      "eq_number": 551,
      "title": "Geodetic Reference System (GRS80/WGS84 Ellipsoid)",
      "domain_id": 28,
      "significance": "a=6378137m, f=1/298.257223563 (WGS84); meridian radius M=a(1\u2212e\u00b2)/(1\u2212e\u00b2sin\u00b2\u03c6)^{3/2}"
    },
    {
      "eq_number": 552,
      "title": "Geoid Undulation (Stokes' Formula)",
      "domain_id": 28,
      "significance": "N = (R/4\u03c0\u03b3)\u222b\u222b \u0394g S(\u03c8) d\u03c3; S(\u03c8) = Stokes function; \u0394g=gravity anomaly; \u03c8=angular distance"
    },
    {
      "eq_number": 553,
      "title": "Plate Motion on a Sphere (Euler Pole Rotation)",
      "domain_id": 28,
      "significance": "v = \u03c9 \u00d7 R; v=linear velocity, \u03c9=angular velocity vector, R=Earth radius vector"
    },
    {
      "eq_number": 554,
      "title": "Geomagnetic Secular Variation (IGRF Model)",
      "domain_id": 28,
      "significance": "B(r,\u03b8,\u03c6,t) = \u2212\u2207[a \u03a3(g_n^m cos m\u03c6 + h_n^m sin m\u03c6)(a/r)^{n+1} P_n^m(cos \u03b8)]"
    },
    {
      "eq_number": 555,
      "title": "Curie Temperature Isotherm (Magnetic Crustal Thickness)",
      "domain_id": 28,
      "significance": "Magnetic minerals become paramagnetic above ~580\u00b0C (magnetite Curie point); ~20-30 km depth"
    },
    {
      "eq_number": 556,
      "title": "Darcy's Law (Groundwater Flow in Porous Media)",
      "domain_id": 31,
      "significance": "Q = \u2212K A (dh/dl); v_Darcy = Q/A = \u2212K \u2207h; K = hydraulic conductivity"
    },
    {
      "eq_number": 557,
      "title": "Dupuit-Forchheimer Assumption (Unconfined Aquifer)",
      "domain_id": 31,
      "significance": "Q = \u2212K h (dh/dx); h=saturated thickness; flow lines approximately horizontal"
    },
    {
      "eq_number": 558,
      "title": "Theis Solution (Well Hydraulics, Confined Aquifer)",
      "domain_id": 31,
      "significance": "s(r,t) = Q/(4\u03c0T) \u222b_u^\u221e (e^{-x}/x) dx; u = r\u00b2S/(4Tt); T=transmissivity, S=storativity"
    },
    {
      "eq_number": 559,
      "title": "Manning Equation (Open Channel Flow)",
      "domain_id": 31,
      "significance": "v = (1/n) R_h^{2/3} S^{1/2}; n=Manning roughness; R_h=hydraulic radius; S=slope"
    },
    {
      "eq_number": 560,
      "title": "Rational Method (Peak Runoff Estimation)",
      "domain_id": 31,
      "significance": "Q_peak = C i A; C=runoff coefficient (0-1); i=rainfall intensity; A=watershed area"
    },
    {
      "eq_number": 561,
      "title": "Richards Equation (Unsaturated Flow in Soils)",
      "domain_id": 31,
      "significance": "\u2202\u03b8/\u2202t = \u2207\u00b7[K(\u03b8) \u2207(\u03c8+z)]; \u03b8=moisture content; \u03c8=pressure head; K(\u03b8)=hydraulic conductivity"
    },
    {
      "eq_number": 562,
      "title": "Horton Infiltration Model",
      "domain_id": 31,
      "significance": "f(t) = f_c + (f\u2080\u2212f_c) e^{\u2212kt}; infiltration capacity decays exponentially"
    },
    {
      "eq_number": 563,
      "title": "Penman-Monteith Evapotranspiration Equation",
      "domain_id": 31,
      "significance": "ET = [\u0394(R_n\u2212G) + \u03c1_a c_p (e_s\u2212e_a)/r_a] / [\u0394 + \u03b3(1+r_s/r_a)]"
    },
    {
      "eq_number": 564,
      "title": "Stomatal Conductance (Jarvis Model)",
      "domain_id": 31,
      "significance": "g_s = g_s_max \u00b7 f\u2081(PAR) \u00b7 f\u2082(T) \u00b7 f\u2083(VPD) \u00b7 f\u2084(CO\u2082) \u00b7 f\u2085(\u03c8_leaf); multiplicative stress functions"
    },
    {
      "eq_number": 565,
      "title": "Linear Wave Theory (Airy Wave, Dispersion Relation)",
      "domain_id": 30,
      "significance": "\u03c9\u00b2 = gk tanh(kh); deep water (kh\u226b1): \u03c9\u00b2=gk, c=g/\u03c9; shallow water (kh\u226a1): \u03c9\u00b2=ghk\u00b2, c=\u221a(gh)"
    },
    {
      "eq_number": 566,
      "title": "Stokes Drift (Mass Transport Under Waves)",
      "domain_id": 30,
      "significance": "U_s = \u00bd a\u00b2 \u03c9 k e^{2kz}; net Lagrangian drift under progressive waves; ~O(\u03b5\u00b2)"
    },
    {
      "eq_number": 567,
      "title": "Significant Wave Height (Sverdrup-Munk-Bretschneider)",
      "domain_id": 30,
      "significance": "H_s = H_{1/3} \u2248 4\u221am\u2080; m\u2080 = \u222b S(f) df (zeroth spectral moment)"
    },
    {
      "eq_number": 568,
      "title": "Tide-Generating Potential (Equilibrium Theory)",
      "domain_id": 30,
      "significance": "V_tide = \u2212(3/2) GM_\u2299 R\u00b2/r\u00b3 (cos\u00b2\u03b8 \u2212 1/3); Laplace's tidal equations govern dynamic response"
    },
    {
      "eq_number": 569,
      "title": "Geostrophic Balance (Ocean Currents)",
      "domain_id": 30,
      "significance": "f v = (1/\u03c1) \u2202p/\u2202x; f u = \u2212(1/\u03c1) \u2202p/\u2202y; f=2\u03a9 sin \u03c6 (Coriolis parameter); large-scale flow"
    },
    {
      "eq_number": 570,
      "title": "Ekman Transport (Wind-Driven Surface Layer)",
      "domain_id": 30,
      "significance": "M_E = \u03c4_wind / (\u03c1 f); net transport 90\u00b0 to right of wind (NH); left (SH)"
    },
    {
      "eq_number": 571,
      "title": "Thermohaline Circulation (Stommel-Arons Model)",
      "domain_id": 30,
      "significance": "Balance of advection, diffusion, and sources/sinks of heat and salt; deep ocean circulation"
    },
    {
      "eq_number": 572,
      "title": "Sverdrup Balance (Wind-Driven Gyre Circulation)",
      "domain_id": 30,
      "significance": "\u03b2 v = f \u2202w/\u2202z + curl_z(\u03c4)/(\u03c1); \u03b2=df/dy; meridional transport from wind stress curl"
    },
    {
      "eq_number": 573,
      "title": "Munk's Western Boundary Current Theory",
      "domain_id": 30,
      "significance": "A_H \u2207\u2074\u03c8 \u2212 \u03b2 \u2202\u03c8/\u2202x = \u2212curl_z(\u03c4)/\u03c1; lateral friction balances \u03b2-effect; Gulf Stream width"
    },
    {
      "eq_number": 574,
      "title": "Sonar Equation (Active, Monostatic)",
      "domain_id": 48,
      "significance": "SL \u2212 2TL + TS = NL \u2212 DI + DT; SL=source level, TL=transmission loss, TS=target strength, NL=noise, DI=directivity index, DT=detection threshold"
    },
    {
      "eq_number": 575,
      "title": "Sound Speed in Seawater (UNESCO/IES-80/CTD)",
      "domain_id": 48,
      "significance": "c(S,T,P) = 1449.2 + 4.6T \u2212 0.055T\u00b2 + 0.00029T\u00b3 + (1.34\u22120.010T)(S\u221235) + 0.016z"
    },
    {
      "eq_number": 576,
      "title": "Acoustic Doppler Current Profiler (ADCP) Principle",
      "domain_id": 48,
      "significance": "v_radial = (c \u0394f)/(2 f\u2080); Doppler shift from scatterers moving with water; 4 beams \u2192 3D velocity"
    },
    {
      "eq_number": 577,
      "title": "Hydrostatic Equation (Atmospheric)",
      "domain_id": 29,
      "significance": "dp/dz = \u2212\u03c1 g; pressure decreases exponentially with height in isothermal atmosphere"
    },
    {
      "eq_number": 578,
      "title": "Ideal Gas Law (Moist Air, Virtual Temperature)",
      "domain_id": 29,
      "significance": "p = \u03c1 R_d T_v; T_v = T (1 + 0.608 q); q=specific humidity; virtual temperature correction"
    },
    {
      "eq_number": 579,
      "title": "Potential Temperature (Adiabatic Reference)",
      "domain_id": 29,
      "significance": "\u03b8 = T (p\u2080/p)^{R_d/c_p}; R_d/c_p \u2248 0.286; conserved under adiabatic vertical displacement"
    },
    {
      "eq_number": 580,
      "title": "Brunt-V\u00e4is\u00e4l\u00e4 Frequency (Atmospheric Stability)",
      "domain_id": 29,
      "significance": "N\u00b2 = (g/\u03b8) d\u03b8/dz; buoyancy oscillation frequency; N\u00b2>0 \u2192 stable oscillation"
    },
    {
      "eq_number": 581,
      "title": "Geostrophic Wind (Pressure Gradient + Coriolis Balance)",
      "domain_id": 29,
      "significance": "u_g = \u2212(1/\u03c1f) \u2202p/\u2202y; v_g = (1/\u03c1f) \u2202p/\u2202x; wind parallel to isobars; f=2\u03a9 sin \u03c6"
    },
    {
      "eq_number": 582,
      "title": "Thermal Wind Equation (Vertical Wind Shear)",
      "domain_id": 29,
      "significance": "\u2202u_g/\u2202z = \u2212(g/fT) \u2202T/\u2202y; \u2202v_g/\u2202z = (g/fT) \u2202T/\u2202x; temperature gradient \u2192 wind shear"
    },
    {
      "eq_number": 583,
      "title": "Rossby Number (Inertial vs Coriolis)",
      "domain_id": 29,
      "significance": "Ro = U/(f L); Ro \u226a 1 \u2192 geostrophic; Ro ~ 1 \u2192 gradient wind; Ro \u226b 1 \u2192 cyclostrophic"
    },
    {
      "eq_number": 584,
      "title": "Rossby Wave Phase Speed (Planetary Waves)",
      "domain_id": 29,
      "significance": "c = \u016b \u2212 \u03b2/k\u00b2; \u03b2 = 2\u03a9 cos \u03c6 / R; westward phase speed relative to mean flow"
    },
    {
      "eq_number": 585,
      "title": "Clausius-Clapeyron (Water Vapor Saturation Pressure)",
      "domain_id": 29,
      "significance": "de_s/dT = L e_s/(R_v T\u00b2); saturated vapor pressure increases ~7%/K near surface"
    },
    {
      "eq_number": 586,
      "title": "Schwarzschild Equation (Radiative Transfer, No Scattering)",
      "domain_id": 29,
      "significance": "dI_\u03bd/ds = \u2212k_\u03bd \u03c1 I_\u03bd + k_\u03bd \u03c1 B_\u03bd(T); absorption + thermal emission along path"
    },
    {
      "eq_number": 587,
      "title": "K\u00f6hler Theory (Cloud Droplet Activation)",
      "domain_id": 29,
      "significance": "S\u22121 = A/r \u2212 B/r\u00b3; A=Kelvin term (curvature); B=solute term (Raoult effect); critical supersaturation at r*"
    },
    {
      "eq_number": 588,
      "title": "Terminal Fall Speed of Cloud/Rain Drops (Stokes/Davies)",
      "domain_id": 29,
      "significance": "v_t = k \u00b7 r\u00b2 (cloud droplets, r<40\u03bcm, Stokes regime); v_t = k' \u00b7 r^{0.5} (rain, r>0.6mm, turbulent)"
    },
    {
      "eq_number": 589,
      "title": "Mie Scattering (Atmospheric Aerosols, Clouds)",
      "domain_id": 29,
      "significance": "Q_ext, Q_sca, Q_abs as function of size parameter x=2\u03c0r/\u03bb and refractive index m"
    },
    {
      "eq_number": 590,
      "title": "Rayleigh Scattering (Molecular Atmosphere)",
      "domain_id": 29,
      "significance": "I_sca \u221d 1/\u03bb\u2074; cross-section \u03c3_R \u221d 1/\u03bb\u2074; blue sky, red sunsets; polarization patterns"
    },
    {
      "eq_number": 591,
      "title": "Lightning Return Stroke Current (Heidler Function / Bruce-Golde)",
      "domain_id": 29,
      "significance": "i(t) = (I\u2080/\u03b7)((t/\u03c4\u2081)^n/(1+(t/\u03c4\u2081)^n)) exp(\u2212t/\u03c4\u2082); I\u2080\u224810-200 kA; \u03c4\u2081\u22481-2\u03bcs, \u03c4\u2082\u224810-100\u03bcs"
    },
    {
      "eq_number": 592,
      "title": "Primitive Equations (Numerical Weather Prediction)",
      "domain_id": 29,
      "significance": "Conservation of momentum (3D), mass (continuity), energy (thermodynamic), moisture, and ideal gas law"
    },
    {
      "eq_number": 593,
      "title": "Nernst Equation (Membrane Equilibrium Potential)",
      "domain_id": 32,
      "significance": "E_ion = (RT/zF) ln([ion]_out/[ion]_in); equilibrium potential for single ion species"
    },
    {
      "eq_number": 594,
      "title": "Goldman-Hodgkin-Katz (GHK) Voltage Equation",
      "domain_id": 32,
      "significance": "V_m = (RT/F) ln[(P_K[K]_o+P_Na[Na]_o+P_Cl[Cl]_i)/(P_K[K]_i+P_Na[Na]_i+P_Cl[Cl]_o)]"
    },
    {
      "eq_number": 595,
      "title": "Hodgkin-Huxley Equations (Action Potential)",
      "domain_id": 32,
      "significance": "C_m dV/dt = \u2212g_K n\u2074(V\u2212E_K) \u2212 g_Na m\u00b3h(V\u2212E_Na) \u2212 g_L(V\u2212E_L) + I_stim"
    },
    {
      "eq_number": 596,
      "title": "FitzHugh-Nagumo Model (Simplified Excitable Dynamics)",
      "domain_id": 32,
      "significance": "dv/dt = v \u2212 v\u00b3/3 \u2212 w + I; dw/dt = \u03b5(v + a \u2212 bw); 2-variable reduction of HH"
    },
    {
      "eq_number": 597,
      "title": "Cable Equation (Neuronal Dendrite/Axon)",
      "domain_id": 32,
      "significance": "\u03bb\u00b2 \u2202\u00b2V/\u2202x\u00b2 = \u03c4_m \u2202V/\u2202t + V; \u03bb=\u221a(r_m/r_i); \u03c4_m=r_m c_m; passive spread along membrane"
    },
    {
      "eq_number": 598,
      "title": "Einstein-Smoluchowski Relation (Molecular Motor Stalling Force)",
      "domain_id": 32,
      "significance": "F_stall = k_B T / \u03b4; \u03b4=step size (~8 nm for kinesin); ~6 pN stall force"
    },
    {
      "eq_number": 599,
      "title": "Bell's Model (Bond Rupture Under Force)",
      "domain_id": 32,
      "significance": "k_off(F) = k\u2080 exp(F \u03b3/k_B T); \u03b3=reactive compliance (~0.1-0.5 nm); slip bond kinetics"
    },
    {
      "eq_number": 600,
      "title": "Hill's Equation (Muscle Force-Velocity Relation)",
      "domain_id": 32,
      "significance": "(F + a)(v + b) = (F\u2080 + a)b; hyperbola; v_max = F\u2080 b/a; a,b constants"
    },
    {
      "eq_number": 601,
      "title": "Huxley Sliding Filament Model (Cross-Bridge Dynamics)",
      "domain_id": 32,
      "significance": "\u2202n(x,t)/\u2202t = f(x)[1\u2212n(x,t)] \u2212 g(x) n(x,t); n=attached cross-bridge probability; x=distortion"
    },
    {
      "eq_number": 602,
      "title": "Monod-Wyman-Changeux (MWC) Model (Allosteric Transitions)",
      "domain_id": 32,
      "significance": "L = [T\u2080]/[R\u2080]; Y = \u03b1(1+\u03b1)^{n-1}/(L + (1+\u03b1)^n); \u03b1=[S]/K_R; cooperative ligand binding"
    },
    {
      "eq_number": 603,
      "title": "Michaelis-Menten Enzyme Kinetics",
      "domain_id": 33,
      "significance": "v = V_max [S] / (K_m + [S]); K_m = (k_{\u22121}+k_cat)/k\u2081; V_max = k_cat [E]_total"
    },
    {
      "eq_number": 604,
      "title": "Transition State Theory (Eyring Equation)",
      "domain_id": 33,
      "significance": "k = (k_B T/h) exp(\u2212\u0394G\u2021/RT) = (k_B T/h) exp(\u0394S\u2021/R) exp(\u2212\u0394H\u2021/RT)"
    },
    {
      "eq_number": 605,
      "title": "Arrhenius Equation (Chemical Reaction Rate)",
      "domain_id": 33,
      "significance": "k = A exp(\u2212E_a/RT); log\u2081\u2080(k\u2082/k\u2081) = (E_a/2.303R)(1/T\u2081\u22121/T\u2082); activation energy from T-dependence"
    },
    {
      "eq_number": 606,
      "title": "Marcus Theory (Electron Transfer Rate)",
      "domain_id": 33,
      "significance": "k_ET = (2\u03c0/\u210f) H_AB\u00b2 (1/\u221a(4\u03c0\u03bb k_B T)) exp[\u2212(\u0394G\u2070+\u03bb)\u00b2/(4\u03bb k_B T)]; Nobel 1992"
    },
    {
      "eq_number": 607,
      "title": "Butler-Volmer Equation (Electrode Kinetics)",
      "domain_id": 33,
      "significance": "j = j\u2080[exp(\u03b1_a F \u03b7/RT) \u2212 exp(\u2212\u03b1_c F \u03b7/RT)]; \u03b7=overpotential; \u03b1_a+\u03b1_c\u22481"
    },
    {
      "eq_number": 608,
      "title": "Beer-Lambert Law (Spectroscopy)",
      "domain_id": 34,
      "significance": "A = \u2212log\u2081\u2080(T) = \u03b5 c L; absorbance, molar absorptivity, concentration, path length"
    },
    {
      "eq_number": 609,
      "title": "F\u00f6rster Resonance Energy Transfer (FRET) Efficiency",
      "domain_id": 32,
      "significance": "E = R\u2080\u2076/(R\u2080\u2076+r\u2076); R\u2080\u2076 \u221d \u03ba\u00b2 \u03a6_D J(\u03bb)/n\u2074; R\u2080~1-10 nm"
    },
    {
      "eq_number": 610,
      "title": "Einstein Rate Equations (Laser Dynamics)",
      "domain_id": 34,
      "significance": "dN\u2082/dt = R_p \u2212 B\u2082\u2081 \u03c1(\u03bd) N\u2082 \u2212 A\u2082\u2081 N\u2082; d\u03c6/dt = B\u2082\u2081 \u03c1(\u03bd)(N\u2082\u2212N\u2081) c' \u2212 \u03c6/\u03c4_c"
    },
    {
      "eq_number": 611,
      "title": "Laser Threshold Condition",
      "domain_id": 34,
      "significance": "g_th = \u03b1_int + (1/2L) ln(1/R\u2081R\u2082); gain must overcome internal loss + mirror transmission"
    },
    {
      "eq_number": 612,
      "title": "Schawlow-Townes Linewidth (Fundamental Laser Linewidth)",
      "domain_id": 34,
      "significance": "\u0394\u03bd = (\u03c0 h\u03bd (\u0394\u03bd_c)\u00b2)/P_out; quantum-limited linewidth; narrower with higher power"
    },
    {
      "eq_number": 613,
      "title": "Mode-Locking Condition (fs/ps Pulses)",
      "domain_id": 34,
      "significance": "T_R = 2L/c (round-trip time); f_rep = 1/T_R; N locked modes \u2192 \u03c4_p = T_R/N \u221d 1/\u0394\u03bd_gain"
    },
    {
      "eq_number": 614,
      "title": "Nonlinear Polarization (\u03c7^{(n)} Expansion)",
      "domain_id": 34,
      "significance": "P_i = \u03b5\u2080[\u03c7^{(1)}_{ij} E_j + \u03c7^{(2)}_{ijk} E_j E_k + \u03c7^{(3)}_{ijkl} E_j E_k E_l + ...]"
    },
    {
      "eq_number": 615,
      "title": "Phase-Matching Condition (Nonlinear Optics)",
      "domain_id": 34,
      "significance": "\u0394k = k_3 \u2212 k_2 \u2212 k_1 = 0 for SHG; n(2\u03c9)=n(\u03c9) required; birefringent or QPM"
    },
    {
      "eq_number": 616,
      "title": "Nonlinear Schr\u00f6dinger Equation (Optical Solitons in Fibers)",
      "domain_id": 34,
      "significance": "i \u2202A/\u2202z \u2212 (\u03b2\u2082/2)\u2202\u00b2A/\u2202t\u00b2 + \u03b3|A|\u00b2A = 0; balance GVD (\u03b2\u2082) and Kerr nonlinearity (\u03b3)"
    },
    {
      "eq_number": 617,
      "title": "Kramers-Kronig Relations (Optical Dispersion)",
      "domain_id": 34,
      "significance": "n(\u03c9)\u22121 = (2/\u03c0)P\u222b\u2080^\u221e \u03c9'\u03ba(\u03c9')/(\u03c9'\u00b2\u2212\u03c9\u00b2)d\u03c9'; causality \u2192 real and imaginary parts of \u03c7 linked"
    },
    {
      "eq_number": 618,
      "title": "Rate Equations for Semiconductor Lasers",
      "domain_id": 34,
      "significance": "dN/dt = \u03b7_i I/qV \u2212 R(N) \u2212 v_g g(N) N_ph; dN_ph/dt = \u0393 v_g g(N) N_ph \u2212 N_ph/\u03c4_ph + \u03b2_sp R_sp"
    },
    {
      "eq_number": 619,
      "title": "Master Equation for Mode-Locked Lasers (Haus)",
      "domain_id": 34,
      "significance": "\u0394A = (g\u2212l + jD) A + (g/\u03a9_g\u00b2 + jD_g) \u2202\u00b2A/\u2202t\u00b2 + (\u03b3\u2212j\u03b4)|A|\u00b2A; Haus master equation"
    },
    {
      "eq_number": 620,
      "title": "Coupled-Mode Theory (Waveguides, Gratings, Resonators)",
      "domain_id": 34,
      "significance": "da_\u03bc/dz = \u2212j \u03a3_\u03ba K_\u03bc\u03ba a_\u03ba exp[j(\u03b2_\u03ba\u2212\u03b2_\u03bc)z]; coupling between waveguide/grating modes"
    },
    {
      "eq_number": 621,
      "title": "Zeeman Effect (Normal + Anomalous)",
      "domain_id": 35,
      "significance": "\u0394E = \u03bc_B g_J m_J B; g_J = 1 + [J(J+1)+S(S+1)\u2212L(L+1)]/[2J(J+1)]; Land\u00e9 g-factor"
    },
    {
      "eq_number": 622,
      "title": "Stark Effect (Linear + Quadratic)",
      "domain_id": 35,
      "significance": "Linear: \u0394E = 3ea\u2080 n (n\u2081\u2212n\u2082) E / 2 (Hydrogen); Quadratic: \u0394E = \u2212\u00bd \u03b1 E\u00b2 (general)"
    },
    {
      "eq_number": 623,
      "title": "Hyperfine Structure (Fermi Contact Interaction)",
      "domain_id": 35,
      "significance": "\u0394E_HFS = (A/2) [F(F+1) \u2212 I(I+1) \u2212 J(J+1)]; A \u221d \u03bc_B \u03bc_N \u27e81/r\u00b3\u27e9 |\u03c8(0)|\u00b2"
    },
    {
      "eq_number": 624,
      "title": "Born-Oppenheimer Approximation (Molecular Hamiltonian Separation)",
      "domain_id": 35,
      "significance": "\u03a8(r,R) \u2248 \u03c8_e(r;R) \u03c7_N(R); electronic Schr\u00f6dinger eq at fixed nuclear geometry; then nuclear motion"
    },
    {
      "eq_number": 625,
      "title": "Franck-Condon Principle (Vibrational Transition Intensities)",
      "domain_id": 35,
      "significance": "I_v'v'' \u221d |\u222b \u03c8_v'* \u03c8_v'' dR|\u00b2; vertical transitions; overlap of vibrational wavefunctions"
    },
    {
      "eq_number": 626,
      "title": "Molecular Rotational Spectroscopy (Rigid Rotor)",
      "domain_id": 35,
      "significance": "E_J = B J(J+1); B = \u210f\u00b2/(2I); I = \u03bc R\u00b2; \u0394J = \u00b11 selection rule \u2192 2B spacing"
    },
    {
      "eq_number": 627,
      "title": "Molecular Vibrational Spectroscopy (Harmonic)",
      "domain_id": 35,
      "significance": "E_v = \u210f\u03c9(v+\u00bd); \u03c9 = \u221a(k/\u03bc); fundamental transition \u03bd\u2080 = \u03c9/(2\u03c0c)"
    },
    {
      "eq_number": 628,
      "title": "Morse Potential (Anharmonic Diatomic)",
      "domain_id": 35,
      "significance": "V(r) = D_e [1 \u2212 e^{\u2212a(r\u2212r_e)}]\u00b2; analytical eigenvalues E_v = \u210f\u03c9(v+\u00bd)\u2212\u210f\u03c9x_e(v+\u00bd)\u00b2"
    },
    {
      "eq_number": 629,
      "title": "Rydberg Formula (Atomic Series Limits)",
      "domain_id": 35,
      "significance": "1/\u03bb = R\u221e/(n\u2081+\u03b4\u2081)\u00b2 \u2212 R\u221e/(n\u2082+\u03b4\u2082)\u00b2; R\u221e=10973731.568157 m\u207b\u00b9; \u03b4=quantum defect"
    },
    {
      "eq_number": 630,
      "title": "Racah Algebra (Angular Momentum Coupling in Complex Atoms)",
      "domain_id": 35,
      "significance": "Wigner 3-j, 6-j, 9-j symbols; recoupling coefficients; matrix elements of tensor operators"
    },
    {
      "eq_number": 631,
      "title": "Newtonian Constitutive Equation (Viscous Fluid)",
      "domain_id": 36,
      "significance": "\u03c4 = \u03bc \u03b3\u0307; \u03c3 = \u2212p I + 2 \u03bc D; D = strain-rate tensor; \u03c4 \u221d shear rate linearly"
    },
    {
      "eq_number": 632,
      "title": "Power-Law Fluid (Ostwald-de Waele Model)",
      "domain_id": 36,
      "significance": "\u03c4 = K \u03b3\u0307^n; \u03b7_app = K \u03b3\u0307^{n-1}; n<1 \u2192 shear-thinning (pseudoplastic); n>1 \u2192 shear-thickening; n=1 \u2192 Newtonian"
    },
    {
      "eq_number": 633,
      "title": "Bingham Plastic (Yield Stress Fluid)",
      "domain_id": 36,
      "significance": "\u03c4 = \u03c4_y + \u03bc_p \u03b3\u0307 for \u03c4 > \u03c4_y; no flow for \u03c4 < \u03c4_y"
    },
    {
      "eq_number": 634,
      "title": "Herschel-Bulkley Model (Yield + Power-Law)",
      "domain_id": 36,
      "significance": "\u03c4 = \u03c4_y + K \u03b3\u0307^n for \u03c4 > \u03c4_y"
    },
    {
      "eq_number": 635,
      "title": "Carreau-Yasuda Model (Shear-Thinning With Zero/Infinite Limits)",
      "domain_id": 36,
      "significance": "\u03b7(\u03b3\u0307) = \u03b7_\u221e + (\u03b7\u2080\u2212\u03b7_\u221e)[1 + (\u03bb \u03b3\u0307)^a]^{(n-1)/a}"
    },
    {
      "eq_number": 636,
      "title": "Maxwell Viscoelastic Model (Liquid)",
      "domain_id": 36,
      "significance": "d\u03b5/dt = (1/E) d\u03c3/dt + \u03c3/\u03b7; relaxation time \u03c4 = \u03b7/E; elastic at short times, viscous at long"
    },
    {
      "eq_number": 637,
      "title": "Kelvin-Voigt Model (Viscoelastic Solid)",
      "domain_id": 36,
      "significance": "\u03c3 = E \u03b5 + \u03b7 d\u03b5/dt; retardation time = \u03b7/E; creep compliance J(t)=[1\u2212e^{\u2212t/\u03c4}]/E"
    },
    {
      "eq_number": 638,
      "title": "Generalized Maxwell / Wiechert Model (Multiple Relaxation Times)",
      "domain_id": 36,
      "significance": "G(t) = G_\u221e + \u03a3_i G_i exp(\u2212t/\u03c4_i); relaxation spectrum H(\u03c4); Prony series"
    },
    {
      "eq_number": 639,
      "title": "Cox-Merz Rule (Steady vs Dynamic Viscosity Equivalence)",
      "domain_id": 36,
      "significance": "\u03b7(\u03b3\u0307) \u2248 |\u03b7*(\u03c9)| when \u03b3\u0307 = \u03c9; empirical equivalence for many polymer melts/solutions"
    },
    {
      "eq_number": 640,
      "title": "Trouton Ratio (Extensional/Shear Viscosity Ratio)",
      "domain_id": 36,
      "significance": "Tr = \u03b7_E / \u03b7; Newtonian: Tr=3; viscoelastic: Tr\u226b3; strain-hardening in extension"
    },
    {
      "eq_number": 641,
      "title": "Amontons-Coulomb Friction Laws",
      "domain_id": 37,
      "significance": "F_f = \u03bc N (macroscopic); independent of apparent contact area and sliding speed (approximately)"
    },
    {
      "eq_number": 642,
      "title": "Archard's Law (Adhesive Wear)",
      "domain_id": 37,
      "significance": "V = k F s / H; k=wear coefficient (~10^{-2} to 10^{-7}); softer material hardness controls"
    },
    {
      "eq_number": 643,
      "title": "Stribeck Curve (Lubrication Regimes)",
      "domain_id": 37,
      "significance": "\u03bc = f(\u03b7 N/p, roughness, geometry); boundary \u2192 mixed \u2192 EHL \u2192 hydrodynamic as speed increases"
    },
    {
      "eq_number": 644,
      "title": "Reynolds Equation (Thin-Film Lubrication)",
      "domain_id": 37,
      "significance": "\u2202/\u2202x[(h\u00b3/\u03b7)\u2202p/\u2202x] + \u2202/\u2202y[(h\u00b3/\u03b7)\u2202p/\u2202y] = 6(U\u2202h/\u2202x + 2\u2202h/\u2202t)"
    },
    {
      "eq_number": 645,
      "title": "Hertzian Contact (Elastic Contact Between Curved Surfaces)",
      "domain_id": 18,
      "significance": "a = (3FR/4E*)^{1/3}; p_max = 3F/(2\u03c0a\u00b2); E* = [(1\u2212\u03bd\u2081\u00b2)/E\u2081+(1\u2212\u03bd\u2082\u00b2)/E\u2082]\u207b\u00b9"
    },
    {
      "eq_number": 646,
      "title": "Elastohydrodynamic Lubrication (EHL) Film Thickness (Hamrock-Dowson)",
      "domain_id": 37,
      "significance": "h_min/R_x = 3.63 U\u2070\u00b7\u2076\u2078 G\u2070\u00b7\u2074\u2079 W\u207b\u2070\u00b7\u2070\u2077\u00b3 (1\u2212e^{\u22120.68k}); U=\u03b7\u2080u/E'R_x, G=\u03b1E', W=w/E'R_x\u00b2"
    },
    {
      "eq_number": 647,
      "title": "Janssen Effect (Pressure Saturation in Silos)",
      "domain_id": 38,
      "significance": "p(z) = (\u03c1 g D / 4 \u03bc_w K) [1 \u2212 exp(\u22124 \u03bc_w K z/D)]; pressure saturates at finite depth"
    },
    {
      "eq_number": 648,
      "title": "Coulomb Yield Criterion (Granular Failure)",
      "domain_id": 38,
      "significance": "\u03c4 = \u03c3 tan \u03c6 + c; \u03c6=internal friction angle (~25-45\u00b0 for sands); c=cohesion (0 for dry sand)"
    },
    {
      "eq_number": 649,
      "title": "Angle of Repose (Granular Pile)",
      "domain_id": 38,
      "significance": "tan \u03c6_r = H_max / R; \u03c6_r \u2248 \u03c6 (internal friction angle); ~30-40\u00b0 for most granular materials"
    },
    {
      "eq_number": 650,
      "title": "Brazil Nut Effect (Granular Convection/Segregation)",
      "domain_id": 38,
      "significance": "Larger particles rise during vibration or shaking due to percolation + convection"
    },
    {
      "eq_number": 651,
      "title": "Bagnold Scaling (Granular Flow Rheology - Inertial)",
      "domain_id": 38,
      "significance": "\u03c4 = a (\u03c1_p d\u00b2) \u03b3\u0307\u00b2 (inertial regime); Bagnold number Ba = \u03c1_p d\u00b2 \u03b3\u0307/\u03b7_f; Ba>450 \u2192 grain inertia dominates"
    },
    {
      "eq_number": 652,
      "title": "\u03bc(I) Rheology (Inertial Number Scaling for Dense Granular Flow)",
      "domain_id": 38,
      "significance": "\u03bc(I) = \u03bc_s + (\u03bc\u2082\u2212\u03bc_s)/(1+I\u2080/I); I = \u03b3\u0307 d/\u221a(p/\u03c1_p); dimensionless inertial number"
    },
    {
      "eq_number": 653,
      "title": "Coulomb Blockade Condition (Single-Electron Transistor)",
      "domain_id": 39,
      "significance": "E_c = e\u00b2/(2C_\u03a3) > k_B T; charging energy must exceed thermal energy for CB to be observed"
    },
    {
      "eq_number": 654,
      "title": "Landauer Formula (Ballistic Conductance)",
      "domain_id": 39,
      "significance": "G = (2e\u00b2/h) \u03a3 T_n; G\u2080 = 2e\u00b2/h \u2248 77.5 \u03bcS (~12.9 k\u03a9); quantized conductance"
    },
    {
      "eq_number": 655,
      "title": "Kondo Effect (Resistance Minimum in Dilute Magnetic Alloys)",
      "domain_id": 39,
      "significance": "R \u221d \u2212ln(T) below Kondo temperature T_K; magnetic impurity spin screened by conduction electrons"
    },
    {
      "eq_number": 656,
      "title": "2D Electron Gas Density of States (Constant)",
      "domain_id": 39,
      "significance": "g_{2D}(E) = m*/(\u03c0\u210f\u00b2) = constant; independent of energy"
    },
    {
      "eq_number": 657,
      "title": "Graphene Dirac Dispersion (Massless 2D Fermions)",
      "domain_id": 39,
      "significance": "E = \u00b1 v_F |k|; v_F \u2248 10\u2076 m/s; linear dispersion near Dirac points K,K'"
    },
    {
      "eq_number": 658,
      "title": "Quantum Confinement (Infinite Well \u2014 Nanowire/Quantum Well)",
      "domain_id": 39,
      "significance": "E_n = n\u00b2\u03c0\u00b2\u210f\u00b2/(2m*L\u00b2); 1D wire; 2D well adds E_{n_x,n_y} terms; 0D dot adds all three"
    },
    {
      "eq_number": 659,
      "title": "Casimir Force (Between Ideal Plates, Nanoscale)",
      "domain_id": 39,
      "significance": "F/A = \u2212\u03c0\u00b2\u210fc/(240 d\u2074); d=separation; attractive; zero-point EM fluctuations"
    },
    {
      "eq_number": 660,
      "title": "DLVO Theory (Colloidal Nanoparticle Stability)",
      "domain_id": 39,
      "significance": "V_total(d) = V_vdW + V_EDL; van der Waals attraction + electric double-layer repulsion"
    },
    {
      "eq_number": 661,
      "title": "Single Qubit State (Bloch Sphere)",
      "domain_id": 40,
      "significance": "|\u03c8\u27e9 = cos(\u03b8/2)|0\u27e9 + e^{i\u03c6} sin(\u03b8/2)|1\u27e9; pure state on Bloch sphere surface"
    },
    {
      "eq_number": 662,
      "title": "Bell States (Maximally Entangled Two-Qubit States)",
      "domain_id": 40,
      "significance": "|\u03a6\u207a\u27e9=(|00\u27e9+|11\u27e9)/\u221a2; |\u03a6\u207b\u27e9=(|00\u27e9\u2212|11\u27e9)/\u221a2; |\u03a8\u207a\u27e9=(|01\u27e9+|10\u27e9)/\u221a2; |\u03a8\u207b\u27e9=(|01\u27e9\u2212|10\u27e9)/\u221a2"
    },
    {
      "eq_number": 663,
      "title": "No-Cloning Theorem",
      "domain_id": 40,
      "significance": "An unknown quantum state cannot be copied perfectly; U|\u03c8\u27e9|0\u27e9 \u2260 |\u03c8\u27e9|\u03c8\u27e9 for all |\u03c8\u27e9"
    },
    {
      "eq_number": 664,
      "title": "Holevo Bound (Classical Information From Qubit)",
      "domain_id": 40,
      "significance": "\u03c7 \u2264 S(\u03c1) \u2212 \u03a3 p_i S(\u03c1_i); at most 1 classical bit extractable per qubit"
    },
    {
      "eq_number": 665,
      "title": "Deutsch-Jozsa Algorithm Speedup",
      "domain_id": 40,
      "significance": "Single-query solution to balanced/constant problem; first clear quantum advantage"
    },
    {
      "eq_number": 666,
      "title": "Grover's Search Algorithm (Quadratic Speedup)",
      "domain_id": 40,
      "significance": "O(\u221aN) quantum search via amplitude amplification; ~\u03c0\u221aN/4 Grover iterations"
    },
    {
      "eq_number": 667,
      "title": "Shor's Factoring Algorithm",
      "domain_id": 40,
      "significance": "Quantum period-finding via QFT \u2192 polynomial-time integer factorization; O((log N)\u00b3); exponentially faster than classical best known"
    },
    {
      "eq_number": 668,
      "title": "Concatenated Quantum Error Correction Threshold Theorem",
      "domain_id": 40,
      "significance": "If gate error < p_th (~10\u207b\u00b2 to 10\u207b\u2074 depending on code), errors can be arbitrarily suppressed"
    },
    {
      "eq_number": 669,
      "title": "Lorenz Equations (Deterministic Chaos)",
      "domain_id": 41,
      "significance": "\u1e8b = \u03c3(y\u2212x); \u1e8f = x(\u03c1\u2212z)\u2212y; \u017c = xy\u2212\u03b2 z; \u03c3=10, \u03b2=8/3, \u03c1=28 \u2192 strange attractor"
    },
    {
      "eq_number": 670,
      "title": "Logistic Map (Period-Doubling Route to Chaos)",
      "domain_id": 41,
      "significance": "x_{n+1} = r x_n (1\u2212x_n); period-doubling bifurcations; chaos at r\u22483.57; Feigenbaum universality"
    },
    {
      "eq_number": 671,
      "title": "Lyapunov Exponent (Chaos Diagnostic)",
      "domain_id": 41,
      "significance": "\u03bb = lim_{t\u2192\u221e} (1/t) ln |\u03b4x(t)/\u03b4x(0)|; \u03bb>0 \u2192 chaos; \u03bb<0 \u2192 stable; \u03bb=0 \u2192 marginal"
    },
    {
      "eq_number": 672,
      "title": "KAM Theorem (Kolmogorov-Arnold-Moser)",
      "domain_id": 41,
      "significance": "Most invariant tori survive small perturbations if frequency ratio is sufficiently irrational"
    },
    {
      "eq_number": 673,
      "title": "Kuramoto Model (Synchronization of Coupled Oscillators)",
      "domain_id": 41,
      "significance": "\u03b8\u0307_i = \u03c9_i + (K/N) \u03a3_j sin(\u03b8_j\u2212\u03b8_i); K>K_c \u2192 phase transition to global synchronization"
    },
    {
      "eq_number": 674,
      "title": "Mandelbrot Set (Fractal Geometry)",
      "domain_id": 41,
      "significance": "z_{n+1} = z_n\u00b2 + c; bounded orbits \u2192 c \u2208 Mandelbrot set; fractal boundary with infinite complexity"
    },
    {
      "eq_number": 675,
      "title": "Bloch Equations (NMR/MRI Signal)",
      "domain_id": 42,
      "significance": "dM/dt = \u03b3 M \u00d7 B \u2212 (M_x i\u0302+M_y j\u0302)/T\u2082 \u2212 (M_z\u2212M\u2080)k\u0302/T\u2081; relaxation toward equilibrium"
    },
    {
      "eq_number": 676,
      "title": "Larmor Frequency (NMR Precession)",
      "domain_id": 42,
      "significance": "\u03c9\u2080 = \u03b3 B\u2080; \u03b3_H/2\u03c0 = 42.577 MHz/T; proton Larmor frequency in clinical MRI"
    },
    {
      "eq_number": 677,
      "title": "Beer-Lambert Law (X-ray/\u03b3-ray Attenuation, CT)",
      "domain_id": 42,
      "significance": "I = I\u2080 e^{\u2212\u03bcx}; \u03bc/p = mass attenuation coefficient; CT: \u03bc(x,y) \u2192 Hounsfield units"
    },
    {
      "eq_number": 678,
      "title": "Radon Transform (CT Image Reconstruction)",
      "domain_id": 42,
      "significance": "p(s,\u03b8) = \u222b_{-\u221e}^\u221e f(s cos\u03b8\u2212t sin\u03b8, s sin\u03b8+t cos\u03b8) dt; projection \u2192 2D image via filtered backprojection"
    },
    {
      "eq_number": 679,
      "title": "Ultrasound Wave Equation (Medical Imaging)",
      "domain_id": 42,
      "significance": "\u2202\u00b2p/\u2202t\u00b2 = c\u00b2\u2207\u00b2p; reflection at tissue interfaces (acoustic impedance mismatch Z=\u03c1c)"
    },
    {
      "eq_number": 680,
      "title": "Attenuation of Ultrasound in Tissue",
      "domain_id": 42,
      "significance": "I(x) = I\u2080 e^{\u2212\u03b1 f^n x}; n\u22481 for most soft tissues; \u03b1\u22480.5-1 dB/(cm\u00b7MHz)"
    },
    {
      "eq_number": 681,
      "title": "Linear-Quadratic (LQ) Model (Radiation Therapy Cell Survival)",
      "domain_id": 43,
      "significance": "S = exp(\u2212\u03b1D \u2212 \u03b2D\u00b2); \u03b1/\u03b2 ratio ~3 Gy for late-responding (CNS, spinal); ~10 Gy for early-responding/acutely responding; D=total dose"
    },
    {
      "eq_number": 682,
      "title": "Bragg Peak (Proton/Ion Beam Depth-Dose)",
      "domain_id": 43,
      "significance": "dE/dx peaks sharply at end of range (Bragg peak); R \u221d E^{1.7\u22121.8}; sharp distal falloff"
    },
    {
      "eq_number": 683,
      "title": "Bethe-Bloch Formula (Stopping Power for Charged Particles)",
      "domain_id": 43,
      "significance": "\u2212\u27e8dE/dx\u27e9 = K z\u00b2 (Z/A)(1/\u03b2\u00b2)[\u00bd ln(2m_e c\u00b2\u03b2\u00b2\u03b3\u00b2T_max/I\u00b2) \u2212 \u03b2\u00b2 \u2212 \u03b4(\u03b2\u03b3)/2]"
    },
    {
      "eq_number": 684,
      "title": "Dosimetry: Cavity Theory (Bragg-Gray / Spencer-Attix)",
      "domain_id": 43,
      "significance": "D_med = (S\u0304/\u03c1)_med^wall \u00b7 D_wall; dose to medium from dose measured in wall/gas cavity"
    },
    {
      "eq_number": 685,
      "title": "MIRD Formalism (Internal Dosimetry)",
      "domain_id": 43,
      "significance": "D\u0304(r_T\u2190r_S) = \u00c3_S \u03a3_i \u0394_i \u03c6_i(r_T\u2190r_S); \u00c3_S=cumulated activity; \u0394_i=mean energy per transition; \u03c6=fraction absorbed"
    },
    {
      "eq_number": 686,
      "title": "Shockley-Queisser Limit (Single-Junction Solar Cell Efficiency)",
      "domain_id": 44,
      "significance": "\u03b7_max \u2248 33.7% for E_g=1.34 eV under AM1.5 spectrum (non-concentrated); detailed balance limit"
    },
    {
      "eq_number": 687,
      "title": "Solar Cell I-V Characteristic (One-Diode Model)",
      "domain_id": 44,
      "significance": "I = I_ph \u2212 I\u2080 [exp(q(V+IR_s)/nk_B T)\u22121] \u2212 (V+IR_s)/R_sh"
    },
    {
      "eq_number": 688,
      "title": "Fill Factor (Solar Cell)",
      "domain_id": 44,
      "significance": "FF = (V_mpp I_mpp)/(V_oc I_sc); \u03b7 = P_max/P_in = FF \u00b7 V_oc \u00b7 J_sc / P_in"
    },
    {
      "eq_number": 689,
      "title": "Betz Limit (Wind Turbine Maximum Efficiency)",
      "domain_id": 44,
      "significance": "C_p_max = 16/27 \u2248 59.3%; maximum fraction of kinetic power extractable from wind"
    },
    {
      "eq_number": 690,
      "title": "Rankine Cycle Efficiency (Steam Power Plant)",
      "domain_id": 44,
      "significance": "\u03b7 = (W_turbine \u2212 W_pump)/Q_in \u2248 1 \u2212 T_c/T_h (ideal Carnot upper bound, real ~30-45%)"
    },
    {
      "eq_number": 691,
      "title": "Brayton Cycle Efficiency (Gas Turbine / Jet Engine)",
      "domain_id": 44,
      "significance": "\u03b7 = 1 \u2212 1/r_p^{(\u03b3\u22121)/\u03b3}; r_p = compressor pressure ratio; \u03b3 = c_p/c_v"
    },
    {
      "eq_number": 692,
      "title": "Nernst Equation (Fuel Cell Open-Circuit Voltage)",
      "domain_id": 44,
      "significance": "E_rev = \u2212\u0394G/(nF); H\u2082/O\u2082 Fuel Cell: E\u2070 = 1.229 V at 25\u00b0C; actual: E = E_rev \u2212 \u03b7_act \u2212 \u03b7_ohm \u2212 \u03b7_conc"
    },
    {
      "eq_number": 693,
      "title": "Parker Spiral (Interplanetary Magnetic Field)",
      "domain_id": 45,
      "significance": "B_r \u221d 1/r\u00b2; B_\u03c6 = \u2212B_r (\u03a9 r sin \u03b8)/v_SW; Archimedean spiral angle; v_SW\u2248400 km/s (slow), ~750 km/s (fast)"
    },
    {
      "eq_number": 694,
      "title": "Chapman-Ferraro Model (Magnetopause Standoff Distance)",
      "domain_id": 45,
      "significance": "Balance of solar wind dynamic pressure with Earth's magnetic pressure: R_MP ~ 10 R_E (subsolar)"
    },
    {
      "eq_number": 695,
      "title": "Alfv\u00e9n Mach Number (Solar Wind \u2014 Magnetosphere Coupling)",
      "domain_id": 45,
      "significance": "M_A = v_SW / v_A; M_A typical solar wind ~5-10; super-Alfv\u00e9nic flow \u2192 bow shock forms"
    },
    {
      "eq_number": 696,
      "title": "Dungey Cycle (Magnetospheric Convection via Reconnection)",
      "domain_id": 45,
      "significance": "Open flux transport from dayside reconnection \u2192 tail lobe \u2192 nightside reconnection \u2192 return flow (2-cell convection)"
    },
    {
      "eq_number": 697,
      "title": "St\u00f8rmer Theory (Charged Particle Motion in Dipole Field)",
      "domain_id": 45,
      "significance": "Allowed/forbidden zones for cosmic ray access; rigidity cutoff P_c = 59.6 cos\u2074 \u03bb / r\u00b2 (GV, dipole approx.)"
    },
    {
      "eq_number": 698,
      "title": "Radiation Belt Diffusion Equation (Fokker-Planck Approach)",
      "domain_id": 45,
      "significance": "\u2202f/\u2202t = L\u00b2 \u2202/\u2202L (D_LL L^{-2} \u2202f/\u2202L) + radial diffusion + sources (CRAND, injections) + losses (wave-particle, atmospheric)"
    },
    {
      "eq_number": 699,
      "title": "Auroral Electron Acceleration (Knight Relation)",
      "domain_id": 45,
      "significance": "j_\u2225 = K (V \u2212 V_c); K = field-aligned conductance; V_c = critical voltage; parallel potential drop above aurora"
    },
    {
      "eq_number": 700,
      "title": "Chapman-Jouguet (CJ) Detonation Theory",
      "domain_id": 46,
      "significance": "Detonation products at sonic condition relative to shock front (M=1); Rayleigh line tangent to Hugoniot at CJ point"
    },
    {
      "eq_number": 701,
      "title": "ZND Model (Zeldovich-Von Neumann-D\u00f6ring Structure)",
      "domain_id": 46,
      "significance": "Lead shock \u2192 von Neumann spike (induction zone, no reaction) \u2192 reaction zone \u2192 CJ plane"
    },
    {
      "eq_number": 702,
      "title": "Rankine-Hugoniot Relations (General Shock Jump Conditions)",
      "domain_id": 46,
      "significance": "\u03c1\u2081 u\u2081 = \u03c1\u2082 u\u2082; p\u2081+\u03c1\u2081u\u2081\u00b2 = p\u2082+\u03c1\u2082u\u2082\u00b2; h\u2081+\u00bdu\u2081\u00b2 = h\u2082+\u00bdu\u2082\u00b2; conservation across any shock or detonation front"
    },
    {
      "eq_number": 703,
      "title": "Mie-Gr\u00fcneisen Equation of State (Solids Under Shock)",
      "domain_id": 46,
      "significance": "p(V,E) = p_ref(V) + (\u03b3(V)/V)[E \u2212 E_ref(V)]; \u03b3(V)/V = Gr\u00fcneisen parameter / volume"
    },
    {
      "eq_number": 704,
      "title": "Hopkinson-Cranz (Cube-Root) Blast Scaling Law",
      "domain_id": 46,
      "significance": "R\u2081/R\u2082 = (W\u2081/W\u2082)^{1/3} at equal overpressure; scaled distance Z = R/W^{1/3}"
    },
    {
      "eq_number": 705,
      "title": "Taylor-Sedov Blast Wave (Point Explosion, Self-Similar Solution)",
      "domain_id": 46,
      "significance": "R(t) = \u03be\u2080 (E/\u03c1\u2080)^{1/5} t^{2/5} (strong shock, spherical); nuclear fireball radius"
    },
    {
      "eq_number": 706,
      "title": "Veselago's Left-Handed Material Condition",
      "domain_id": 47,
      "significance": "\u03b5 < 0 and \u03bc < 0 simultaneously \u2192 \u00f1 < 0 (negative index); reversed Snell's law, reversed Doppler, reversed Cherenkov"
    },
    {
      "eq_number": 707,
      "title": "Pendry's Perfect Lens (Subwavelength Imaging)",
      "domain_id": 47,
      "significance": "n = \u22121, \u03bc = \u03b5 = \u22121 \u2192 amplifies evanescent waves \u2192 subwavelength resolution; no diffraction limit"
    },
    {
      "eq_number": 708,
      "title": "Transformation Optics (Cloaking / Invisibility)",
      "domain_id": 47,
      "significance": "g'^{\u03bc\u03bd} = \u039b^\u03bc_\u03b1 \u039b^\u03bd_\u03b2 g^{\u03b1\u03b2} where \u039b relates virtual to physical space; coordinate transformation \u2192 anisotropic \u03b5,\u03bc"
    },
    {
      "eq_number": 709,
      "title": "Effective Medium Theory (Maxwell Garnett / Bruggeman)",
      "domain_id": 47,
      "significance": "MG: (\u03b5_eff\u2212\u03b5_h)/(\u03b5_eff+2\u03b5_h) = f (\u03b5_i\u2212\u03b5_h)/(\u03b5_i+2\u03b5_h); Bruggeman: f(\u03b5_i\u2212\u03b5_eff)/(\u03b5_i+2\u03b5_eff) + (1\u2212f)(\u03b5_h\u2212\u03b5_eff)/(\u03b5_h+2\u03b5_eff)=0"
    },
    {
      "eq_number": 710,
      "title": "NTU-Effectiveness Method (Heat Exchanger Design)",
      "domain_id": 49,
      "significance": "\u03b5 = Q/Q_max; NTU = UA/C_min; \u03b5 = f(NTU, C_r, flow arrangement); C_r = C_min/C_max"
    },
    {
      "eq_number": 711,
      "title": "Natural Frequency of a Cantilever Beam",
      "domain_id": 49,
      "significance": "f_n = (\u03b2_n L)\u00b2/(2\u03c0 L\u00b2) \u221a(EI/\u03c1A); \u03b2\u2081L=1.875, \u03b2\u2082L=4.694, \u03b2\u2083L=7.855 for cantilever"
    },
    {
      "eq_number": 712,
      "title": "PID Control Law (Feedback Control)",
      "domain_id": 49,
      "significance": "u(t) = K_p e(t) + K_i \u222b\u2080\u1d57 e(\u03c4)d\u03c4 + K_d de/dt; e(t)=setpoint \u2212 measurement"
    },
    {
      "eq_number": 713,
      "title": "Nyquist Stability Criterion",
      "domain_id": 49,
      "significance": "N = Z \u2212 P; encirclements of \u22121 point determine closed-loop stability from open-loop transfer function"
    },
    {
      "eq_number": 714,
      "title": "Young-Laplace Equation (Capillary Pressure, Droplets/Bubbles)",
      "domain_id": 25,
      "significance": "\u0394p = \u03b3 (1/R\u2081 + 1/R\u2082); \u0394p = 2\u03b3/R (spherical); \u0394p = 4\u03b3/R for soap bubble (2 interfaces)"
    },
    {
      "eq_number": 715,
      "title": "Kelvin Equation (Curvature + Vapor Pressure)",
      "domain_id": 25,
      "significance": "ln(P/P_sat) = 2\u03b3 V_m/(r R T); concave meniscus (r<0) \u2192 condensation below P_sat"
    },
    {
      "eq_number": 716,
      "title": "Washburn Equation (Capillary Rise Dynamics)",
      "domain_id": 25,
      "significance": "h(t) = \u221a(\u03b3 R cos \u03b8 t/(2\u03b7)); Lucas-Washburn; \u221at dependence for capillary imbibition"
    },
    {
      "eq_number": 717,
      "title": "Poiseuille Law (Microfluidic Channel Flow)",
      "domain_id": 36,
      "significance": "Q = (\u0394p w h\u00b3)/(12 \u03b7 L) [1 \u2212 0.63 h/w] for rectangular channel (h\u226aw); ~h\u00b3 dependence"
    },
    {
      "eq_number": 718,
      "title": "Stribeck Curve \u2014 Empirical Friction-Speed-Load Relation",
      "domain_id": 37,
      "significance": "\u03bc = \u03bc_b + (\u03bc_h\u2212\u03bc_b) / [1 + (\u03b7 N/p)^m]; boundary \u2192 mixed \u2192 hydrodynamic transition"
    },
    {
      "eq_number": 719,
      "title": "Zener-Hollomon Parameter (Hot Deformation)",
      "domain_id": 21,
      "significance": "Z = \u03b5\u0307 exp(Q/RT); flow stress \u03c3 = f(Z); Z unifies temperature and strain-rate effects"
    },
    {
      "eq_number": 720,
      "title": "Ashby Deformation Mechanism Maps",
      "domain_id": 21,
      "significance": "\u03c3/G vs T/T_m plot with boundaries between plasticity, power-law creep, diffusional flow, etc."
    },
    {
      "eq_number": 721,
      "title": "Tabor Parameter (Indentation Representative Strain)",
      "domain_id": 21,
      "significance": "\u03b5_rep \u2248 0.2 tan \u03b2; \u03b2 = indenter angle; uniaxial stress-strain relation from hardness"
    },
    {
      "eq_number": 722,
      "title": "Lode Parameter (Stress Triaxiality for Ductile Fracture)",
      "domain_id": 21,
      "significance": "\u03b7 = \u03c3_m/\u03c3_v; \u03c3_m=hydrostatic stress; \u03c3_v=von Mises stress; \u03b7>1/3 needed for void growth"
    },
    {
      "eq_number": 723,
      "title": "Dislocation Density Evolution (Kocks-Mecking Model)",
      "domain_id": 21,
      "significance": "d\u03c1/d\u03b5 = k\u2081 \u221a\u03c1 \u2212 k\u2082 \u03c1; storage (hardening) vs dynamic recovery (annihilation); Stage II\u2192III"
    },
    {
      "eq_number": 724,
      "title": "Bauschinger Effect (Kinematic Hardening)",
      "domain_id": 21,
      "significance": "Yield stress in reverse loading lower than forward loading; dislocation back-stress accumulation"
    },
    {
      "eq_number": 725,
      "title": "Schottky Diode I-V (Thermionic Emission Model)",
      "domain_id": 23,
      "significance": "J = A* T\u00b2 exp(\u2212q\u03c6_Bn/k_B T) [exp(qV/k_B T)\u22121]; A* = Richardson constant modified for effective mass"
    },
    {
      "eq_number": 726,
      "title": "Solar Cell Quantum Efficiency (External/Internal)",
      "domain_id": 44,
      "significance": "EQE(\u03bb) = (J_sc(\u03bb)/q) / \u03a6(\u03bb); IQE(\u03bb) = EQE(\u03bb) / (1\u2212R(\u03bb)\u2212T(\u03bb))"
    },
    {
      "eq_number": 727,
      "title": "Detailed Balance Limit (Tandem/Multijunction Solar Cells)",
      "domain_id": 44,
      "significance": "\u03b7_max \u2192 45.7% (2-junction), 51.3% (3-junction), 68.2% (infinite junctions) at 1 sun; ~86.8% at max concentration"
    },
    {
      "eq_number": 728,
      "title": "Tandem Solar Cell Current-Matching Condition",
      "domain_id": 44,
      "significance": "J_sc_top = J_sc_bottom (series-connected); otherwise limited by lower subcell current"
    },
    {
      "eq_number": 729,
      "title": "Thermoelectric Figure of Merit (ZT)",
      "domain_id": 44,
      "significance": "ZT = S\u00b2 \u03c3 T / \u03ba; S=Seebeck coefficient; \u03c3=electrical conductivity; \u03ba=thermal conductivity"
    },
    {
      "eq_number": 730,
      "title": "Seebeck Effect (Thermoelectric Voltage)",
      "domain_id": 44,
      "significance": "\u0394V = \u2212\u222b S(T) dT; V_oc = S \u0394T for small \u0394T; S\u221dk_B/e (\u224886 \u03bcV/K per k_B/e)"
    },
    {
      "eq_number": 731,
      "title": "Peltier Effect (Thermoelectric Heat Pumping)",
      "domain_id": 44,
      "significance": "Q\u0307 = \u03a0 I; \u03a0 = S T (Kelvin relation); heat absorbed/released at junction"
    },
    {
      "eq_number": 732,
      "title": "Electrochemical Overpotential Components (Fuel Cell/Battery)",
      "domain_id": 44,
      "significance": "V_cell = E_rev \u2212 \u03b7_act \u2212 \u03b7_ohm \u2212 \u03b7_conc; \u03b7_act from Butler-Volmer; \u03b7_ohm=IR; \u03b7_conc=(RT/nF)ln(1\u2212j/j_L)"
    },
    {
      "eq_number": 733,
      "title": "Peukert's Law (Battery Capacity vs Discharge Rate)",
      "domain_id": 44,
      "significance": "C = I^k t (k>1 for lead-acid, k\u22481.1-1.3); capacity decreases at higher discharge rates"
    },
    {
      "eq_number": 734,
      "title": "Ragone Plot (Energy vs Power Density)",
      "domain_id": 44,
      "significance": "Specific energy (Wh/kg) vs specific power (W/kg); batteries, fuel cells, capacitors, flywheels occupy different regions"
    },
    {
      "eq_number": 735,
      "title": "Magnetorheological Fluid (Bingham Plastic with Field-Dependent Yield)",
      "domain_id": 36,
      "significance": "\u03c4 = \u03c4_y(B) + \u03b7 \u03b3\u0307; yield stress controllable via applied magnetic field; ~50-100 kPa max"
    },
    {
      "eq_number": 736,
      "title": "Electrorheological Fluid (Field-Dependent Viscosity)",
      "domain_id": 36,
      "significance": "\u03b7(E) = \u03b7\u2080 + \u03b1 E^n; viscosity increases with electric field; Winslow effect"
    },
    {
      "eq_number": 737,
      "title": "Phononic Crystal Band Gap (Bragg Scattering of Sound)",
      "domain_id": 47,
      "significance": "\u03a9(k+G)=\u03a9(k); periodic elastic constants \u2192 band gaps for acoustic/elastic waves"
    },
    {
      "eq_number": 738,
      "title": "NSM Semantic Primes Explication",
      "domain_id": 51,
      "significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
    },
    {
      "eq_number": 738,
      "title": "NSM Semantic Primes Explication",
      "domain_id": 51,
      "significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
    },
    {
      "eq_number": 738,
      "title": "NSM Semantic Primes Explication",
      "domain_id": 51,
      "significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
    },
    {
      "eq_number": 738,
      "title": "NSM Semantic Primes Explication",
      "domain_id": 51,
      "significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
    },
    {
      "eq_number": 738,
      "title": "NSM Semantic Primes Explication",
      "domain_id": 51,
      "significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
    },
    {
      "eq_number": 738,
      "title": "NSM Semantic Primes Explication",
      "domain_id": 51,
      "significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
    },
    {
      "eq_number": 738,
      "title": "NSM Semantic Primes Explication",
      "domain_id": 51,
      "significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
    },
    {
      "eq_number": 738,
      "title": "NSM Semantic Primes Explication",
      "domain_id": 51,
      "significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
    },
    {
      "eq_number": 738,
      "title": "NSM Semantic Primes Explication",
      "domain_id": 51,
      "significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
    },
    {
      "eq_number": 738,
      "title": "NSM Semantic Primes Explication",
      "domain_id": 51,
      "significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
    },
    {
      "eq_number": 739,
      "title": "Cognitive Load Matrix (Invariant-Enhanced)",
      "domain_id": 51,
      "significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
    },
    {
      "eq_number": 739,
      "title": "Cognitive Load Matrix (Invariant-Enhanced)",
      "domain_id": 51,
      "significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
    },
    {
      "eq_number": 739,
      "title": "Cognitive Load Matrix (Invariant-Enhanced)",
      "domain_id": 51,
      "significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
    },
    {
      "eq_number": 739,
      "title": "Cognitive Load Matrix (Invariant-Enhanced)",
      "domain_id": 51,
      "significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
    },
    {
      "eq_number": 739,
      "title": "Cognitive Load Matrix (Invariant-Enhanced)",
      "domain_id": 51,
      "significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
    },
    {
      "eq_number": 739,
      "title": "Cognitive Load Matrix (Invariant-Enhanced)",
      "domain_id": 51,
      "significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
    },
    {
      "eq_number": 739,
      "title": "Cognitive Load Matrix (Invariant-Enhanced)",
      "domain_id": 51,
      "significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
    },
    {
      "eq_number": 739,
      "title": "Cognitive Load Matrix (Invariant-Enhanced)",
      "domain_id": 51,
      "significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
    },
    {
      "eq_number": 739,
      "title": "Cognitive Load Matrix (Invariant-Enhanced)",
      "domain_id": 51,
      "significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
    },
    {
      "eq_number": 739,
      "title": "Cognitive Load Matrix (Invariant-Enhanced)",
      "domain_id": 51,
      "significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
    },
    {
      "eq_number": 740,
      "title": "Evolutionary Operator (Universal)",
      "domain_id": 51,
      "significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
    },
    {
      "eq_number": 740,
      "title": "Evolutionary Operator (Universal)",
      "domain_id": 51,
      "significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
    },
    {
      "eq_number": 740,
      "title": "Evolutionary Operator (Universal)",
      "domain_id": 51,
      "significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
    },
    {
      "eq_number": 740,
      "title": "Evolutionary Operator (Universal)",
      "domain_id": 51,
      "significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
    },
    {
      "eq_number": 740,
      "title": "Evolutionary Operator (Universal)",
      "domain_id": 51,
      "significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
    },
    {
      "eq_number": 740,
      "title": "Evolutionary Operator (Universal)",
      "domain_id": 51,
      "significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
    },
    {
      "eq_number": 740,
      "title": "Evolutionary Operator (Universal)",
      "domain_id": 51,
      "significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
    },
    {
      "eq_number": 740,
      "title": "Evolutionary Operator (Universal)",
      "domain_id": 51,
      "significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
    },
    {
      "eq_number": 740,
      "title": "Evolutionary Operator (Universal)",
      "domain_id": 51,
      "significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
    },
    {
      "eq_number": 740,
      "title": "Evolutionary Operator (Universal)",
      "domain_id": 51,
      "significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
    },
    {
      "eq_number": 741,
      "title": "Hutter Prize Compression Equation",
      "domain_id": 51,
      "significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
    },
    {
      "eq_number": 741,
      "title": "Hutter Prize Compression Equation",
      "domain_id": 51,
      "significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
    },
    {
      "eq_number": 741,
      "title": "Hutter Prize Compression Equation",
      "domain_id": 51,
      "significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
    },
    {
      "eq_number": 741,
      "title": "Hutter Prize Compression Equation",
      "domain_id": 51,
      "significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
    },
    {
      "eq_number": 741,
      "title": "Hutter Prize Compression Equation",
      "domain_id": 51,
      "significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
    },
    {
      "eq_number": 741,
      "title": "Hutter Prize Compression Equation",
      "domain_id": 51,
      "significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
    },
    {
      "eq_number": 741,
      "title": "Hutter Prize Compression Equation",
      "domain_id": 51,
      "significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
    },
    {
      "eq_number": 741,
      "title": "Hutter Prize Compression Equation",
      "domain_id": 51,
      "significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
    },
    {
      "eq_number": 741,
      "title": "Hutter Prize Compression Equation",
      "domain_id": 51,
      "significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
    },
    {
      "eq_number": 741,
      "title": "Hutter Prize Compression Equation",
      "domain_id": 51,
      "significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
    },
    {
      "eq_number": 742,
      "title": "Semantic Compression Operator",
      "domain_id": 51,
      "significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
    },
    {
      "eq_number": 742,
      "title": "Semantic Compression Operator",
      "domain_id": 51,
      "significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
    },
    {
      "eq_number": 742,
      "title": "Semantic Compression Operator",
      "domain_id": 51,
      "significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
    },
    {
      "eq_number": 742,
      "title": "Semantic Compression Operator",
      "domain_id": 51,
      "significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
    },
    {
      "eq_number": 742,
      "title": "Semantic Compression Operator",
      "domain_id": 51,
      "significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
    },
    {
      "eq_number": 742,
      "title": "Semantic Compression Operator",
      "domain_id": 51,
      "significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
    },
    {
      "eq_number": 742,
      "title": "Semantic Compression Operator",
      "domain_id": 51,
      "significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
    },
    {
      "eq_number": 742,
      "title": "Semantic Compression Operator",
      "domain_id": 51,
      "significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
    },
    {
      "eq_number": 742,
      "title": "Semantic Compression Operator",
      "domain_id": 51,
      "significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
    },
    {
      "eq_number": 742,
      "title": "Semantic Compression Operator",
      "domain_id": 51,
      "significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
    },
    {
      "eq_number": 743,
      "title": "Prime-to-Byte Mapping",
      "domain_id": 51,
      "significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
    },
    {
      "eq_number": 743,
      "title": "Prime-to-Byte Mapping",
      "domain_id": 51,
      "significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
    },
    {
      "eq_number": 743,
      "title": "Prime-to-Byte Mapping",
      "domain_id": 51,
      "significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
    },
    {
      "eq_number": 743,
      "title": "Prime-to-Byte Mapping",
      "domain_id": 51,
      "significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
    },
    {
      "eq_number": 743,
      "title": "Prime-to-Byte Mapping",
      "domain_id": 51,
      "significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
    },
    {
      "eq_number": 743,
      "title": "Prime-to-Byte Mapping",
      "domain_id": 51,
      "significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
    },
    {
      "eq_number": 743,
      "title": "Prime-to-Byte Mapping",
      "domain_id": 51,
      "significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
    },
    {
      "eq_number": 743,
      "title": "Prime-to-Byte Mapping",
      "domain_id": 51,
      "significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
    },
    {
      "eq_number": 743,
      "title": "Prime-to-Byte Mapping",
      "domain_id": 51,
      "significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
    },
    {
      "eq_number": 743,
      "title": "Prime-to-Byte Mapping",
      "domain_id": 51,
      "significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
    },
    {
      "eq_number": 744,
      "title": "Context as Cognitive Load Function",
      "domain_id": 51,
      "significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
    },
    {
      "eq_number": 744,
      "title": "Context as Cognitive Load Function",
      "domain_id": 51,
      "significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
    },
    {
      "eq_number": 744,
      "title": "Context as Cognitive Load Function",
      "domain_id": 51,
      "significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
    },
    {
      "eq_number": 744,
      "title": "Context as Cognitive Load Function",
      "domain_id": 51,
      "significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
    },
    {
      "eq_number": 744,
      "title": "Context as Cognitive Load Function",
      "domain_id": 51,
      "significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
    },
    {
      "eq_number": 744,
      "title": "Context as Cognitive Load Function",
      "domain_id": 51,
      "significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
    },
    {
      "eq_number": 744,
      "title": "Context as Cognitive Load Function",
      "domain_id": 51,
      "significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
    },
    {
      "eq_number": 744,
      "title": "Context as Cognitive Load Function",
      "domain_id": 51,
      "significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
    },
    {
      "eq_number": 744,
      "title": "Context as Cognitive Load Function",
      "domain_id": 51,
      "significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
    },
    {
      "eq_number": 744,
      "title": "Context as Cognitive Load Function",
      "domain_id": 51,
      "significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
    },
    {
      "eq_number": 745,
      "title": "Gap Adaptation Equation",
      "domain_id": 51,
      "significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
    },
    {
      "eq_number": 745,
      "title": "Gap Adaptation Equation",
      "domain_id": 51,
      "significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
    },
    {
      "eq_number": 745,
      "title": "Gap Adaptation Equation",
      "domain_id": 51,
      "significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
    },
    {
      "eq_number": 745,
      "title": "Gap Adaptation Equation",
      "domain_id": 51,
      "significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
    },
    {
      "eq_number": 745,
      "title": "Gap Adaptation Equation",
      "domain_id": 51,
      "significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
    },
    {
      "eq_number": 745,
      "title": "Gap Adaptation Equation",
      "domain_id": 51,
      "significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
    },
    {
      "eq_number": 745,
      "title": "Gap Adaptation Equation",
      "domain_id": 51,
      "significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
    },
    {
      "eq_number": 745,
      "title": "Gap Adaptation Equation",
      "domain_id": 51,
      "significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
    },
    {
      "eq_number": 745,
      "title": "Gap Adaptation Equation",
      "domain_id": 51,
      "significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
    },
    {
      "eq_number": 745,
      "title": "Gap Adaptation Equation",
      "domain_id": 51,
      "significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
    },
    {
      "eq_number": 746,
      "title": "Unified Semantic Compression Equation",
      "domain_id": 51,
      "significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
    },
    {
      "eq_number": 746,
      "title": "Unified Semantic Compression Equation",
      "domain_id": 51,
      "significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
    },
    {
      "eq_number": 746,
      "title": "Unified Semantic Compression Equation",
      "domain_id": 51,
      "significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
    },
    {
      "eq_number": 746,
      "title": "Unified Semantic Compression Equation",
      "domain_id": 51,
      "significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
    },
    {
      "eq_number": 746,
      "title": "Unified Semantic Compression Equation",
      "domain_id": 51,
      "significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
    },
    {
      "eq_number": 746,
      "title": "Unified Semantic Compression Equation",
      "domain_id": 51,
      "significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
    },
    {
      "eq_number": 746,
      "title": "Unified Semantic Compression Equation",
      "domain_id": 51,
      "significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
    },
    {
      "eq_number": 746,
      "title": "Unified Semantic Compression Equation",
      "domain_id": 51,
      "significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
    },
    {
      "eq_number": 746,
      "title": "Unified Semantic Compression Equation",
      "domain_id": 51,
      "significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
    },
    {
      "eq_number": 746,
      "title": "Unified Semantic Compression Equation",
      "domain_id": 51,
      "significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
    },
    {
      "eq_number": 747,
      "title": "Gap-Dependent Prime Activation",
      "domain_id": 51,
      "significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
    },
    {
      "eq_number": 747,
      "title": "Gap-Dependent Prime Activation",
      "domain_id": 51,
      "significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
    },
    {
      "eq_number": 747,
      "title": "Gap-Dependent Prime Activation",
      "domain_id": 51,
      "significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
    },
    {
      "eq_number": 747,
      "title": "Gap-Dependent Prime Activation",
      "domain_id": 51,
      "significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
    },
    {
      "eq_number": 747,
      "title": "Gap-Dependent Prime Activation",
      "domain_id": 51,
      "significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
    },
    {
      "eq_number": 747,
      "title": "Gap-Dependent Prime Activation",
      "domain_id": 51,
      "significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
    },
    {
      "eq_number": 747,
      "title": "Gap-Dependent Prime Activation",
      "domain_id": 51,
      "significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
    },
    {
      "eq_number": 747,
      "title": "Gap-Dependent Prime Activation",
      "domain_id": 51,
      "significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
    },
    {
      "eq_number": 747,
      "title": "Gap-Dependent Prime Activation",
      "domain_id": 51,
      "significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
    },
    {
      "eq_number": 747,
      "title": "Gap-Dependent Prime Activation",
      "domain_id": 51,
      "significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
    },
    {
      "eq_number": 748,
      "title": "Prime Compression Matrix",
      "domain_id": 51,
      "significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
    },
    {
      "eq_number": 748,
      "title": "Prime Compression Matrix",
      "domain_id": 51,
      "significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
    },
    {
      "eq_number": 748,
      "title": "Prime Compression Matrix",
      "domain_id": 51,
      "significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
    },
    {
      "eq_number": 748,
      "title": "Prime Compression Matrix",
      "domain_id": 51,
      "significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
    },
    {
      "eq_number": 748,
      "title": "Prime Compression Matrix",
      "domain_id": 51,
      "significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
    },
    {
      "eq_number": 748,
      "title": "Prime Compression Matrix",
      "domain_id": 51,
      "significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
    },
    {
      "eq_number": 748,
      "title": "Prime Compression Matrix",
      "domain_id": 51,
      "significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
    },
    {
      "eq_number": 748,
      "title": "Prime Compression Matrix",
      "domain_id": 51,
      "significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
    },
    {
      "eq_number": 748,
      "title": "Prime Compression Matrix",
      "domain_id": 51,
      "significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
    },
    {
      "eq_number": 748,
      "title": "Prime Compression Matrix",
      "domain_id": 51,
      "significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
    },
    {
      "eq_number": 749,
      "title": "Matrix-Vector Compression",
      "domain_id": 51,
      "significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
    },
    {
      "eq_number": 749,
      "title": "Matrix-Vector Compression",
      "domain_id": 51,
      "significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
    },
    {
      "eq_number": 749,
      "title": "Matrix-Vector Compression",
      "domain_id": 51,
      "significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
    },
    {
      "eq_number": 749,
      "title": "Matrix-Vector Compression",
      "domain_id": 51,
      "significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
    },
    {
      "eq_number": 749,
      "title": "Matrix-Vector Compression",
      "domain_id": 51,
      "significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
    },
    {
      "eq_number": 749,
      "title": "Matrix-Vector Compression",
      "domain_id": 51,
      "significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
    },
    {
      "eq_number": 749,
      "title": "Matrix-Vector Compression",
      "domain_id": 51,
      "significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
    },
    {
      "eq_number": 749,
      "title": "Matrix-Vector Compression",
      "domain_id": 51,
      "significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
    },
    {
      "eq_number": 749,
      "title": "Matrix-Vector Compression",
      "domain_id": 51,
      "significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
    },
    {
      "eq_number": 749,
      "title": "Matrix-Vector Compression",
      "domain_id": 51,
      "significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
    },
    {
      "eq_number": 750,
      "title": "Invariant Load with Prime Activation",
      "domain_id": 51,
      "significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
    },
    {
      "eq_number": 750,
      "title": "Invariant Load with Prime Activation",
      "domain_id": 51,
      "significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
    },
    {
      "eq_number": 750,
      "title": "Invariant Load with Prime Activation",
      "domain_id": 51,
      "significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
    },
    {
      "eq_number": 750,
      "title": "Invariant Load with Prime Activation",
      "domain_id": 51,
      "significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
    },
    {
      "eq_number": 750,
      "title": "Invariant Load with Prime Activation",
      "domain_id": 51,
      "significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
    },
    {
      "eq_number": 750,
      "title": "Invariant Load with Prime Activation",
      "domain_id": 51,
      "significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
    },
    {
      "eq_number": 750,
      "title": "Invariant Load with Prime Activation",
      "domain_id": 51,
      "significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
    },
    {
      "eq_number": 750,
      "title": "Invariant Load with Prime Activation",
      "domain_id": 51,
      "significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
    },
    {
      "eq_number": 750,
      "title": "Invariant Load with Prime Activation",
      "domain_id": 51,
      "significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
    },
    {
      "eq_number": 750,
      "title": "Invariant Load with Prime Activation",
      "domain_id": 51,
      "significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
    },
    {
      "eq_number": 751,
      "title": "Gap Threshold Function",
      "domain_id": 51,
      "significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
    },
    {
      "eq_number": 751,
      "title": "Gap Threshold Function",
      "domain_id": 51,
      "significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
    },
    {
      "eq_number": 751,
      "title": "Gap Threshold Function",
      "domain_id": 51,
      "significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
    },
    {
      "eq_number": 751,
      "title": "Gap Threshold Function",
      "domain_id": 51,
      "significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
    },
    {
      "eq_number": 751,
      "title": "Gap Threshold Function",
      "domain_id": 51,
      "significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
    },
    {
      "eq_number": 751,
      "title": "Gap Threshold Function",
      "domain_id": 51,
      "significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
    },
    {
      "eq_number": 751,
      "title": "Gap Threshold Function",
      "domain_id": 51,
      "significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
    },
    {
      "eq_number": 751,
      "title": "Gap Threshold Function",
      "domain_id": 51,
      "significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
    },
    {
      "eq_number": 751,
      "title": "Gap Threshold Function",
      "domain_id": 51,
      "significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
    },
    {
      "eq_number": 751,
      "title": "Gap Threshold Function",
      "domain_id": 51,
      "significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
    },
    {
      "eq_number": 752,
      "title": "Hutter Prize Penalty with Invariants",
      "domain_id": 51,
      "significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
    },
    {
      "eq_number": 752,
      "title": "Hutter Prize Penalty with Invariants",
      "domain_id": 51,
      "significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
    },
    {
      "eq_number": 752,
      "title": "Hutter Prize Penalty with Invariants",
      "domain_id": 51,
      "significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
    },
    {
      "eq_number": 752,
      "title": "Hutter Prize Penalty with Invariants",
      "domain_id": 51,
      "significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
    },
    {
      "eq_number": 752,
      "title": "Hutter Prize Penalty with Invariants",
      "domain_id": 51,
      "significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
    },
    {
      "eq_number": 752,
      "title": "Hutter Prize Penalty with Invariants",
      "domain_id": 51,
      "significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
    },
    {
      "eq_number": 752,
      "title": "Hutter Prize Penalty with Invariants",
      "domain_id": 51,
      "significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
    },
    {
      "eq_number": 752,
      "title": "Hutter Prize Penalty with Invariants",
      "domain_id": 51,
      "significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
    },
    {
      "eq_number": 752,
      "title": "Hutter Prize Penalty with Invariants",
      "domain_id": 51,
      "significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
    },
    {
      "eq_number": 752,
      "title": "Hutter Prize Penalty with Invariants",
      "domain_id": 51,
      "significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
    },
    {
      "eq_number": 753,
      "title": "Gap Adaptation Dynamics",
      "domain_id": 51,
      "significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
    },
    {
      "eq_number": 753,
      "title": "Gap Adaptation Dynamics",
      "domain_id": 51,
      "significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
    },
    {
      "eq_number": 753,
      "title": "Gap Adaptation Dynamics",
      "domain_id": 51,
      "significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
    },
    {
      "eq_number": 753,
      "title": "Gap Adaptation Dynamics",
      "domain_id": 51,
      "significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
    },
    {
      "eq_number": 753,
      "title": "Gap Adaptation Dynamics",
      "domain_id": 51,
      "significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
    },
    {
      "eq_number": 753,
      "title": "Gap Adaptation Dynamics",
      "domain_id": 51,
      "significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
    },
    {
      "eq_number": 753,
      "title": "Gap Adaptation Dynamics",
      "domain_id": 51,
      "significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
    },
    {
      "eq_number": 753,
      "title": "Gap Adaptation Dynamics",
      "domain_id": 51,
      "significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
    },
    {
      "eq_number": 753,
      "title": "Gap Adaptation Dynamics",
      "domain_id": 51,
      "significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
    },
    {
      "eq_number": 753,
      "title": "Gap Adaptation Dynamics",
      "domain_id": 51,
      "significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
    },
    {
      "eq_number": 754,
      "title": "Prime Conservation Theorem (Spectral Entropy Bound)",
      "domain_id": 51,
      "significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
    },
    {
      "eq_number": 754,
      "title": "Prime Conservation Theorem (Spectral Entropy Bound)",
      "domain_id": 51,
      "significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
    },
    {
      "eq_number": 754,
      "title": "Prime Conservation Theorem (Spectral Entropy Bound)",
      "domain_id": 51,
      "significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
    },
    {
      "eq_number": 754,
      "title": "Prime Conservation Theorem (Spectral Entropy Bound)",
      "domain_id": 51,
      "significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
    },
    {
      "eq_number": 754,
      "title": "Prime Conservation Theorem (Spectral Entropy Bound)",
      "domain_id": 51,
      "significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
    },
    {
      "eq_number": 754,
      "title": "Prime Conservation Theorem (Spectral Entropy Bound)",
      "domain_id": 51,
      "significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
    },
    {
      "eq_number": 754,
      "title": "Prime Conservation Theorem (Spectral Entropy Bound)",
      "domain_id": 51,
      "significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
    },
    {
      "eq_number": 754,
      "title": "Prime Conservation Theorem (Spectral Entropy Bound)",
      "domain_id": 51,
      "significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
    },
    {
      "eq_number": 754,
      "title": "Prime Conservation Theorem (Spectral Entropy Bound)",
      "domain_id": 51,
      "significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
    },
    {
      "eq_number": 754,
      "title": "Prime Conservation Theorem (Spectral Entropy Bound)",
      "domain_id": 51,
      "significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
    },
    {
      "eq_number": 755,
      "title": "Invariant Preservation Theorem",
      "domain_id": 51,
      "significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
    },
    {
      "eq_number": 755,
      "title": "Invariant Preservation Theorem",
      "domain_id": 51,
      "significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
    },
    {
      "eq_number": 755,
      "title": "Invariant Preservation Theorem",
      "domain_id": 51,
      "significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
    },
    {
      "eq_number": 755,
      "title": "Invariant Preservation Theorem",
      "domain_id": 51,
      "significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
    },
    {
      "eq_number": 755,
      "title": "Invariant Preservation Theorem",
      "domain_id": 51,
      "significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
    },
    {
      "eq_number": 755,
      "title": "Invariant Preservation Theorem",
      "domain_id": 51,
      "significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
    },
    {
      "eq_number": 755,
      "title": "Invariant Preservation Theorem",
      "domain_id": 51,
      "significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
    },
    {
      "eq_number": 755,
      "title": "Invariant Preservation Theorem",
      "domain_id": 51,
      "significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
    },
    {
      "eq_number": 755,
      "title": "Invariant Preservation Theorem",
      "domain_id": 51,
      "significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
    },
    {
      "eq_number": 755,
      "title": "Invariant Preservation Theorem",
      "domain_id": 51,
      "significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
    },
    {
      "eq_number": 756,
      "title": "Matrix Evolution Learning Rule",
      "domain_id": 51,
      "significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
    },
    {
      "eq_number": 756,
      "title": "Matrix Evolution Learning Rule",
      "domain_id": 51,
      "significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
    },
    {
      "eq_number": 756,
      "title": "Matrix Evolution Learning Rule",
      "domain_id": 51,
      "significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
    },
    {
      "eq_number": 756,
      "title": "Matrix Evolution Learning Rule",
      "domain_id": 51,
      "significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
    },
    {
      "eq_number": 756,
      "title": "Matrix Evolution Learning Rule",
      "domain_id": 51,
      "significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
    },
    {
      "eq_number": 756,
      "title": "Matrix Evolution Learning Rule",
      "domain_id": 51,
      "significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
    },
    {
      "eq_number": 756,
      "title": "Matrix Evolution Learning Rule",
      "domain_id": 51,
      "significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
    },
    {
      "eq_number": 756,
      "title": "Matrix Evolution Learning Rule",
      "domain_id": 51,
      "significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
    },
    {
      "eq_number": 756,
      "title": "Matrix Evolution Learning Rule",
      "domain_id": 51,
      "significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
    },
    {
      "eq_number": 756,
      "title": "Matrix Evolution Learning Rule",
      "domain_id": 51,
      "significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
    },
    {
      "eq_number": 757,
      "title": "Cross-Linguistic Compression Equation",
      "domain_id": 51,
      "significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
    },
    {
      "eq_number": 757,
      "title": "Cross-Linguistic Compression Equation",
      "domain_id": 51,
      "significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
    },
    {
      "eq_number": 757,
      "title": "Cross-Linguistic Compression Equation",
      "domain_id": 51,
      "significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
    },
    {
      "eq_number": 757,
      "title": "Cross-Linguistic Compression Equation",
      "domain_id": 51,
      "significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
    },
    {
      "eq_number": 757,
      "title": "Cross-Linguistic Compression Equation",
      "domain_id": 51,
      "significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
    },
    {
      "eq_number": 757,
      "title": "Cross-Linguistic Compression Equation",
      "domain_id": 51,
      "significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
    },
    {
      "eq_number": 757,
      "title": "Cross-Linguistic Compression Equation",
      "domain_id": 51,
      "significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
    },
    {
      "eq_number": 757,
      "title": "Cross-Linguistic Compression Equation",
      "domain_id": 51,
      "significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
    },
    {
      "eq_number": 757,
      "title": "Cross-Linguistic Compression Equation",
      "domain_id": 51,
      "significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
    },
    {
      "eq_number": 757,
      "title": "Cross-Linguistic Compression Equation",
      "domain_id": 51,
      "significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
    },
    {
      "eq_number": 758,
      "title": "Language-Specific Gap Function",
      "domain_id": 51,
      "significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
    },
    {
      "eq_number": 758,
      "title": "Language-Specific Gap Function",
      "domain_id": 51,
      "significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
    },
    {
      "eq_number": 758,
      "title": "Language-Specific Gap Function",
      "domain_id": 51,
      "significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
    },
    {
      "eq_number": 758,
      "title": "Language-Specific Gap Function",
      "domain_id": 51,
      "significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
    },
    {
      "eq_number": 758,
      "title": "Language-Specific Gap Function",
      "domain_id": 51,
      "significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
    },
    {
      "eq_number": 758,
      "title": "Language-Specific Gap Function",
      "domain_id": 51,
      "significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
    },
    {
      "eq_number": 758,
      "title": "Language-Specific Gap Function",
      "domain_id": 51,
      "significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
    },
    {
      "eq_number": 758,
      "title": "Language-Specific Gap Function",
      "domain_id": 51,
      "significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
    },
    {
      "eq_number": 758,
      "title": "Language-Specific Gap Function",
      "domain_id": 51,
      "significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
    },
    {
      "eq_number": 758,
      "title": "Language-Specific Gap Function",
      "domain_id": 51,
      "significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
    },
    {
      "eq_number": 759,
      "title": "0-AVMR: Square Shell Identity",
      "domain_id": 51,
      "significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
    },
    {
      "eq_number": 759,
      "title": "0-AVMR: Square Shell Identity",
      "domain_id": 51,
      "significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
    },
    {
      "eq_number": 759,
      "title": "0-AVMR: Square Shell Identity",
      "domain_id": 51,
      "significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
    },
    {
      "eq_number": 759,
      "title": "0-AVMR: Square Shell Identity",
      "domain_id": 51,
      "significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
    },
    {
      "eq_number": 759,
      "title": "0-AVMR: Square Shell Identity",
      "domain_id": 51,
      "significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
    },
    {
      "eq_number": 759,
      "title": "0-AVMR: Square Shell Identity",
      "domain_id": 51,
      "significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
    },
    {
      "eq_number": 759,
      "title": "0-AVMR: Square Shell Identity",
      "domain_id": 51,
      "significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
    },
    {
      "eq_number": 759,
      "title": "0-AVMR: Square Shell Identity",
      "domain_id": 51,
      "significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
    },
    {
      "eq_number": 759,
      "title": "0-AVMR: Square Shell Identity",
      "domain_id": 51,
      "significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
    },
    {
      "eq_number": 760,
      "title": "0-AVMR: Tip Coordinate Map",
      "domain_id": 51,
      "significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
    },
    {
      "eq_number": 760,
      "title": "0-AVMR: Tip Coordinate Map",
      "domain_id": 51,
      "significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
    },
    {
      "eq_number": 760,
      "title": "0-AVMR: Tip Coordinate Map",
      "domain_id": 51,
      "significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
    },
    {
      "eq_number": 760,
      "title": "0-AVMR: Tip Coordinate Map",
      "domain_id": 51,
      "significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
    },
    {
      "eq_number": 760,
      "title": "0-AVMR: Tip Coordinate Map",
      "domain_id": 51,
      "significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
    },
    {
      "eq_number": 760,
      "title": "0-AVMR: Tip Coordinate Map",
      "domain_id": 51,
      "significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
    },
    {
      "eq_number": 760,
      "title": "0-AVMR: Tip Coordinate Map",
      "domain_id": 51,
      "significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
    },
    {
      "eq_number": 760,
      "title": "0-AVMR: Tip Coordinate Map",
      "domain_id": 51,
      "significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
    },
    {
      "eq_number": 760,
      "title": "0-AVMR: Tip Coordinate Map",
      "domain_id": 51,
      "significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
    },
    {
      "eq_number": 761,
      "title": "0-AVMR: Interaction Score",
      "domain_id": 51,
      "significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
    },
    {
      "eq_number": 761,
      "title": "0-AVMR: Interaction Score",
      "domain_id": 51,
      "significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
    },
    {
      "eq_number": 761,
      "title": "0-AVMR: Interaction Score",
      "domain_id": 51,
      "significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
    },
    {
      "eq_number": 761,
      "title": "0-AVMR: Interaction Score",
      "domain_id": 51,
      "significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
    },
    {
      "eq_number": 761,
      "title": "0-AVMR: Interaction Score",
      "domain_id": 51,
      "significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
    },
    {
      "eq_number": 761,
      "title": "0-AVMR: Interaction Score",
      "domain_id": 51,
      "significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
    },
    {
      "eq_number": 761,
      "title": "0-AVMR: Interaction Score",
      "domain_id": 51,
      "significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
    },
    {
      "eq_number": 761,
      "title": "0-AVMR: Interaction Score",
      "domain_id": 51,
      "significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
    },
    {
      "eq_number": 761,
      "title": "0-AVMR: Interaction Score",
      "domain_id": 51,
      "significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
    },
    {
      "eq_number": 762,
      "title": "0-AVMR: Genetic Transduction",
      "domain_id": 51,
      "significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
    },
    {
      "eq_number": 762,
      "title": "0-AVMR: Genetic Transduction",
      "domain_id": 51,
      "significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
    },
    {
      "eq_number": 762,
      "title": "0-AVMR: Genetic Transduction",
      "domain_id": 51,
      "significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
    },
    {
      "eq_number": 762,
      "title": "0-AVMR: Genetic Transduction",
      "domain_id": 51,
      "significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
    },
    {
      "eq_number": 762,
      "title": "0-AVMR: Genetic Transduction",
      "domain_id": 51,
      "significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
    },
    {
      "eq_number": 762,
      "title": "0-AVMR: Genetic Transduction",
      "domain_id": 51,
      "significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
    },
    {
      "eq_number": 762,
      "title": "0-AVMR: Genetic Transduction",
      "domain_id": 51,
      "significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
    },
    {
      "eq_number": 762,
      "title": "0-AVMR: Genetic Transduction",
      "domain_id": 51,
      "significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
    },
    {
      "eq_number": 762,
      "title": "0-AVMR: Genetic Transduction",
      "domain_id": 51,
      "significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
    },
    {
      "eq_number": 763,
      "title": "0-AVMR: Genetic Entropy Bound",
      "domain_id": 51,
      "significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
    },
    {
      "eq_number": 763,
      "title": "0-AVMR: Genetic Entropy Bound",
      "domain_id": 51,
      "significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
    },
    {
      "eq_number": 763,
      "title": "0-AVMR: Genetic Entropy Bound",
      "domain_id": 51,
      "significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
    },
    {
      "eq_number": 763,
      "title": "0-AVMR: Genetic Entropy Bound",
      "domain_id": 51,
      "significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
    },
    {
      "eq_number": 763,
      "title": "0-AVMR: Genetic Entropy Bound",
      "domain_id": 51,
      "significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
    },
    {
      "eq_number": 763,
      "title": "0-AVMR: Genetic Entropy Bound",
      "domain_id": 51,
      "significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
    },
    {
      "eq_number": 763,
      "title": "0-AVMR: Genetic Entropy Bound",
      "domain_id": 51,
      "significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
    },
    {
      "eq_number": 763,
      "title": "0-AVMR: Genetic Entropy Bound",
      "domain_id": 51,
      "significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
    },
    {
      "eq_number": 763,
      "title": "0-AVMR: Genetic Entropy Bound",
      "domain_id": 51,
      "significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
    },
    {
      "eq_number": 764,
      "title": "0-AMMR: Shell Partition of Computation",
      "domain_id": 51,
      "significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
    },
    {
      "eq_number": 764,
      "title": "0-AMMR: Shell Partition of Computation",
      "domain_id": 51,
      "significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
    },
    {
      "eq_number": 764,
      "title": "0-AMMR: Shell Partition of Computation",
      "domain_id": 51,
      "significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
    },
    {
      "eq_number": 764,
      "title": "0-AMMR: Shell Partition of Computation",
      "domain_id": 51,
      "significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
    },
    {
      "eq_number": 764,
      "title": "0-AMMR: Shell Partition of Computation",
      "domain_id": 51,
      "significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
    },
    {
      "eq_number": 764,
      "title": "0-AMMR: Shell Partition of Computation",
      "domain_id": 51,
      "significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
    },
    {
      "eq_number": 764,
      "title": "0-AMMR: Shell Partition of Computation",
      "domain_id": 51,
      "significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
    },
    {
      "eq_number": 764,
      "title": "0-AMMR: Shell Partition of Computation",
      "domain_id": 51,
      "significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
    },
    {
      "eq_number": 764,
      "title": "0-AMMR: Shell Partition of Computation",
      "domain_id": 51,
      "significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
    },
    {
      "eq_number": 765,
      "title": "0-AMMR: Shell Coordinate System",
      "domain_id": 51,
      "significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
    },
    {
      "eq_number": 765,
      "title": "0-AMMR: Shell Coordinate System",
      "domain_id": 51,
      "significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
    },
    {
      "eq_number": 765,
      "title": "0-AMMR: Shell Coordinate System",
      "domain_id": 51,
      "significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
    },
    {
      "eq_number": 765,
      "title": "0-AMMR: Shell Coordinate System",
      "domain_id": 51,
      "significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
    },
    {
      "eq_number": 765,
      "title": "0-AMMR: Shell Coordinate System",
      "domain_id": 51,
      "significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
    },
    {
      "eq_number": 765,
      "title": "0-AMMR: Shell Coordinate System",
      "domain_id": 51,
      "significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
    },
    {
      "eq_number": 765,
      "title": "0-AMMR: Shell Coordinate System",
      "domain_id": 51,
      "significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
    },
    {
      "eq_number": 765,
      "title": "0-AMMR: Shell Coordinate System",
      "domain_id": 51,
      "significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
    },
    {
      "eq_number": 765,
      "title": "0-AMMR: Shell Coordinate System",
      "domain_id": 51,
      "significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
    },
    {
      "eq_number": 766,
      "title": "0-AMMR: Additive Shell Interaction",
      "domain_id": 51,
      "significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
    },
    {
      "eq_number": 766,
      "title": "0-AMMR: Additive Shell Interaction",
      "domain_id": 51,
      "significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
    },
    {
      "eq_number": 766,
      "title": "0-AMMR: Additive Shell Interaction",
      "domain_id": 51,
      "significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
    },
    {
      "eq_number": 766,
      "title": "0-AMMR: Additive Shell Interaction",
      "domain_id": 51,
      "significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
    },
    {
      "eq_number": 766,
      "title": "0-AMMR: Additive Shell Interaction",
      "domain_id": 51,
      "significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
    },
    {
      "eq_number": 766,
      "title": "0-AMMR: Additive Shell Interaction",
      "domain_id": 51,
      "significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
    },
    {
      "eq_number": 766,
      "title": "0-AMMR: Additive Shell Interaction",
      "domain_id": 51,
      "significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
    },
    {
      "eq_number": 766,
      "title": "0-AMMR: Additive Shell Interaction",
      "domain_id": 51,
      "significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
    },
    {
      "eq_number": 766,
      "title": "0-AMMR: Additive Shell Interaction",
      "domain_id": 51,
      "significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
    },
    {
      "eq_number": 767,
      "title": "0-AMMR: Temporal-Genetic Transduction",
      "domain_id": 51,
      "significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
    },
    {
      "eq_number": 767,
      "title": "0-AMMR: Temporal-Genetic Transduction",
      "domain_id": 51,
      "significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
    },
    {
      "eq_number": 767,
      "title": "0-AMMR: Temporal-Genetic Transduction",
      "domain_id": 51,
      "significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
    },
    {
      "eq_number": 767,
      "title": "0-AMMR: Temporal-Genetic Transduction",
      "domain_id": 51,
      "significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
    },
    {
      "eq_number": 767,
      "title": "0-AMMR: Temporal-Genetic Transduction",
      "domain_id": 51,
      "significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
    },
    {
      "eq_number": 767,
      "title": "0-AMMR: Temporal-Genetic Transduction",
      "domain_id": 51,
      "significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
    },
    {
      "eq_number": 767,
      "title": "0-AMMR: Temporal-Genetic Transduction",
      "domain_id": 51,
      "significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
    },
    {
      "eq_number": 767,
      "title": "0-AMMR: Temporal-Genetic Transduction",
      "domain_id": 51,
      "significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
    },
    {
      "eq_number": 767,
      "title": "0-AMMR: Temporal-Genetic Transduction",
      "domain_id": 51,
      "significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
    },
    {
      "eq_number": 768,
      "title": "0-AMMR: Information Capacity Bound",
      "domain_id": 51,
      "significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
    },
    {
      "eq_number": 768,
      "title": "0-AMMR: Information Capacity Bound",
      "domain_id": 51,
      "significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
    },
    {
      "eq_number": 768,
      "title": "0-AMMR: Information Capacity Bound",
      "domain_id": 51,
      "significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
    },
    {
      "eq_number": 768,
      "title": "0-AMMR: Information Capacity Bound",
      "domain_id": 51,
      "significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
    },
    {
      "eq_number": 768,
      "title": "0-AMMR: Information Capacity Bound",
      "domain_id": 51,
      "significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
    },
    {
      "eq_number": 768,
      "title": "0-AMMR: Information Capacity Bound",
      "domain_id": 51,
      "significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
    },
    {
      "eq_number": 768,
      "title": "0-AMMR: Information Capacity Bound",
      "domain_id": 51,
      "significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
    },
    {
      "eq_number": 768,
      "title": "0-AMMR: Information Capacity Bound",
      "domain_id": 51,
      "significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
    },
    {
      "eq_number": 768,
      "title": "0-AMMR: Information Capacity Bound",
      "domain_id": 51,
      "significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
    },
    {
      "eq_number": 769,
      "title": "0-AMMR: RG Flow Shell Preservation",
      "domain_id": 51,
      "significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
    },
    {
      "eq_number": 769,
      "title": "0-AMMR: RG Flow Shell Preservation",
      "domain_id": 51,
      "significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
    },
    {
      "eq_number": 769,
      "title": "0-AMMR: RG Flow Shell Preservation",
      "domain_id": 51,
      "significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
    },
    {
      "eq_number": 769,
      "title": "0-AMMR: RG Flow Shell Preservation",
      "domain_id": 51,
      "significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
    },
    {
      "eq_number": 769,
      "title": "0-AMMR: RG Flow Shell Preservation",
      "domain_id": 51,
      "significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
    },
    {
      "eq_number": 769,
      "title": "0-AMMR: RG Flow Shell Preservation",
      "domain_id": 51,
      "significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
    },
    {
      "eq_number": 769,
      "title": "0-AMMR: RG Flow Shell Preservation",
      "domain_id": 51,
      "significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
    },
    {
      "eq_number": 769,
      "title": "0-AMMR: RG Flow Shell Preservation",
      "domain_id": 51,
      "significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
    },
    {
      "eq_number": 769,
      "title": "0-AMMR: RG Flow Shell Preservation",
      "domain_id": 51,
      "significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
    },
    {
      "eq_number": 770,
      "title": "Graph Laplacian Spectral Decomposition",
      "domain_id": 51,
      "significance": "Spectral decomposition of graph Laplacian for graph-native computation. Enables eigenvector-based graph operations."
    },
    {
      "eq_number": 770,
      "title": "Graph Laplacian Spectral Decomposition",
      "domain_id": 51,
      "significance": "Spectral decomposition of graph Laplacian for graph-native computation. Enables eigenvector-based graph operations."
    },
    {
      "eq_number": 770,
      "title": "Graph Laplacian Spectral Decomposition",
      "domain_id": 51,
      "significance": "Spectral decomposition of graph Laplacian for graph-native computation. Enables eigenvector-based graph operations."
    },
    {
      "eq_number": 770,
      "title": "Graph Laplacian Spectral Decomposition",
      "domain_id": 51,
      "significance": "Spectral decomposition of graph Laplacian for graph-native computation. Enables eigenvector-based graph operations."
    },
    {
      "eq_number": 770,
      "title": "Graph Laplacian Spectral Decomposition",
      "domain_id": 51,
      "significance": "Spectral decomposition of graph Laplacian for graph-native computation. Enables eigenvector-based graph operations."
    },
    {
      "eq_number": 770,
      "title": "Graph Laplacian Spectral Decomposition",
      "domain_id": 51,
      "significance": "Spectral decomposition of graph Laplacian for graph-native computation. Enables eigenvector-based graph operations."
    },
    {
      "eq_number": 770,
      "title": "Graph Laplacian Spectral Decomposition",
      "domain_id": 51,
      "significance": "Spectral decomposition of graph Laplacian for graph-native computation. Enables eigenvector-based graph operations."
    },
    {
      "eq_number": 770,
      "title": "Graph Laplacian Spectral Decomposition",
      "domain_id": 51,
      "significance": "Spectral decomposition of graph Laplacian for graph-native computation. Enables eigenvector-based graph operations."
    },
    {
      "eq_number": 771,
      "title": "Graph Attention Mechanism",
      "domain_id": 51,
      "significance": "Attention-based message passing for graph neural networks. Enables context-aware graph operations."
    },
    {
      "eq_number": 771,
      "title": "Graph Attention Mechanism",
      "domain_id": 51,
      "significance": "Attention-based message passing for graph neural networks. Enables context-aware graph operations."
    },
    {
      "eq_number": 771,
      "title": "Graph Attention Mechanism",
      "domain_id": 51,
      "significance": "Attention-based message passing for graph neural networks. Enables context-aware graph operations."
    },
    {
      "eq_number": 771,
      "title": "Graph Attention Mechanism",
      "domain_id": 51,
      "significance": "Attention-based message passing for graph neural networks. Enables context-aware graph operations."
    },
    {
      "eq_number": 771,
      "title": "Graph Attention Mechanism",
      "domain_id": 51,
      "significance": "Attention-based message passing for graph neural networks. Enables context-aware graph operations."
    },
    {
      "eq_number": 771,
      "title": "Graph Attention Mechanism",
      "domain_id": 51,
      "significance": "Attention-based message passing for graph neural networks. Enables context-aware graph operations."
    },
    {
      "eq_number": 771,
      "title": "Graph Attention Mechanism",
      "domain_id": 51,
      "significance": "Attention-based message passing for graph neural networks. Enables context-aware graph operations."
    },
    {
      "eq_number": 771,
      "title": "Graph Attention Mechanism",
      "domain_id": 51,
      "significance": "Attention-based message passing for graph neural networks. Enables context-aware graph operations."
    },
    {
      "eq_number": 772,
      "title": "Graph Convolution",
      "domain_id": 51,
      "significance": "Spectral graph convolution for graph-native processing. Enables convolution operations on graph-structured data."
    },
    {
      "eq_number": 772,
      "title": "Graph Convolution",
      "domain_id": 51,
      "significance": "Spectral graph convolution for graph-native processing. Enables convolution operations on graph-structured data."
    },
    {
      "eq_number": 772,
      "title": "Graph Convolution",
      "domain_id": 51,
      "significance": "Spectral graph convolution for graph-native processing. Enables convolution operations on graph-structured data."
    },
    {
      "eq_number": 772,
      "title": "Graph Convolution",
      "domain_id": 51,
      "significance": "Spectral graph convolution for graph-native processing. Enables convolution operations on graph-structured data."
    },
    {
      "eq_number": 772,
      "title": "Graph Convolution",
      "domain_id": 51,
      "significance": "Spectral graph convolution for graph-native processing. Enables convolution operations on graph-structured data."
    },
    {
      "eq_number": 772,
      "title": "Graph Convolution",
      "domain_id": 51,
      "significance": "Spectral graph convolution for graph-native processing. Enables convolution operations on graph-structured data."
    },
    {
      "eq_number": 772,
      "title": "Graph Convolution",
      "domain_id": 51,
      "significance": "Spectral graph convolution for graph-native processing. Enables convolution operations on graph-structured data."
    },
    {
      "eq_number": 772,
      "title": "Graph Convolution",
      "domain_id": 51,
      "significance": "Spectral graph convolution for graph-native processing. Enables convolution operations on graph-structured data."
    },
    {
      "eq_number": 773,
      "title": "WGSL Vector Swizzle Operation",
      "domain_id": 51,
      "significance": "GPU vector component swizzling for efficient parallel processing. Enables flexible vector manipulation on GPU."
    },
    {
      "eq_number": 773,
      "title": "WGSL Vector Swizzle Operation",
      "domain_id": 51,
      "significance": "GPU vector component swizzling for efficient parallel processing. Enables flexible vector manipulation on GPU."
    },
    {
      "eq_number": 773,
      "title": "WGSL Vector Swizzle Operation",
      "domain_id": 51,
      "significance": "GPU vector component swizzling for efficient parallel processing. Enables flexible vector manipulation on GPU."
    },
    {
      "eq_number": 773,
      "title": "WGSL Vector Swizzle Operation",
      "domain_id": 51,
      "significance": "GPU vector component swizzling for efficient parallel processing. Enables flexible vector manipulation on GPU."
    },
    {
      "eq_number": 773,
      "title": "WGSL Vector Swizzle Operation",
      "domain_id": 51,
      "significance": "GPU vector component swizzling for efficient parallel processing. Enables flexible vector manipulation on GPU."
    },
    {
      "eq_number": 773,
      "title": "WGSL Vector Swizzle Operation",
      "domain_id": 51,
      "significance": "GPU vector component swizzling for efficient parallel processing. Enables flexible vector manipulation on GPU."
    },
    {
      "eq_number": 773,
      "title": "WGSL Vector Swizzle Operation",
      "domain_id": 51,
      "significance": "GPU vector component swizzling for efficient parallel processing. Enables flexible vector manipulation on GPU."
    },
    {
      "eq_number": 773,
      "title": "WGSL Vector Swizzle Operation",
      "domain_id": 51,
      "significance": "GPU vector component swizzling for efficient parallel processing. Enables flexible vector manipulation on GPU."
    },
    {
      "eq_number": 774,
      "title": "WGSL Workgroup Synchronization",
      "domain_id": 51,
      "significance": "Barrier synchronization for GPU workgroups. Ensures correct parallel execution order."
    },
    {
      "eq_number": 774,
      "title": "WGSL Workgroup Synchronization",
      "domain_id": 51,
      "significance": "Barrier synchronization for GPU workgroups. Ensures correct parallel execution order."
    },
    {
      "eq_number": 774,
      "title": "WGSL Workgroup Synchronization",
      "domain_id": 51,
      "significance": "Barrier synchronization for GPU workgroups. Ensures correct parallel execution order."
    },
    {
      "eq_number": 774,
      "title": "WGSL Workgroup Synchronization",
      "domain_id": 51,
      "significance": "Barrier synchronization for GPU workgroups. Ensures correct parallel execution order."
    },
    {
      "eq_number": 774,
      "title": "WGSL Workgroup Synchronization",
      "domain_id": 51,
      "significance": "Barrier synchronization for GPU workgroups. Ensures correct parallel execution order."
    },
    {
      "eq_number": 774,
      "title": "WGSL Workgroup Synchronization",
      "domain_id": 51,
      "significance": "Barrier synchronization for GPU workgroups. Ensures correct parallel execution order."
    },
    {
      "eq_number": 774,
      "title": "WGSL Workgroup Synchronization",
      "domain_id": 51,
      "significance": "Barrier synchronization for GPU workgroups. Ensures correct parallel execution order."
    },
    {
      "eq_number": 774,
      "title": "WGSL Workgroup Synchronization",
      "domain_id": 51,
      "significance": "Barrier synchronization for GPU workgroups. Ensures correct parallel execution order."
    },
    {
      "eq_number": 775,
      "title": "WGSL Shared Memory Reduction",
      "domain_id": 51,
      "significance": "Parallel reduction in GPU shared memory. Enables efficient aggregation across workgroup."
    },
    {
      "eq_number": 775,
      "title": "WGSL Shared Memory Reduction",
      "domain_id": 51,
      "significance": "Parallel reduction in GPU shared memory. Enables efficient aggregation across workgroup."
    },
    {
      "eq_number": 775,
      "title": "WGSL Shared Memory Reduction",
      "domain_id": 51,
      "significance": "Parallel reduction in GPU shared memory. Enables efficient aggregation across workgroup."
    },
    {
      "eq_number": 775,
      "title": "WGSL Shared Memory Reduction",
      "domain_id": 51,
      "significance": "Parallel reduction in GPU shared memory. Enables efficient aggregation across workgroup."
    },
    {
      "eq_number": 775,
      "title": "WGSL Shared Memory Reduction",
      "domain_id": 51,
      "significance": "Parallel reduction in GPU shared memory. Enables efficient aggregation across workgroup."
    },
    {
      "eq_number": 775,
      "title": "WGSL Shared Memory Reduction",
      "domain_id": 51,
      "significance": "Parallel reduction in GPU shared memory. Enables efficient aggregation across workgroup."
    },
    {
      "eq_number": 775,
      "title": "WGSL Shared Memory Reduction",
      "domain_id": 51,
      "significance": "Parallel reduction in GPU shared memory. Enables efficient aggregation across workgroup."
    },
    {
      "eq_number": 775,
      "title": "WGSL Shared Memory Reduction",
      "domain_id": 51,
      "significance": "Parallel reduction in GPU shared memory. Enables efficient aggregation across workgroup."
    },
    {
      "eq_number": 776,
      "title": "Vector Append Operation",
      "domain_id": 51,
      "significance": "Dynamic vector appending for incremental processing. Enables streaming vector operations."
    },
    {
      "eq_number": 776,
      "title": "Vector Append Operation",
      "domain_id": 51,
      "significance": "Dynamic vector appending for incremental processing. Enables streaming vector operations."
    },
    {
      "eq_number": 776,
      "title": "Vector Append Operation",
      "domain_id": 51,
      "significance": "Dynamic vector appending for incremental processing. Enables streaming vector operations."
    },
    {
      "eq_number": 776,
      "title": "Vector Append Operation",
      "domain_id": 51,
      "significance": "Dynamic vector appending for incremental processing. Enables streaming vector operations."
    },
    {
      "eq_number": 776,
      "title": "Vector Append Operation",
      "domain_id": 51,
      "significance": "Dynamic vector appending for incremental processing. Enables streaming vector operations."
    },
    {
      "eq_number": 776,
      "title": "Vector Append Operation",
      "domain_id": 51,
      "significance": "Dynamic vector appending for incremental processing. Enables streaming vector operations."
    },
    {
      "eq_number": 776,
      "title": "Vector Append Operation",
      "domain_id": 51,
      "significance": "Dynamic vector appending for incremental processing. Enables streaming vector operations."
    },
    {
      "eq_number": 776,
      "title": "Vector Append Operation",
      "domain_id": 51,
      "significance": "Dynamic vector appending for incremental processing. Enables streaming vector operations."
    },
    {
      "eq_number": 777,
      "title": "Vector Concatenation",
      "domain_id": 51,
      "significance": "Efficient vector concatenation for batch processing. Enables combining multiple vectors."
    },
    {
      "eq_number": 777,
      "title": "Vector Concatenation",
      "domain_id": 51,
      "significance": "Efficient vector concatenation for batch processing. Enables combining multiple vectors."
    },
    {
      "eq_number": 777,
      "title": "Vector Concatenation",
      "domain_id": 51,
      "significance": "Efficient vector concatenation for batch processing. Enables combining multiple vectors."
    },
    {
      "eq_number": 777,
      "title": "Vector Concatenation",
      "domain_id": 51,
      "significance": "Efficient vector concatenation for batch processing. Enables combining multiple vectors."
    },
    {
      "eq_number": 777,
      "title": "Vector Concatenation",
      "domain_id": 51,
      "significance": "Efficient vector concatenation for batch processing. Enables combining multiple vectors."
    },
    {
      "eq_number": 777,
      "title": "Vector Concatenation",
      "domain_id": 51,
      "significance": "Efficient vector concatenation for batch processing. Enables combining multiple vectors."
    },
    {
      "eq_number": 777,
      "title": "Vector Concatenation",
      "domain_id": 51,
      "significance": "Efficient vector concatenation for batch processing. Enables combining multiple vectors."
    },
    {
      "eq_number": 777,
      "title": "Vector Concatenation",
      "domain_id": 51,
      "significance": "Efficient vector concatenation for batch processing. Enables combining multiple vectors."
    },
    {
      "eq_number": 778,
      "title": "Vector Append with Capacity Growth",
      "domain_id": 51,
      "significance": "Amortized O(1) append with geometric capacity growth. Optimizes memory allocation."
    },
    {
      "eq_number": 778,
      "title": "Vector Append with Capacity Growth",
      "domain_id": 51,
      "significance": "Amortized O(1) append with geometric capacity growth. Optimizes memory allocation."
    },
    {
      "eq_number": 778,
      "title": "Vector Append with Capacity Growth",
      "domain_id": 51,
      "significance": "Amortized O(1) append with geometric capacity growth. Optimizes memory allocation."
    },
    {
      "eq_number": 778,
      "title": "Vector Append with Capacity Growth",
      "domain_id": 51,
      "significance": "Amortized O(1) append with geometric capacity growth. Optimizes memory allocation."
    },
    {
      "eq_number": 778,
      "title": "Vector Append with Capacity Growth",
      "domain_id": 51,
      "significance": "Amortized O(1) append with geometric capacity growth. Optimizes memory allocation."
    },
    {
      "eq_number": 778,
      "title": "Vector Append with Capacity Growth",
      "domain_id": 51,
      "significance": "Amortized O(1) append with geometric capacity growth. Optimizes memory allocation."
    },
    {
      "eq_number": 778,
      "title": "Vector Append with Capacity Growth",
      "domain_id": 51,
      "significance": "Amortized O(1) append with geometric capacity growth. Optimizes memory allocation."
    },
    {
      "eq_number": 778,
      "title": "Vector Append with Capacity Growth",
      "domain_id": 51,
      "significance": "Amortized O(1) append with geometric capacity growth. Optimizes memory allocation."
    },
    {
      "eq_number": 779,
      "title": "Graph Vector Append",
      "domain_id": 51,
      "significance": "Appending vectors to graph nodes for incremental graph updates. Enables dynamic graph processing."
    },
    {
      "eq_number": 779,
      "title": "Graph Vector Append",
      "domain_id": 51,
      "significance": "Appending vectors to graph nodes for incremental graph updates. Enables dynamic graph processing."
    },
    {
      "eq_number": 779,
      "title": "Graph Vector Append",
      "domain_id": 51,
      "significance": "Appending vectors to graph nodes for incremental graph updates. Enables dynamic graph processing."
    },
    {
      "eq_number": 779,
      "title": "Graph Vector Append",
      "domain_id": 51,
      "significance": "Appending vectors to graph nodes for incremental graph updates. Enables dynamic graph processing."
    },
    {
      "eq_number": 779,
      "title": "Graph Vector Append",
      "domain_id": 51,
      "significance": "Appending vectors to graph nodes for incremental graph updates. Enables dynamic graph processing."
    },
    {
      "eq_number": 779,
      "title": "Graph Vector Append",
      "domain_id": 51,
      "significance": "Appending vectors to graph nodes for incremental graph updates. Enables dynamic graph processing."
    },
    {
      "eq_number": 779,
      "title": "Graph Vector Append",
      "domain_id": 51,
      "significance": "Appending vectors to graph nodes for incremental graph updates. Enables dynamic graph processing."
    },
    {
      "eq_number": 779,
      "title": "Graph Vector Append",
      "domain_id": 51,
      "significance": "Appending vectors to graph nodes for incremental graph updates. Enables dynamic graph processing."
    },
    {
      "eq_number": 780,
      "title": "HNSW Hierarchical Navigable Small World",
      "domain_id": 51,
      "significance": "Graph-based approximate nearest neighbor search. Combines probability skip list with navigable small world graphs for fast vector similarity search."
    },
    {
      "eq_number": 780,
      "title": "HNSW Hierarchical Navigable Small World",
      "domain_id": 51,
      "significance": "Graph-based approximate nearest neighbor search. Combines probability skip list with navigable small world graphs for fast vector similarity search."
    },
    {
      "eq_number": 780,
      "title": "HNSW Hierarchical Navigable Small World",
      "domain_id": 51,
      "significance": "Graph-based approximate nearest neighbor search. Combines probability skip list with navigable small world graphs for fast vector similarity search."
    },
    {
      "eq_number": 780,
      "title": "HNSW Hierarchical Navigable Small World",
      "domain_id": 51,
      "significance": "Graph-based approximate nearest neighbor search. Combines probability skip list with navigable small world graphs for fast vector similarity search."
    },
    {
      "eq_number": 780,
      "title": "HNSW Hierarchical Navigable Small World",
      "domain_id": 51,
      "significance": "Graph-based approximate nearest neighbor search. Combines probability skip list with navigable small world graphs for fast vector similarity search."
    },
    {
      "eq_number": 780,
      "title": "HNSW Hierarchical Navigable Small World",
      "domain_id": 51,
      "significance": "Graph-based approximate nearest neighbor search. Combines probability skip list with navigable small world graphs for fast vector similarity search."
    },
    {
      "eq_number": 780,
      "title": "HNSW Hierarchical Navigable Small World",
      "domain_id": 51,
      "significance": "Graph-based approximate nearest neighbor search. Combines probability skip list with navigable small world graphs for fast vector similarity search."
    },
    {
      "eq_number": 781,
      "title": "Approximate Nearest Neighbor Search",
      "domain_id": 51,
      "significance": "Efficient vector search with slight accuracy penalty for massive speedup. Uses HNSW for O(log N) search complexity."
    },
    {
      "eq_number": 781,
      "title": "Approximate Nearest Neighbor Search",
      "domain_id": 51,
      "significance": "Efficient vector search with slight accuracy penalty for massive speedup. Uses HNSW for O(log N) search complexity."
    },
    {
      "eq_number": 781,
      "title": "Approximate Nearest Neighbor Search",
      "domain_id": 51,
      "significance": "Efficient vector search with slight accuracy penalty for massive speedup. Uses HNSW for O(log N) search complexity."
    },
    {
      "eq_number": 781,
      "title": "Approximate Nearest Neighbor Search",
      "domain_id": 51,
      "significance": "Efficient vector search with slight accuracy penalty for massive speedup. Uses HNSW for O(log N) search complexity."
    },
    {
      "eq_number": 781,
      "title": "Approximate Nearest Neighbor Search",
      "domain_id": 51,
      "significance": "Efficient vector search with slight accuracy penalty for massive speedup. Uses HNSW for O(log N) search complexity."
    },
    {
      "eq_number": 781,
      "title": "Approximate Nearest Neighbor Search",
      "domain_id": 51,
      "significance": "Efficient vector search with slight accuracy penalty for massive speedup. Uses HNSW for O(log N) search complexity."
    },
    {
      "eq_number": 781,
      "title": "Approximate Nearest Neighbor Search",
      "domain_id": 51,
      "significance": "Efficient vector search with slight accuracy penalty for massive speedup. Uses HNSW for O(log N) search complexity."
    },
    {
      "eq_number": 782,
      "title": "Proximity Graph Edge Probability",
      "domain_id": 51,
      "significance": "Probability of edge creation based on vector proximity in HNSW. Controls graph connectivity."
    },
    {
      "eq_number": 782,
      "title": "Proximity Graph Edge Probability",
      "domain_id": 51,
      "significance": "Probability of edge creation based on vector proximity in HNSW. Controls graph connectivity."
    },
    {
      "eq_number": 782,
      "title": "Proximity Graph Edge Probability",
      "domain_id": 51,
      "significance": "Probability of edge creation based on vector proximity in HNSW. Controls graph connectivity."
    },
    {
      "eq_number": 782,
      "title": "Proximity Graph Edge Probability",
      "domain_id": 51,
      "significance": "Probability of edge creation based on vector proximity in HNSW. Controls graph connectivity."
    },
    {
      "eq_number": 782,
      "title": "Proximity Graph Edge Probability",
      "domain_id": 51,
      "significance": "Probability of edge creation based on vector proximity in HNSW. Controls graph connectivity."
    },
    {
      "eq_number": 782,
      "title": "Proximity Graph Edge Probability",
      "domain_id": 51,
      "significance": "Probability of edge creation based on vector proximity in HNSW. Controls graph connectivity."
    },
    {
      "eq_number": 782,
      "title": "Proximity Graph Edge Probability",
      "domain_id": 51,
      "significance": "Probability of edge creation based on vector proximity in HNSW. Controls graph connectivity."
    },
    {
      "eq_number": 783,
      "title": "Property Graph Traversal",
      "domain_id": 51,
      "significance": "Efficient traversal of property graphs with nodes, edges, and properties. Basis for graph query languages like Cypher and GSQL."
    },
    {
      "eq_number": 783,
      "title": "Property Graph Traversal",
      "domain_id": 51,
      "significance": "Efficient traversal of property graphs with nodes, edges, and properties. Basis for graph query languages like Cypher and GSQL."
    },
    {
      "eq_number": 783,
      "title": "Property Graph Traversal",
      "domain_id": 51,
      "significance": "Efficient traversal of property graphs with nodes, edges, and properties. Basis for graph query languages like Cypher and GSQL."
    },
    {
      "eq_number": 783,
      "title": "Property Graph Traversal",
      "domain_id": 51,
      "significance": "Efficient traversal of property graphs with nodes, edges, and properties. Basis for graph query languages like Cypher and GSQL."
    },
    {
      "eq_number": 783,
      "title": "Property Graph Traversal",
      "domain_id": 51,
      "significance": "Efficient traversal of property graphs with nodes, edges, and properties. Basis for graph query languages like Cypher and GSQL."
    },
    {
      "eq_number": 783,
      "title": "Property Graph Traversal",
      "domain_id": 51,
      "significance": "Efficient traversal of property graphs with nodes, edges, and properties. Basis for graph query languages like Cypher and GSQL."
    },
    {
      "eq_number": 783,
      "title": "Property Graph Traversal",
      "domain_id": 51,
      "significance": "Efficient traversal of property graphs with nodes, edges, and properties. Basis for graph query languages like Cypher and GSQL."
    },
    {
      "eq_number": 784,
      "title": "Graph Pattern Matching",
      "domain_id": 51,
      "significance": "Pattern-based query in graph databases. Enables complex relationship queries like Cypher's MATCH clause."
    },
    {
      "eq_number": 784,
      "title": "Graph Pattern Matching",
      "domain_id": 51,
      "significance": "Pattern-based query in graph databases. Enables complex relationship queries like Cypher's MATCH clause."
    },
    {
      "eq_number": 784,
      "title": "Graph Pattern Matching",
      "domain_id": 51,
      "significance": "Pattern-based query in graph databases. Enables complex relationship queries like Cypher's MATCH clause."
    },
    {
      "eq_number": 784,
      "title": "Graph Pattern Matching",
      "domain_id": 51,
      "significance": "Pattern-based query in graph databases. Enables complex relationship queries like Cypher's MATCH clause."
    },
    {
      "eq_number": 784,
      "title": "Graph Pattern Matching",
      "domain_id": 51,
      "significance": "Pattern-based query in graph databases. Enables complex relationship queries like Cypher's MATCH clause."
    },
    {
      "eq_number": 784,
      "title": "Graph Pattern Matching",
      "domain_id": 51,
      "significance": "Pattern-based query in graph databases. Enables complex relationship queries like Cypher's MATCH clause."
    },
    {
      "eq_number": 784,
      "title": "Graph Pattern Matching",
      "domain_id": 51,
      "significance": "Pattern-based query in graph databases. Enables complex relationship queries like Cypher's MATCH clause."
    },
    {
      "eq_number": 785,
      "title": "Multi-Model Query Integration",
      "domain_id": 51,
      "significance": "Unified querying across document, key-value, and graph models. Enables ArangoDB-style multi-model databases."
    },
    {
      "eq_number": 785,
      "title": "Multi-Model Query Integration",
      "domain_id": 51,
      "significance": "Unified querying across document, key-value, and graph models. Enables ArangoDB-style multi-model databases."
    },
    {
      "eq_number": 785,
      "title": "Multi-Model Query Integration",
      "domain_id": 51,
      "significance": "Unified querying across document, key-value, and graph models. Enables ArangoDB-style multi-model databases."
    },
    {
      "eq_number": 785,
      "title": "Multi-Model Query Integration",
      "domain_id": 51,
      "significance": "Unified querying across document, key-value, and graph models. Enables ArangoDB-style multi-model databases."
    },
    {
      "eq_number": 785,
      "title": "Multi-Model Query Integration",
      "domain_id": 51,
      "significance": "Unified querying across document, key-value, and graph models. Enables ArangoDB-style multi-model databases."
    },
    {
      "eq_number": 785,
      "title": "Multi-Model Query Integration",
      "domain_id": 51,
      "significance": "Unified querying across document, key-value, and graph models. Enables ArangoDB-style multi-model databases."
    },
    {
      "eq_number": 785,
      "title": "Multi-Model Query Integration",
      "domain_id": 51,
      "significance": "Unified querying across document, key-value, and graph models. Enables ArangoDB-style multi-model databases."
    },
    {
      "eq_number": 786,
      "title": "Parallel Graph Processing",
      "domain_id": 51,
      "significance": "Native parallel graph engine for real-time analytics. TigerGraph-style parallel processing for massive graphs."
    },
    {
      "eq_number": 786,
      "title": "Parallel Graph Processing",
      "domain_id": 51,
      "significance": "Native parallel graph engine for real-time analytics. TigerGraph-style parallel processing for massive graphs."
    },
    {
      "eq_number": 786,
      "title": "Parallel Graph Processing",
      "domain_id": 51,
      "significance": "Native parallel graph engine for real-time analytics. TigerGraph-style parallel processing for massive graphs."
    },
    {
      "eq_number": 786,
      "title": "Parallel Graph Processing",
      "domain_id": 51,
      "significance": "Native parallel graph engine for real-time analytics. TigerGraph-style parallel processing for massive graphs."
    },
    {
      "eq_number": 786,
      "title": "Parallel Graph Processing",
      "domain_id": 51,
      "significance": "Native parallel graph engine for real-time analytics. TigerGraph-style parallel processing for massive graphs."
    },
    {
      "eq_number": 786,
      "title": "Parallel Graph Processing",
      "domain_id": 51,
      "significance": "Native parallel graph engine for real-time analytics. TigerGraph-style parallel processing for massive graphs."
    },
    {
      "eq_number": 786,
      "title": "Parallel Graph Processing",
      "domain_id": 51,
      "significance": "Native parallel graph engine for real-time analytics. TigerGraph-style parallel processing for massive graphs."
    },
    {
      "eq_number": 787,
      "title": "Shockwave Alignment and Relaxation",
      "domain_id": 51,
      "significance": "Quasi-charged cells align under shockwave, propagate charge symmetrically, then dissipate and relax. Four-phase cycle: anisotropic \u2192 shock_aligned \u2192 discharge \u2192 relaxed."
    },
    {
      "eq_number": 787,
      "title": "Shockwave Alignment and Relaxation",
      "domain_id": 51,
      "significance": "Quasi-charged cells align under shockwave, propagate charge symmetrically, then dissipate and relax. Four-phase cycle: anisotropic \u2192 shock_aligned \u2192 discharge \u2192 relaxed."
    },
    {
      "eq_number": 787,
      "title": "Shockwave Alignment and Relaxation",
      "domain_id": 51,
      "significance": "Quasi-charged cells align under shockwave, propagate charge symmetrically, then dissipate and relax. Four-phase cycle: anisotropic \u2192 shock_aligned \u2192 discharge \u2192 relaxed."
    },
    {
      "eq_number": 787,
      "title": "Shockwave Alignment and Relaxation",
      "domain_id": 51,
      "significance": "Quasi-charged cells align under shockwave, propagate charge symmetrically, then dissipate and relax. Four-phase cycle: anisotropic \u2192 shock_aligned \u2192 discharge \u2192 relaxed."
    },
    {
      "eq_number": 787,
      "title": "Shockwave Alignment and Relaxation",
      "domain_id": 51,
      "significance": "Quasi-charged cells align under shockwave, propagate charge symmetrically, then dissipate and relax. Four-phase cycle: anisotropic \u2192 shock_aligned \u2192 discharge \u2192 relaxed."
    },
    {
      "eq_number": 787,
      "title": "Shockwave Alignment and Relaxation",
      "domain_id": 51,
      "significance": "Quasi-charged cells align under shockwave, propagate charge symmetrically, then dissipate and relax. Four-phase cycle: anisotropic \u2192 shock_aligned \u2192 discharge \u2192 relaxed."
    },
    {
      "eq_number": 788,
      "title": "Phonon Force Law",
      "domain_id": 51,
      "significance": "Phonon correlation structure for self-healing. Force decays exponentially with Manhattan distance and oscillates with coherence period."
    },
    {
      "eq_number": 788,
      "title": "Phonon Force Law",
      "domain_id": 51,
      "significance": "Phonon correlation structure for self-healing. Force decays exponentially with Manhattan distance and oscillates with coherence period."
    },
    {
      "eq_number": 788,
      "title": "Phonon Force Law",
      "domain_id": 51,
      "significance": "Phonon correlation structure for self-healing. Force decays exponentially with Manhattan distance and oscillates with coherence period."
    },
    {
      "eq_number": 788,
      "title": "Phonon Force Law",
      "domain_id": 51,
      "significance": "Phonon correlation structure for self-healing. Force decays exponentially with Manhattan distance and oscillates with coherence period."
    },
    {
      "eq_number": 788,
      "title": "Phonon Force Law",
      "domain_id": 51,
      "significance": "Phonon correlation structure for self-healing. Force decays exponentially with Manhattan distance and oscillates with coherence period."
    },
    {
      "eq_number": 788,
      "title": "Phonon Force Law",
      "domain_id": 51,
      "significance": "Phonon correlation structure for self-healing. Force decays exponentially with Manhattan distance and oscillates with coherence period."
    },
    {
      "eq_number": 789,
      "title": "Cartesian Phonon Prime Integration",
      "domain_id": 51,
      "significance": "256\u00d7256 Cartesian coordinate space with 16-bit fixed addressing and Manhattan distance metric for hardware-efficient phonon transport."
    },
    {
      "eq_number": 789,
      "title": "Cartesian Phonon Prime Integration",
      "domain_id": 51,
      "significance": "256\u00d7256 Cartesian coordinate space with 16-bit fixed addressing and Manhattan distance metric for hardware-efficient phonon transport."
    },
    {
      "eq_number": 789,
      "title": "Cartesian Phonon Prime Integration",
      "domain_id": 51,
      "significance": "256\u00d7256 Cartesian coordinate space with 16-bit fixed addressing and Manhattan distance metric for hardware-efficient phonon transport."
    },
    {
      "eq_number": 789,
      "title": "Cartesian Phonon Prime Integration",
      "domain_id": 51,
      "significance": "256\u00d7256 Cartesian coordinate space with 16-bit fixed addressing and Manhattan distance metric for hardware-efficient phonon transport."
    },
    {
      "eq_number": 789,
      "title": "Cartesian Phonon Prime Integration",
      "domain_id": 51,
      "significance": "256\u00d7256 Cartesian coordinate space with 16-bit fixed addressing and Manhattan distance metric for hardware-efficient phonon transport."
    },
    {
      "eq_number": 789,
      "title": "Cartesian Phonon Prime Integration",
      "domain_id": 51,
      "significance": "256\u00d7256 Cartesian coordinate space with 16-bit fixed addressing and Manhattan distance metric for hardware-efficient phonon transport."
    },
    {
      "eq_number": 790,
      "title": "Phonon Load Dissipation",
      "domain_id": 51,
      "significance": "Phonon energy dissipates through discrete steps. Full dissipation returns cell to relaxed zero-load state."
    },
    {
      "eq_number": 790,
      "title": "Phonon Load Dissipation",
      "domain_id": 51,
      "significance": "Phonon energy dissipates through discrete steps. Full dissipation returns cell to relaxed zero-load state."
    },
    {
      "eq_number": 790,
      "title": "Phonon Load Dissipation",
      "domain_id": 51,
      "significance": "Phonon energy dissipates through discrete steps. Full dissipation returns cell to relaxed zero-load state."
    },
    {
      "eq_number": 790,
      "title": "Phonon Load Dissipation",
      "domain_id": 51,
      "significance": "Phonon energy dissipates through discrete steps. Full dissipation returns cell to relaxed zero-load state."
    },
    {
      "eq_number": 790,
      "title": "Phonon Load Dissipation",
      "domain_id": 51,
      "significance": "Phonon energy dissipates through discrete steps. Full dissipation returns cell to relaxed zero-load state."
    },
    {
      "eq_number": 790,
      "title": "Phonon Load Dissipation",
      "domain_id": 51,
      "significance": "Phonon energy dissipates through discrete steps. Full dissipation returns cell to relaxed zero-load state."
    },
    {
      "eq_number": 791,
      "title": "Shock Aligned Contact Energy",
      "domain_id": 51,
      "significance": "Positive aligned charge and contact coupling produce positive contact energy during shock-forced propagation."
    },
    {
      "eq_number": 791,
      "title": "Shock Aligned Contact Energy",
      "domain_id": 51,
      "significance": "Positive aligned charge and contact coupling produce positive contact energy during shock-forced propagation."
    },
    {
      "eq_number": 791,
      "title": "Shock Aligned Contact Energy",
      "domain_id": 51,
      "significance": "Positive aligned charge and contact coupling produce positive contact energy during shock-forced propagation."
    },
    {
      "eq_number": 791,
      "title": "Shock Aligned Contact Energy",
      "domain_id": 51,
      "significance": "Positive aligned charge and contact coupling produce positive contact energy during shock-forced propagation."
    },
    {
      "eq_number": 791,
      "title": "Shock Aligned Contact Energy",
      "domain_id": 51,
      "significance": "Positive aligned charge and contact coupling produce positive contact energy during shock-forced propagation."
    },
    {
      "eq_number": 791,
      "title": "Shock Aligned Contact Energy",
      "domain_id": 51,
      "significance": "Positive aligned charge and contact coupling produce positive contact energy during shock-forced propagation."
    },
    {
      "eq_number": 792,
      "title": "Photonic Spectral Witness",
      "domain_id": 51,
      "significance": "Spectral amplitudes encoded into optical mode amplitudes. Photon-count distribution recovers scalar observable \u03a9[u]."
    },
    {
      "eq_number": 792,
      "title": "Photonic Spectral Witness",
      "domain_id": 51,
      "significance": "Spectral amplitudes encoded into optical mode amplitudes. Photon-count distribution recovers scalar observable \u03a9[u]."
    },
    {
      "eq_number": 792,
      "title": "Photonic Spectral Witness",
      "domain_id": 51,
      "significance": "Spectral amplitudes encoded into optical mode amplitudes. Photon-count distribution recovers scalar observable \u03a9[u]."
    },
    {
      "eq_number": 792,
      "title": "Photonic Spectral Witness",
      "domain_id": 51,
      "significance": "Spectral amplitudes encoded into optical mode amplitudes. Photon-count distribution recovers scalar observable \u03a9[u]."
    },
    {
      "eq_number": 792,
      "title": "Photonic Spectral Witness",
      "domain_id": 51,
      "significance": "Spectral amplitudes encoded into optical mode amplitudes. Photon-count distribution recovers scalar observable \u03a9[u]."
    },
    {
      "eq_number": 792,
      "title": "Photonic Spectral Witness",
      "domain_id": 51,
      "significance": "Spectral amplitudes encoded into optical mode amplitudes. Photon-count distribution recovers scalar observable \u03a9[u]."
    },
    {
      "eq_number": 793,
      "title": "Pair-Bonded Shockwave Propagation",
      "domain_id": 51,
      "significance": "Two quasi-charged cells form temporary bond during shock alignment, enabling symmetric charge transfer before relaxation."
    },
    {
      "eq_number": 793,
      "title": "Pair-Bonded Shockwave Propagation",
      "domain_id": 51,
      "significance": "Two quasi-charged cells form temporary bond during shock alignment, enabling symmetric charge transfer before relaxation."
    },
    {
      "eq_number": 793,
      "title": "Pair-Bonded Shockwave Propagation",
      "domain_id": 51,
      "significance": "Two quasi-charged cells form temporary bond during shock alignment, enabling symmetric charge transfer before relaxation."
    },
    {
      "eq_number": 793,
      "title": "Pair-Bonded Shockwave Propagation",
      "domain_id": 51,
      "significance": "Two quasi-charged cells form temporary bond during shock alignment, enabling symmetric charge transfer before relaxation."
    },
    {
      "eq_number": 793,
      "title": "Pair-Bonded Shockwave Propagation",
      "domain_id": 51,
      "significance": "Two quasi-charged cells form temporary bond during shock alignment, enabling symmetric charge transfer before relaxation."
    },
    {
      "eq_number": 793,
      "title": "Pair-Bonded Shockwave Propagation",
      "domain_id": 51,
      "significance": "Two quasi-charged cells form temporary bond during shock alignment, enabling symmetric charge transfer before relaxation."
    },
    {
      "eq_number": 794,
      "title": "Phonon-Mediated Information Transport",
      "domain_id": 51,
      "significance": "Information encoded in phonon packets propagates through lattice. Lossy transport preserves spectral structure not exact state."
    },
    {
      "eq_number": 794,
      "title": "Phonon-Mediated Information Transport",
      "domain_id": 51,
      "significance": "Information encoded in phonon packets propagates through lattice. Lossy transport preserves spectral structure not exact state."
    },
    {
      "eq_number": 794,
      "title": "Phonon-Mediated Information Transport",
      "domain_id": 51,
      "significance": "Information encoded in phonon packets propagates through lattice. Lossy transport preserves spectral structure not exact state."
    },
    {
      "eq_number": 794,
      "title": "Phonon-Mediated Information Transport",
      "domain_id": 51,
      "significance": "Information encoded in phonon packets propagates through lattice. Lossy transport preserves spectral structure not exact state."
    },
    {
      "eq_number": 794,
      "title": "Phonon-Mediated Information Transport",
      "domain_id": 51,
      "significance": "Information encoded in phonon packets propagates through lattice. Lossy transport preserves spectral structure not exact state."
    },
    {
      "eq_number": 794,
      "title": "Phonon-Mediated Information Transport",
      "domain_id": 51,
      "significance": "Information encoded in phonon packets propagates through lattice. Lossy transport preserves spectral structure not exact state."
    },
    {
      "eq_number": 795,
      "title": "\u0394\u03c6\u03b3K\u03bb Compression Law",
      "domain_id": 51,
      "significance": "Compression-domain instance of GCCL with separate fields for transform pressure (\u03b3) and cost paid (K). Corrected from \u0394\u03c6\u03b3\u03bb which overloaded \u03b3."
    },
    {
      "eq_number": 795,
      "title": "\u0394\u03c6\u03b3K\u03bb Compression Law",
      "domain_id": 51,
      "significance": "Compression-domain instance of GCCL with separate fields for transform pressure (\u03b3) and cost paid (K). Corrected from \u0394\u03c6\u03b3\u03bb which overloaded \u03b3."
    },
    {
      "eq_number": 795,
      "title": "\u0394\u03c6\u03b3K\u03bb Compression Law",
      "domain_id": 51,
      "significance": "Compression-domain instance of GCCL with separate fields for transform pressure (\u03b3) and cost paid (K). Corrected from \u0394\u03c6\u03b3\u03bb which overloaded \u03b3."
    },
    {
      "eq_number": 795,
      "title": "\u0394\u03c6\u03b3K\u03bb Compression Law",
      "domain_id": 51,
      "significance": "Compression-domain instance of GCCL with separate fields for transform pressure (\u03b3) and cost paid (K). Corrected from \u0394\u03c6\u03b3\u03bb which overloaded \u03b3."
    },
    {
      "eq_number": 795,
      "title": "\u0394\u03c6\u03b3K\u03bb Compression Law",
      "domain_id": 51,
      "significance": "Compression-domain instance of GCCL with separate fields for transform pressure (\u03b3) and cost paid (K). Corrected from \u0394\u03c6\u03b3\u03bb which overloaded \u03b3."
    },
    {
      "eq_number": 796,
      "title": "Goxel Scalar Sub-Manifold",
      "domain_id": 51,
      "significance": "N-space shape inhabiting geometric volume, expressed as bounded scalar sub-manifold. Admitted only through declared projection, audit, and receipt gates."
    },
    {
      "eq_number": 796,
      "title": "Goxel Scalar Sub-Manifold",
      "domain_id": 51,
      "significance": "N-space shape inhabiting geometric volume, expressed as bounded scalar sub-manifold. Admitted only through declared projection, audit, and receipt gates."
    },
    {
      "eq_number": 796,
      "title": "Goxel Scalar Sub-Manifold",
      "domain_id": 51,
      "significance": "N-space shape inhabiting geometric volume, expressed as bounded scalar sub-manifold. Admitted only through declared projection, audit, and receipt gates."
    },
    {
      "eq_number": 796,
      "title": "Goxel Scalar Sub-Manifold",
      "domain_id": 51,
      "significance": "N-space shape inhabiting geometric volume, expressed as bounded scalar sub-manifold. Admitted only through declared projection, audit, and receipt gates."
    },
    {
      "eq_number": 796,
      "title": "Goxel Scalar Sub-Manifold",
      "domain_id": 51,
      "significance": "N-space shape inhabiting geometric volume, expressed as bounded scalar sub-manifold. Admitted only through declared projection, audit, and receipt gates."
    },
    {
      "eq_number": 797,
      "title": "Model Genome Encoding",
      "domain_id": 51,
      "significance": "Compact generative encoding of model family with codon\u2192gene\u2192chromosome\u2192genome\u2192phenotype hierarchy."
    },
    {
      "eq_number": 797,
      "title": "Model Genome Encoding",
      "domain_id": 51,
      "significance": "Compact generative encoding of model family with codon\u2192gene\u2192chromosome\u2192genome\u2192phenotype hierarchy."
    },
    {
      "eq_number": 797,
      "title": "Model Genome Encoding",
      "domain_id": 51,
      "significance": "Compact generative encoding of model family with codon\u2192gene\u2192chromosome\u2192genome\u2192phenotype hierarchy."
    },
    {
      "eq_number": 797,
      "title": "Model Genome Encoding",
      "domain_id": 51,
      "significance": "Compact generative encoding of model family with codon\u2192gene\u2192chromosome\u2192genome\u2192phenotype hierarchy."
    },
    {
      "eq_number": 797,
      "title": "Model Genome Encoding",
      "domain_id": 51,
      "significance": "Compact generative encoding of model family with codon\u2192gene\u2192chromosome\u2192genome\u2192phenotype hierarchy."
    },
    {
      "eq_number": 798,
      "title": "Kinetic Operation Token (KOT)",
      "domain_id": 51,
      "significance": "Accounting layer for action cost. Every transformation pays and leaves a trace. Prevents free transformations."
    },
    {
      "eq_number": 798,
      "title": "Kinetic Operation Token (KOT)",
      "domain_id": 51,
      "significance": "Accounting layer for action cost. Every transformation pays and leaves a trace. Prevents free transformations."
    },
    {
      "eq_number": 798,
      "title": "Kinetic Operation Token (KOT)",
      "domain_id": 51,
      "significance": "Accounting layer for action cost. Every transformation pays and leaves a trace. Prevents free transformations."
    },
    {
      "eq_number": 798,
      "title": "Kinetic Operation Token (KOT)",
      "domain_id": 51,
      "significance": "Accounting layer for action cost. Every transformation pays and leaves a trace. Prevents free transformations."
    },
    {
      "eq_number": 798,
      "title": "Kinetic Operation Token (KOT)",
      "domain_id": 51,
      "significance": "Accounting layer for action cost. Every transformation pays and leaves a trace. Prevents free transformations."
    },
    {
      "eq_number": 799,
      "title": "Bounded Lawful Surface",
      "domain_id": 51,
      "significance": "Set of transitions and phenotypes that can be expressed, replayed, checked, budgeted, and receipted under declared constraints."
    },
    {
      "eq_number": 799,
      "title": "Bounded Lawful Surface",
      "domain_id": 51,
      "significance": "Set of transitions and phenotypes that can be expressed, replayed, checked, budgeted, and receipted under declared constraints."
    },
    {
      "eq_number": 799,
      "title": "Bounded Lawful Surface",
      "domain_id": 51,
      "significance": "Set of transitions and phenotypes that can be expressed, replayed, checked, budgeted, and receipted under declared constraints."
    },
    {
      "eq_number": 799,
      "title": "Bounded Lawful Surface",
      "domain_id": 51,
      "significance": "Set of transitions and phenotypes that can be expressed, replayed, checked, budgeted, and receipted under declared constraints."
    },
    {
      "eq_number": 799,
      "title": "Bounded Lawful Surface",
      "domain_id": 51,
      "significance": "Set of transitions and phenotypes that can be expressed, replayed, checked, budgeted, and receipted under declared constraints."
    },
    {
      "eq_number": 800,
      "title": "Genotype-Phenotype Split",
      "domain_id": 51,
      "significance": "Separation of internal encoding from outward expression. Prevents projection from being mistaken for source object."
    },
    {
      "eq_number": 800,
      "title": "Genotype-Phenotype Split",
      "domain_id": 51,
      "significance": "Separation of internal encoding from outward expression. Prevents projection from being mistaken for source object."
    },
    {
      "eq_number": 800,
      "title": "Genotype-Phenotype Split",
      "domain_id": 51,
      "significance": "Separation of internal encoding from outward expression. Prevents projection from being mistaken for source object."
    },
    {
      "eq_number": 800,
      "title": "Genotype-Phenotype Split",
      "domain_id": 51,
      "significance": "Separation of internal encoding from outward expression. Prevents projection from being mistaken for source object."
    },
    {
      "eq_number": 800,
      "title": "Genotype-Phenotype Split",
      "domain_id": 51,
      "significance": "Separation of internal encoding from outward expression. Prevents projection from being mistaken for source object."
    },
    {
      "eq_number": 801,
      "title": "Mixture Primitive Combination",
      "domain_id": 51,
      "significance": "Multiple coding families (DNA, codons, proteins, ambiguity, etc.) can be mixed only under explicit decoder, residual, KOT, scale, projection, and receipt rules."
    },
    {
      "eq_number": 801,
      "title": "Mixture Primitive Combination",
      "domain_id": 51,
      "significance": "Multiple coding families (DNA, codons, proteins, ambiguity, etc.) can be mixed only under explicit decoder, residual, KOT, scale, projection, and receipt rules."
    },
    {
      "eq_number": 801,
      "title": "Mixture Primitive Combination",
      "domain_id": 51,
      "significance": "Multiple coding families (DNA, codons, proteins, ambiguity, etc.) can be mixed only under explicit decoder, residual, KOT, scale, projection, and receipt rules."
    },
    {
      "eq_number": 801,
      "title": "Mixture Primitive Combination",
      "domain_id": 51,
      "significance": "Multiple coding families (DNA, codons, proteins, ambiguity, etc.) can be mixed only under explicit decoder, residual, KOT, scale, projection, and receipt rules."
    },
    {
      "eq_number": 801,
      "title": "Mixture Primitive Combination",
      "domain_id": 51,
      "significance": "Multiple coding families (DNA, codons, proteins, ambiguity, etc.) can be mixed only under explicit decoder, residual, KOT, scale, projection, and receipt rules."
    },
    {
      "eq_number": 802,
      "title": "Layered Mountain Model",
      "domain_id": 51,
      "significance": "GCCL sits over layered state mountains: NUVMAP (address), AVMR (vector evolution), AMMR (commit history), O-AMMR (orthogonal projection), GCCL-Rep (transition rope)."
    },
    {
      "eq_number": 802,
      "title": "Layered Mountain Model",
      "domain_id": 51,
      "significance": "GCCL sits over layered state mountains: NUVMAP (address), AVMR (vector evolution), AMMR (commit history), O-AMMR (orthogonal projection), GCCL-Rep (transition rope)."
    },
    {
      "eq_number": 802,
      "title": "Layered Mountain Model",
      "domain_id": 51,
      "significance": "GCCL sits over layered state mountains: NUVMAP (address), AVMR (vector evolution), AMMR (commit history), O-AMMR (orthogonal projection), GCCL-Rep (transition rope)."
    },
    {
      "eq_number": 802,
      "title": "Layered Mountain Model",
      "domain_id": 51,
      "significance": "GCCL sits over layered state mountains: NUVMAP (address), AVMR (vector evolution), AMMR (commit history), O-AMMR (orthogonal projection), GCCL-Rep (transition rope)."
    },
    {
      "eq_number": 802,
      "title": "Layered Mountain Model",
      "domain_id": 51,
      "significance": "GCCL sits over layered state mountains: NUVMAP (address), AVMR (vector evolution), AMMR (commit history), O-AMMR (orthogonal projection), GCCL-Rep (transition rope)."
    },
    {
      "eq_number": 803,
      "title": "Wavefront Emission",
      "domain_id": 51,
      "significance": "State changes emit wavefronts that propagate through resonant field with amplitude, frequency, phase, position, and decay."
    },
    {
      "eq_number": 803,
      "title": "Wavefront Emission",
      "domain_id": 51,
      "significance": "State changes emit wavefronts that propagate through resonant field with amplitude, frequency, phase, position, and decay."
    },
    {
      "eq_number": 803,
      "title": "Wavefront Emission",
      "domain_id": 51,
      "significance": "State changes emit wavefronts that propagate through resonant field with amplitude, frequency, phase, position, and decay."
    },
    {
      "eq_number": 803,
      "title": "Wavefront Emission",
      "domain_id": 51,
      "significance": "State changes emit wavefronts that propagate through resonant field with amplitude, frequency, phase, position, and decay."
    },
    {
      "eq_number": 804,
      "title": "MOIM Behavioral Fingerprint",
      "domain_id": 51,
      "significance": "Objects become behavioral points across identity, conservation, transformation, scaling, and dynamics axes."
    },
    {
      "eq_number": 804,
      "title": "MOIM Behavioral Fingerprint",
      "domain_id": 51,
      "significance": "Objects become behavioral points across identity, conservation, transformation, scaling, and dynamics axes."
    },
    {
      "eq_number": 804,
      "title": "MOIM Behavioral Fingerprint",
      "domain_id": 51,
      "significance": "Objects become behavioral points across identity, conservation, transformation, scaling, and dynamics axes."
    },
    {
      "eq_number": 804,
      "title": "MOIM Behavioral Fingerprint",
      "domain_id": 51,
      "significance": "Objects become behavioral points across identity, conservation, transformation, scaling, and dynamics axes."
    },
    {
      "eq_number": 805,
      "title": "Universal Binding Manifold",
      "domain_id": 51,
      "significance": "Binding affinity surface for conceptual relationships with energy-based binding strength."
    },
    {
      "eq_number": 805,
      "title": "Universal Binding Manifold",
      "domain_id": 51,
      "significance": "Binding affinity surface for conceptual relationships with energy-based binding strength."
    },
    {
      "eq_number": 805,
      "title": "Universal Binding Manifold",
      "domain_id": 51,
      "significance": "Binding affinity surface for conceptual relationships with energy-based binding strength."
    },
    {
      "eq_number": 805,
      "title": "Universal Binding Manifold",
      "domain_id": 51,
      "significance": "Binding affinity surface for conceptual relationships with energy-based binding strength."
    },
    {
      "eq_number": 806,
      "title": "Info Bottleneck Principle",
      "domain_id": 51,
      "significance": "Optimal neural compression: minimize mutual information with input while maximizing with output."
    },
    {
      "eq_number": 806,
      "title": "Info Bottleneck Principle",
      "domain_id": 51,
      "significance": "Optimal neural compression: minimize mutual information with input while maximizing with output."
    },
    {
      "eq_number": 806,
      "title": "Info Bottleneck Principle",
      "domain_id": 51,
      "significance": "Optimal neural compression: minimize mutual information with input while maximizing with output."
    },
    {
      "eq_number": 806,
      "title": "Info Bottleneck Principle",
      "domain_id": 51,
      "significance": "Optimal neural compression: minimize mutual information with input while maximizing with output."
    },
    {
      "eq_number": 807,
      "title": "Free Energy Principle",
      "domain_id": 51,
      "significance": "Variational self-organization invariant: systems minimize free energy by minimizing surprise."
    },
    {
      "eq_number": 807,
      "title": "Free Energy Principle",
      "domain_id": 51,
      "significance": "Variational self-organization invariant: systems minimize free energy by minimizing surprise."
    },
    {
      "eq_number": 807,
      "title": "Free Energy Principle",
      "domain_id": 51,
      "significance": "Variational self-organization invariant: systems minimize free energy by minimizing surprise."
    },
    {
      "eq_number": 807,
      "title": "Free Energy Principle",
      "domain_id": 51,
      "significance": "Variational self-organization invariant: systems minimize free energy by minimizing surprise."
    },
    {
      "eq_number": 808,
      "title": "Predictive Coding",
      "domain_id": 51,
      "significance": "Hierarchical prediction error update: predictions drive learning and inference."
    },
    {
      "eq_number": 808,
      "title": "Predictive Coding",
      "domain_id": 51,
      "significance": "Hierarchical prediction error update: predictions drive learning and inference."
    },
    {
      "eq_number": 808,
      "title": "Predictive Coding",
      "domain_id": 51,
      "significance": "Hierarchical prediction error update: predictions drive learning and inference."
    },
    {
      "eq_number": 808,
      "title": "Predictive Coding",
      "domain_id": 51,
      "significance": "Hierarchical prediction error update: predictions drive learning and inference."
    },
    {
      "eq_number": 809,
      "title": "Onsager Reciprocity",
      "domain_id": 51,
      "significance": "Coupled transport symmetry law: cross-coupling coefficients are symmetric."
    },
    {
      "eq_number": 809,
      "title": "Onsager Reciprocity",
      "domain_id": 51,
      "significance": "Coupled transport symmetry law: cross-coupling coefficients are symmetric."
    },
    {
      "eq_number": 809,
      "title": "Onsager Reciprocity",
      "domain_id": 51,
      "significance": "Coupled transport symmetry law: cross-coupling coefficients are symmetric."
    },
    {
      "eq_number": 809,
      "title": "Onsager Reciprocity",
      "domain_id": 51,
      "significance": "Coupled transport symmetry law: cross-coupling coefficients are symmetric."
    },
    {
      "eq_number": 810,
      "title": "Jarzynski Equality",
      "domain_id": 51,
      "significance": "Non-equilibrium work-extraction relation connects work fluctuations to free energy difference."
    },
    {
      "eq_number": 810,
      "title": "Jarzynski Equality",
      "domain_id": 51,
      "significance": "Non-equilibrium work-extraction relation connects work fluctuations to free energy difference."
    },
    {
      "eq_number": 810,
      "title": "Jarzynski Equality",
      "domain_id": 51,
      "significance": "Non-equilibrium work-extraction relation connects work fluctuations to free energy difference."
    },
    {
      "eq_number": 810,
      "title": "Jarzynski Equality",
      "domain_id": 51,
      "significance": "Non-equilibrium work-extraction relation connects work fluctuations to free energy difference."
    },
    {
      "eq_number": 811,
      "title": "DNA Linking Number",
      "domain_id": 51,
      "significance": "Topological constraint on circular DNA: linking number equals twist plus writhe."
    },
    {
      "eq_number": 811,
      "title": "DNA Linking Number",
      "domain_id": 51,
      "significance": "Topological constraint on circular DNA: linking number equals twist plus writhe."
    },
    {
      "eq_number": 811,
      "title": "DNA Linking Number",
      "domain_id": 51,
      "significance": "Topological constraint on circular DNA: linking number equals twist plus writhe."
    },
    {
      "eq_number": 811,
      "title": "DNA Linking Number",
      "domain_id": 51,
      "significance": "Topological constraint on circular DNA: linking number equals twist plus writhe."
    },
    {
      "eq_number": 812,
      "title": "Cavity Persistence",
      "domain_id": 51,
      "significance": "Topological information processing metric: persistence of topological features."
    },
    {
      "eq_number": 812,
      "title": "Cavity Persistence",
      "domain_id": 51,
      "significance": "Topological information processing metric: persistence of topological features."
    },
    {
      "eq_number": 812,
      "title": "Cavity Persistence",
      "domain_id": 51,
      "significance": "Topological information processing metric: persistence of topological features."
    },
    {
      "eq_number": 812,
      "title": "Cavity Persistence",
      "domain_id": 51,
      "significance": "Topological information processing metric: persistence of topological features."
    },
    {
      "eq_number": 813,
      "title": "Hill Regulation",
      "domain_id": 51,
      "significance": "Nonlinear saturation feedback: sigmoid functions used throughout OTOM."
    },
    {
      "eq_number": 813,
      "title": "Hill Regulation",
      "domain_id": 51,
      "significance": "Nonlinear saturation feedback: sigmoid functions used throughout OTOM."
    },
    {
      "eq_number": 813,
      "title": "Hill Regulation",
      "domain_id": 51,
      "significance": "Nonlinear saturation feedback: sigmoid functions used throughout OTOM."
    },
    {
      "eq_number": 813,
      "title": "Hill Regulation",
      "domain_id": 51,
      "significance": "Nonlinear saturation feedback: sigmoid functions used throughout OTOM."
    },
    {
      "eq_number": 814,
      "title": "Wilson-Cowan Equations",
      "domain_id": 51,
      "significance": "Mean-field neural population dynamics for cognitive load modeling."
    },
    {
      "eq_number": 814,
      "title": "Wilson-Cowan Equations",
      "domain_id": 51,
      "significance": "Mean-field neural population dynamics for cognitive load modeling."
    },
    {
      "eq_number": 814,
      "title": "Wilson-Cowan Equations",
      "domain_id": 51,
      "significance": "Mean-field neural population dynamics for cognitive load modeling."
    },
    {
      "eq_number": 814,
      "title": "Wilson-Cowan Equations",
      "domain_id": 51,
      "significance": "Mean-field neural population dynamics for cognitive load modeling."
    },
    {
      "eq_number": 815,
      "title": "Turing Morphogenesis",
      "domain_id": 51,
      "significance": "Spontaneous symmetry breaking for pattern formation on manifolds."
    },
    {
      "eq_number": 815,
      "title": "Turing Morphogenesis",
      "domain_id": 51,
      "significance": "Spontaneous symmetry breaking for pattern formation on manifolds."
    },
    {
      "eq_number": 815,
      "title": "Turing Morphogenesis",
      "domain_id": 51,
      "significance": "Spontaneous symmetry breaking for pattern formation on manifolds."
    },
    {
      "eq_number": 815,
      "title": "Turing Morphogenesis",
      "domain_id": 51,
      "significance": "Spontaneous symmetry breaking for pattern formation on manifolds."
    },
    {
      "eq_number": 816,
      "title": "Mass Number Admissibility Gate",
      "domain_id": 51,
      "significance": "Three-layer Mass Number structure: Admissible (A), Residual (R), Boundary (\u03b5). Core rule: A \u2264 threshold * (R + \u03b5)."
    },
    {
      "eq_number": 816,
      "title": "Mass Number Admissibility Gate",
      "domain_id": 51,
      "significance": "Three-layer Mass Number structure: Admissible (A), Residual (R), Boundary (\u03b5). Core rule: A \u2264 threshold * (R + \u03b5)."
    },
    {
      "eq_number": 816,
      "title": "Mass Number Admissibility Gate",
      "domain_id": 51,
      "significance": "Three-layer Mass Number structure: Admissible (A), Residual (R), Boundary (\u03b5). Core rule: A \u2264 threshold * (R + \u03b5)."
    },
    {
      "eq_number": 817,
      "title": "Admissible Reduction Packet",
      "domain_id": 51,
      "significance": "Layer 1 of Mass Number: records concrete reduction achieved by modeling move. Must be grounded in surface feature/invariant."
    },
    {
      "eq_number": 817,
      "title": "Admissible Reduction Packet",
      "domain_id": 51,
      "significance": "Layer 1 of Mass Number: records concrete reduction achieved by modeling move. Must be grounded in surface feature/invariant."
    },
    {
      "eq_number": 817,
      "title": "Admissible Reduction Packet",
      "domain_id": 51,
      "significance": "Layer 1 of Mass Number: records concrete reduction achieved by modeling move. Must be grounded in surface feature/invariant."
    },
    {
      "eq_number": 818,
      "title": "Residual Risk Receipt",
      "domain_id": 51,
      "significance": "Layer 2 of Mass Number: records what remains unreduced after move. Must be inspectable and bounded."
    },
    {
      "eq_number": 818,
      "title": "Residual Risk Receipt",
      "domain_id": 51,
      "significance": "Layer 2 of Mass Number: records what remains unreduced after move. Must be inspectable and bounded."
    },
    {
      "eq_number": 818,
      "title": "Residual Risk Receipt",
      "domain_id": 51,
      "significance": "Layer 2 of Mass Number: records what remains unreduced after move. Must be inspectable and bounded."
    },
    {
      "eq_number": 819,
      "title": "Boundary Marker (\u03b5 Guard)",
      "domain_id": 51,
      "significance": "Layer 3 of Mass Number: ensures denominator never zero. Carries threshold for admissibility decisions."
    },
    {
      "eq_number": 819,
      "title": "Boundary Marker (\u03b5 Guard)",
      "domain_id": 51,
      "significance": "Layer 3 of Mass Number: ensures denominator never zero. Carries threshold for admissibility decisions."
    },
    {
      "eq_number": 819,
      "title": "Boundary Marker (\u03b5 Guard)",
      "domain_id": 51,
      "significance": "Layer 3 of Mass Number: ensures denominator never zero. Carries threshold for admissibility decisions."
    },
    {
      "eq_number": 820,
      "title": "NaNMass Doctrine",
      "domain_id": 51,
      "significance": "Apparent infinity is diagnostic, not destination. NaNMass means coordinate system failed to close mass."
    },
    {
      "eq_number": 820,
      "title": "NaNMass Doctrine",
      "domain_id": 51,
      "significance": "Apparent infinity is diagnostic, not destination. NaNMass means coordinate system failed to close mass."
    },
    {
      "eq_number": 820,
      "title": "NaNMass Doctrine",
      "domain_id": 51,
      "significance": "Apparent infinity is diagnostic, not destination. NaNMass means coordinate system failed to close mass."
    },
    {
      "eq_number": 821,
      "title": "Closure Path to Metric",
      "domain_id": 51,
      "significance": "Mass becomes distance only through admissibility closure. Raw mass \u2192 pseudometric \u2192 zero-distance quotient \u2192 metric."
    },
    {
      "eq_number": 821,
      "title": "Closure Path to Metric",
      "domain_id": 51,
      "significance": "Mass becomes distance only through admissibility closure. Raw mass \u2192 pseudometric \u2192 zero-distance quotient \u2192 metric."
    },
    {
      "eq_number": 821,
      "title": "Closure Path to Metric",
      "domain_id": 51,
      "significance": "Mass becomes distance only through admissibility closure. Raw mass \u2192 pseudometric \u2192 zero-distance quotient \u2192 metric."
    },
    {
      "eq_number": 822,
      "title": "Erd\u0151s Forced-Pattern Model",
      "domain_id": 51,
      "significance": "If system is large enough, disorder cannot remain pure. Organized substructure must appear."
    },
    {
      "eq_number": 822,
      "title": "Erd\u0151s Forced-Pattern Model",
      "domain_id": 51,
      "significance": "If system is large enough, disorder cannot remain pure. Organized substructure must appear."
    },
    {
      "eq_number": 822,
      "title": "Erd\u0151s Forced-Pattern Model",
      "domain_id": 51,
      "significance": "If system is large enough, disorder cannot remain pure. Organized substructure must appear."
    },
    {
      "eq_number": 823,
      "title": "General-Position Convexity Forcing",
      "domain_id": 51,
      "significance": "Points in general position: when does convex n-gon become unavoidable?"
    },
    {
      "eq_number": 823,
      "title": "General-Position Convexity Forcing",
      "domain_id": 51,
      "significance": "Points in general position: when does convex n-gon become unavoidable?"
    },
    {
      "eq_number": 823,
      "title": "General-Position Convexity Forcing",
      "domain_id": 51,
      "significance": "Points in general position: when does convex n-gon become unavoidable?"
    },
    {
      "eq_number": 824,
      "title": "Cup-Cap Monotonicity",
      "domain_id": 51,
      "significance": "Geometry converted to ordered subsequences. Convexity becomes pattern of slope changes."
    },
    {
      "eq_number": 824,
      "title": "Cup-Cap Monotonicity",
      "domain_id": 51,
      "significance": "Geometry converted to ordered subsequences. Convexity becomes pattern of slope changes."
    },
    {
      "eq_number": 824,
      "title": "Cup-Cap Monotonicity",
      "domain_id": 51,
      "significance": "Geometry converted to ordered subsequences. Convexity becomes pattern of slope changes."
    },
    {
      "eq_number": 825,
      "title": "Probabilistic Existence Method",
      "domain_id": 51,
      "significance": "Do not construct directly. Show random object avoids bad event with positive probability."
    },
    {
      "eq_number": 825,
      "title": "Probabilistic Existence Method",
      "domain_id": 51,
      "significance": "Do not construct directly. Show random object avoids bad event with positive probability."
    },
    {
      "eq_number": 825,
      "title": "Probabilistic Existence Method",
      "domain_id": 51,
      "significance": "Do not construct directly. Show random object avoids bad event with positive probability."
    },
    {
      "eq_number": 826,
      "title": "Extremal Density Threshold",
      "domain_id": 51,
      "significance": "Maximum possible density before forbidden structure is forced."
    },
    {
      "eq_number": 826,
      "title": "Extremal Density Threshold",
      "domain_id": 51,
      "significance": "Maximum possible density before forbidden structure is forced."
    },
    {
      "eq_number": 826,
      "title": "Extremal Density Threshold",
      "domain_id": 51,
      "significance": "Maximum possible density before forbidden structure is forced."
    },
    {
      "eq_number": 827,
      "title": "Sidon Additive Collision",
      "domain_id": 51,
      "significance": "Integers as collision surfaces. Forbidden equality becomes overlap in additive address space."
    },
    {
      "eq_number": 827,
      "title": "Sidon Additive Collision",
      "domain_id": 51,
      "significance": "Integers as collision surfaces. Forbidden equality becomes overlap in additive address space."
    },
    {
      "eq_number": 827,
      "title": "Sidon Additive Collision",
      "domain_id": 51,
      "significance": "Integers as collision surfaces. Forbidden equality becomes overlap in additive address space."
    },
    {
      "eq_number": 828,
      "title": "Order-Type Signature Function",
      "domain_id": 51,
      "significance": "Coordinates discarded. Only orientation signatures kept for convexity encoding."
    },
    {
      "eq_number": 828,
      "title": "Order-Type Signature Function",
      "domain_id": 51,
      "significance": "Coordinates discarded. Only orientation signatures kept for convexity encoding."
    },
    {
      "eq_number": 828,
      "title": "Order-Type Signature Function",
      "domain_id": 51,
      "significance": "Coordinates discarded. Only orientation signatures kept for convexity encoding."
    },
    {
      "eq_number": 829,
      "title": "Strain121 Temperature Limit",
      "domain_id": 51,
      "significance": "Absolute biological temperature limit: 122\u00b0C (395K) protein denaturation wall."
    },
    {
      "eq_number": 829,
      "title": "Strain121 Temperature Limit",
      "domain_id": 51,
      "significance": "Absolute biological temperature limit: 122\u00b0C (395K) protein denaturation wall."
    },
    {
      "eq_number": 830,
      "title": "Diatom Stiffness Limit",
      "domain_id": 51,
      "significance": "Silica shells approach inorganic material limits. \u03ba_T \u2248 2.7\u00d710^-11 Pa^-1."
    },
    {
      "eq_number": 830,
      "title": "Diatom Stiffness Limit",
      "domain_id": 51,
      "significance": "Silica shells approach inorganic material limits. \u03ba_T \u2248 2.7\u00d710^-11 Pa^-1."
    },
    {
      "eq_number": 831,
      "title": "Vibrio Natriegens Replication Speed",
      "domain_id": 51,
      "significance": "Absolute biological replication speed limit: 10-15 minute doubling time."
    },
    {
      "eq_number": 831,
      "title": "Vibrio Natriegens Replication Speed",
      "domain_id": 51,
      "significance": "Absolute biological replication speed limit: 10-15 minute doubling time."
    },
    {
      "eq_number": 832,
      "title": "Pyrococcus Pressure-Volume Work",
      "domain_id": 51,
      "significance": "P\u00b7\u0394V > kT prevents protein unfolding. Obligate piezophile stability condition."
    },
    {
      "eq_number": 832,
      "title": "Pyrococcus Pressure-Volume Work",
      "domain_id": 51,
      "significance": "P\u00b7\u0394V > kT prevents protein unfolding. Obligate piezophile stability condition."
    },
    {
      "eq_number": 833,
      "title": "Desulforudis Energy Flux",
      "domain_id": 51,
      "significance": "Deep biosphere champion: 10^-15 W/cell energy flux, 1000-year division time."
    },
    {
      "eq_number": 833,
      "title": "Desulforudis Energy Flux",
      "domain_id": 51,
      "significance": "Deep biosphere champion: 10^-15 W/cell energy flux, 1000-year division time."
    },
    {
      "eq_number": 834,
      "title": "Landauer Limit",
      "domain_id": 51,
      "significance": "Minimum energy per bit erasure: E = kT ln(2)."
    },
    {
      "eq_number": 834,
      "title": "Landauer Limit",
      "domain_id": 51,
      "significance": "Minimum energy per bit erasure: E = kT ln(2)."
    },
    {
      "eq_number": 835,
      "title": "Resonant Cavity Q-Factor Limit",
      "domain_id": 51,
      "significance": "Material damping prevents infinite Q. Q_max \u2248 100 for biological tissue."
    },
    {
      "eq_number": 835,
      "title": "Resonant Cavity Q-Factor Limit",
      "domain_id": 51,
      "significance": "Material damping prevents infinite Q. Q_max \u2248 100 for biological tissue."
    },
    {
      "eq_number": 836,
      "title": "Turing Pattern Growth Limit",
      "domain_id": 51,
      "significance": "Finite nutrient flux prevents infinite growth in reaction-diffusion systems."
    },
    {
      "eq_number": 836,
      "title": "Turing Pattern Growth Limit",
      "domain_id": 51,
      "significance": "Finite nutrient flux prevents infinite growth in reaction-diffusion systems."
    },
    {
      "eq_number": 837,
      "title": "Navier-Stokes Blow-up Rejection",
      "domain_id": 51,
      "significance": "Evolutionary rejection of blow-up: infinite vorticity, zero compressibility, zero viscosity, infinite energy."
    },
    {
      "eq_number": 837,
      "title": "Navier-Stokes Blow-up Rejection",
      "domain_id": 51,
      "significance": "Evolutionary rejection of blow-up: infinite vorticity, zero compressibility, zero viscosity, infinite energy."
    },
    {
      "eq_number": 838,
      "title": "Thermococcus Pressure Adaptability",
      "domain_id": 51,
      "significance": "Widest pressure-range organism: 1 atm to 130 MPa adaptive flexibility."
    },
    {
      "eq_number": 838,
      "title": "Thermococcus Pressure Adaptability",
      "domain_id": 51,
      "significance": "Widest pressure-range organism: 1 atm to 130 MPa adaptive flexibility."
    },
    {
      "eq_number": 839,
      "title": "Thermus Moderate Thermophily",
      "domain_id": 51,
      "significance": "Moderate thermophile: 50-80\u00b0C (Taq polymerase source)."
    },
    {
      "eq_number": 839,
      "title": "Thermus Moderate Thermophily",
      "domain_id": 51,
      "significance": "Moderate thermophile: 50-80\u00b0C (Taq polymerase source)."
    },
    {
      "eq_number": 840,
      "title": "E. Coli Replication Reference",
      "domain_id": 51,
      "significance": "Baseline replication efficiency: 20 minutes optimal doubling, 4.6M bp genome."
    },
    {
      "eq_number": 840,
      "title": "E. Coli Replication Reference",
      "domain_id": 51,
      "significance": "Baseline replication efficiency: 20 minutes optimal doubling, 4.6M bp genome."
    },
    {
      "eq_number": 841,
      "title": "Rotational Phase Encoding",
      "domain_id": 51,
      "significance": "4-bit \u03c0 field encodes 16 rotational states (22.5\u00b0 resolution) for geometric information flow."
    },
    {
      "eq_number": 842,
      "title": "Chiral Alignment Coupling",
      "domain_id": 51,
      "significance": "Alignment strength between rotational states: A = cos(\u0394\u03b8). Determines information flow."
    },
    {
      "eq_number": 843,
      "title": "Manifold Blit Equation",
      "domain_id": 51,
      "significance": "Hardware-accelerated manifold update: M_{k+1} = Quant_LLM( J_DAG[ M_k \u2295 (\u03a8_q \u2297 R_RT) ] )."
    },
    {
      "eq_number": 844,
      "title": "Blitter Accumulation",
      "domain_id": 51,
      "significance": "Saturating bitwise accumulation: saturate(M_k + \u03b4) for discrete Picard integral."
    },
    {
      "eq_number": 845,
      "title": "Quantum Walk Amplitude",
      "domain_id": 51,
      "significance": "Discrete diffusion for quadratic convergence: A_{t+1} = (A_t \u2297 K) / 4."
    },
    {
      "eq_number": 846,
      "title": "Anisotropic Torsion Flow",
      "domain_id": 51,
      "significance": "\u2202_t \u03d5 = \u2207_i(M^ij \u2207_j \u03b4F/\u03b4\u03d5) - \u03c3 \u2202\u03d5/\u2202I_lock for manifold evolution."
    },
    {
      "eq_number": 847,
      "title": "Interlocking Energy",
      "domain_id": 51,
      "significance": "I_lock = w(1 - cos(k\u00b7frustration)) for recursive deposition snagging."
    },
    {
      "eq_number": 848,
      "title": "Spike Sync TVI",
      "domain_id": 51,
      "significance": "Temporal Variant Index for spike trains: coarse-grained timing/rate/pattern/collapse."
    },
    {
      "eq_number": 849,
      "title": "Coarse-Graining Rule",
      "domain_id": 51,
      "significance": "Quantize time into bins: t_bin = floor(t / \u0394t) for jitter tolerance."
    },
    {
      "eq_number": 850,
      "title": "Soliton Phase Singularity",
      "domain_id": 51,
      "significance": "Phase winding number +1 around soliton center: topological charge = vortex."
    }
  ]
}