Research-Stack/3-Mathematical-Models/MATH_MODEL_MAP.tsv

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1#Model_NameFamilyEquationVariablesPurposeLocationImplementedStatusCross_RefsDomain_TypeBind_Class
21Intrinsic_Load_LICognitive LoadL_I(x) = -Σ_{b=0}^{255} p(b|x) log₂ p(b|x)p(b|x) = empirical byte distributionShannon entropy of byte distribution; irreducible complexitycore/intrinsic/specs/COGNITIVE_LOAD_FUNCTIONS_SPEC.mdPython2,71,40LAYER_A_COMPRESSIONinformational_bind
32Extraneous_Load_LECognitive LoadL_E(x) = BPB(x,w_prior) - BPB*(x) = (1/n)Σ_i log₂(P_w*(x_i)/P_wprior(x_i))BPB=bits-per-byte, w_prior=prior weights, w*=optimalCost of architectural mismatch; penalty for suboptimal routingcore/intrinsic/specs/COGNITIVE_LOAD_FUNCTIONS_SPEC.mdPython1,3,6,8LAYER_A_COMPRESSIONinformational_bind
43Germane_Load_LGCognitive LoadL_G(x,t) = Σ_{s=1}^{S} γ^s · ΔL_E(x_s,t+1) ≈ τ·L_E·log(S+1)/log(S_max+1)γ=temporal discount, τ=trust, S=engram supportProductive learning effort reducing future extraneous loadcore/intrinsic/specs/COGNITIVE_LOAD_FUNCTIONS_SPEC.mdPython2,6,73LAYER_B_ROUTINGinformational_bind
54Routing_Load_LRCognitive LoadL_R(x) = Σ_j c_j·1[f_j computed] + Σ_{l=1}^{D(x)} log₂|M_l|d=9 feature dims, c_j=feature cost, D=tree depth, M_l=candidatesComputational cost of classification and method selectioncore/intrinsic/specs/COGNITIVE_LOAD_FUNCTIONS_SPEC.mdPython6,16-23LAYER_B_ROUTINGinformational_bind
65Memory_Load_LMCognitive LoadL_M(x) = log₂|E| + α·1[hit] + β + λ·|E|/|E_max||E|=engram store size, α/β=retrieval/update costs, λ=eviction pressureBurden of storing, retrieving, and updating routing memorycore/intrinsic/specs/COGNITIVE_LOAD_FUNCTIONS_SPEC.mdPython6,10,30LAYER_B_ROUTINGinformational_bind
76Total_Cognitive_LoadCognitive LoadL_total = λI·l̂I + λE·l̂E - λG·l̂G + λR·l̂R + λM·l̂M (Σλ=1, λG≤λE)5 normalized load components with weighting coefficientsAggregate processing burden; combines all load classescore/intrinsic/specs/COGNITIVE_LOAD_FUNCTIONS_SPEC.mdPython1,2,3,4,5,7,8,9,74LAYER_A_COMPRESSIONinformational_bind
87Cognitive_EfficiencyCognitive Loadη(x) = l̂I(x) / (l̂I + l̂E + l̂R + l̂M + ε)Ratio of intrinsic to total normalized loadRouting efficiency (1=perfect, 0=maximum waste)core/intrinsic/specs/COGNITIVE_LOAD_FUNCTIONS_SPEC.mdPython6LAYER_A_COMPRESSIONinformational_bind
98Regret_Adjusted_LoadCognitive LoadL_ρ(x) = L_total(x) · (1 + ρ(x)/ρ_max)ρ=BPB regret, ρ_max=max regretLoad penalized by historical performance on similar inputscore/intrinsic/specs/COGNITIVE_LOAD_FUNCTIONS_SPEC.mdPython2,6LAYER_A_COMPRESSIONinformational_bind
109Basin_Conditional_LoadCognitive LoadL(x|B) = L_I(x) + L_E(x|B) + L_R^B(x)w_B=basin-optimal weightsLoad conditioned on attractor basin membershipcore/intrinsic/specs/COGNITIVE_LOAD_FUNCTIONS_SPEC.mdPython1,2,4,12LAYER_A_COMPRESSIONinformational_bind
1110MoE_Predictor_DistributionCognitive LoadP_w(x_i|x_{<i}) = Σ_{j=1}^{k} w_j · P_{m_j}(x_i|x_{<i})k=predictors, w∈Δ^{k-1} simplexCombines multiple compression predictors into mixed distributioncore/intrinsic/specs/COGNITIVE_LOAD_FUNCTIONS_SPEC.mdPython1,2LAYER_A_COMPRESSIONinformational_bind
1211Pressure_PilingKDA PhysicsP(i) = P₀ · χ^i (χ ≈ 1.63)P₀=1.0 GPa, i=shock iteration, χ=Dodecahedral Piling GainSequential shock pressure amplification via Mach stem focusingcore/intrinsic/formalisms/9_KDA_Equation_Manifest.mdDocumented12,15LAYER_E_VERIFICATIONthermodynamic_bind
1312Hugoniot_TemperatureKDA PhysicsT_peak = T₀ · (P_peak/P₀)^0.65T₀=293.15K, γ≈1.66, exponent=0.65Non-isentropic temperature jump across shock fronts; predicts ~13,446Kcore/intrinsic/formalisms/9_KDA_Equation_Manifest.mdDocumented11,14,51LAYER_G_ENERGYthermodynamic_bind
1413Pressure_IonizationKDA Physicsα(P) = 1 - exp(-k(P - P_MIT))P_MIT≈138-155 GPa, k=rate constantInsulator-to-metal transition via electronic band overlapcore/intrinsic/formalisms/9_KDA_Equation_Manifest.mdDocumented15LAYER_G_ENERGYthermodynamic_bind
1514Energy_Recovery_EfficiencyKDA Physicsη_net = (W_rec - W_erasure) / W_in; W_erasure ≥ k_B·T·ln2 per bitk_B=Boltzmann, T=T_peak=13,446KMaxwell's Demon energy recovery after erasure costcore/intrinsic/formalisms/9_KDA_Equation_Manifest.mdDocumented12,54,55LAYER_E_VERIFICATIONthermodynamic_bind
1615Q_FactorKDA PhysicsQ = (E_flash + E_enthalpy + E_recovered - W_demon) / (E_work + E_loss) > 1.0Target Q≈1.05Global energy balance; net gain thresholdcore/intrinsic/formalisms/9_KDA_Equation_Manifest.mdDocumented11,12,13,14LAYER_E_VERIFICATIONthermodynamic_bind
1717Rotational_AlignmentGWL Rotationg = cos(Δθ·2π/16) · cos(Δφ·π/8) · (1 - 2|χ_i - χ_j|)θ∈{0..15}, φ∈{0..7}, χ∈{0,1}Frame orientation compatibilitydocs/gwl/GWL_ROTATIONAL_COUPLING_AND_LOCAL_INTERACTION_LAW_V1.mdRust16,25LAYER_C_TOPOLOGYgeometric_bind
1818Spatial_ProximityGWL Rotationh = exp(-|Δp|²/(2σ²)) · 1_{|Δp|<r_max}σ=coupling length, r_max=coupling radiusDistance decay functiondocs/gwl/GWL_ROTATIONAL_COUPLING_AND_LOCAL_INTERACTION_LAW_V1.mdRust16,25LAYER_C_TOPOLOGYgeometric_bind
1919Interaction_ForceGWL RotationF_ij = w_ij · (a_j - a_i) · Δp_ij / |Δp_ij|a=activation, Δp=position deltaActivation flow between neighborsdocs/gwl/GWL_ROTATIONAL_COUPLING_AND_LOCAL_INTERACTION_LAW_V1.mdRust16,20,21,22LAYER_H_ALGEBRAgeometric_bind
2020Energy_FunctionGWL RotationE(f) = -½ Σ_{i,j} w_ij · a_i · a_j + Σ_i V(a_i)V(a)=potential (e.g., λa⁴)Frame field energy landscapedocs/gwl/GWL_ROTATIONAL_COUPLING_AND_LOCAL_INTERACTION_LAW_V1.mdRust19,21,22,23LAYER_G_ENERGYgeometric_bind
2121Frame_Evolution_DiscreteGWL Rotationθ_i(t+1) = θ_i(t) + α_θ·F_{i,θ} + ξ_θα=learning rate, ξ=noiseDiscrete-time state updatedocs/gwl/GWL_ROTATIONAL_COUPLING_AND_LOCAL_INTERACTION_LAW_V1.mdRust19,22LAYER_H_ALGEBRAgeometric_bind
2222Frame_Evolution_ContinuousGWL Rotationdf_i/dt = -α ∇_{f_i} E(f) + ξ(t)α=step size, ξ(t)=Langevin noiseContinuous dynamics; gradient descent on energydocs/gwl/GWL_ROTATIONAL_COUPLING_AND_LOCAL_INTERACTION_LAW_V1.mdRust20,21,23LAYER_H_ALGEBRAgeometric_bind
2323Energy_Monotonicity_TheoremGWL RotationdE/dt = -α Σ_i |∇_{f_i} E|² ≤ 0Proves energy decreases monotonically; convergence guaranteedocs/gwl/GWL_ROTATIONAL_COUPLING_AND_LOCAL_INTERACTION_LAW_V1.mdRust20,22LAYER_H_ALGEBRAgeometric_bind
2424Temporal_WeightGWL Temporalw_ij^(τ) = cos(2π(τ_j - τ_i)/16)τ∈{0..15} (4 bits)Temporal phase coupling strengthdocs/gwl/GWL_TEMPORAL_DIMENSION_AND_T_VARIABLE_FORMALISM_V1.mdDocumented16,25,26LAYER_C_TOPOLOGYgeometric_bind
2525Complete_Weight_5FactorGWL Temporalw_ij = cos(Δθ·22.5°)·cos(Δφ·22.5°)·cos(2πΔτ/16)·(1-2|Δχ|)·exp(-|Δp|²/2σ²)θ,φ,τ,χ,Δp combinedFull spatial+temporal+rotational+chiral+proximity couplingdocs/gwl/GWL_TEMPORAL_DIMENSION_AND_T_VARIABLE_FORMALISM_V1.mdDocumented16,17,18,24,26LAYER_C_TOPOLOGYgeometric_bind
2626Temporal_ForceGWL TemporalF_ij^(τ) = w_ij^(τ) · (a_j - a_i) · sgn(τ_j - τ_i)Information flow from past to futuredocs/gwl/GWL_TEMPORAL_DIMENSION_AND_T_VARIABLE_FORMALISM_V1.mdDocumented24,25,27LAYER_C_TOPOLOGYgeometric_bind
2727Temporal_EvolutionGWL Temporalτ_i(t+1) = τ_i(t) + α_τ·F_{i,τ} + ω₀ω₀=intrinsic temporal frequencyTemporal phase evolution with intrinsic oscillatordocs/gwl/GWL_TEMPORAL_DIMENSION_AND_T_VARIABLE_FORMALISM_V1.mdDocumented26,28LAYER_C_TOPOLOGYgeometric_bind
2828Temporal_StabilityGWL Temporald/dt(τ_j - τ_i) = 0 for all i,jLocked temporal attractor conditiondocs/gwl/GWL_TEMPORAL_DIMENSION_AND_T_VARIABLE_FORMALISM_V1.mdDocumented27,29LAYER_D_INVARIANTSgeometric_bind
2929Temporal_EntropyGWL TemporalH_τ = -Σ_{k=0}^{15} p(τ=k) log₂ p(τ=k)0≤H_τ≤4 bitsMeasure of temporal disorder (0=synchronized, 4=max)docs/gwl/GWL_TEMPORAL_DIMENSION_AND_T_VARIABLE_FORMALISM_V1.mdDocumented28LAYER_C_TOPOLOGYgeometric_bind
3030Mu_Seed_CardinalityGWL State Space|S_μ| = 8×8×16×16×8×8×2×16×8×4 = 8,589,934,592 ≈ 2^3310 independent fieldsComplete enumeration of local mu-seed configuration spacedocs/gwl/GWL_TOTAL_STATE_SPACE_AND_CARDINALITY_LEDGER_V1.mdDocumented31,32LAYER_D_INVARIANTSgeometric_bind
3131Fractal_OccupancyGWL State Space|P_occ| = ρ_occ · N^{d_H} (d_H ≈ 2.7268)N=64 → |P_occ|≈84,000 vs 262,144 rawEffective addressable positions in Menger sponge latticedocs/gwl/GWL_TOTAL_STATE_SPACE_AND_CARDINALITY_LEDGER_V1.mdDocumented30,32,49LAYER_D_INVARIANTSgeometric_bind
3232Total_Formal_State_SpaceGWL State Space|S_total| ≈ 2^{5,900,000}μ-seed: 2,772,000 bits + edges: 2,016,000 + rotation: 924,000 + boot: 8,000 + registers: 133,000Upper bound on all possible GWPL configurationsdocs/gwl/GWL_TOTAL_STATE_SPACE_AND_CARDINALITY_LEDGER_V1.mdDocumented30,31,33,5LAYER_B_ROUTINGgeometric_bind
3333Reachable_State_SpaceGWL State Space|S_reachable| ≈ |S_total| / 10^{29}Constraints: geometric 10^6, convergence 10^{12}, energy 10^3, chirality 10^2, TTM 10^6State space satisfying all physical and computational constraintsdocs/gwl/GWL_TOTAL_STATE_SPACE_AND_CARDINALITY_LEDGER_V1.mdDocumented32,59LAYER_A_COMPRESSIONgeometric_bind
3434Wave_Packet_StateGWL ThroatP = (ΔV, Δt, π, τ, χ, C, A)7-tuple: volume, time, rotation, temporal, chirality, confidence, activationComplete state of computational packet traversing throatdocs/gwl/GWL_WAVE_PACKET_THROAT_ARCHITECTURE_V1.mdDocumented35,36LAYER_C_TOPOLOGYgeometric_bind
3535Throat_ConditionGWL ThroatT_throat = {(i,j) | Φ_topo(i,j) >> Φ_metric(i,j)}topological adjacency >> metric distanceDefines non-local transport corridordocs/gwl/GWL_WAVE_PACKET_THROAT_ARCHITECTURE_V1.mdDocumented34,36,37LAYER_C_BRAIDgeometric_bind
3636Multi_Factor_Coupling_WeightGWL Throatw_ij = w_p · w_π · w_τ · w_χ · w_topo · w_σ6 coupling factors (spatial, rotational, temporal, chirality, topological, stress)Comprehensive coupling evaluation during packet propagationdocs/gwl/GWL_WAVE_PACKET_THROAT_ARCHITECTURE_V1.mdDocumented25,34,35LAYER_C_BRAIDgeometric_bind
3737Holonomy_AccumulationGWL ThroatHol(γ_loop) = ∮_γ T(p) dpT(p)=torsion tensor, γ=closed loopPhase accumulated when transporting vector around closed loopdocs/gwl/GWL_WAVE_PACKET_THROAT_ARCHITECTURE_V1.mdDocumented38,78LAYER_C_BRAIDgeometric_bind
3838Non_Euclidean_DistanceGWL Throatd_N = path_length(γ_ij) + curvature_penalty(κ) + torsion_cost(T); d_T = d_E + λ_N·d_NEuclidean + topological combinedTopology-aware distance for routing through multi-manifold structuresdocs/gwl/GWL_MANIFOLD_INTERSECTION_JOIN_AND_TOPOLOGICAL_ADDRESSING_V1.mdDocumented37,63LAYER_C_TOPOLOGYgeometric_bind
3939Trixal_AxesThermodynamicTrixalAxes = (thermal, work, irreversibility), each ∈ [0,1]; |axes| = √(th² + w² + ir²)3D thermodynamic phase spacePhase space coordinates for process trackingcore/gwl-vm/src/thermo/mod.rsRust40,42,50LAYER_G_ENERGYthermodynamic_bind
4040Shannon_EntropyThermodynamicH = -Σ_b p(b) log₂ p(b), where p(b)=count(b)/lenByte distribution, H ∈ [0,8] bits/byteInformation content measurementcore/gwl-vm/src/thermo/entropy_engine.rsRust1,39,41,42,43LAYER_G_ENERGYthermodynamic_bind
4141Kolmogorov_EstimateThermodynamicK_est = (8 - H) / 8From Shannon entropyRough estimate of structure via compressibilitycore/gwl-vm/src/thermo/entropy_engine.rsRust40,42LAYER_G_ENERGYthermodynamic_bind
4242Thermodynamic_EntropyThermodynamicS_thermo = H + K_est · 0.1Shannon + Kolmogorov contributionCombined information thermodynamicscore/gwl-vm/src/thermo/entropy_engine.rsRust40,41,43,55LAYER_G_ENERGYthermodynamic_bind
4343Entropy_GradientThermodynamicdS/dt = (S_current - S_previous) / ΔtTime derivativeTracks rate of entropy change over timecore/gwl-vm/src/thermo/entropy_engine.rsRust42,55LAYER_H_ALGEBRAthermodynamic_bind
4444Mutual_Information_ExtractedThermodynamicMI = H_initial - H_currentDifference in Shannon entropyWork performed by compression; information extractedcore/gwl-vm/src/thermo/entropy_engine.rsRust40,71LAYER_A_COMPRESSIONthermodynamic_bind
4545Carnot_EfficiencyThermodynamicη_Carnot = 1 - T_cold / T_hotAbsolute temperaturesMaximum theoretical thermodynamic efficiencycore/gwl-vm/src/thermo/mod.rsRust46,51,54LAYER_B_ROUTINGthermodynamic_bind
4646Work_ExtractionThermodynamicW_actual = Q_absorbed · η_Carnot · 0.770% of Carnot limitModel computation as thermodynamic work extraction cyclecore/gwl-vm/src/thermo/heat_engine.rsRust45,55LAYER_C_TOPOLOGYthermodynamic_bind
4747Irreversibility_MetricThermodynamicscore = (entropy_production + path_asymmetry + time_reversal_violation) / 3Three irreversibility componentsQuantifies thermodynamic irreversibility of process trajectorycore/gwl-vm/src/thermo/process_shape.rsRust48,49LAYER_G_ENERGYthermodynamic_bind
4848Thermodynamic_LengthThermodynamicL_thermo = Σ_i distance_i · (1 + irreversibility_i)Weighted by dissipationDissipative trajectory length accounting for thermodynamic costcore/gwl-vm/src/thermo/process_shape.rsRust47,49LAYER_B_ROUTINGthermodynamic_bind
4949Thermodynamic_DepthThermodynamicdepth = entropy_production · ln(time_steps)Combined dissipation × trajectory lengthComplexity measurecore/gwl-vm/src/thermo/mod.rsRust31,47,48LAYER_G_ENERGYthermodynamic_bind
5050Stamp_CodeThermodynamicSHA256(axes || traj_hash || hardware_entropy || timing_jitter || process_nonce)Cryptographic hash of thermodynamic stateUnique non-reproducible process fingerprintcore/gwl-vm/src/thermo/trixalating_stamp.rsRust39,42LAYER_B_ROUTINGthermodynamic_bind
5151Arrhenius_Temperature_FactorInformatic StressAF = exp(E_a / (k_B · T))E_a=activation energy (eV), k_B=8.617e-5 eV/KFailure rate temperature accelerationcore/gwl-vm/src/thermo/informatic_stress.rsRust12,45,52,59LAYER_G_ENERGYthermodynamic_bind
5252Blacks_Equation_EMInformatic StressEM_risk = J^n · exp(E_a / (k_B · T)) / 10^{12}J=current density, n=exponent (2.0)Electromigration failure riskcore/gwl-vm/src/thermo/informatic_stress.rsRust51,59LAYER_G_ENERGYthermodynamic_bind
5353Coffin_Manson_FatigueInformatic StressCM_damage = (ΔT / ΔT_threshold)^m · 10^{-8}, m≈1.9Solder/interconnect fatigue exponentAccumulates damage from thermal cyclingcore/gwl-vm/src/thermo/informatic_stress.rsRust56,59LAYER_G_ENERGYthermodynamic_bind
5454Landauer_LimitInformatic StressW_erasure ≥ k_B · T · ln(2) ≈ 2.87e-21 J/bit at 300KMinimum energy per bit erasedThermodynamic lower bound on computationcore/gwl-vm/src/thermo/informatic_stress.rsRust14,45,55LAYER_A_COMPRESSIONthermodynamic_bind
5555Entropy_Generation_RateInformatic StressdS/dt = power_dissipation / (k_B · T · ln 2) [bits/s]Power dissipation rateRate of entropy generation from computational dissipationcore/gwl-vm/src/thermo/informatic_stress.rsRust14,42,43,46,54LAYER_E_VERIFICATIONthermodynamic_bind
5656BitFlip_Gradient_5DInformatic Stress|G| = (D1/120 + D2/5000 + D3/1000 + D4/100 + D5/5000) / 5D1=T_ambient, D2=V_jitter(nV), D3=φ_leak(μA), D4=ε_SEU(Hz), D5=dt_clock(ppm)Normalized 5D hardware stress gradientcore/gwl-vm/src/thermo/informatic_stress.rsRust53,57,59LAYER_D_INVARIANTSthermodynamic_bind
5757SEU_BitFlip_RateInformatic StressBFR = ε_SEU · 2^{(T-25)/10} · (1 + V_jitter/1000) · 3600 [flips/hr]SEU rate, temperature, voltage jitterEstimated bit-flips per hour from hardware conditionscore/gwl-vm/src/thermo/informatic_stress.rsRust56,59LAYER_G_ENERGYthermodynamic_bind
5858Stress_DecayInformatic Stressstress(t) = stress_0 · e^{-t/300} + intensity · (1 - e^{-t/300})5-minute half-lifeStress accumulation with exponential decaycore/gwl-vm/src/thermo/informatic_stress.rsRust59LAYER_G_ENERGYthermodynamic_bind
5959Remaining_Useful_LifeInformatic StressRUL = MTBF / (AF · (1 + fatigue·0.01 + thermal_fatigue·0.1))MTBF=10^9/FIT_rate, AF=acceleration factorHardware lifetime prediction under current stresscore/gwl-vm/src/thermo/informatic_stress.rsRust33,51,52,53,56,57,58LAYER_G_ENERGYthermodynamic_bind
6060Homeostatic_Stress_InjectionInformatic Stresssurprise = -ln(margin), margin = 1 - stress_magnitude; regret = max(0, stress - 0.5)Optimal stress ≈ 0.5Converts stress margin to surprise/regret for homeostatic controllercore/gwl-vm/src/thermo/informatic_stress.rsRust58,73LAYER_E_VERIFICATIONthermodynamic_bind
6161Exact_Int_to_FP_CastInformatic Stressint_bits ≤ fp_mantissa_bits + 1 (signed) or ≤ fp_mantissa_bits + 1 (unsigned)fp_mantissa=23 (f32) or 52 (f64)Proves safe integer-to-floating-point conversioncore/gwl-vm/src/thermo/informatic_stress.rsRust62,63LAYER_D_INVARIANTSthermodynamic_bind
6262Safe_Narrowing_ProofInformatic Stresscan_safely_narrow(src_bits, signed, dst_mantissa) → boolDelegates to exact cast checkProves double→single precision narrowing is safecore/gwl-vm/src/thermo/informatic_stress.rsRust61,63LAYER_D_INVARIANTSthermodynamic_bind
6363RISCV_Instruction_LatencyInformatic Stressfdiv.d=33, fdiv.s=19 → penalty=0.737; fadd.d=4, fadd.s=4 → penalty=0.0SiFive P550 measurementsPer-instruction latency for dispatch scoringcore/gwl-vm/src/thermo/informatic_stress.rsRust61,62LAYER_D_INVARIANTSthermodynamic_bind
6464Photon_EnergyQCL EnergyE = hc/λ = 1.2398 / λ_μm [eV]λ in μmWavelength-to-energy conversioncore/gwl-vm/src/thermo/qcl_energy_model.rsRust+Python65,70LAYER_G_ENERGYphysical_bind
6565Subband_SpacingQCL EnergyΔE = E_upper - E_lower; E_upper=hc/λ_min, E_lower=hc/λ_maxQCL emission range [λ_min, λ_max]Energy difference between subbands in conduction bandcore/gwl-vm/src/thermo/qcl_energy_model.rsRust+Python64,66,67LAYER_G_ENERGYphysical_bind
6666Cascade_GainQCL EnergyG = photons_per_e⁻ × n_wells; photons_per_e⁻ = ⌊E_electron / ΔE⌋E_electron=1.0 eV, n_wells=50Total photon amplification per cascading electroncore/gwl-vm/src/thermo/qcl_energy_model.rsRust+Python65,67,68LAYER_G_ENERGYphysical_bind
6767Temperature_TuningQCL Energyλ(T) = λ₀ + α · (T - T₀); α=5e-6 /K (GaAs/AlGaAs)dλ/dT ≈ 0.3 nm/KWavelength shift due to thermal expansion of quantum wellscore/gwl-vm/src/thermo/qcl_energy_model.rsRust+Python12,64,68LAYER_G_ENERGYphysical_bind
6868Injection_EfficiencyQCL Energyη = (0.5 + window_bonus) · spacing_eff · (1 - stress_penalty); clamped to [0,1]window_bonus=0.3 if in atmospheric windowDispatch efficiency at given stress levelcore/gwl-vm/src/thermo/qcl_energy_model.rsRust+Python65,66,67,69LAYER_G_ENERGYphysical_bind
6969Atmospheric_WindowsQCL EnergyWindows: (3,5), (8,12), (16,20) μm; transmission=1.0 inside, exp(-dist·0.5) outsideMWIR, LWIR, Far-IR bandsLossless carrier propagation bandscore/gwl-vm/src/thermo/qcl_energy_model.rsRust+Python68,70LAYER_G_ENERGYphysical_bind
7070Tuning_RangeQCL Energyν = 10⁴/λ [cm⁻¹]; DFB: ±7.5 cm⁻¹, EC: ±200 cm⁻¹; λ_min=10⁴/ν_max, λ_max=10⁴/ν_minCenter wavelength, tuning modeSpectral tuning capabilitycore/gwl-vm/src/thermo/qcl_energy_model.rsRust+Python64,69LAYER_G_ENERGYphysical_bind
7171Mutual_Information_SignalMI SignalMI(x) = baseline_bpb(x) - actual_bpb(x)Bits-per-byte differenceMeasures structural density in datacore/intrinsic/formalisms/ene_mi_signal.pyPython1,2,44,72,73LAYER_A_COMPRESSIONinformational_bind
7272kNN_MI_PredictionMI SignalMI_pred = Σ_i (w_i · MI_i · S_i) / Σ_i (w_i · S_i); w_i = 1/(d_i + ε)k-nearest neighbors in feature spaceLocal MI estimation from similar inputscore/intrinsic/formalisms/ene_mi_signal.pyPython71,73,74LAYER_B_ROUTINGinformational_bind
7373Surprise_MetricMI Signalsurprise = log(1 + |MI_actual - MI_predicted|)Log-scaled absolute differenceLearning trigger; prevents blowupcore/intrinsic/formalisms/ene_mi_signal.pyPython71,72,60LAYER_B_ROUTINGinformational_bind
7474Structure_YieldMI Signalρ(x) = MI(x) / (cost(x) + ε)Information per compute costROI of computation; high MI + low cost = valuablecore/intrinsic/formalisms/ene_mi_signal.pyPython6,71,72LAYER_A_COMPRESSIONinformational_bind
7575Weighted_Feature_DistanceMI Signald(z₁,z₂) = √Σ_i w_i · ((z₁_i - z₂_i) / s_i)²9-dim feature vector, learned weights, online scalesScale-normalized weighted distance in MI feature spacecore/intrinsic/formalisms/ene_mi_signal.pyPython4,72LAYER_B_ROUTINGinformational_bind
7676DAG_Force_EquilibriumDAG ForceΣ F_in = Σ F_out at every nodeDiscrete analogue of ∇·σ = 0Structural synthesis force balancecore/PTOS_FRAMEWORK/PAPER/paper.texTeX77,78LAYER_C_BRAIDgeometric_bind
7777Constitutive_LawDAG Forceσ = C : ε; ε = ½(∇u + ∇u^T)C=stiffness tensor, u=displacementStress-strain relationship; linear elasticitycore/PTOS_FRAMEWORK/PAPER/paper.texTeX76LAYER_C_BRAIDgeometric_bind
7878DAG_Global_ValidityDAG ForceV(G) = ∧_{v∈V} (ΣF_in = ΣF_out)Universal quantification over all nodesDAG constraint closure; proof by inductioncore/PTOS_FRAMEWORK/PAPER/paper.texTeX37,76,77LAYER_C_BRAIDgeometric_bind
7979Cosine_SimilarityBracket Braidcos = x · x_ref / (‖x‖ · ‖x_ref‖)Dot product of normalized vectorsReference solution alignmentaudit/benchmarks/benchmark_bracket_braid_sb.pyPython80LAYER_K_SIGNALgeometric_bind
8080Gradient_AlignmentBracket Braidalignment = ∇g_i · ∇g_j / (‖∇g_i‖ · ‖∇g_j‖)Cosine of gradient angleGradient coherence between bracketsaudit/benchmarks/benchmark_bracket_braid_sb.pyPython79,81LAYER_K_SIGNALgeometric_bind
8181Phase_AccumulationBracket Braidphase += Σ y · dxWork integral along trajectoryCumulative phase along computational pathaudit/benchmarks/benchmark_bracket_braid_sb.pyPython37,80LAYER_K_SIGNALgeometric_bind
8282Metric_Tensor_From_CircumferencesGWL Riemannian Geometryg_tt = M², g_tp = 0, g_pp = (N·cos(θ))²; a=C_eq/(2π), c=C_mer/(2π), f=(a-c)/a, e²=2f-f², N=a/√(1-e²sin²θ), M=a(1-e²)/(1-e²sin²θ)^(3/2)C_eq=equatorial circumference, C_mer=meridional circumference, θ=latitudeDerive metric tensor from measured circumferences (oblate spheroid)core/lean/geoweird/GWLKernel.leanLean83,84,87LAYER_C_TOPOLOGYgeometric_bind
8383Line_ElementGWL Riemannian Geometryds² = g_tt·dθ² + 2·g_tp·dθ·dφ + g_pp·dφ²g_ij=metric tensor, dθ/dφ=differentialsCompute infinitesimal distance on 2D Riemannian surfacecore/lean/geoweird/GWLKernel.leanLean82LAYER_C_TOPOLOGYgeometric_bind
8484Christoffel_Symbols_2DGWL ConnectionGamma^k_ij = ½·g^kl·(∂_i g_jl + ∂_j g_il - ∂_l g_ij)g=metric, gInv=inverse metric, dg=metric derivativesConnection coefficients for geodesic integrationcore/lean/geoweird/GWLKernel.leanLean82,85LAYER_C_TOPOLOGYgeometric_bind
8585Geodesic_Step_Symplectic_EulerGWL Geodesic Integrationa^θ = -(Gamma^θ_tt·v_θ² + 2·Gamma^θ_tp·v_θ·v_φ + Gamma^θ_pp·v_φ²); v' = v + a·dt; x' = x + v'·dtposition=(θ,φ), velocity=(v_θ,v_φ), Gamma=Christoffel, dt=stepSingle-step geodesic integration (symplectic Euler)core/lean/geoweird/GWLKernel.leanLean83,84LAYER_C_TOPOLOGYgeometric_bind
8686Stereographic_Chart_TransitionGWL Coordinate Chartsu = 2·tan(θ/2)·cos(φ), v = 2·tan(θ/2)·sin(φ); θ = 2·atan(√(u²+v²)/2), φ = atan2(v,u); select when |cos(θ)| < 0.01theta, phi, u, vAvoid pole singularities via adaptive coordinate chartcore/lean/geoweird/GWL_Integrated.leanLean85LAYER_C_TOPOLOGYgeometric_bind
8787Chirality_AlgebraGWL Chiral InteractionD+D→D, L+L→L, D+L→W(COLLAPSE→W); chiralityToTernary: D→Active, L→Active, W→LatentChirality∈{D,L,W}, GammaMode, TernaryStateChirality propagation through Gamma transform operationscore/lean/geoweird/GWL_Integrated.leanLeanLAYER_C_TOPOLOGYLAYER_C_TOPOLOGYgeometric_bind
8888BLINK_GATE_Ternary_ClockGWL Ternary Stateφ = (τ mod T)/T, T=942/1000; ternary: φ<1/3→Q, φ<2/3→A, else L; φ' = (φ+1/4) mod 1 if surprise>thresholdτ=time, surprise, threshold, T=0.942sTernary clock action with phase modulationcore/lean/geoweird/GWLKernel.leanLean90LAYER_C_TOPOLOGYcontrol_bind
8989Geodesic_Step_VerletGWL Geodesic Integration (Integrated)Same acceleration as M85 + Verlet velocity update + chart transition checkintegratorMethod∈{SymplecticEuler,Verlet,AdaptiveVerlet}Real geodesic integration with pole singularity handlingcore/lean/geoweird/GWL_Integrated.leanLean84,85,86LAYER_C_TOPOLOGYgeometric_bind
9090Waveprobe_Risk_FunctionWaveprobe Controlrisk(s) = (1 + γ·(1-cos(θ)))/d² + η·h; γ=3, η=0.8d=distance, θ=torsion angle, h=heatCombined risk metric for hysteretic controlcore/lean/geoweird/WaveprobeKernel.leanLean91,92LAYER_F_CONTROLcontrol_bind
9191Waveprobe_Heat_EvolutionWaveprobe Controlh' = α·h + β·a; α=0.95, β=0.2h=heat, a=excitation value (E1=1,E2=2,E3=3,E4=4)Heat accumulation with exponential decaycore/lean/geoweird/WaveprobeKernel.leanLean90LAYER_F_CONTROLcontrol_bind
9292Waveprobe_Mode_HysteresisWaveprobe Controllocked→caution if risk<B_recover=2; caution→locked if risk≥B_lock=5; normal→caution if risk≥B_warn=3B_recover=2, B_warn=3, B_lock=5Hysteretic mode transitions with risk barrierscore/lean/geoweird/WaveprobeKernel.leanLean90LAYER_F_CONTROLcontrol_bind
9394Universe_Collision_ConsensusEmergence SystemcanCollide: dim_a==dim_b AND type_a!=type_b; strength=min(vol_a,vol_b)/vol_intersection; invariant: κ_a·κ_b>0→SphereLike, <0→HyperbolicLiketwo universes with manifolds, types, axiomsCompute intersection of computational universes for witness generationcore/lean/geoweird/EmergenceSystem.leanLean93LAYER_E_VERIFICATIONcontrol_bind
9495Shannon_Type_Entropy_SwarmSwarm CoordinationH = -Σ(p_i·log₂(p_i)) where p_i=w_i/Σw; confidence update: reinforced=min(conf+0.1,1.0), decayed=conf·0.95learned types, weights, confidence threshold=0.8Measure uncertainty in agent's universe type assignmentcore/lean/geoweird/SwarmCoordination.leanLean94LAYER_B_ROUTINGcontrol_bind
9596Mean_Curvature_From_ConstraintsConstraint Geometrykappa = (cyclicRatio - treeRatio)·2.0; range [-2,2]cyclicRatio=count(Cyclic)/total, treeRatio=count(Hierarchical)/totalEstimate curvature from constraint topologycore/lean/geoweird/ConstraintGeometry.leanLean97LAYER_C_TOPOLOGYgeometric_bind
9697Universe_Type_ScoringConstraint GeometryscoreEuclidean=flatness+translation+noTime+linearGrowth; scoreHyperbolic=negativeCurve+exponential+treeLike; scoreSpherical=positiveCurve+compact+rotation; scoreLorentzian=time+causal+mixedMetric; each capped at 1.0constraint features (curvature, symmetry, causality, etc.)Score how well each universe type fits domain constraintscore/lean/geoweird/ConstraintGeometry.leanLean96LAYER_C_TOPOLOGYgeometric_bind
9798Homeostatic_Pressure_DynamicsHomeostatic Controls_t = α·surprise_t + β·regret_t; p_{t+1} = γ·p_t + s_t; surprise=-log(P_actual); regret=max(0,log(P_optimal)-log(P_actual))p=pressure, λ=canal width, s=stress, γ=decay∈(0,1), α,β weightsPressure-driven exploration-exploitation controldocs/HOMEOSTATIC_CONTROL_SPEC.texLaTeX99,100LAYER_B_ROUTINGcontrol_bind
9899Canal_DeformationHomeostatic Controlλ_t = λ₀·(σ + (1-σ)·e^{-ξ·p_t}); λ_t ∈ [λ₀·σ, λ₀]λ₀=base canal width, σ∈(0,1)=min fraction, ξ=sensitivityAdaptive selectivity controlled by pressuredocs/HOMEOSTATIC_CONTROL_SPEC.texLaTeX98LAYER_B_ROUTINGcontrol_bind
99100Decision_Reweighting_HomeostaticHomeostatic Controlscore_t = log(P_expected) - λ_t·s_tλ_t from canal deformation, s_t=stressReduce penalty on high-stress paths when pressure risesdocs/HOMEOSTATIC_CONTROL_SPEC.texLaTeX98,99LAYER_B_ROUTINGcontrol_bind
100101Homeostatic_EquilibriumHomeostatic Control(1-γ)·p* = s(p*); stability: |γ + s'(p*)| < 1p*=fixed point, γ=decay, s'=stress derivativeStability condition for homeostatic fixed pointdocs/HOMEOSTATIC_CONTROL_SPEC.texLaTeX98LAYER_B_ROUTINGcontrol_bind
101102Tree FiddyCombinatorial AnalysisL_max(k) = TREE(k) where k=3k=labels, L_max=maximum sequence lengthTheoretical upper bound for tree sequences under homeomorphic embedding; state space pruning shortcut via Kruskal's theoremdocs/semantics/TREE_FIDDY.mdLean30,32,33,95LAYER_D_INVARIANTSgeometric_bind
102103BHOCSBounded Hierarchical Cryptographic Spacedepth ≤ TREE(3); OuterMMR = H(InnerMMR.hash || InnerMMR.summary); NUVMAP = (u,v) = (distance·1000, spectral_index)depth=nesting depth, InnerMMR=orthogonal basis, OuterMMR=nested commitment, NUVMAP=compressed projectionBounded hierarchical orthogonal cryptographic space via Tree Fiddy × MMR-on-MMR × NUVMAP crossbreed; finite but unbounded verifiable hierarchy; GCL fractal fold surface support via nested MMR hierarchydocs/semantics/BHOCS.mdLean102,112,113,637,638,639,640,641,642,643,644,645,646,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230LAYER_D_INVARIANTSgeometric_bind
103102Hutter_Shape_EquationInformation TheoryH(W) = H₀ - I_local - I_structural - I_semantic - I_syntactic - I_cross; H₀=8.0, H(W)≈0.92 bpb (115 MB); I(W)=ΣI_k≈7.08 bpbH₀=uniform baseline (8 bpb), I_k=mutual information wedges per layerHierarchical entropy decomposition for compressiondocs/geometry/HUTTER_SHAPE_EQUATION.mdDocumented103LAYER_A_COMPRESSIONinformational_bind
104103Hutter_Manifold_DeformationInformation TheoryΔ_t = M_{t+1} - M_t; v(x,t) = Φ_{t+1}(x) - Φ_t(x); H(x,t)=‖v(x,t)‖²; ε=½(∇v+(∇v)^T)M_t=manifold state at release t, Φ_t=embedding, v=velocity, ε=strainModel Hutter corpus evolution as tensor field deformationdocs/roadmap/HUTTER_MANIFOLD_DYNAMICS.mdDocumented102,104LAYER_J_DYNAMICSinformational_bind
105104SHA256_Field_EquationsInformation TheoryΔ_manifold = μ_{t+1} - μ_t; ΔΣ = Σ_{t+1} - Σ_t; SHA256(μ_t‖Σ_t) → chain_hash; chunk_residual = f - μ_tμ_t=manifold mean, Σ_t=covariance, f=chunk featureTrack Wikipedia knowledge evolution as hash-chained manifold deltasdocs/roadmap/SHA256_FIELD_EQUATIONS.mdDocumented102,103LAYER_J_DYNAMICSinformational_bind
106105Dyson_Swarm_Geodesic_EquationDyson Swarm Geodesicsd²x^μ/dτ² + Gamma^μ_νλ(dx^ν/dτ)(dx^λ/dτ) = η^μ(τ); <η^μ(τ)η^ν(τ')> = 2D·δ^μν·δ(τ-τ')x^μ=position in data space, Q=QUBO matrix, Gamma=Christoffel, η=thermal noiseModel carrier state waves as geodesics on QUBO-derived data manifolddocs/geometry/DYSON_SWARM_GEODESIC_EQUATIONS_2026-04-09.mdDocumented84,85,106LAYER_C_TOPOLOGYgeometric_bind
107106Metric_From_QUBO_HessianDyson Swarm Geodesicsg_μν(x) = δ_μν + α·∂²E/∂x^μ∂x^ν = δ_μν + α·Q_μνQ=QUBO matrix, α=scaling, δ=Kronecker deltaDefine Riemannian metric from QUBO energy landscapedocs/geometry/DYSON_SWARM_GEODESIC_EQUATIONS_2026-04-09.mdDocumented105LAYER_C_TOPOLOGYgeometric_bind
108107Langevin_Geodesic_DampingDyson Swarm Geodesicsd²x^μ/dτ² + Gamma^μ_νλ(dx^ν/dτ)(dx^λ/dτ) = -γ(dx^μ/dτ) + √(2D)·ξ^μ(τ)γ=damping coefficient, D=diffusion, ξ=Gaussian white noiseDamped stochastic geodesic with thermal frictiondocs/geometry/DYSON_SWARM_GEODESIC_EQUATIONS_2026-04-09.mdDocumented105LAYER_C_TOPOLOGYgeometric_bind
109108Alcubierre_Information_MetricVirtual AlcubierredI² = -dτ² + (dH - β·dτ)²; β = v_eff·f·Ω; v_eff = v_local/(1-φ)τ=proper time, H=entropy, β=shift vector, φ=foam score, Ω=opcode couplingInformation-theoretic model for search acceleration via negative curvaturedocs/geometry/Virtual_Alcubierre_Information_Metric_Proof.mdDocumented83,98LAYER_G_ENERGYgeometric_bind
110110Ray_Casting_Braid_StepBraid TopologyR(s) = G_new + s·d⃗; collision: ‖R(s) - P_j‖ < T; D_topo = ∫_L σ(s)·P(s) dsG_new=new point, d⃗=direction, P_j=existing peak, T=threshold, σ=coherence, P=purityGeometric parent selection for Merkle braiding via raycastingdocs/formal_spec/RAYCAST_MMR_BRAID_TOPOLOGY.mdDocumented111,112LAYER_C_BRAIDgeometric_bind
111111Fast_Inverse_Square_RootBraid Topology0x5f3759df bit-hack for 1/d calculation; DIAT constraint: y_inv·y_sqrt ≈ C² where C=2^{16}k=index, y_inv=1/√k, y_sqrt=√k, 16.16 fixed-pointNear-constant-time intersection calculationdocs/formal_spec/THETA_TAN_RAYCAST_AMMR.mdDocumented110LAYER_C_BRAIDgeometric_bind
112112UVMAP_ProjectionBraid Topology32-bit packed texel: (v<<16)|u; U-axis: distance-based albedo (t×1000); V-axis: spectral frequency index (k from DIAT)t=distance, k=frequency indexEncode raycast intersections for stochastic UV map renderingdocs/formal_spec/THETA_TAN_RAYCAST_AMMR.mdDocumented110LAYER_C_BRAIDgeometric_bind
113113Boltzmann_Transport_Equation_ThetaTaNTheta-TaN Phonon Physicsκ_ph^{αβ} = (1/V)·Σ_{pq} C_V(pq)·v_{pq}^α·F_{pq}^β; C_V(pq)=ℏω·∂n₀/∂T; v_{pq}^α=∂ω/∂q_αC_V=mode heat capacity, v=phonon group velocity, F=mean free displacementModel lattice thermal conductivity in θ-phase TaN via phonon BTEdocs/formal_spec/THETA_TAN_RAYCAST_AMMR.mdDocumented114LAYER_G_ENERGYphysical_bind
114114Phonon_Scattering_MatrixTheta-TaN Phonon Physicsτ_{pq}^{-1} = τ_{3ph}^{-1} + τ_{4ph}^{-1} + τ_{ph-iso}^{-1} + τ_{ph-el}^{-1}3-phonon, 4-phonon, isotope, electron-phonon scattering channelsTotal phonon scattering rate as sum of independent channelsdocs/formal_spec/THETA_TAN_RAYCAST_AMMR.mdDocumented113LAYER_G_ENERGYphysical_bind
115115Tensor_Field_QUBONon-Euclidean UV QUBOF_u = -stress_xx·du - stress_xy·dv, F_v = -stress_yx·du - stress_yy·dv; wave = amplitude·e^{-dist·0.1}·cos(phase+dist); spin_i = 1 if wave>0 else 0tensor=[stress_xx,stress_xy,stress_yx,stress_yy] per grid pointMap QUBO to tensor field with wave-like carrier propagationaudit/benchmarks/benchmark_tensor_carrier.rsRust116,117LAYER_C_TOPOLOGYgeometric_bind
116116Non_Euclidean_UV_DistanceNon-Euclidean UV QUBOHyperbolic (Poincare): d = acosh(1 + 2·|p1-p2|²/((1-|p1|²)(1-|p2|²))); Spherical: d = acos(dot(p1,p2)); energy uses non-Euclidean distanceUV points, geometry type∈{Euclidean,Hyperbolic,Spherical}Map QUBO to non-Euclidean surface where frustration=curvatureaudit/benchmarks/benchmark_non_euclidean_uv.rsRust15,115LAYER_C_TOPOLOGYgeometric_bind
117117PGA_Cl301_NFold_MechanicsGeometric Algebra16D algebra: scalar(1)+vector(4)+bivector(6)+trivector(4)+pseudoscalar(1); B_i=I·e_i; SE(3) motor: M=R+ε(½·t·R); CFA: h_i^{l+1}=h_i^l+Σ_j M_{ij}^l·h_j^l·M_{ji}^lPGA multivectors, motors, curvature κ, torsion τUnified framework for protein structures using projective geometric algebradocs/geometry/GEOMETRIC_FIRST_LANGUAGE_Protein_NFold_Mechanics.mdDocumented84,96,105LAYER_H_ALGEBRAgeometric_bind
118118Sine_Gordon_EquationGeometric Algebrad²θ/dt² - c²·d²θ/ds² + (m²c⁴/ℏ²)·sin(θ) = 0θ=phase, c=wave speed, m=mass, ℏ=reduced Planck constantSoliton dynamics in protein backbonedocs/geometry/GEOMETRIC_FIRST_LANGUAGE_Protein_NFold_Mechanics.mdDocumented117LAYER_H_ALGEBRAgeometric_bind
119119Curvature_Torsion_Coupling_NFoldGeometric Algebraκ_n = f(κ_{n-1}, τ_{n-1}, S_n), τ_n = g(κ_n, τ_{n-1}, S_n); Γ^λ_{μν} = {Levi-Civita}^λ_{μν} + K^λ_{μν} (contortion)κ=curvature, τ=torsion, S_n=substrate properties, K=contortion tensorCurvature-torsion coupling in N-fold mechanicsdocs/geometry/GEOMETRIC_FIRST_LANGUAGE_Protein_NFold_Mechanics.mdDocumented117LAYER_H_ALGEBRAgeometric_bind
120120Hormone_Half_Life_to_Decay_RateHormone Derivationremaining = 2^{-t_sec/(T_half_min·60)}; decay_rate = 1 - remaining; Ebbinghaus: R(t) = e^{-t/S} where S = T_half/ln(2)T_half=biological half-life (min), pulse_interval_sec=10.0Convert biological half-life to computational decay ratelab/hormone_derivation.pyPython98,59LAYER_B_ROUTINGcontrol_bind
121121Logit_Z_NormalizationHormone Derivationlogit(x) = log(x/(1-x)); z = (logit(x) - mean_logit) / std_logitx∈[0,1], mean_logit, std_logit from calibrationTransform bounded concentration to z-scorelab/hormone_derivation.pyPython120LAYER_B_ROUTINGcontrol_bind
122122Dynamic_Amplification_FactorFEA Semi-TruckDAF = 1.0 + braking_factor·0.5 (1.0-1.5); load_steer = base_load·DAF; load_drive = base_load·(1-braking·0.2); patch area = load/tire_pressureGVWR, axle loads, tire pressure, braking factorMoving load simulation with dynamic amplificationlab/physics_fea/semi_truck_physics.pyPythonLAYER_L_APPLICATIONLAYER_L_APPLICATIONphysical_bind
123123Voxel_Key_EncodingHeerich Voxelkey = ((x+512)&0x3FF)<<20 | ((y+512)&0x3FF)<<10 | ((z+512)&0x3FF); inverse: decode bit fields; range [-512,511] per axis, 30-bitx,y,z integersFast 30-bit integer address for n-space positionstools/heerich_model.pyPython124LAYER_I_ENCODINGgeometric_bind
124124Microvoxel_Seed_4Byte_EncodingMicrovoxel Seed32-bit: delta_p[9:0]|region[13:10]|gamma[18:14]|activation[22:19]|polarity[26:23]|confidence[30:27]|flag[31]; switch: eff<0.8→EXCLUDE, <1.2→EXPLORE, ≥1.2→PROMOTEdelta_p, region, gamma, activation, polarity, confidenceCompressed self-evolving program via ternary switch encodinglab/microvoxel_seed_self_evolving_encoder.pyPython1,123LAYER_I_ENCODINGcontrol_bind
125125DCVN_Verification_Invariant_SurvivalDCVN Verification4 invariants: completeness(c), consistency(s), freshness(f), provenance(p); survival mask: bit set if value≥threshold; participation: 4→FULL(1.0), ≥2→PARTIAL(0.5), ≥1→OBSERVER(0.1), 0→ABSENT(0.0); variance=var([c,s,f,p])/4c,s,f,p∈[0,1], thresholdsGraduated participation based on invariant health in verification swarmlab/consensus_net/verification_core.pyPython10,11LAYER_E_VERIFICATIONcontrol_bind
126126Watanabe_Total_CorrelationTensor Field AnalysisTC = ΣH(X_i) - H(X_1,...,X_n); KSG estimator for MI; Kolmogorov: len(compressed)/len(original); formula complexity: 0.4·kolmogorov + 0.4·entropy/8 + 0.2·CVfeature vectors, variant formulas, Hutter submission dataRigorous tensor field analysis of Hutter variantslab/hutter/tensor_field_analysis_rigorous.pyPython102,103LAYER_A_COMPRESSIONinformational_bind
127127Neural_Delta_GCL_Field_EquationCompressionx = DeltaGCL(m); q_θ(z|x) = N(μ_θ(x), diag(σ²_θ(x))); z = μ_θ(x) + σ_θ(x) ⊙ ε, ε~N(0,I); x̂ = g_φ(z); L = D(x,x̂) + β·KL(q_θ(z|x)||N(0,I)); R_total = R_ΔGCL·R_neuralm=raw metadata, x=Delta GCL compressed, z=latent (64-dim), f_θ=VAE encoder, g_φ=decoder, D=reconstruction loss, KL=KL divergence, β=1e-3, R_ΔGCL∈[0.01,0.08], R_neural∈[0.3,0.5]Canonical VAE-style neural compression on Delta GCL; encoder regularized toward standard normal prior; safety: verify x̂≈x and DeltaGCLDecode(x̂) preserves invariant before commit; Delta GCL=lawful base codec, neural=learned transport compressor, verifier=semantic authoritydocs/papers/NEURAL_COMPRESSION_ON_DELTA_GCL.mdDocumented127LAYER_A_COMPRESSIONinformational_bind
128128Riemann_Zeta_Pair_CorrelationNumber TheoryR₂(x) = 1 (sin(πx)/(πx))²R₂=pair correlation, x=normalized spacingMontgomery-Odlyzko law: Riemann zeta zero spacings correspond to GUE eigenvalue spacings; verified to billions of zerosdocs/math/cross_domain_structures.mdDocumented128LAYER_A_COMPRESSIONinformational_bind
129129Zeta_As_Partition_FunctionStatistical Mechanicsζ(s) = Σ_{n=1}^∞ n^{-s} ↔ Z(β) = Σ_n e^{-βE_n}s=complex variable, β=inverse temperature, E_n=energy levelsAnalytic number theory ↔ equilibrium statistical mechanics; Bost-Connes model; phase transition at β=1 mirrors pole at s=1docs/math/cross_domain_structures.mdDocumented129LAYER_G_ENERGYthermodynamic_bind
130130Gutzwiller_Trace_FormulaQuantum Chaosd(E) = d̄(E) + (1/πℏ) Σ_p Σ_{r=1}^∞ (A_{p,r}/√|det(M_p^r I)|) cos(rS_p(E)/ℏ rμπ_p/2)d=density of states, E=energy, ℏ=Planck constant, p=periodic orbit, r=repetition, A=amplitude, M=monodromy, S=action, μ=Maslov indexBridges classical periodic orbits to quantum spectral properties; structural parallel to explicit formula of prime number theory; motivates Hilbert-Pólya conjecturedocs/math/cross_domain_structures.mdDocumented130LAYER_C_TOPOLOGYgeometric_bind
131131Explicit_Formula_Prime_NumberNumber Theoryψ(x) = x Σ_ρ x^ρ/ρ log(2π) 1/2 log(1x^{2})ψ=Chebyshev function, x=real variable, ρ=Riemann zeta zerosExplicit formula of prime number theory; structural parallel to Gutzwiller trace formula; connects quantum chaos to additive number theorydocs/math/cross_domain_structures.mdDocumented131LAYER_A_COMPRESSIONinformational_bind
132132Heat_EquationPhysics∂T/∂t = α ∇²TT=temperature, t=time, α=thermal diffusivity, ∇²=LaplacianCaloric theory of heat (superseded) but equation remains valid; demonstrates persistence of mathematical structure despite physical theory replacementdocs/math/cross_domain_structures.mdDocumented132LAYER_G_ENERGYphysical_bind
133133Wave_EquationPhysics∇²E - (1/c²)∂²E/∂t² = 0E=electric field, c=speed of light, ∇²=Laplacian, t=timeLuminiferous ether theory (superseded) but equation remains valid; demonstrates persistence of mathematical structure despite physical theory replacementdocs/math/cross_domain_structures.mdDocumented133LAYER_C_TOPOLOGYgeometric_bind
134134Euler_Divergent_SeriesNumber Theory1 + 2 + 3 + 4 + ... = -1/12ζ(-1) = -1/12 via analytic continuationHeuristic result validated through string theory and Casimir effect; exemplifies paramathematical mathematics that obeys physical laws despite appearing "wrong"docs/math/cross_domain_structures.mdDocumented134LAYER_A_COMPRESSIONinformational_bind
135135P_Adic_MetricNumber Theory|x + y|_p ≤ max(|x|_p, |y|_p)|x|_p=p-adic absolute value, p=primeNon-Archimedean metric for Planck-scale physics; strong triangle inequality; enables adelic formulations bridging real and p-adic worldsdocs/math/cross_domain_structures.mdDocumented135LAYER_D_INVARIANTSgeometric_bind
136127Relation_Sieve_5_SymbolCache Sievepack 5 2-bit symbols into 10-bit: sig=(T<<8)|(D<<6)|(C<<4)|(A<<2)|R; torsion: value<t0→00,<t1→01,<t2→10,else 11; coherence inverted: ≥c2→00,≥c1→01,≥c0→10,else 11; classify: REJECT if T=11,A=11,C=11,(T≥10&C≥10),(D=11&A≥10),(R=11&C≥10); HOLD if any=10 or C∈{01,10}; else PASST=torsion, D=drift, C=coherence, A=angular momentum, R=radiusStructural classification of cache manifolds into PASS/HOLD/REJECTtools/cache_sieve_exploration.pyPython115,128LAYER_D_INVARIANTScontrol_bind
137128Proxy_Extraction_From_ManifoldCache Sievetorsion = Σ|τ_i|(i<32)·100; drift = |φ_corr|·255; coherence = 255-torsion; angmom from geometry plugin; radius = |radius-1.0|·255τ=torsion gradient samples, φ_corr=correlationManifold features to discrete scalarstools/cache_sieve_exploration.pyPython127LAYER_D_INVARIANTScontrol_bind
138130Recursive_StructureHyperFabricP_{m+1} = R(P_m) ∩ C_mR=refinement operator, C_m=cooling constraintHierarchical structure refinement with constraint filteringtools/lean/Semantics/Semantics/HachimojiPipeline.leanLean129LAYER_C_TOPOLOGYgeometric_bind
139132Loss_FunctionUpdate_RuleL = λ₁ N_unresolved + λ₂ N_revisited + λ₃ N_degenerate + λ₄ T_updateλ=weights, N=counts, T=timeOptimization objective for search processtools/lean/Semantics/Semantics/HachimojiPipeline.leanLean130LAYER_B_ROUTINGinformational_bind
140133Continuity_EquationTetris_Manifold∂_t ρ + ∇·(ρv) = 0ρ=density, v=velocityMass conservation in fluid-like systemstools/lean/Semantics/Semantics/HachimojiPipeline.leanLean134LAYER_C_TOPOLOGYphysical_bind
141134Momentum_BalanceTetris_Manifoldρ(∂_t v + v·∇v) = -∇p + ∇·σ - ρ∇G + f_alignρ=density, v=velocity, p=pressure, σ=stress, G=potential, f_align=alignment forceNewton's second law for continuum mechanicstools/lean/Semantics/Semantics/HachimojiPipeline.leanLean133,135LAYER_C_TOPOLOGYphysical_bind
142136Nutrient_StateUpdate_RuleN = (N_local, N_indexed, N_committed), decay_rateThree-tier nutrient: local (fast weak), indexed (stronger reusable), committed (slow reserve)Resource allocation with temporal hierarchytools/lean/Semantics/Semantics/HachimojiPipeline.leanLean137,138LAYER_B_ROUTINGcontrol_bind
143137Nutrient_Gain_LawUpdate_RuleΔN = α₁ U_crc + α₂ U_btree + α₃ U_ammrα=weights, U=utility from CRC, B-tree, AMMR supportNutrient accumulation from validated successtools/lean/Semantics/Semantics/HachimojiPipeline.leanLean136,138LAYER_B_ROUTINGcontrol_bind
144138Nutrient_Decay_LawUpdate_RuleN(t+1) = (1-λ)N(t) - cost, λ_C < λ_I < λ_Lλ=decay rate, cost=signaling/duty expenditureStructural shedding/pruning like leaves falling from branchestools/lean/Semantics/Semantics/HachimojiPipeline.leanLean136,137LAYER_B_ROUTINGcontrol_bind
145139Total_Stress_TensorHyperFabricΣ = Σ_phase + Σ_elastic + Σ_torsional + Σ_lockingΣ_phase=phase stress, Σ_elastic=fold-back stress, Σ_torsional=torsion stress, Σ_locking=interlocking stressCombined stress from multiple sources in field dynamicstools/lean/Semantics/Semantics/HachimojiPipeline.leanLean129LAYER_G_ENERGYphysical_bind
146140Energy_Dissipation_RateHyperFabricdF/dt = (F_current - F_previous)/dtF=energy, dt=time stepRate of energy decrease; dF/dt ≤ 0 for dissipative systemstools/lean/Semantics/Semantics/HachimojiPipeline.leanLean139LAYER_G_ENERGYthermodynamic_bind
147129SEISMIC_Shell_DetectionTopological Encoder0.35 ≤ φ_corr < 0.47; PHI-offset retry: φ_offset = φ_corr·PHI mod 1.0; SHA256 pattern hash: phi_weighted_profile[i]=τ_i·PHI^{-i/100}; hash=SHA256(φ_corr:path_length:Σphi_weighted_profile); orientation=(φ_corr·PHI) mod 2πφ_corr, τ=torsion, PHI=golden ratioDetect SEISMIC torsion shells for "The Wall" constructiontools/topological_encoder.pyPython19,37LAYER_I_ENCODINGgeometric_bind
148130Half_Mobius_Closure_IntegralTopological Encoder∮τ·ds = π; solve: accumulated = Σ(τ_i·step_size); return path_length s when accumulated ≥ πτ=torsion gradient samples, step_size, φ_corr=correlationHalf-twist closure condition for topologically stable walltools/topological_encoder.pyPython41,129LAYER_C_BRAIDgeometric_bind
149131Regret_Field_Blink_CycleKDA ControlR_magnitude = surprise·(predicted_bpb - actual_bpb)/actual_bpb; blink_duration = baseline_ms + (regret_ms - baseline_ms)·R_magnitude ∈ [500ms,700ms]; decay: R *= e^{-λ·dt}, λ=2.0surprise, predicted/actual bpb, blink baseline/regret, decay lambdaAdaptive blink cycle control via regret field magnitudelab/CONCEPT_STAGE/2_KDA_Control_Kernel_Source.pyPython88,90,25LAYER_F_CONTROLcontrol_bind
150132Hugoniot_Shock_PhysicsKDA ControlRankine-Hugoniot relations: T-P relationship for shock compression; transient vs equilibrium temperature; E_kinetic = ½·I·ω² (plasma flywheel); E_harvested = E_stored·efficiencyPressure(GPa), temperature(K), moment of inertia, RPMThermodynamic state computation for plasma substratelab/CONCEPT_STAGE/2_KDA_Control_Kernel_Source.pyPython25,98LAYER_F_CONTROLcontrol_bind
151133Waveprobe_Binning_FunctionsWaveprobe ControlbinDistance: d<0.5→near(0), d<1→mid(1), else far(2); binTorsion: |sin(θ)|<1/3→aligned(0), <2/3→skewed(1), else twisted(2); binHeat: h<2→cool(0), h<5→warm(1), else hot(2)d=distance, θ=torsion angle, h=heat3-category binning for LUT policy indexingcore/lean/geoweird/WaveprobeKernel.leanLean90LAYER_F_CONTROLcontrol_bind
152134Waveprobe_LUT_PolicyWaveprobe ControlbaseLUT: 3×3×3 lookup mapping (distBin,torsionBin,heatBin)→Excitation; applyMode: normal=pass, caution=cap E2, locked=force E1; nextMode: hysteretic transitions by risk vs B_recover/B_warn/B_lock3 bins × 3 bins × 3 bins = 27 entriesBase control policy with mode overridecore/lean/geoweird/WaveprobeKernel.leanLean90,133LAYER_F_CONTROLcontrol_bind
153135Parallel_Transport_WritheNon-Euclidean Geometryproject to oblique 2D: (x+z·cos(π/4)·0.5, y+z·cos(π/4)·0.5); deltas=consecutive differences; wr = Σ(ax·by - ay·bx)/(n-1)history of nd_points, window W=16, dox=doy=cos(π/4)·0.5Parallel-transport writhe in curved PCA manifoldtools/geometry_noneuclidean.pyPython35,51LAYER_C_TOPOLOGYgeometric_bind
154136NE_Path_ValidationGeometry VerifierPHI-weighted distance: d = √(Σ w_i·(a_i-b_i)²) where w_i=PHI^{-i}; metric continuity: max_jump>5.0→fail; writhe bound: |writhe|>2.0→fail; parallel transport: |sim-prev_sim|>1.5→instability14D concept vectors, PHI weights, path pointsValidate paths through non-Euclidean concept spacetools/geometry_verifier.pyPython37,51,135LAYER_C_TOPOLOGYgeometric_bind
155137Hormone_Concentration_NormalizationHormone Derivationnormalized = (value - species_min)/(species_max - species_min); modulation = effect_size·(1 - CV_species); CV = std/mean across speciesconcentration values, species min/max, effect sizes, cross-species varianceNormalize biological concentrations to computational 0-1 rangelab/hormone_derivation.pyPython120LAYER_B_ROUTINGcontrol_bind
156138Thermal_Finality_BEABEA Thermo Bridgeconfidence = min(bea_confidence + avg_irreversibility/n, 1.0); time_to_reverse = exp(total_irreversibility·100); weight = work_done/(1+entropy)·(1-0.5·irreversibility)bea_confidence, irreversibility array, work_done, entropyIntegrate thermodynamic process tracking with BEA consensuscore/gwl-vm/src/thermo/bea_thermo_bridge.rsRust21,11,27LAYER_E_VERIFICATIONthermodynamic_bind
157139Entropy_Generation_Rate_InformaticInformatic StressdS/dt = power_dissipation/(k_B·T·ln(2)) [bits/s]; k_B=8.617e-5 eV/K; LANDAUER_LIMIT_300K=2.87e-21 J/bitpower_dissipation, temperature, Boltzmann constantRate of entropy generation from computational dissipationcore/gwl-vm/src/thermo/informatic_stress.rsRust55,14,54LAYER_G_ENERGYthermodynamic_bind
158140Remaining_Useful_Life_PredictionInformatic StressRUL = MTBF/(AF·(1+fatigue·0.01+thermal_fatigue·0.1)); MTBF=10^9/FIT_rate; AF=Arrhenius acceleration factorMTBF, acceleration factor, fatigue, thermal fatigueHardware lifetime prediction under current stresscore/gwl-vm/src/thermo/informatic_stress.rsRust51,52,53,56,57,58LAYER_G_ENERGYthermodynamic_bind
159141PBACS_1bit_TransportPBACS Signalb_t = 1[v_t + e_{t-1} > θ_t]; e_t = v_t + e_{t-1} - b_tv_t=input, e_t=error accumulator, θ_t=threshold1-bit noise-shaped encoding with error feedbackdocs/semantics/PBACS_CANONICAL_SIGNAL_ARCHITECTURE.mdSpec142,143,144,145LAYER_K_SIGNALcontrol_bind
160142PBACS_Phi_TraversalPBACS SignalΦ_{t+1} = Φ_t + 106070 (mod 2^32); idx = (Φ_t >> n) ⊕ MSB_flipΦ=accumulator, 106070=φ·2^16, MSB=mirror bitDeterministic φ-based scheduling for uniform coveragedocs/semantics/PBACS_CANONICAL_SIGNAL_ARCHITECTURE.mdSpec141,143LAYER_F_CONTROLcontrol_bind
161143PBACS_Void_Mask_LUTPBACS Signalθ_t = LUT_void[idx]; attest = popcount(LUT[i] ∧ deviation)LUT_void=8Kbit blue noise mask, idx=traversal indexStructured thresholding replaces arithmetic transformsdocs/semantics/PBACS_CANONICAL_SIGNAL_ARCHITECTURE.mdSpec141,142,145LAYER_K_SIGNALcontrol_bind
162144PBACS_SLUQ_StressPBACS Signala_{t+1} = a_t - (a_t >> 6) + α|e_t| + β|μ| + γ·attesta=stress accumulator, e=residual, μ=density mismatchAdaptive stress accumulation with decaydocs/semantics/PBACS_CANONICAL_SIGNAL_ARCHITECTURE.mdSpec141,146LAYER_F_CONTROLcontrol_bind
163145PBACS_CMYK_RoutingPBACS Signals_t = a_t >> 14; π(s_t) ∈ {K,C,M,Y} for [0,1,2,3]s=2-bit state, K=fast, C=monitor, M=verify, Y=prunePolicy-based adaptive routing with 4-state hysteresisdocs/semantics/PBACS_CANONICAL_SIGNAL_ARCHITECTURE.mdSpec144LAYER_B_ROUTINGcontrol_bind
164146PBACS_BracketedDIATPBACS Signal = ⟨l,u,v,g_l,g_u⟩ with g_l + g_u = u - ll,u=bounds, v=value, g=gap, scaleConstraint-preserving interval arithmetic with gap conservationtools/lean/Semantics/Semantics/BracketedCalculus.leanLean141,147LAYER_D_INVARIANTScontrol_bind
165147PBACS_Unified_UpdatePBACS SignalX_{t+1} = F(X_t) where X = (Φ,e,a,s,b,)Unified state vector with 8-step canonical loopComplete PBACS signal transport with all 5 layersdocs/semantics/PBACS_CANONICAL_SIGNAL_ARCHITECTURE.mdSpec141,142,143,144,145,146LAYER_F_CONTROLcontrol_bind
166148LUT_as_DSPFPGA Signalφ_corr = LUT_void[(Φ_acc >> n) ⊕ MSB_flip]Φ_acc=32-bit accumulator, MSB_flip=mirror detectionLUT-based DSP replacing floating-point with table lookupsdocs/semantics/LUT_AS_DSP_EQUATION.mdSpec142,149LAYER_K_SIGNALcontrol_bind
167149Mirror_LUT_UnifiedFPGA Signal*(q,s,t) = ⌊φ^{α·ℋ(q,s) + β·𝒯(t)}·2^n⌋ ⊕ (q)q=query, s=state, t=time, =hash, =foldCross-domain mirror LUT unificationdocs/semantics/MIRROR_LUT_EQUATIONS.mdSpec142,148LAYER_K_SIGNALcontrol_bind
168150Adaptive_1bit_CMYKFPGA Signalstress_t = α|e_t| + β|μ| + γ·popcount; s_t = a_t >> 141-bit encoder + CMYK stress routing mergedAdaptive 1-bit transport with stress-based state machinedocs/semantics/ADAPTIVE_1BIT_CMYK_MERGED.mdSpec141,145,147LAYER_K_SIGNALcontrol_bind
169151Extended_Cognitive_Load_v2Cognitive LoadL_cog(x) = L_comp(x) + λ_D·E_dist(x); L_comp = λ_H·H(x) + λ_B·B(x) + λ_R·R(x); E_dist = μ_κ·Δκ(x) + μ_Θ·ΔΘ(x) + μ_Λ·ΔΛ(x) + μ_ρ·Δρ(x)H=entropy, B=branching, R=recursion depth, Δκ=substrate mismatch, ΔΘ=temporal mismatch, ΔΛ=structural mismatch, Δρ=saturation imbalancePressure-adaptive cognitive load: complexity + distortion energy to maintain coherencedocs/semantics/PBACS_COGNITIVE_LOAD_REVISION.mdSpec6,7,8,152,153LAYER_A_COMPRESSIONinformational_bind
170153Release_Index_RManifold DynamicsR(x) = E_dist(x) / (ρ_avail(x) + ε)E_dist=distortion energy, ρ_avail=effective available resource flow, ε=small constantCritical threshold: R<1 gradual adaptation, R≥1 discontinuous reconfiguration/throat/resetdocs/semantics/PBACS_COGNITIVE_LOAD_REVISION.mdSpec151,152LAYER_F_CONTROLcontrol_bind
171154PBACS_Stress_v2PBACS Signalstress_t = α|e_t| + β|μ̂_t - μ*_t| + γ·attest_t + δ·Ê_dist(x_t)e=residual, μ=density mismatch, attest=attestation, Ê_dist=stored manifold mismatch energyUpgraded stress: reacts to signal AND stored manifold mismatch energydocs/semantics/PBACS_COGNITIVE_LOAD_REVISION.mdSpec144,151,145LAYER_F_CONTROLcontrol_bind
172155Cognitive_Load_Revised_AdmissibilityCognitive LoadL_cog(x) ≤ L_max ∧ R(x) < 1; violation modes: L>L_max→NoSurface, R≥1→Forced Transition/Throat/ResetL_max=maximum load, R=release index, NoSurface/Throat=failure modesTwo-constraint admissibility: load AND release index must both be satisfieddocs/semantics/PBACS_COGNITIVE_LOAD_REVISION.mdSpec151,153LAYER_E_VERIFICATIONcontrol_bind
173156Normalized_Cognitive_Load_v2Cognitive LoadL̂_cog(x) = λ_H·Ĥ(x) + λ_B·B̂(x) + λ_R·R̂(x) + λ_D·Ê_dist(x); Ê_dist = μ_κ·Δ̂κ + μ_Θ·Δ̂Θ + μ_Λ·Δ̂Λ + μ_ρ·Δ̂ρAll components normalized [0,1], λ weights sum to 1Operational normalized form for simulation and thresholdingdocs/semantics/PBACS_COGNITIVE_LOAD_REVISION.mdSpec151LAYER_A_COMPRESSIONinformational_bind
174157Phonon_Graph_ForcePhonon GraphF(i, j) = e^(-d/127) * cos(2πd/127); d = |i - j|d=distance between positions, 127=φ⁷≈29.03→period, e^(-d/127)=dampingDamped oscillating force law holding engrams together; self-contained, 32 charsdocs/semantics/PHONON_GRAPH_EQUATION.mdSpec6,152LAYER_K_SIGNALphysical_bind
175158Phonon_Damage_TracePhonon GraphDamagePattern(i) = F(i, i₀) ⊕ F(i, i₁) ⊕ ... ⊕ F(i, iₙ)i₀...iₙ=reference positions, ⊕=XOR of force signaturesJunk DNA pattern: repeating phonon force signature attached to code sectionsdocs/semantics/PHONON_DAMAGE_TRACE.mdSpec157LAYER_E_VERIFICATIONinformational_bind
176159Phonon_Rederivation_MechanismPhonon GraphHowDamaged = argmin_D ||ObservedPattern - ExpectedPattern(D)||; D ∈ {bit_flip, shift, truncation, swap}D=damage type, ExpectedPattern(D)=phonon signature under damage model DPost-hoc damage reconstruction: compare observed to expected under each damage modeldocs/semantics/PHONON_REDERIVATION.mdSpec157,158LAYER_E_VERIFICATIONcontrol_bind
177160Emoji_Machine_CoreEmoji MachineLUT[s] → (emit, next); next = emit; s,emit,next ∈ UnicodeLUT=lookup table, s=current state/codepoint, emit=output codepointQuine-like self-referential machine: output IS next state addressdocs/semantics/EMOJI_MACHINE.mdSpec⚠️-LAYER_K_SIGNALcontrol_bind
178161Emoji_Machine_BoundedEmoji MachineLUT[s] → (emit, next); next = emit; s,emit,next ∈ Fin 65536 (BMP only)LUT=64K entries, s=16-bit state index, emit=output codepointFinite version: restricted to Basic Multilingual Plane (16-bit)docs/semantics/EMOJI_MACHINE.mdSpec160LAYER_K_SIGNALcontrol_bind
179162Emoji_Machine_Recovery_ClaimEmoji Machine∀ positions p in trajectory T: LUT[p] must equal the unique successor in TT=trajectory/cycle, p=position in state space, LUT[p]=required transitionStructural constraint: valid trajectories enforce their own consistencydocs/semantics/EMOJI_MACHINE.mdSpec⚠️160,161LAYER_E_VERIFICATIONcontrol_bind
180163Prime_Hash_WatermarkCombined SystemEvery P bits: embed prime π where π = HashToPrime(H(chunk)); H=SHA256, chunk=preceding P bitsP=period (≈127=φ⁷), π=prime encoding hash, chunk=bit segmentPeriodic prime watermarks for integrity verification; primes encode hash of preceding streamdocs/semantics/PRIME_HASH_WATERMARK.mdSpec⚠️157,6LAYER_E_VERIFICATIONcontrol_bind
181164Cartesian_Map_EncodingEncoding SystemSymbol(s) → (x, y) ∈ Fin W × Fin H; W×H = 2^16 = 65536s=symbol index, (x,y)=Cartesian coordinates, W=width, H=heightFixed 2D grid replaces UTF-8: 16-bit address space, direct indexing, no parsingdocs/semantics/CARTESIAN_MAP_ENCODING.mdSpec160,161LAYER_K_SIGNALcontrol_bind
182165Unified_Cartesian_State_MachineCartesian System(xₜ₊₁, yₜ₊₁) = LUT[xₜ, yₜ]; x,y ∈ Fin 256LUT=64K entry table, (x,y)=current state, emit=next stateSelf-contained state machine: output coordinate IS next addressdocs/semantics/CARTESIAN_PHONON_PRIME_INTEGRATION.mdSpec164,152LAYER_K_SIGNALcontrol_bind
183166Cartesian_Manhattan_DistanceCartesian Systemd_M(cᵢ, cⱼ) = |xᵢ - xⱼ| + |yᵢ - yⱼ|cᵢ,cⱼ=coordinates, d_M=Manhattan distanceHardware-efficient distance: no multiply, no sqrtdocs/semantics/CARTESIAN_PHONON_PRIME_INTEGRATION.mdSpec157,165LAYER_C_TOPOLOGYgeometric_bind
184167Cartesian_Phonon_Force_LawCartesian SystemF(cᵢ,cⱼ) = e^(-d_M/127) * cos(2πd_M/127); d_M=|xᵢ-xⱼ|+|yᵢ-yⱼ|d_M=Manhattan distance, 127=phonon periodPhonon correlation on 2D grid using Manhattan metricdocs/semantics/CARTESIAN_PHONON_PRIME_INTEGRATION.mdSpec157,166LAYER_K_SIGNALphysical_bind
185168Neighbor_Consensus_RecoveryCartesian SystemLUT_repaired[x,y] = mode{LUT[x±1,y], LUT[x,y±1]}mode=majority vote, neighbors=4-connected (N,S,E,W)Self-healing: repair damaged cell via neighbor consensusdocs/semantics/CARTESIAN_PHONON_PRIME_INTEGRATION.mdSpec165,167LAYER_E_VERIFICATIONcontrol_bind
186169Error_Dimension_CorrectionCartesian Systeme = ErrorType × Magnitude; CorrectionSpace = (x, y, e) ∈ Fin 256 × Fin 256 × Fin Ne=error coordinate, N=error type spaceErrors induce new dimensional axis; damage becomes navigable coordinatedocs/semantics/ERROR_DIMENSION_CORRECTION.mdSpec⚠️165,168LAYER_E_VERIFICATIONcontrol_bind
187170Original_Coordinate_Neighborhood_RecoveryCartesian SystemN(x,y) = {(x±1,y), (x,y±1), (x±1,y±1)}; Recovery: ∀n∈N(x,y), Verify(n) → Consensus(x,y)(x,y)=original coordinate, N=8-connected neighborhood, Verify=attestation checkUse original coordinate to locate neighborhood; verify neighbors to recover damaged celldocs/semantics/NEIGHBORHOOD_RECOVERY.mdSpec165,168LAYER_E_VERIFICATIONcontrol_bind
188171Waveprobe_Overlap_Energy_QUBOWaveprobe QUBOE(s) = ⟨ψ_past|P̂|ψ_past⟩ = |⟨ψ_curr|ψ_past⟩|²; P̂ = |ψ_curr⟩⟨ψ_curr||ψ⟩∈ℂⁿ state vector, P̂=projector, s=candidate stateOverlap energy between active engram state and past witness trace (QUBO objective)docs/specs/waveprobe_qubo_spec.texSpec172,173,90LAYER_F_CONTROLcontrol_bind
189172Waveprobe_QUBO_MatrixWaveprobe QUBOQ_ij = c̄_i · c_j; objective: maximize x†Qx over candidate states in search window Wc_i=complex coefficients of |ψ_curr⟩, x=candidate selection vector, W=search windowQUBO matrix form of local Waveprobe selection problemdocs/specs/waveprobe_qubo_spec.texSpec171LAYER_F_CONTROLcontrol_bind
190173Waveprobe_Phase_Lock_CoherenceWaveprobe QUBOφ(s,x) = w_e·φ_e + w_r·φ_r + w_d·φ_d; w_e=0.4, w_r=0.3, w_d=0.3φ_e=normalized Shannon entropy, φ_r=byte repetition rate, φ_d=4-gram dictionary potentialFused phase-lock coherence scalar governing Waveprobe overlap magnitudedocs/specs/waveprobe_qubo_spec.texSpec171,131LAYER_A_COMPRESSIONinformational_bind
191174Waveprobe_Bell_BoundWaveprobe QUBO|⟨O_AB⟩| ≤ 2; violations (>2) confirm non-classical selection advantageO_AB=causal order observable between candidate arms A,BBell-like bound distinguishing classical matching from quantum-inspired Waveprobe selectiondocs/specs/waveprobe_qubo_spec.texSpec171LAYER_E_VERIFICATIONcontrol_bind
192175Waveprobe_BPB_ConservationWaveprobe QUBOBPB(x, s_probe) ≤ BPB(x, s_local); injection admitted iff local bits-per-byte does not increaseBPB=bits-per-byte, s_probe=injected state, s_local=current local stateInformation-conservative injection rule preventing spoiler termination of compression tracedocs/specs/waveprobe_qubo_spec.texSpec171,2LAYER_D_INVARIANTSinformational_bind
193554Boltzmann_State_WeightingState WeightingP_i = e^{-\Delta G_i / RT} / \sum e^{-\Delta G_j / RT}ΔG_i=relative energy, R=gas constant, T=temperatureWeighted average of multiple candidate states based on local stability/energytools/lean/Semantics/Semantics/Extensions/ManifoldBlit.leanDocumented141,412LAYER_M_LEAN_SEMANTICSstate_bind
194555Rubric_as_Reward_RaRAgent LogicR(τ) = Σ w_j · f_judge(τ, r_j)τ=agent trajectory, r_j=rubric criteria, w_j=weightsTraining autonomous agents via semantic judge models and group relative policy optimizationinfra/access_control/orchestrator.pyDocumented6,10,412LAYER_M_LEAN_SEMANTICSagent_bind
195556Global_Metric_Learning_GMLMetric Learningd_M^2(x_i,x_j) = (x_i-x_j)^T M (x_i-x_j)M=PSD matrix, L=decomposition (M=L^TL), S=similar, R=tripletsOptimizing the distance metric directly on the PSD manifold to improve manifold-to-manifold mappingcore/field_solver_emulator.pyDocumented38,75,412LAYER_C_TOPOLOGYmetric_bind
196557Equation_Chain_YECIdentity Logic0 -> 1 -> X -> 1 -> 00=Balance, 1=Identity, X=TransformationStructural lifecycle for mathematical reasoning and invariant verificationtools/lean/Semantics/Semantics/Quantization.leanDocumented532,412LAYER_M_LEAN_SEMANTICSlogic_bind
197558Differential_Spectral_Correction_DSCSignal CorrectionxL' = xL + s · (xL - yL)xL=current low-freq, yL=predicted clean low-freq, s=scalerCorrecting spectral bias and SNR-t drift in iterative reconstruction and signal transportcore/src/dsp_neuromorphic_translation.rsDocumented141,142,425LAYER_K_SIGNALsignal_bind
198559Autogenetic_Update_RuleAgent EvolutionA_{t+1} = A_t + η ∇_A [M_t + Φ]η=meta-learning rate, M_t=meta-objective, Φ=homeostasisMathematical framework for autonomous self-modification of agent protocols and logicinfra/access_control/orchestrator.pyDocumented555,412,101LAYER_M_LEAN_SEMANTICSagent_bind
199182Hardy-WeinbergPopulation Genetics$p^2 + 2pq + q^2 = 1$Genetic state closure conditiondocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
200183Arrhenius EqThermodynamics$k = A e^{-E_a/RT}$Thermal metabolic deformation tensordocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
201184Fick's LawThermodynamics$J = -D \nabla \phi$Laplace-Beltrami diffusion operatordocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
202185RNA Folding ΔGBiophysics$\Delta G = \sum E_{stack} + E_{loop}$Thermodynamic information topologydocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
203186Central Dogma ODEGenetics$\dot{m} = \alpha_m - \delta_m m$Cellular state production driftdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
204187Hill RegulationControl Theory$f(X) = X^n / (K^n + X^n)$Nonlinear saturation feedbackdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
205188Waddington PotentialDevelopmental Biology$V(x) = x^4/4 - bx^2/2 - ax$Epigenetic landscape bifurcationdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
206189Turing MorphogenesisControl Theory$\partial_t u = \Delta_{LB} u + f(u,v)$u,v=morphogen concentrations, Δ_LB=Laplace-Beltrami, f=reactionSpontaneous symmetry breakingdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
207190Replicator EqEvolutionary Dynamics$\dot{x}_i = x_i(f_i - \bar{f})$Evolutionary game theory flowdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
208191Social Force ModelBehavioral Dynamics$F = -\nabla V_{soc}$Collective human trajectory manifolddocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
209192Free Energy PrincipleInformation Theory$F = \mathbb{E}_q [\ln q - \ln p]$Variational self-organization invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
210193Fisher's TheoremPopulation Genetics$\dot{M} = \text{Var}_A(w)$Population fitness increase identitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
211194Neutral Theory (Kimura)Evolutionary Dynamics$k = v$Genomic drift baseline lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
212195Wilson-Cowan EqControl Theory$\dot{E} = -E + S(wE - wI + P)$Mean-field neural population dynamicsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
213196Wolff's Law EquilibriumBiomechanics$[T, H] = 0$Biomechanical structural optimizationdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
214197Onsager ReciprocityThermodynamics$L_{ij} = L_{ji}$Coupled transport symmetry lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
215198Jarzynski EqualityThermodynamics$\langle e^{-\beta W} \rangle = e^{-\beta \Delta F}$Non-equilibrium information physicsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
216199Flux Balance (FBA)Metabolism$S \cdot v = 0$Metabolic steady-state constraintdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
217200Gierer-MeinhardtDevelopmental Biology$\dot{a} = \rho a^2/i - \mu a + \sigma$Morphogenetic pattern formation enginedocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
218201Price EquationEvolutionary Dynamics$\Delta \bar{z} = \text{cov}(w,z)/\bar{w} + E[w \Delta z]/\bar{w}$General law of selection and transmissiondocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
219202Quasispecies EqEvolutionary Dynamics$\dot{x}_i = (w_i q_i - \bar{w})x_i + \sum w_{ij} x_j$Molecular evolution error thresholddocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
220203May's StabilityEcology$s \sqrt{nC} < 1$Ecosystem diversity-stability constraintdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
221204MaxEnt ProductionThermodynamics$\sigma = \sum J_k X_k \to \text{max}$Thermodynamic network steady-state lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
222205Wright-Fisher DriftPopulation Genetics$\text{Var}(\Delta p) = p(1-p)/N$Stochastic genetic sampling invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
223206Daisyworld HomeostasisEcology$\dot{w} = w(\beta x - \gamma)$Planetary-scale regulatory feedbackdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
224207MTE Master EqBiophysics$I = i_0 M^{3/4} e^{-E/kT}$Thermodynamic constraint on life ratesdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
225208Lifespan ScalingBiophysics$t_L \propto M^{1/4}$Biological time invariant across speciesdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
226209Simplicial CliqueTopology$n\text{-simplex} = \text{clique}(n+1)$High-dimensional neural connectivity unitdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
227210Cavity PersistenceTopology$\Delta \beta_k = \text{birth} - \text{death}$Topological information processing metricdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
228211Reproductive EffortLife History$Et/M \approx \text{const}$Lifetime energy efficiency invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
229212Radical Pair EqQuantum Biology$\dot{\rho} = -i[H,\rho] - \sum k_i \{P_i, \rho\}$Quantum magnetoreception dynamicsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
230213Exciton TransferQuantum Biology$H = \sum \epsilonm\rangle\langle mdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
231214Proton TunnelingQuantum Biology$k \approx \nu \exp(-2 \int \sqrt{2m(V-E)}/\hbar)$Quantum-induced DNA mutation ratedocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
232215Genetic ToggleSynthetic Biology$\dot{u} = \alpha_1/(1+v^\beta) - u$Synthetic bistable memory switchdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
233216The RepressilatorSynthetic Biology$\dot{m}_i = -m_i + \alpha/(1+p_j^n)$Synthetic genetic oscillator circuitdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
234217Feed-Forward LoopSynthetic Biology$Z = f(X, Y)$Biological network logic gate motifdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
235218Competitive ExclusionEcology$dN/dt = rN(K-N-\alpha M)/K$Gause's law of niche partitioningdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
236219Allee EffectEcology$\dot{N} = rN(N/A-1)(1-N/K)$Population resilience thresholddocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
237220Island BiogeographyEcology$dS/dt = I - E$Species equilibrium on islandsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
238221Marginal Value ThmEcology$g'(T) = g(T)/(T+\tau)$Optimal foraging stay-time lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
239222Hamilton's RuleEvolutionary Dynamics$rB > C$Kin selection altruism identitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
240223Poiseuille's LawPhysiology$Q = \Delta P \pi r^4 / 8\eta L$Cardiovascular flow rate lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
241224Starling's LawPhysiology$SV \propto EDV$Cardiac contractility invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
242225Fick PrinciplePhysiology$CO = VO_2 / \Delta [O_2]$Cardiac output measurement lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
243226SA:V Scaling LawBiophysics$SA/V \propto 1/L$Bergmann/Allen climate adaptation ruledocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
244227Cope's RuleEvolutionary Dynamics$S_t = S_0 e^{kt}$Macroevolutionary body size trenddocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
245228Clonal SelectionImmunology$\dot{B} = (r(A) - d)B$Lymphocyte clone expansion lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
246229Viral KineticsEpidemiology$T-I-V$ systemViral load and target cell dynamicsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
247230Circadian OscillatorNeuroscienceVan der Pol eqSelf-sustained biological rhythmsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
248231Muscle Force-VelBiomechanics$(F+a)(v+b) = \text{const}$Hill's hyperbolic muscle lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
249232Behavioral AttractorNeuroscienceLorenz systemNonlinear behavioral state flowdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
250233Gompertz-MakehamDemography$\mu(x) = \alpha e^{\beta x} + \lambda$Fundamental law of human mortalitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
251234Cancer InvasionOncology$T-N-L$ systemGatenby's evolutionary oncology modeldocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
252235Izhikevich NeuronNeuroscienceHybrid spiking eqEfficient cortical firing dynamicsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
253236Kuramoto SynchronyNeuroscience$r =(1/N) \sum e^{i\theta}docs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
254237Pipe Model TheoryBotany$A_{parent} = \sum A_{daughters}$Botanical area-preserving branching lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
255238Integrated InformationNeuroscience$\Phi = D_{KL} [ p(X)docs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
256239Neuronal WorkspaceNeuroscience$S = \text{sigmoid}(W_{asc} \Phi(W_{desc}))$GNW global broadcast gating lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
257240Objective ReductionQuantum Biology$\tau \approx \hbar / E_G$Orch-OR quantum consciousness eventdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
258241Sonar RangingBiophysics$R = c \Delta t / 2$Bio-acoustic distance invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
259242Auditory FilterPerception$g(t) = a t^{n-1} e^{-2\pi bt}$Gammatone cochlear processing modeldocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
260243Kinematic ReplicationDevelopmental Biology$N \propto \int \sigma(v, \text{shape}) dt$Xenobot self-assembly probabilitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
261244Vicsek SwarmingBehavioral Dynamics$\theta_{t+1} = \langle \theta \rangle + \Delta \theta$Collective motion phase transition lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
262245Lévy FlightBehavioral Dynamics$P(l) \sim l^{-\mu}$Superdiffusive animal search invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
263246Cable EquationBiophysics$\lambda^2 \partial^2 V / \partial x^2 = \tau \dot{V} + V$Passive electrical signal decay lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
264247Membrane TensionBiophysics$\Delta P = 2\gamma / R$Young-Laplace curvature-pressure lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
265248Osmotic PressureBiophysics$\Pi = icRT$Van 't Hoff protocell internal pressuredocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
266249Info BottleneckInformation Theory$\min I(X;Z) - \beta I(Z;Y)$Optimal neural compression principledocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
267250Predictive CodingInformation Theory$\dot{r} \propto U^T(I - f(Ur))$Hierarchical prediction error updatedocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
268251Weber-Fechner LawPerception$S = k \ln(I/I_0)$Logarithmic perception scaling lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
269252Stevens' Power LawPerception$S = k I^a$Modality-specific stimulus scalingdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
270253WBE BranchingBiophysics$\beta = n^{-1/2}, \gamma = n^{-1/3}$Fractal vascular network ratiosdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
271254Energy InvariantBiophysics$E_{total} \approx \text{const}$Universal lifetime energy budget lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
272255ROS DamageBiophysics$\dot{D} = k \Phi_{ROS} - R$Mitochondrial oxidative aging modeldocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
273256Maturity InvariantLife History$\alpha \cdot M \approx \text{const}$Charnov's maturity-mortality lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
274257Fecundity RatioLife History$b/M \approx \text{const}$Reproductive effort life-history invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
275258Hill EquationBiophysics$Y = [L]^n / (K_d + [L]^n)$Molecular cooperativity lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
276259Adair EquationBiophysics$Y = \sum i K_i L^i / n(1 + \sum K_i L^i)$Stepwise thermodynamic binding lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
277260MWC AllosteryBiophysics$Y = f(\alpha, L, c)$Concerted symmetry-state transitiondocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
278261KNF AllosteryBiophysics$Y = f(\alpha, K_{int})$Sequential induced-fit binding lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
279262Opponent TheoryPerception$RG = L - 2M, BY = (L+M) - S$LMS cone-to-opponent channel lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
280263Retinex DesignatorPerception$R = \log(I/I_{sur})$Land's color constancy invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
281264Lateral InhibitionPerception$r_p = e_p - \sum k_j r_j$Hartline-Ratliff edge enhancement lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
282265CIELAB MappingPerception$f(t) = t^{1/3}$Perceptually uniform color scalingdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
283266Morpho-TransformDevelopmental Biology$x' = f(x, y), y' = g(x, y)$D'Arcy Thompson coordinate mappingdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
284267Murray's LawBiophysics$r_0^3 = \sum r_i^3$Optimal vascular branching invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
285268DNA Linking NumTopology$Lk = Tw + Wr$Topological constraint on circular DNAdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
286269Brain AllometryNeuroscience$E = k S^\alpha$Brain-body mass power law scalingdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
287270EQ IndexNeuroscience$EQ = E / k S^{2/3}$Encephalization quotient metricdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
288271Mendelian SumGenetics$(p+q)^2 = 1$Genotypic probability distribution lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
289272Morgan LinkageGenetics$RF = (Rec/Total) \times 100$centiMorgan genetic distance lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
290273Liebig's LawEcology$Y = \min(k_i R_i)$Limiting factor growth invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
291274Shelford ToleranceEcology$P(x) \propto \exp(-\Delta x^2/2\sigma^2)$Gaussian environmental performance lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
292275Polygenic CLTGenetics$X = \sum g_i + \epsilon$Additive trait convergence identitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
293276Nernst PotentialBiophysics$E = (RT/zF) \ln(C_{out}/C_{in})$Ion-specific reversal potential lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
294277GHK EquationBiophysics$V_m = f(P_i, C_i)$Resting membrane potential invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
295278Donnan ProductBiophysics$[K]_{in}[Cl]_{in} = [K]_{out}[Cl]_{out}$Passive ion distribution equilibriumdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
296279Gibbs-Duhem EqBiophysics$\sum n_i d\mu_i = 0$Cellular chemical potential couplingdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
297280Second Bio-LawBiophysics$\Delta S_{tot} > 0$Biological entropy export requirementdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
298281French Flag ModelDevelopmental Biology$C(x) \to \{B, W, R\}$Positional information threshold lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
299282SDD GradientDevelopmental Biology$C(x) = C_0 e^{-x/\lambda}$Source-Diffusion-Degradation steady statedocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
300283Lewis's LawBiophysics$A_n \propto 1 + \alpha(n-6)$Cell area-neighbor topological lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
301284Aboav-Weaire LawBiophysics$m(n) = 5 + 6/n$Neighbor-neighbor topology invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
302285Growth DilutionEcology$\dot{C} = -C \nabla \cdot V$Advective scaling on growing manifoldsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
303286Redfield RatioBiogeochemistry$C:N:P = 106:16:1$Biogeochemical stoichiometry invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
304287Holling ResponseEcology$f(N) = aN / (1+ahN)$Predator-prey functional response lawsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
305288Eco-ConnectanceEcology$C = L / S^2$Food web structural complexity metricdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
306289Taylor's LawEcology$\sigma^2 = a \mu^b$Population density variance scaling lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
307290Logistic MapEcology$x_{n+1} = rx(1-x)$Discrete population chaos transitiondocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
308291Lotka's InvariantDemography$\int e^{-ra} p(a) m(a) da = 1$Stable population age distribution lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
309292Tetz's LawEvolutionary Biology$t_{death} \leftarrow q(t) \ge q_{max}$Pangenome alteration lifespan limitdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
310293Survival LimitDemography$P(S) \to 0$Stretched exponential mortality plateaudocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
311294Genomic EntropyGenetics$H = -\sum p \log p$DNA/RNA sequence information contentdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
312295Codon HammingGenetics$d_H = \sum [b_i \neq b_j]$Genetic mutation cost metricdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
313296Bio-CapacityGenetics$C = 1 - H(p)$Maximum sustainable replication ratedocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
314297Error CatastropheEvolutionary Biology$p_{max} \approx \ln(\sigma)/L$Information persistence limit lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
315298Bio-HamiltonianEvolutionary Biology$H = p \cdot f(x,u,t)$Pontryagin life-history optimizationdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
316299Requisite VarietyControl Theory$V_{sys} \ge V_{env}$Ashby's homeostatic stability lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
317300Bio-ReinforcementNeuroscience$\Delta V = r - V$Integral RL optimal control updatedocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
318301Pareto RobustnessEvolutionary Biology$Dist(R, P) \to 0$Performance-reliability trade-off lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
319302Limit Cycle ThmDynamical SystemsPoincaré-BendixsonRhythmic robustness necessity lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
320303Phase SingularityDynamical Systems$\text{Amp} \to 0$ at $S^*$Winfree's topological clock stoppingdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
321304Self-Assembly ΔGBiophysics$\Delta G = \Delta H - T\Delta S$Hydrophobic structure formation lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
322305CMC ThresholdBiophysics$C > CMC$Micelle formation phase transitiondocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
323306DNA Tile LogicBiophysics$G_a \equiv G_b > T$Algorithmic self-assembly matchingdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
324307Hawk-Dove ESSEvolutionary Biology$p = V/C$Evolutionary game mixed strategydocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
325308Van Valen's LawEvolutionary Biology$\ln N = -kt + C$Constant extinction probability lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
326309Scale-Free DistNetwork Theory$P(k) \propto k^{-\gamma}$Biological network structural invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
327310Preferential AttNetwork Theory$w_i \propto k_i$Network hub formation mechanismdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
328311Neutrality RuleEvolutionary Biology$sdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
329312Adami ComplexityEvolutionary Biology$C = L - H$Genomic information measuredocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
330313Regulatory LawGenetics$R \propto N^2$Quadratic scaling of transcription factorsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
331314Revelle FactorBiogeochemistry$\beta = \Delta pCO_2 / \Delta DIC$Oceanic chemical buffer capacity lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
332315Remineral RatioBiogeochemistry$O:C \approx 1.3$Redfield-Kester remineralization invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
333316Small-World LawNetwork Theory$C(\beta) \sim (1-\beta)^3$Local clustering in modular networksdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
334317Modularity QNetwork Theory$Q = \sum (e_{ii} - a_i^2)$Network functional division metricdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
335318HOT PrincipleComplexity Theory$\min \sum P_i L_i$Robustness-fragility optimization lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
336319Complexity LawComplexity Theory$\sigma^2(t) = \sigma^2(0) + 2Dt$McShea's spontaneous complexity growthdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
337320GK-SwitchBiophysics$x = G(v_1, v_2, J_1, J_2)$Zeroth-order ultrasensitivity lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
338321Mitotic OscillatorCell Biology$\dot{u}, \dot{v}$ (Tyson)Cell cycle limit cycle dynamicsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
339322PER-CRY FeedbackCircadian Biology$Rate = K^n / (K^n + R^n)$Molecular circadian clock repressiondocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
340323Keller-SegelDevelopmental Biology$J = \chi u \nabla c$Chemotactic advection-diffusion lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
341324Epigenetic ClockEpigenetics$Age = \sum \beta_i \cdot DNAm_i$Horvath's methylation-based aging lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
342325Kinetic ProofreadBiophysics$\eta \approx (\eta_{eq})^N$Energy-driven biological error correctiondocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
343326Biodiversity NumEcology$\theta = 2 J_m \nu$Hubbell's unified neutral theory metricdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
344327Hick's LawCognitive Science$RT = a + b \log(n+1)$Decision time vs choice complexity lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
345328Fitts's LawCognitive Science$MT = a + b \log(A/W + 1)$Motor control speed-accuracy invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
346329Zipf's LawInformation Theory$P(r) \propto r^{-s}$Codeword/abundance power law scalingdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
347330Laughlin's LawNeuroscience$Cost \propto \text{Capacity}$Metabolic efficiency of neural informationdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
348331Hebb's LawNeuroscience$\Delta w = \eta xy$Fundamental associative learning ruledocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
349332Oja's RuleNeuroscience$\Delta w = \eta(xy - y^2 w)$Stable PCA-based synaptic plasticitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
350333Hopfield EnergyNeuroscience$E = -0.5 \sum w_{ij} s_i s_j$Neural attractor memory stability lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
351334Critical Power LawComplexity Theory$P(s) \sim s^{-\tau}$Self-organized criticality invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
352335Broken StickEcology$E(R_j) = \frac{1}{n} \sum 1/i$Null model for species abundancedocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
353336Niche BreadthEcology$B = 1 / \sum p_i^2$Levins' specialization-diversity indexdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
354337Niche OverlapEcology$M_{jk} = \sum p_j p_k / \sum p_j^2$Competitive impact asymmetry lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
355338Motor EfficiencyBiophysics$\eta_{th} = fl / \Delta \mu$Thermodynamic Brownian ratchet lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
356339Parrondo ParadoxGame Theory$L_1 + L_2 \to W$Winning-by-switching evolution lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
357340Somite Size LawDevelopmental Biology$S = v \cdot T$Clock-and-wavefront segmentation invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
358341Morphogen ScalingDevelopmental Biology$\lambda \propto L$Expansion-repression scale invariancedocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
359342MDDR Growth LawDevelopmental Biology$\dot{M}/M = \text{const}$Morphogen-dependent cell division ruledocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
360343Anfinsen's DogmaBiophysics$G_{native} = \min(G)$Protein native state global minimumdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
361344Levinthal SpaceBiophysics$\Omega = m^n$Conformational search complexitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
362345Folding LandscapeBiophysics$P_i = e^{-E_i/kT} / Z$Boltzmann conformation distributiondocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
363346Contact OrderBiophysics$CO = (1/LN) \sum \Delta S$Protein folding rate topological lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
364347Fermat's Path LawBiophysics$\sin \theta / v = \text{const}$Optimal animal trail refraction lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
365348Max Flux PrincipleMetabolism$\max \mathbf{c}^T \mathbf{v}$Metabolic network optimization objectivedocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
366349Max Power LawThermodynamics$P = \eta \Phi \to \max$Lotka's principle of self-organizationdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
367350Least Action LawPhysics$\delta \int (T-V) dt = 0$Euler-Lagrange population trajectorydocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
368351Action FunctionalPhysics$S = \int L dt$Cumulative trajectory cost metricdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
369352CSD AutocorrComplexity Theory$\alpha \to 1$ as $\lambda \to 0$Tipping point early warning signaldocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
370353CSD VarianceComplexity Theory$Var \propto 1 / (1-\alpha^2)$Noise amplification near instabilitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
371354Recovery RateEcology$\lambda = -1/\tau$Speed of return to stable equilibriumdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
372355Resilience BasinEcology$Depth, Width$Geometrical stability of attractorsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
373356Horton Number LawBiophysics$N_k = R_B^{K-k}$Branch count geometric series lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
374357Horton Length LawBiophysics$L_k = L_1 R_L^{k-1}$Branch length geometric series lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
375358WBE ExponentBiophysics$\alpha = 3/4$Metabolic scaling fractal dimensiondocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
376359Heart Rate LawBiophysics$HR \propto M^{-1/4}$Quarter-power heart rate scaling lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
377360Blood Volume LawBiophysics$V_b \propto M^1$Isometric blood mass invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
378361Sensing LimitBiophysics$\delta c/c \sim (Dac\tau)^{-1/2}$Berg-Purcell chemoreception bounddocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
379362Signaling SNRBiophysics$SNR \approx \Delta c^2 Dac\tau$Bialek's physical limit of detectorsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
380363Positional NoiseBiophysics$\Delta x \approx (\delta c/c) /\nabla c/cdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
381364Oregonator BZChemical Physics$x, y, z$ (non-equilibrium)Chemical oscillator kinetics lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
382365Firefly SynchronyNeuroscience$\dot{\theta} = \omega + A \sin(\Delta \theta)$Pulse-coupled oscillator entrainmentdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
383366Bio-ContinuityBiophysics$\dot{u} + \nabla \cdot (Vu) = F$General conservation of biological massdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
384367Strouhal NumberBiomechanics$St = fA / U$Propulsive efficiency invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
385368Froude NumberBiomechanics$Fr = v^2 / gL$Terrestrial gait transition invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
386369Huxley Muscle LawBiomechanics$\dot{n} = f(1-n) - gn$Myosin cross-bridge attachment kineticsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
387370Monod EquationMicrobiology$\mu = \mu_{max} S / (K_s + S)$Nutrient-limited microbial growth lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
388371Pirt's LawMicrobiology$q_s = \mu / Y_G + m$Maintenance energy partitioning lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
389372Verhulst LogisticEcology$\dot{P} = rP(1-P/K)$Population growth with carrying capacitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
390373Gompertz GrowthEcology$\dot{V} = r V \ln(K/V)$Asymmetric sigmoidal biomass accumulationdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
391374MCA Control CoeffMetabolism$C^J_v = \partial \ln J / \partial \ln v$System-level metabolic sensitivitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
392375MCA SummationMetabolism$\sum C^J_{v_i} = 1$Conservation of metabolic control lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
393376Perfect AdaptControl Theory$\dot{m} = k_R R - k_B B \phi(A)$Barkai-Leibler robustness invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
394377Demand RuleGenetics$D \to 1 \implies \text{Activator}$Savageau's regulatory logic selectiondocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
395378Place TheoryPerception$m \ddot{x} + \beta \dot{x} + \kappa x = F$Helmholtz cochlear resonance lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
396379Traveling WavePerception$\phi = \omega t - \int k dx$Békésy's cochlear wave phase invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
397380Tonotopic MapPerception$f = A(10^{ax} - K)$Greenwood frequency-position functiondocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
398381Cochlear AmpBiophysics$\dot{z} = (\mu+i\omega)z -zdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
399382R* TheoryEcology$R^* = Kd / (\mu_{max} - d)$Resource-ratio competition equilibriumdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
400383SM CorrelationDemography$\ln \alpha \approx \ln K - c\beta$Initial mortality vs aging rate lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
401384Vitality DecayDemography$V(t) = V_0(1 - Bt)$Homeostatic energy reserve declinedocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
402385Mortality PlateauDemography$\mu(x) \to s$ as $x \to \infty$Late-life mortality deceleration lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
403386Bark ScalePerception$z = 13 \arctan(k f) + \dots$Perceptual auditory filter rate lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
404387Crit BandwidthPerception$\Delta f = 25 + 75 [1 + 1.4 f^2]^{0.69}$Ear energy integration bandwidth lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
405388Equal LoudnessPerception$L_p = f(Phons, f)$Phon-to-SPL perceptual intensity lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
406389Niche HypervolumeEcology$H = \{ \mathbf{x} \mid L_i \le x_i \le U_i \}$Hutchinson's fundamental niche lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
407390Nernst-PlanckBiophysics$J = -D(\nabla c + \frac{ze}{kT} c \nabla \phi)$Charged ion transport invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
408391Richness ScalingEcology$\ln S = -E/kT + C$MTE biodiversity-temperature lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
409392FHN ExcitabilityNeuroscience$\dot{v}, \dot{w}$ (FitzHugh)Simplified excitable system lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
410393Swift-HohenbergPattern Formation$\dot{u} = ru - (1+\nabla^2)^2 u$Universal pattern formation invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
411394Tissue StiffnessBiomechanics$E \propto \rho^n$Gibson-Ashby density-stiffness scalingdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
412395Cytoskeletal FBiomechanics$F = -kx$Hookean elastic restoring force lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
413396Spiral VogelBotany$\theta = n\psi, r = c\sqrt{n}$Biological spiral floret arrangementdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
414397Golden AngleBotany$\psi \approx 137.5^\circ$Optimal packing angular invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
415398Hofmeister RuleBotany$\max Dist(P_{new}, P_{old})$Primordium placement growth axiomdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
416399Muller's RatchetEvolutionary Biology$n_0 = N e^{-\lambda/s}$deleterious mutation accumulation lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
417401Neutral DiversityEvolutionary Biology$\theta = 4 N_e u$Mutation-drift equilibrium invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
418402Reynolds NumberFluid Dynamics$Re = \rho u L / \mu$Inertial-viscous flow regime lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
419403Peclet NumberFluid Dynamics$Pe = u L / D$Advective-diffusive transport ratiodocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
420404Darcy's LawFluid Dynamics$v = -(\kappa/\mu) \nabla P$Interstitial fluid flux invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
421405Starling EqPhysiology$J_v = f(\Delta P, \Delta \pi)$Capillary-tissue filtration rate lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
422406Handicap PrincipleEvolutionary Biology$w = f(a, p, q)$Costly signaling fitness lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
423407Honesty ConditionEvolutionary Biology$\partial^2 w / \partial a \partial q > 0$Marginal cost stability invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
424408Honest EquilibriumEvolutionary Biology$P^*[A^*(q)] = q$Perceptual-quality identity lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
425409Schwan EquationBiophysics$V_m = 1.5 E R \cos \theta$Induced membrane potential lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
426410Cole-Cole EqBiophysics$\varepsilon^* = \varepsilon_\infty + \dots$Tissue dielectric relaxation invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
427411Dispersion LawPhysics$\alpha, \beta, \gamma$ regionsFrequency-dependent tissue impedancedocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
428412RNA CombinatorsSynthetic Biology$Kxy=x, Sxyz=(xz)(yz)$Ribosome Turing completeness proofdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
429413BioBrick LogicSynthetic Biology$f(A,B) \to C, type(A)=type(C)$Idempotent genetic assembly lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
430414Genetic LoadEvolutionary Biology$V_{cell} = I_{load} R_{meta}$Ohm's law metabolic burden analogydocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
431415Critical DepthEcology$Z_{cr} \propto I_0 / k I_c$Sverdrup's bloom initiation lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
432416Particle SinkingOceanography$v \propto (\rho_p - \rho_f) R^2$Stokes' marine snow export invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
433417Q10 RuleBiophysics$Q_{10} = (R_2/R_1)^{10/\Delta T}$Thermal biological rate sensitivitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
434418Base SaturationBiogeochemistry$\%Sat_i = C_i / CEC \times 100$Albrecht's soil chemistry ratiosdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
435419Hyphal FlowMycology$\partial_t n + v \partial_x n = bn$Schnepf-Roose fungal mining kineticsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
436420Terraced BarrelBotany$\mu = \min(I_S, \tilde{I})$Global physical growth constraint lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
437421Reliability LawEngineering$P(t) = 1 - (1-e^{-kt})^n$Redundant system aging invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
438422Reaction PropChemistry$a_j = c_j h_j$Stochastic event probability densitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
439423Gillespie StepStochastic Processes$\tau = (1/a_0) \ln(1/r)$Discrete event-waiting time invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
440424Master EquationStatistical Mechanics$\dot{P} = \sum [aP_{pre} - aP_{post}]$State probability density flow lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
441425Masking SlopePerception$S_2 \approx 24 + 230/f - 0.2L$Zwicker's upward spread of maskingdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
442426SMR PriorityNeuroscience$SMR = L_{sig} - L_{mask}$Informational saliency filtering lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
443427Specific LoudnessPerception$N' = k (E/E_0)^{0.23}$Auditory power-law intensity invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
444428STDP LawNeuroscience$\Delta w \propto \exp(-\Delta t/\tau)$Timing-dependent synaptic plasticitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
445429Trophic 10% RuleEcology$P_n = 0.1 P_{n-1}$Energy transfer attenuation invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
446430Slender-Body FFluid Dynamics$f = -\partial_t(mv) - U \partial_x(mv)$Lighthill's reactive swimming forcedocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
447431DVM FitnessMarine Biology$F(z) = g(z,t) - \mu(z,t)$Migration depth optimization lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
448432Swim ResponseMarine Biology$w = w_{max} \tanh(\alpha \Delta I)$Light-dependent vertical speeddocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
449433Turbulent EncounterFluid Dynamics$E = \pi R^2 \sqrt{\sum v_i^2} C$Rothschild-Osborn foraging lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
450434Patch ResidenceEcology$f'(t^*) = f(t^*)/(T+t^*)$MVT optimal stay-time invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
451435Input MatchingForaging Theory$N_i/\sum N = R_i/\sum R$Ideal Free Distribution (IFD) lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
452436Fitness EquiEvolutionary Biology$F_i(N_i) = F_j(N_j)$Payoff equilibration in social groupsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
453437V-Formation UpwashAerodynamics$v \propto \Gamma / r$Aerodynamic vortex-capture lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
454438Induced Drag LawAerodynamics$D_i \propto L^2 / \rho V^2$Formation flight drag reductiondocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
455439Flight EfficiencyAerodynamics$Range \times 1.71$Collective aerodynamic range extensiondocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
456440Reproduction NumDemography$R_0 = \beta / \gamma$Basic disease transmission invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
457441Herd ImmunityEpidemiology$HIT = 1 - 1/R_0$Contagion resistance threshold lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
458442SIR DynamicsEpidemiology$\dot{S}, \dot{I}, \dot{R}$Compartmental infectious disease modeldocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
459443NDZ ModelEcology$\dot{C} = \nabla \cdot [D \nabla C + vC/b]$Nutrient depletion zone kineticsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
460444Root UptakePlant Physiology$F = I_{max} \Delta c / (K_m + \Delta c)$Root surface nutrient flux lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
461445Root FractalPlant Physiology$N(\epsilon) \propto \epsilon^{-D}$Root system space-filling invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
462446Constructal LawPhysics$d_1/d_0 = n^{-1/3}$Bejan's optimal flow configuration lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
463447Muscle MechanicsBiomechanics$(F+a)(v+b) = \text{const}$Hill's 3-element contractile modeldocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
464448Square-Cube LawBiophysics$SA \propto L^2, V \propto L^3$Geometric scaling limit on organism sizedocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
465449Fung's LawBiophysics$\sigma \propto e^{\epsilon^2}$Exponential strain-stiffening invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
466450Alveolar LaplacePhysiology$P = 2\gamma / r$Lung stability surface-tension lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
467451Ventricular WallPhysiology$\sigma = Pr / 2h$Cardiac stress-thickness invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
468452Process SMetabolism$\dot{S} \propto (S_{max} - S)$Homeostatic sleep pressure lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
469453Process CMetabolism$H^{\pm} = \text{mean} \pm A\cos(\omega t)$Circadian drive threshold invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
470454Aschoff's RuleChronobiology$\tau(I) = \tau_0 \pm k \log I$Internal clock period scaling lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
471455Lack's PrincipleEcology$W = n \cdot P(n)$Optimal reproductive clutch size lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
472456Smith-FretwellEvolutionary Biology$W = (R/s) f(s)$Offspring size-number trade-off lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
473457Repro ScalingEvolutionary Biology$R \propto M^{3/4}$Life-history resource allocation lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
474458Dunbar's LawSocial Neuroscience$\log N \propto \log CR$Social brain group-size limit lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
475459Relationship LawSocial Science$R = N(N-1)/2$Quadratic growth of social linksdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
476460Brain CurvatureNeuroscience$\log E \propto (\log S)^2$Brain-body curvilinear scaling lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
477461Glottal BernoulliBiophysics$P_g = P_s - 0.5 \rho v^2$Vocal fold aerodynamic suction lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
478462Source-FilterSpeech Science$P(z) = S(z)V(z)R(z)$Vocal production linear system modeldocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
479463Pitch ScalingSpeech Science$f_0 \propto M^{-0.4}$Fletcher's optimal communication pitchdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
480464VTL ScalingSpeech Science$VTL \propto M^{1/3}$Vocal tract geometric scaling invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
481465Tissue FluenceBiophysics$\frac{1}{c}\dot{\Phi} = D \nabla^2 \Phi - \mu_a \Phi + S$Light transport diffusion approximationdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
482466Luciferase LawMolecular Biology$v = V_{max} [S] / (K_m + [S])$Bioluminescence kinetic emission ratedocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
483467Beer-LambertChemistry$I = I_0 e^{-\mu_a z}$Light intensity attenuation in tissuedocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
484468Cole's ParadoxEvolutionary Biology$m_a = m_p + S/s$Annual vs perennial fitness thresholddocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
485469Maturity RatioLife History$L_{\alpha} / L_{\infty} \approx 0.65$Stearns' size-at-maturity invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
486470Allocation LawLife History$T = R + S + G$Fundamental biological energy trade-offdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
487471Euler-Lotka EqDemography$\sum e^{-rx} l_x m_x = 1$Universal fitness and growth identitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
488472Rescorla-WagnerPsychology$\Delta V = \alpha \beta (\lambda - \sum V)$Prediction-error based learning lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
489473Cognitive LévyCognitive Science$P(l) \sim l^{-\mu}$Heavy-tailed information search lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
490474Cognitive MVTForaging Theory$R'(t^*) = R(t^*)/(t^*+\tau)$Optimal category-switching invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
491475SAM ProbabilityEvolutionary Biology$P(iQ) \propto S(Q, i)$docs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
492476Gouy-StodolaThermodynamics$I = T_0 S_{gen}$Metabolic lost-work exergy destructiondocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
493477MinEnt ProdThermodynamics$\dot{S}_{gen} \to \min$Prigogine's steady-state stability lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
494478MaxEnt ProdThermodynamics$\dot{S}_{gen} \to \max$Ziegler's far-from-equilibrium drivedocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
495479Useful WorkThermodynamics$W_{actual} = W_{max} - I$Thermodynamic metabolic efficiency lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
496480Corner's LawEcology$A_{la} = \alpha A_{cs}^\beta$Stem-leaf coordinative architecturedocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
497481Pipe ModelBotany$A(z) = c W_L(z)$Botanical vascular cross-section lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
498482Cavitation LawFluid Dynamics$PLC = f(\psi, \psi_{50})$Xylem hydraulic vulnerability invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
499483Species-Area LawEcology$S = c A^z$Arrhenius richness scaling invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
500484Cell PrestressBiomechanics$G \approx k \sigma_0$Tensegrity-based stiffness tuning lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
501485Reciprocal YieldAgriculture$1/w = a + bd$Shinozaki-Kira biomass saturation lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
502486Noble ModelCardiac Physiology$C_m \dot{V} = -\sum I_{ion}$First cardiac action potential modeldocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
503487Gating DynamicsNeuroscience$\dot{x} = \alpha_x(1-x) - \beta_x x$Noble ion channel state transitionsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
504488Inward RectifierBiophysics$g_{K1} = f(V)$Noble voltage-dependent K-conductancedocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
505489Stevens' 3/2 LawPerception$N_{out} \propto N_{in}^{3/2}$Cortical dimensionality expansion lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
506490White Matter LawNeuroscience$V_w \propto V_g^{4/3}$Neural wiring volume scaling invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
507491Rall's 3/2 LawNeuroscience$\sum d_d^{1.5} = d_p^{1.5}$Dendritic impedance matching invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
508492Synaptic InvariantNeuroscience$Syn / Pair \approx 1$Sparse connectivity discriminatory ruledocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
509493Multi-Hit LawRadiation Biology$P \approx 1 - e^{-kt^n}$Knudson's oncogenesis probability lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
510494MCA ElasticityMetabolism$\epsilon^v_s = \partial \ln v / \partial \ln s$Local enzyme-metabolite sensitivitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
511495Connectivity ThmNetwork Theory$\sum C^J \epsilon = 0$Flux control-elasticity link identitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
512496Price SelectionEvolutionary Biology$S = Cov(w, z) / \bar{w}$Fitness-trait covariance selection lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
513497Reichardt DetectVision Science$R = I_1 I_2' - I_1' I_2$Biological motion correlation lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
514498ACO TransitionChemistry$P \propto \tau^\alpha \eta^\beta$Probabilistic ant-trail following lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
515499Pheromone LawChemical Ecology$\tau \leftarrow (1-\rho)\tau + \Delta \tau$Evaporation-deposition optimization lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
516500Donachie RuleCell Biology$M_{init} / n_{ori} \approx \text{const}$DNA replication initiation invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
517501Cell Size LawCell Biology$S = S_0 2^{(C+D)/\tau}$Cooper-Helmstetter size-growth lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
518502Adder PrincipleCell Biology$V_{div} = V_{birth} + \Delta V$Incremental cellular volume addition lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
519503Wright's GradientEvolutionary Biology$\dot{q} = \frac{q(1-q)}{2\bar{w}} \nabla \bar{w}$Evolutionary landscape ascent lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
520504Mean FitnessEvolutionary Biology$\bar{w} = \sum p_i^2 w_{ii} + \dots$Adaptive landscape value identitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
521505SBT DriftEvolutionary Biology$4N_e s < 1$Shifting balance exploratory conditiondocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
522506Amari Neural FieldNeuroscience$\tau \dot{u} = -u + \int wf(u) + I$Continuous population activity lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
523507Mexican Hat KernelNeuroscience$w(x) = \text{Exc} - \text{Inh}$Local-excitation lateral-inhibition lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
524508Sigmoid ActivityNeuroscience$f(u) = 1/(1+e^{-\beta(u-h)})$Nonlinear population response invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
525509Shell Spiral LawBiophysics$r = a e^{b\theta}$Logarithmic gnomonic growth invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
526510Mass Action LawChemistry$Rate = k [A] [B]$Fundamental biological kinetic lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
527511Equilibrium InvariantThermodynamics$K_{eq} = [P]/[R]$Thermodynamic steady-state identitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
528512Malthusian LawDemography$P(t) = P_0 e^{rt}$Unlimited exponential growth invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
529513Hayflick LimitCell Biology$L_n = L_0 - n\Delta L$Replicative telomere shortening lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
530514Senescence RuleGerontology$L_n \le L_{crit}$Critical mass cell division arrest lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
531515Sheldon SpectrumOceanography$B(M) \propto M^0$Constant biomass per size class lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
532516Inverse Mass NEcology$N(M) \propto M^{-1}$Mass-abundance scaling invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
533517Productivity LawEcology$P(M) \propto M^{-1/4}$Size-dependent biological production ratedocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
534518Fisher FGM PotentialEvolutionary Biology$w(z) \propto e^{-zdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
535519Beneficial ProbEvolutionary Biology$P_a \approx 1 - \Phi(r\sqrt{n}/2d)$Fisher's geometric mutation lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
536520Small Mutation LawEvolutionary Biology$r \to 0 \implies P_a \to 0.5$Gradualism in high-dimensional systemsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
537521Complexity CostEvolutionary Biology$P_a \propto 1/\sqrt{n}$Adaptation slowdown with trait countdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
538522Gene Family LawGenetics$P(i) \propto i^{-\gamma}$Paralog size power law distributiondocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
539523Functional ScaleEvolutionary Biology$N_c \propto G^\alpha$Functional category non-linear scalingdocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
540524BDIM DynamicsEvolutionary Biology$\dot{n}_i = f(\lambda, \delta)$Birth-death-innovation genome driftdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
541525Margalef IndexEcology$D = (S-1) / \ln N$Sample-size corrected species richnessdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
542526Shannon IndexEcology$H' = -\sum p_i \ln p_i$Information-theoretic community uncertaintydocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
543527Info-StabilityInformation Theory$Flow \propto 1/Info$Margalef's stability-complexity lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
544528Info-SheddingInformation Theory$\dot{D} \propto -Stress$Diversity loss as energy-saving strategydocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
545529Drake's RuleEvolutionary Biology$u \cdot G \approx 0.003$Universal genomic mutation fidelity lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
546530Drift-BarrierEvolutionary Biology$\log(N_e u) \sim \log G$Non-coding DNA expansion scaling lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
547531Minimal GenomeGenetics$G_{min} = N_{inf} + N_{met}$Theoretical gene count floor for lifedocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
548532Effective InfoInformation Theory$C = G(1-R)$Redundancy-weighted genomic complexitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
549533Constrained TEEThermodynamics$TEE = BMR + (1-C)PAEE$Pontzer's metabolic budget reallocationdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
550534Metabolic CeilingMetabolism$TEE_{max} \approx 2.5 BMR$Alimentary limit on long-term endurancedocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
551535Metabolic ScopeMetabolism$PAL = TEE / BMR$Sustainable energy throughput invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
552536Droop EquationMicrobiology$\mu(Q) \propto 1 - Q_0/Q$Internal nutrient quota growth lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
553537Herbert's LawEcology$Q = 1/Y = \text{const}$Constant cell composition invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
554538Quota DynamicsEcology$\dot{Q} = \rho(S) - \mu Q$Decoupled uptake-growth kineticsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
555539Homeostatic EqPhysiology$y = c x^{1/H}$Sterner-Elser nutrient regulation lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
556540Damuth's LawEcology$N \propto M^{-3/4}$Population density-mass scaling lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
557541Unified MetabMetabolism$B \propto M^{3/4} e^{-E/kT}$Temperature-mass unified scaling lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
558542Minimum VolumeBiophysics$V_{cell} \ge \sum V_{mach}$Physical floor for autonomous lifedocs/MATH_MODEL_MAP.mdDocumentedLAYER_C_TOPOLOGYgeometric_bind
559543Locomotion SpeedBiomechanics$V \propto M^{1/6}$Bejan's universal movement lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
560544Movement FreqBiomechanics$f \propto M^{-1/6}$Universal stride/stroke frequency lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
561545Diffusion SpeedBiophysics$t \approx x^2 / 2D$Passive transport speed limit lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_L_APPLICATIONphysical_bind
562546Rubisco LimitPlant Physiology$A_c = f(V_{cmax}, C_c, O)$Calvin cycle carboxylation capacitydocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
563547RuBP RegenPlant Physiology$A_j \approx J/4$Electron transport regeneration lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
564548Ball-Berry LawPlant Physiology$g_s = g_0 + m Ah/C$Stomatal conductance regulation ruledocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
565549Intrinsic WUEPlant Physiology$iWUE = A_n / g_s$Carbon-water compromise efficiencydocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
566550Light ResponsePlant Physiology$P_n = \frac{\alpha I P_{max}}{\alpha I + P_{max}}$Photosynthesis-irradiance saturation lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
567551Reed-Frost LawEpidemiology$C_{t+1} = S_t(1 - q^{C_t})$Chain-binomial infection spread lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
568552Trophic WaveEcology$\dot{\Phi} + \nabla \cdot (K \Phi) = -\mu \Phi$Continuous biomass flow spectrum lawdocs/MATH_MODEL_MAP.mdDocumentedLAYER_A_COMPRESSIONinformational_bind
569553Trophic KineticEcology$K = P/B$Biomass transfer velocity invariantdocs/MATH_MODEL_MAP.mdDocumentedLAYER_F_CONTROLcontrol_bind
570554Boltzmann State WeightingStatistical Mechanics$P_i = \frac{e^{-\Delta G_i / RT}}{\sum_j e^{-\Delta G_j / RT}}$Multi-conformer weighting for state predictiondocs/MATH_MODEL_MAP.mdDocumentedLAYER_G_ENERGYthermodynamic_bind
571555Rubric-as-Reward (RaR)Machine Learning$R(\tau) = \sum w_j \cdot f_{judge}(\tau, r_j)$Trajectory-based semantic reward for agent trainingdocs/MATH_MODEL_MAP.mdDocumentedLAYER_M_LEAN_SEMANTICSagent_bind
572556Global Metric Learning (GML)Machine Learning$d_M^2 = (x_i - x_j)^T L^T L (x_i - x_j)$Mahalanobis metric optimization on PSD manifoldsdocs/MATH_MODEL_MAP.mdDocumentedLAYER_M_LEAN_SEMANTICSmetric_bind
573557Equation Chain (YEC)Logic$0 \to 1 \to X \to 1 \to 0$Structural lifecycle of mathematical identity and balancedocs/MATH_MODEL_MAP.mdDocumentedLAYER_M_LEAN_SEMANTICSlogic_bind
574558Differential Spectral Correction (DSC)Signal Processing$x_L' = x_L + s \cdot (x_L - y_L)$Mitigating SNR-t bias via wavelet-domain low-frequency adjustmentdocs/MATH_MODEL_MAP.mdDocumentedLAYER_K_SIGNALsignal_bind
575559Autogenetic Update RuleAgent Logic$\mathcal{A}_{t+1} = \mathcal{A}_t + \eta \nabla_{\mathcal{A}} [\mathcal{M}_t + \Phi]$Recursive self-modification of agent architecture and meta-objectivesdocs/MATH_MODEL_MAP.mdDocumentedLAYER_M_LEAN_SEMANTICSagent_bind
576560Anisotropy_GradientCompressionAnisotropy_i = Var(P(transition | feature_i)) / Mean(P(transition | feature_i))P=transition probability, feature_i=feature indexQuantifying directional bias in feature-byte transitions for Hutter Prize optimization.data/germane/research/hutter/ANISOTROPIC_FEATURE_GENOME.mdHINT⚠️-LAYER_A_COMPRESSIONinformational_bind
577561Procedural_Mirror_SeedCompressionseed = (entropy * φ + spectral_density * π + log(size+1) / φ) mod 1.0 * 2^64φ=golden ratio, π=pi, S=sizeSelecting regime-based LUT strategies from global corpus metrics.data/germane/research/hutter/ANISOTROPIC_FEATURE_GENOME.mdHINT⚠️-LAYER_A_COMPRESSIONinformational_bind
578562Energy_Aware_SchedulingThermodynamicenergy(block) = -log2(P(block) / max(P(all_blocks)))P=block probabilityOrdering speculative decompressor forks by thermodynamic cost.data/germane/research/hutter/ENERGY_AWARE_DECOMPRESSION.mdHINT⚠️-LAYER_G_ENERGYthermodynamic_bind
579563Surprise_Regret_ScoreCognitive LoadScore = log(P) - λ · (α·surprise + β·regret)P=probability, λ=sensitivity, α,β=weightsHomeostatic decision gating for the Waveprobe kernel.data/germane/research/hutter/COGNITIVE_THERMO_README.mdHINT⚠️-LAYER_A_COMPRESSIONinformational_bind
580564aiscroll/ricci_flow_and_turbulenceManifold Evolution∂_t g_ij = -2 Ric_ij + μ T_ijg_ij=metric, Ric=curvature, T=turbulenceRicci flow with turbulence forcingdata/substrate_index.dbManual82,84,37LAYER_J_DYNAMICSgeometric_bind
581567Manifold_LittleLawManifold NetworkingL = λ · WL=average queue length, λ=arrival rate, W=average waiting timeLittle's Law applied to manifold queue statestools/lean/Semantics/Semantics/ManifoldNetworking.leanLean566LAYER_B_ROUTINGinformational_bind
582568Manifold_TokenBucketManifold Networkingtokens_{t+1} = min(bucketSize, tokens_t + rate·Δt - cost)tokens=current tokens, rate=refill rate, cost=consume cost, Δt=time deltaToken bucket mathematics for information density conservationtools/lean/Semantics/Semantics/ManifoldNetworking.leanLean567,570LAYER_A_COMPRESSIONinformational_bind
583569Manifold_AIMDManifold Networkingcwnd_{t+1} = cwnd_t + additiveIncrease (no congestion) OR cwnd_t · multiplicativeDecrease (congestion)cwnd=congestion window, additiveIncrease=increment per RTT, multiplicativeDecrease=fraction on lossAIMD congestion control adapted to manifold topologytools/lean/Semantics/Semantics/ManifoldNetworking.leanLean566,568LAYER_F_CONTROLcontrol_bind
584570Manifold_CUBICManifold Networkingcwnd(t) = C·(t - K)³ + wmaxK = cube_root(wmax·(1-β)/C), wmax=window before congestion, β=decrease factor, C=scaling constantCUBIC congestion control with cube root growth functiontools/lean/Semantics/Semantics/ManifoldNetworking.leanLean566,569LAYER_F_CONTROLcontrol_bind
585571Manifold_RoutingManifold Routingcost(path) = Σ_i (curvature_i·α + torsion_i·β + density_i·γ)curvature=manifold curvature, torsion=manifold torsion, density=information density, α,β,γ=weightsNon-normal manifold routing based on geometric topology tools/lean/Semantics/Semantics/ManifoldNetworking.leanLean566LAYER_C_TOPOLOGYgeometric_bind
586572Flat_Limit_TestManifold Networking Testκ=0 ∧ τ=0 ∧ paths=1 → ordinary_queue_behaviorκ=curvature, τ=torsionTest that flat manifold routing reduces to ordinary queue behaviortools/lean/Semantics/Semantics/FlatLimitTest.leanLean566,571LAYER_C_TOPOLOGYgeometric_bind
587573Conservation_TestManifold Networking Testtokens_{t+1} ≥ 0 ∧ tokens_{t+1} ≤ bucketSizetokens=current tokens, bucketSize=max capacityTest that information density does not silently vanish or duplicatetools/lean/Semantics/Semantics/ConservationTest.leanLean568LAYER_A_COMPRESSIONinformational_bind
588574Congestion_Stability_TestManifold Networking Testcwnd_t ≤ cwnd_0 ∧ cwnd_t > 0cwnd=congestion window, cwnd_0=initial windowTest that AIMD and CUBIC remain bounded under high loadtools/lean/Semantics/Semantics/CongestionStabilityTest.leanLean569,570LAYER_F_CONTROLcontrol_bind
589575Fairness_TestManifold Networking Testdensity_low ≥ threshold_mindensity_low=minimum path density, threshold_min=minimum acceptable densityTest that multiple manifold paths do not starve low-density routestools/lean/Semantics/Semantics/FairnessTest.leanLean571LAYER_B_ROUTINGinformational_bind
590576Phase_Ordering_TestManifold Networking Testphase_{t+1} ≥ phase_tphase=manifold phase flagTest that phase-based flow does not create invalid causal shortcutstools/lean/Semantics/Semantics/PhaseOrderingTest.leanLean571LAYER_D_INVARIANTScontrol_bind
591577Adversarial_Topology_TestManifold Networking Testκ > κ_critical τ > τ_critical → refuse/saturate/renormalizeκ=curvature, τ=torsion, κ_critical,τ_critical=thresholdsTest that extreme curvature/torsion routes are refused, saturated, or renormalizedtools/lean/Semantics/Semantics/AdversarialTopologyTest.leanLean571LAYER_C_TOPOLOGYgeometric_bind
592578PIST_Mass_CalculationPISTmass = a · b = t · (2k+1 - t)k=shell index, t=offset, a=distance to lower square, b=distance to upper squareQuadratic hyperbola index measuring tension within shell intervaltools/lean/Semantics/Semantics/PIST.leanLean579,580LAYER_C_TOPOLOGYgeometric_bind
593579PIST_Shell_CoordinatePISTn = k² + t, where 0 ≤ t ≤ 2k+1k=shell index, t=offset, n=natural numberInterval-local coordinate representation between consecutive squarestools/lean/Semantics/Semantics/PIST.leanLean578,580LAYER_C_TOPOLOGYgeometric_bind
594580PIST_Mirror_InvolutionPISTmirror(c) = {k, 2k+1-t} where mass(mirror(c)) = mass(c)c=coordinate, k=shell index, t=offsetSymmetry operation preserving mass within shelltools/lean/Semantics/Semantics/PIST.leanLean578LAYER_D_INVARIANTSgeometric_bind
595581PIST_Phase_ClassificationPISTphase(c) = grounded if mass=0 else seismicc=coordinate, mass=PIST massDiscrete phase flag based on zero vs positive masstools/lean/Semantics/Semantics/PIST.leanLean578,579LAYER_F_CONTROLcontrol_bind
596582PIST_Resonance_RelationPISTResonant(x,y) ↔ mass(x) = mass(y)x,y=coordinatesEquivalence relation for states with equal hyperbola indextools/lean/Semantics/Semantics/PIST.leanLean578LAYER_B_ROUTINGinformational_bind
597583PIST_Lyapunov_FunctionalPISTpotential(S) = mass(pos) + frictionS=state, pos=coordinate, friction=loss registerEnergy-like functional guaranteeing strict descent for lawful transitionstools/lean/Semantics/Semantics/PIST.leanLean578,581LAYER_G_ENERGYthermodynamic_bind
598584PIST_Hyperbola_IndexPISThyperbolaIndex(n) = a(n) · b(n) where a(n)=n-k², b(n)=(k+1)²-n, k=√nn=natural number, k=floor(sqrt(n))Symmetric square-gap tension measure tools/lean/Semantics/Semantics/PISTMachine.leanLean578,579LAYER_C_TOPOLOGYgeometric_bind
599585PIST_Normalized_TensionPISTρ(n) = a(n) / (2k+1) where k=√nn=natural number, k=floor(sqrt(n)), a=distance to lower squareTension ratio normalized by shell width [0,1]tools/lean/Semantics/Semantics/PISTMachine.leanLean578,584LAYER_G_ENERGYthermodynamic_bind
600586PIST_Zero_Mass_TheoremPISTmass(c) = 0 ↔ t=0 t=2k+1c=coordinate, t=offset, k=shell indexCharacterization of zero mass at shell endpointstools/lean/Semantics/Semantics/PIST.leanLean578,579LAYER_D_INVARIANTSgeometric_bind
601587PIST_Positive_Mass_TheoremPIST0 < mass(c) ↔ 0 < t ∧ t < 2k+1c=coordinate, t=offset, k=shell indexPositive mass equivalent to being strictly inside shelltools/lean/Semantics/Semantics/PIST.leanLean578,586LAYER_D_INVARIANTSgeometric_bind
602588PIST_Shell_EndpointsPISTlower(k) = {k,0}, upper(k) = {k,2k+1}k=shell indexLeft and right shell boundary coordinatestools/lean/Semantics/Semantics/PIST.leanLean578,579LAYER_C_TOPOLOGYgeometric_bind
603589PIST_LogEntryPISTLogEntry = {before, after, move, preservedMass}before/after=coordinates, move=transition type, preservedMass=booleanAppend-only history tracking for state transitionstools/lean/Semantics/Semantics/PIST.leanLean582,583LAYER_B_ROUTINGinformational_bind
604590PIST_State_PenalizePISTfriction_{t+1} = friction_t + penaltyfriction=loss register, penalty=nonnegativeIncrease friction on rejection with penalty accumulationtools/lean/Semantics/Semantics/PIST.leanLean583,589LAYER_F_CONTROLcontrol_bind
605591PIST_State_ResonanceJumpPISTpos_{t+1} = target, mass(pos_t) = mass(target)pos=coordinate, target=resonant coordinateJump to resonant coordinate preserving masstools/lean/Semantics/Semantics/PIST.leanLean582,583,589LAYER_B_ROUTINGinformational_bind
606593PIST_Kernel_StepPISTstep(S,R) = ground(project(assimilate(S,bind)),R)S=state, R=reality, bind=candidate, project=normalizer, ground=selectorOne-step evolution with projection and groundingtools/lean/Semantics/Semantics/PIST.leanLean583,595LAYER_H_ALGEBRAcontrol_bind
607594PIST_Kernel_Strict_DescentPISTpotential(step(S,R)) < potential(S) for nonterminal SS=state, R=nonterminalNonterminal steps strictly decrease Lyapunov potentialtools/lean/Semantics/Semantics/PIST.leanLean583,593LAYER_G_ENERGYthermodynamic_bind
608595PIST_ProjectorPISTproject(project(S)) = project(S)S=state, project=idempotent normalizerIdempotent projection operator for normalizationtools/lean/Semantics/Semantics/PIST.leanLean593,594LAYER_D_INVARIANTScontrol_bind
609596PIST_GrounderPISTground : State → Reality → StateS=state, R=external realityExternal grounding operator choosing next state from lawful candidatestools/lean/Semantics/Semantics/PIST.leanLean593,595LAYER_F_CONTROLcontrol_bind
610597PIST_Transition_PreservesMassPISTS.pos.mass = T.next.pos.massS=state, T=transitionWhether transition preserves shell mass at active coordinatetools/lean/Semantics/Semantics/PIST.leanLean591,598LAYER_D_INVARIANTSgeometric_bind
611598PIST_Transition_StrictlyDecreasesPISTT.next.potential < S.potentialS=state, T=transitionWhether transition strictly decreases Lyapunov potentialtools/lean/Semantics/Semantics/PIST.leanLean594,597LAYER_G_ENERGYthermodynamic_bind
612599PIST_LawfulMove_LinearPISTfriction unchanged, same shell k, t±1 stepfriction=loss register, k=shell index, t=offsetOne-step linear move within same shelltools/lean/Semantics/Semantics/PIST.leanLean597,598LAYER_C_TOPOLOGYgeometric_bind
613600PIST_LawfulMove_CrystallizedPISTtarget.mass=0, friction=0target=coordinate, friction=loss registerCrystallization to zero-mass endpoint with zero frictiontools/lean/Semantics/Semantics/PIST.leanLean588,599LAYER_D_INVARIANTSgeometric_bind
614601PIST_ClassifyPhase_With_ThresholdPISTphase = grounded if m=0 else drift if ρ<α else seismicm=mass, ρ=tension ratio, α=thresholdPhase classifier with drift band for low tensiontools/lean/Semantics/Semantics/PISTMachine.leanLean581,585LAYER_F_CONTROLcontrol_bind
615602PIST_Mirror_Preserves_Mass_TheoremPISThyperbolaIndex(mirror(n)) = hyperbolaIndex(n)n=natural number, mirror=involutionFormal proof that mirror operation preserves hyperbola indextools/lean/Semantics/Semantics/PISTMachine.leanLean580,584LAYER_D_INVARIANTSgeometric_bind
616603PIST_Zero_Mass_Square_TheoremPISThyperbolaIndex(n)=0 ↔ (√n)²=nn=natural numberZero mass iff n is a perfect squaretools/lean/Semantics/Semantics/PISTMachine.leanLean584,586LAYER_D_INVARIANTSgeometric_bind
617605PIST_BlitterStepPISTM_{k+1} = M_k ⊕ (fa+fb)>>16M=manifold, fa,fb=vector field components, ⊕=bitwise XORDiscrete Picard integral via bitwise accumulation (O(1) hardware)tools/lean/Semantics/Semantics/PistBridge.leanLean604LAYER_I_ENCODINGcontrol_bind
618607PIST_SISS_ScatterPISTv_out = R(s_ij)·v_inv=vector, R=reflection matrix, s_ij=tile seamScattering operator across tile boundariestools/lean/Semantics/Semantics/PistBridge.leanLean606LAYER_C_BRAIDgeometric_bind
619609PIST_Torsional_BetaStepPISTΔq3 = η·error, Δq1 = η·(q2-q1)·dt, energy = ||q1-q2||+||error||q=quaternions, η=learning rate, dt=timestep, error=target-q3Torsional beta step with backpropagation energytools/lean/Semantics/Semantics/TorsionalPIST.leanLean608LAYER_H_ALGEBRAgeometric_bind
620610PIST_Torsional_RGFlowPISTrecurse until energy < threshold, max depth Nenergy=error norm, threshold=convergence criterionRicci flow-style descent on torsional energy landscapetools/lean/Semantics/Semantics/TorsionalPIST.leanLean608,609LAYER_J_DYNAMICSgeometric_bind
621611Quaternion_AdditionQuaternion Algebraq1 + q2 = (w1+w2, x1+x2, y1+y2, z1+z2)q=w+xi+yj+zk, component-wise additionQuaternion addition operationtools/lean/Semantics/Semantics/Quaternion.leanLean612,613,614,615LAYER_H_ALGEBRAalgebraic_bind
622612Quaternion_SubtractionQuaternion Algebraq1 - q2 = (w1-w2, x1-x2, y1-y2, z1-z2)q=w+xi+yj+zk, component-wise subtractionQuaternion subtraction operationtools/lean/Semantics/Semantics/Quaternion.leanLean611,613,614,615LAYER_H_ALGEBRAalgebraic_bind
623613Quaternion_MultiplicationQuaternion Algebraq1·q2 = (w1w2-x1x2-y1y2-z1z2, w1x2+x1w2+y1z2-z1y2, w1y2-x1z2+y1w2+z1x2, w1z2+x1y2-y1x2+z1w2)q=w+xi+yj+zk, Hamiltonian productNon-commutative Hamiltonian multiplicationtools/lean/Semantics/Semantics/Quaternion.leanLean611,612,614,615,617LAYER_H_ALGEBRAalgebraic_bind
624614Quaternion_DotProductQuaternion Algebraq1·q2 = w1w2 + x1x2 + y1y2 + z1z2q=w+xi+yj+zk, scalar dot productEuclidean inner product in R⁴tools/lean/Semantics/Semantics/Quaternion.leanLean611,612,613,615,617LAYER_H_ALGEBRAalgebraic_bind
625615Quaternion_NormApproximationQuaternion Algebra||q|| ≈ max(|w|,|x|,|y|,|z|) + (3/8)·Σothersq=w+xi+yj+zk, 4D octagonal normApproximate quaternion norm for hardware efficiencytools/lean/Semantics/Semantics/Quaternion.leanLean611,612,613,614,617LAYER_G_ENERGYalgebraic_bind
626616Quaternion_ConjugateQuaternion Algebraq* = w - xi - yj - zkq=w+xi+yj+zk, sign flip of vector partQuaternion conjugation operationtools/lean/Semantics/Semantics/Quaternion.leanLean611,612,613,614,615,617LAYER_H_ALGEBRAalgebraic_bind
627617Quaternion_ScalarMultiplicationQuaternion Algebras·q = (sw, sx, sy, sz)s=scalar, q=w+xi+yj+zkScalar multiplication of quaterniontools/lean/Semantics/Semantics/Quaternion.leanLean611,612,613,614,615,616LAYER_H_ALGEBRAalgebraic_bind
628618Quaternion_Extension_TheoremQuaternion Algebraext: q1=q2 ↔ w1=w2 ∧ x1=x2 ∧ y1=y2 ∧ z1=z2q=w+xi+yj+zkExtensionality theorem for quaternion equalitytools/lean/Semantics/Semantics/Quaternion.leanLean611,612,613,614,615,616,617LAYER_D_INVARIANTSalgebraic_bind
629619Q16_16_Saturated_From_NatFixed-Point ArithmeticsatFromNat(n) = clamp(n/65536, -32768, 32767)n=natural number, scale=65536Saturating conversion from natural to Q16.16tools/lean/Semantics/Semantics/Q16_16.leanLean620,621,622,623,624,625,626,627,628,629,630,631,632LAYER_I_ENCODINGalgebraic_bind
630620Q16_16_AdditionFixed-Point Arithmetica + b = (a.val + b.val) with saturationa,b=Q16.16 fixed-pointSaturating addition for fixed-point arithmetictools/lean/Semantics/Semantics/Q16_16.leanLean619,621,622,623,624,625,626,627,628,629,630,631,632LAYER_I_ENCODINGalgebraic_bind
631621Q16_16_SubtractionFixed-Point Arithmetica - b = (a.val - b.val) with saturationa,b=Q16.16 fixed-pointSaturating subtraction for fixed-point arithmetictools/lean/Semantics/Semantics/Q16_16.leanLean619,620,622,623,624,625,626,627,628,629,630,631,632LAYER_I_ENCODINGalgebraic_bind
632622Q16_16_MultiplicationFixed-Point Arithmetica × b = (a.val × b.val) >> 16 with saturationa,b=Q16.16 fixed-pointSaturating multiplication with 16-bit right shifttools/lean/Semantics/Semantics/Q16_16.leanLean619,620,621,623,624,625,626,627,628,629,630,631,632LAYER_I_ENCODINGalgebraic_bind
633623Q16_16_DivisionFixed-Point Arithmetica / b = (a.val << 16) / b.val with saturationa,b=Q16.16 fixed-point, b≠0Saturating division with 16-bit left shifttools/lean/Semantics/Semantics/Q16_16.leanLean619,620,621,622,624,625,626,627,628,629,630,631,632LAYER_I_ENCODINGalgebraic_bind
634624Q16_16_AbsoluteValueFixed-Point Arithmetic|a| = if a.val < 0x80000000 then a else -aa=Q16.16 fixed-pointAbsolute value with sign handlingtools/lean/Semantics/Semantics/Q16_16.leanLean619,620,621,622,623,625,626,627,628,629,630,631,632LAYER_I_ENCODINGalgebraic_bind
635625Q16_16_NegationFixed-Point Arithmetic-a = -a.val with saturationa=Q16.16 fixed-pointSaturating negation operationtools/lean/Semantics/Semantics/Q16_16.leanLean619,620,621,622,623,624,626,627,628,629,630,631,632LAYER_I_ENCODINGalgebraic_bind
636626Q16_16_FromNatFixed-Point ArithmeticofNat(n) = n × 65536n=natural number, scale=65536Convert natural number to Q16.16 fixed-pointtools/lean/Semantics/Semantics/Q16_16.leanLean619,620,621,622,623,624,627,628,629,630,631,632LAYER_I_ENCODINGalgebraic_bind
637627Q16_16_ToNatFixed-Point ArithmetictoNat(q) = q.val / 65536q=Q16.16 fixed-point, scale=65536Convert Q16.16 fixed-point to natural numbertools/lean/Semantics/Semantics/Q16_16.leanLean619,620,621,622,623,624,626,628,629,630,631,632LAYER_I_ENCODINGalgebraic_bind
638628Q16_16_FromFracFixed-Point ArithmeticofFrac(num,denom) = (num × 65536) / denomnum,denom=natural numbersConvert fraction to Q16.16 fixed-pointtools/lean/Semantics/Semantics/Q16_16.leanLean619,620,621,622,623,624,625,626,627,629,630,631,632LAYER_I_ENCODINGalgebraic_bind
639629Q16_16_LessThanFixed-Point Arithmetica < b ↔ a.val < b.vala,b=Q16.16 fixed-pointComparison: less thantools/lean/Semantics/Semantics/Q16_16.leanLean619,620,621,622,623,624,625,626,627,628,630,631,632LAYER_D_INVARIANTSalgebraic_bind
640630Q16_16_LessEqualFixed-Point Arithmetica ≤ b ↔ a.val ≤ b.vala,b=Q16.16 fixed-pointComparison: less than or equaltools/lean/Semantics/Semantics/Q16_16.leanLean619,620,621,622,623,624,625,626,627,628,629,631,632LAYER_D_INVARIANTSalgebraic_bind
641631Q16_16_GreaterThanFixed-Point Arithmetica > b ↔ a.val > b.vala,b=Q16.16 fixed-pointComparison: greater thantools/lean/Semantics/Semantics/Q16_16.leanLean619,620,621,622,623,624,625,626,627,628,629,630,632LAYER_D_INVARIANTSalgebraic_bind
642632Q16_16_Equality_FixedPointFixed-Point Arithmetica = b ↔ a.val = b.vala,b=Q16.16 fixed-pointComparison: equality checktools/lean/Semantics/Semantics/Q16_16.leanLean619,620,621,622,623,624,625,626,627,628,629,630,631LAYER_D_INVARIANTSalgebraic_bind
643633Q16_16_MinimumFixed-Point Arithmeticmin(a,b) = if a.val < b.val then a else ba,b=Q16.16 fixed-pointMinimum of two fixed-point valuestools/lean/Semantics/Semantics/Q16_16.leanLean619,620,621,622,623,624,625,626,627,628,629,630,631,632,634LAYER_D_INVARIANTSalgebraic_bind
644634Q16_16_MaximumFixed-Point Arithmeticmax(a,b) = if a.val > b.val then a else ba,b=Q16.16 fixed-pointMaximum of two fixed-point valuestools/lean/Semantics/Semantics/Q16_16.leanLean619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,635LAYER_D_INVARIANTSalgebraic_bind
645635Q16_16_ClampFixed-Point Arithmeticclamp(x,lo,hi) = max(lo, min(x,hi))x,lo,hi=Q16.16 fixed-pointClamp value to range [lo,hi]tools/lean/Semantics/Semantics/Q16_16.leanLean619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634LAYER_D_INVARIANTSalgebraic_bind
646636Q16_16_SquareRootFixed-Point Arithmetic√x via Newton-Raphson iterationx=Q16.16 fixed-point, x≥0Approximate square root using Newton's methodtools/lean/Semantics/Semantics/Q16_16.leanLean619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635LAYER_G_ENERGYalgebraic_bind
647637PTOSManifest_StructureDeltaGCL Compression{version, timestamp, checksum, payload}version=UInt8, timestamp=UInt64, checksum=UInt32, payload=List UInt8PTOS manifest structure for delta compressiontools/lean/Semantics/Semantics/DeltaGCLCompression.leanLean638,639,640,641,642,643LAYER_A_COMPRESSIONinformational_bind
648638DeltaEncoding_StructureDeltaGCL Compression{deltaFlag, changedFields, fieldDeltas}deltaFlag=Bool, changedFields=List String, fieldDeltas=List UInt8Delta encoding structure for manifest changestools/lean/Semantics/Semantics/DeltaGCLCompression.leanLean637,639,640,641,642,643LAYER_A_COMPRESSIONinformational_bind
649639ComputeDelta_DeltaGCLDeltaGCL Compressiondelta = current ⊖ previouscurrent,previous=PTOSManifest, ⊖=delta operationCompute delta between two PTOS manifeststools/lean/Semantics/Semantics/DeltaGCLCompression.leanLean637,638,640,641,642,643LAYER_A_COMPRESSIONinformational_bind
650640ApplyPTOSDictionaryDeltaGCL Compressionencode(manifest) = 4-byte dictionary encodingmanifest=PTOSManifestPTOS dictionary compression to 4 bytestools/lean/Semantics/Semantics/DeltaGCLCompression.leanLean637,639,641,642,643LAYER_A_COMPRESSIONinformational_bind
651641EncodeCodon_VariableLengthDeltaGCL Compressionencode(codon) = codon if known else 0xFF||codoncodon=String, 0xFF=escape byteVariable-length codon encoding with escape mechanismtools/lean/Semantics/Semantics/DeltaGCLCompression.leanLean637,638,639,640,642,643LAYER_A_COMPRESSIONinformational_bind
652642DeltaGCLSequence_StructureDeltaGCL Compression{marker, bytes, fieldCodes}marker=Char ('F' or 'D'), bytes=List UInt8, fieldCodes=List UInt8Delta GCL sequence structuretools/lean/Semantics/Semantics/DeltaGCLCompression.leanLean637,638,639,640,641,643LAYER_A_COMPRESSIONinformational_bind
653643EncodeToDeltaGCLDeltaGCL Compressionencode(manifest, previous) = DeltaGCLSequence(manifest, previous)manifest=PTOSManifest, previous=Option PTOSManifestEncode manifest to Delta GCL format with optional previoustools/lean/Semantics/Semantics/DeltaGCLCompression.leanLean637,638,639,640,641,642,644,645,646LAYER_A_COMPRESSIONinformational_bind
654644CompressionStats_StructureDeltaGCL Compression{originalLength, compressedLength, reduction}originalLength=Nat, compressedLength=Nat, reduction=NatCompression statistics structuretools/lean/Semantics/Semantics/DeltaGCLCompression.leanLean637,638,639,640,641,642,643,645,646LAYER_A_COMPRESSIONinformational_bind
655645CompressionStats_CalculationDeltaGCL Compressionstats(original, deltaGCL) = {len(original), len(deltaGCL), len(original)-len(deltaGCL)}original=String, deltaGCL=DeltaGCLSequenceCalculate compression statisticstools/lean/Semantics/Semantics/DeltaGCLCompression.leanLean637,638,639,640,641,642,643,644,646LAYER_A_COMPRESSIONinformational_bind
656646ComputeDelta_Identical_TheoremDeltaGCL CompressioncomputeDelta(m,m) = {deltaFlag=false, changedFields=[], fieldDeltas=[]}m=PTOSManifestTheorem: delta of identical manifests is emptytools/lean/Semantics/Semantics/DeltaGCLCompression.leanLean637,638,639,640,641,642,643,644,645LAYER_D_INVARIANTSinformational_bind
657647IntNode_StructureBraid Field Theory{val, left, right}val=Int, left/right=IntNodeInteger node structure for braid topologytools/lean/Semantics/Semantics/BraidField.leanLean648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666LAYER_C_TOPOLOGYgeometric_bind
658648IntNode_AdditionBraid Field Theoryadd(a,b) = IntNode with merged structurea,b=IntNodeRecursive addition of integer nodestools/lean/Semantics/Semantics/BraidField.leanLean647,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666LAYER_H_ALGEBRAalgebraic_bind
659649BettiCycle_StructureBraid Field Theory{cycle, index}cycle=List IntNode, index=NatBetti cycle structure for homology computationtools/lean/Semantics/Semantics/BraidField.leanLean647,648,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666LAYER_C_BRAIDgeometric_bind
660650BettiCycleSet_StructureBraid Field Theory{cycles}cycles=List BettiCycleSet of Betti cyclestools/lean/Semantics/Semantics/BraidField.leanLean647,648,649,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666LAYER_C_BRAIDgeometric_bind
661652Mountain_StructureBraid Field Theorynode(height, apex, base, inner)height=Nat, apex=IntNode, base=List IntNode, inner=MMRMountain structure for braid topologytools/lean/Semantics/Semantics/BraidField.leanLean647,648,649,650,651,653,654,655,656,657,658,659,660,661,662,663,664,665,666LAYER_C_TOPOLOGYgeometric_bind
662653Mountain_HeightBraid Field Theoryheight(m) = tree heightm=MountainHeight of mountain structuretools/lean/Semantics/Semantics/BraidField.leanLean647,648,649,650,651,652,654,655,656,657,658,659,660,661,662,663,664,665,666LAYER_C_TOPOLOGYgeometric_bind
663654Mountain_ApexBraid Field Theoryapex(m) = root nodem=MountainApex (root) of mountain structuretools/lean/Semantics/Semantics/BraidField.leanLean647,648,649,650,651,652,653,655,656,657,658,659,660,661,662,663,664,665,666LAYER_C_TOPOLOGYgeometric_bind
664655Mountain_BaseBraid Field Theorybase(m) = leaf nodesm=MountainBase (leaves) of mountain structuretools/lean/Semantics/Semantics/BraidField.leanLean647,648,649,650,651,652,653,654,656,657,658,659,660,661,662,663,664,665,666LAYER_C_TOPOLOGYgeometric_bind
665656Mountain_InnerBraid Field Theoryinner(m) = MMR subtreem=MountainInner MMR structuretools/lean/Semantics/Semantics/BraidField.leanLean647,648,649,650,651,652,653,654,655,657,658,659,660,661,662,663,664,665,666LAYER_C_TOPOLOGYgeometric_bind
666657Mountain_MergeBraid Field Theorymerge(m1,m2) = combined mountainm1,m2=MountainRecursive merge of mountain structurestools/lean/Semantics/Semantics/BraidField.leanLean647,648,649,650,651,652,653,654,655,656,658,659,660,661,662,663,664,665,666LAYER_C_BRAIDgeometric_bind
667658MMR_AppendBraid Field Theoryappend(mmr,m) = mmr ++ mmmr=MMR, m=MountainAppend mountain to MMR (Multi-Mountain Range)tools/lean/Semantics/Semantics/BraidField.leanLean647,648,649,650,651,652,653,654,655,656,657,659,660,661,662,663,664,665,666LAYER_C_BRAIDgeometric_bind
668659PISTField_StructureBraid Field Theory{scale, mmr, mergeDebt, isStable, burden, geometry, adaptation, protection}scale=Nat, mmr=MMR, mergeDebt=Nat, isStable=Bool, burden/geometry/adaptation/protection=Q16_16PIST field structure with cost metricstools/lean/Semantics/Semantics/BraidField.leanLean647,648,649,650,651,652,653,654,655,656,657,658,660,661,662,663,664,665,666LAYER_C_TOPOLOGYgeometric_bind
669660BurdenCost_FunctionBraid Field TheoryburdenCost(load,target,metric) = |load-target|/targetload,target=Nat, metric=MetricBurden cost based on load deviation from targettools/lean/Semantics/Semantics/BraidField.leanLean647,648,649,650,651,652,653,654,655,656,657,658,659,661,662,663,664,665,666LAYER_B_ROUTINGcontrol_bind
670661GeometryCost_FunctionBraid Field TheorygeometryCost(curvature,ideal,metric) = |curvature-ideal|/idealcurvature,ideal=Nat, metric=MetricGeometry cost based on curvature deviationtools/lean/Semantics/Semantics/BraidField.leanLean647,648,649,650,651,652,653,654,655,656,657,658,659,660,662,663,664,665,666LAYER_C_TOPOLOGYcontrol_bind
671662AdaptationCost_FunctionBraid Field TheoryadaptationCost(current,optimal,metric) = |current-optimal|/optimalcurrent,optimal=Nat, metric=MetricAdaptation cost based on deviation from optimaltools/lean/Semantics/Semantics/BraidField.leanLean647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,663,664,665,666LAYER_F_CONTROLcontrol_bind
672663ProtectionCost_FunctionBraid Field TheoryprotectionCost(safety,threshold,metric) = max(0, threshold-safety)/thresholdsafety,threshold=Nat, metric=MetricProtection cost based on safety margin violationtools/lean/Semantics/Semantics/BraidField.leanLean647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,664,665,666LAYER_E_VERIFICATIONcontrol_bind
673664ComputePIST_FunctionBraid Field TheorycomputePIST(scale,mmr,mergeDebt,isStable) = PISTField with computed costsscale=Nat, mmr=MMR, mergeDebt=Nat, isStable=BoolCompute PIST field with all cost metricstools/lean/Semantics/Semantics/BraidField.leanLean647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,665,666LAYER_C_TOPOLOGYgeometric_bind
674665SpherionState_StructureBraid Field Theory{uvScale, mmr, mergeDebt, isStable, currentPIST, void}uvScale=Nat, mmr=MMR, mergeDebt=Nat, isStable=Bool, currentPIST=PISTField, void=BettiCycleSetSpherion state for topological evolutiontools/lean/Semantics/Semantics/BraidField.leanLean647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,666LAYER_J_DYNAMICSgeometric_bind
675666SpherionState_InitBraid Field Theoryinit(uvScale) = initial SpherionStateuvScale=NatInitialize Spherion state with UV scaletools/lean/Semantics/Semantics/BraidField.leanLean647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665LAYER_J_DYNAMICSgeometric_bind
676667Genome_StructureAdaptation Theory{muBin, rhoBin, cBin, mBin, neBin, sigBin}muBin,rhoBin,cBin,mBin,neBin,sigBin=Fin 4Genome structure for adaptive parameterstools/lean/Semantics/Semantics/Adaptation.leanLean668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689LAYER_F_CONTROLcontrol_bind
677668Genome_MuQAdaptation TheorymuQ(g) = 0x41·(muBin+1)g=Genome, muBin=Fin 4Mu parameter quantization (0x41 scale)tools/lean/Semantics/Semantics/Adaptation.leanLean667,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689LAYER_F_CONTROLcontrol_bind
678669Genome_RhoQAdaptation TheoryrhoQ(g) = 0x2000·(rhoBin+1)g=Genome, rhoBin=Fin 4Rho parameter quantization (0x2000 scale)tools/lean/Semantics/Semantics/Adaptation.leanLean667,668,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689LAYER_F_CONTROLcontrol_bind
679670Genome_CFacAdaptation TheorycFac(g) = 0x2000·(cBin+1)g=Genome, cBin=Fin 4C parameter quantization (0x2000 scale)tools/lean/Semantics/Semantics/Adaptation.leanLean667,668,669,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689LAYER_F_CONTROLcontrol_bind
680671Genome_MFacAdaptation TheorymFac(g) = 0x2000·(mBin+1)g=Genome, mBin=Fin 4M parameter quantization (0x2000 scale)tools/lean/Semantics/Semantics/Adaptation.leanLean667,668,669,670,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689LAYER_F_CONTROLcontrol_bind
681672Genome_NeRawAdaptation TheoryneRaw(g) = 0x4000·(neBin+1)g=Genome, neBin=Fin 4Ne parameter quantization (0x4000 scale)tools/lean/Semantics/Semantics/Adaptation.leanLean667,668,669,670,671,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689LAYER_F_CONTROLcontrol_bind
682673Genome_SigQAdaptation TheorysigQ(g) = 65536 + 0x4000·(sigBin+1)g=Genome, sigBin=Fin 4Sigma parameter quantization with offsettools/lean/Semantics/Semantics/Adaptation.leanLean667,668,669,670,671,672,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689LAYER_F_CONTROLcontrol_bind
683674Genome_NeEffAdaptation TheoryneEff(g) = neRaw(g)·(1 - cFac(g)/65536)g=GenomeEffective Ne with cFac modulationtools/lean/Semantics/Semantics/Adaptation.leanLean667,668,669,670,671,672,673,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689LAYER_F_CONTROLcontrol_bind
684675Genome_IsLawfulAdaptation TheoryisLawful(g) = all bins ≤ 3g=GenomeCheck if genome bins are in valid rangetools/lean/Semantics/Semantics/Adaptation.leanLean667,668,669,670,671,672,673,674,676,677,678,679,680,681,682,683,684,685,686,687,688,689LAYER_D_INVARIANTScontrol_bind
685676Genome_BetaStepAdaptation TheorybetaStep(g) = increment bins with overflow wrapg=GenomeOne-step genome mutation (beta step)tools/lean/Semantics/Semantics/Adaptation.leanLean667,668,669,670,671,672,673,674,675,677,678,679,680,681,682,683,684,685,686,687,688,689LAYER_F_CONTROLcontrol_bind
686677Genome_IsScaleCoherentAdaptation TheoryisScaleCoherent(g) = muQ·rhoQ·cFac·mFac·neEff·sigQ in valid rangeg=GenomeCheck if genome parameters are scale-coherenttools/lean/Semantics/Semantics/Adaptation.leanLean667,668,669,670,671,672,673,674,675,676,678,679,680,681,682,683,684,685,686,687,688,689LAYER_D_INVARIANTScontrol_bind
687678FlowAudit_StructureAdaptation Theory{init, steps, final, eventuallyLawful, finalLawful}init=Bool, steps=Nat, final=Genome, eventuallyLawful=Bool, finalLawful=BoolFlow audit structure for genome evolutiontools/lean/Semantics/Semantics/Adaptation.leanLean667,668,669,670,671,672,673,674,675,676,677,679,680,681,682,683,684,685,686,687,688,689LAYER_E_VERIFICATIONcontrol_bind
688679FlowAudit_FunctionAdaptation TheoryflowAudit(g,steps) = audit after n stepsg=Genome, steps=NatRun flow audit for n genome stepstools/lean/Semantics/Semantics/Adaptation.leanLean667,668,669,670,671,672,673,674,675,676,677,678,680,681,682,683,684,685,686,687,688,689LAYER_E_VERIFICATIONcontrol_bind
689680FinalLawfulEqEventuallyLawful_TheoremAdaptation TheoryfinalLawful = eventuallyLawful by constructionGenome evolution auditTheorem: final and eventual lawfulness are equaltools/lean/Semantics/Semantics/Adaptation.leanLean667,668,669,670,671,672,673,674,675,676,677,678,679LAYER_D_INVARIANTScontrol_bind
690681VecN_ZeroDynamic Canal TheoryVecN.zero = zero vector of length nn=Nat, VecN n = Nat → Q16_16Zero vector in n-dimensional spacetools/lean/Semantics/Semantics/DynamicCanal.leanLean682,683,684,685,686LAYER_H_ALGEBRAalgebraic_bind
691682VecN_AddDynamic Canal TheoryvecAdd(a,b) = component-wise additiona,b=VecN nVector addition in n-dimensional spacetools/lean/Semantics/Semantics/DynamicCanal.leanLean681,683,684,685,686LAYER_H_ALGEBRAalgebraic_bind
692683VecN_SubDynamic Canal TheoryvecSub(a,b) = component-wise subtractiona,b=VecN nVector subtraction in n-dimensional spacetools/lean/Semantics/Semantics/DynamicCanal.leanLean681,682,684,685,686LAYER_H_ALGEBRAalgebraic_bind
693684VecN_L1NormDynamic Canal TheoryvecL1(v) = Σ_i |v_i|v=VecN nL1 norm (Manhattan distance)tools/lean/Semantics/Semantics/DynamicCanal.leanLean681,682,683,685,686LAYER_G_ENERGYalgebraic_bind
694685VecN_MaxAbsDynamic Canal TheoryvecMaxAbs(v) = max_i |v_i|v=VecN nMaximum absolute value componenttools/lean/Semantics/Semantics/DynamicCanal.leanLean681,682,683,684,686LAYER_D_INVARIANTSalgebraic_bind
695686VecN_DotProductDynamic Canal TheoryvecDot(a,b) = Σ_i a_i·b_ia,b=VecN nEuclidean dot product in n-dimensional spacetools/lean/Semantics/Semantics/DynamicCanal.leanLean681,682,683,684,685LAYER_H_ALGEBRAalgebraic_bind
696687DIAT_StructureDynamic Canal Theory{shell, offset}shell=Nat, offset=NatDIAT (Distance Interval-Aware Type) structuretools/lean/Semantics/Semantics/DynamicCanal.leanLean688,689,690LAYER_C_TOPOLOGYgeometric_bind
697688DIAT_IntegerSqrtDynamic Canal Theoryisqrt(n) = floor(√n)n=UInt32Integer square root via Newton-Raphsontools/lean/Semantics/Semantics/DynamicCanal.leanLean687,689,690LAYER_G_ENERGYalgebraic_bind
698689DIAT_EncodeDynamic Canal Theoryencode(n) = {shell=√n, offset=n-shell²}n=UInt32Encode natural number to DIAT coordinatestools/lean/Semantics/Semantics/DynamicCanal.leanLean687,688,690LAYER_I_ENCODINGalgebraic_bind
699690DIAT_ShellWidthDynamic Canal TheoryshellWidth(d) = 2·shell+1d=DIATWidth of shell intervaltools/lean/Semantics/Semantics/DynamicCanal.leanLean687,688,689LAYER_C_TOPOLOGYgeometric_bind
700691DIAT_NormADynamic Canal TheorynormA(d) = offset/shellWidthd=DIATNormalized offset within shelltools/lean/Semantics/Semantics/DynamicCanal.leanLean687,688,689,690LAYER_G_ENERGYalgebraic_bind
701692DynamicCanalLambdaDynamic Canal Theoryλ(p,pressure) = saturation + (1-σ)·pressure·e^(-pressure)p=KernelParams, pressure=Q16_16, σ=deformabilityDynamic canal lambda function for pressure responsetools/lean/Semantics/Semantics/DynamicCanal.leanLean693,694,695,696LAYER_F_CONTROLcontrol_bind
702693CanalComplianceDynamic Canal Theorycompliance(p,pressure) = λ(p,pressure)·pressurep=KernelParams, pressure=Q16_16Canal compliance as function of pressuretools/lean/Semantics/Semantics/DynamicCanal.leanLean692,694,695,696LAYER_F_CONTROLcontrol_bind
703694CanalWidthDynamic Canal Theorywidth(p,base,pressure) = base·compliance(p,pressure)p=KernelParams, base=Q16_16, pressure=Q16_16Canal width response to pressuretools/lean/Semantics/Semantics/DynamicCanal.leanLean692,693,695,696LAYER_C_TOPOLOGYgeometric_bind
704695EdgeEval_StructureDynamic Canal Theory{edge, surprise, regret, stress}edge=Edge, surprise/regret/stress=Q16_16Edge evaluation with cognitive metricstools/lean/Semantics/Semantics/DynamicCanal.leanLean696,697,698,699,700,701,702,703,704LAYER_E_VERIFICATIONcontrol_bind
705696SurpriseOfDynamic Canal Theorysurprise(ev) = -log(P(ev))ev=EdgeEval, P=probabilitySurprise metric for edge evaluationtools/lean/Semantics/Semantics/DynamicCanal.leanLean695,697,698,699,700,701,702,703,704LAYER_G_ENERGYthermodynamic_bind
706697RegretOfDynamic Canal Theoryregret(ev) = max(0, optimal - actual)ev=EdgeEval, optimal=best possible, actual=achievedRegret metric for decision qualitytools/lean/Semantics/Semantics/DynamicCanal.leanLean695,696,698,699,700,701,702,703,704LAYER_G_ENERGYthermodynamic_bind
707698StressOfEvalDynamic Canal Theorystress(p,ev) = λ·surprise + μ·regretp=KernelParams, ev=EdgeEval, λ,μ=weightsStress metric combining surprise and regrettools/lean/Semantics/Semantics/DynamicCanal.leanLean695,696,697,699,700,701,702,703,704LAYER_G_ENERGYthermodynamic_bind
708699EdgeScoreDynamic Canal Theoryscore(p,lane,ev) = -stress(p,ev) - bandwidthCostp=KernelParams, lane=Lane, ev=EdgeEvalEdge scoring for lane selectiontools/lean/Semantics/Semantics/DynamicCanal.leanLean695,696,697,698,700,701,702,703,704LAYER_B_ROUTINGcontrol_bind
709700ClassifyRegimeDynamic Canal Theoryregime = classify(p,lane,chosen)p=KernelParams, lane=Lane, chosen=EdgeClassify flow regime based on edge choicetools/lean/Semantics/Semantics/DynamicCanal.leanLean695,696,697,698,699,701,702,703,704LAYER_F_CONTROLcontrol_bind
710701Q16_16_Add_Total_TheoremDynamic Canal Theory∃c, Q16_16.add(a,b) = ca,b=Q16_16Totality theorem for Q16.16 additiontools/lean/Semantics/Semantics/DynamicCanal.leanLean702,703,704,705LAYER_D_INVARIANTSalgebraic_bind
711702Q16_16_Sub_Total_TheoremDynamic Canal Theory∃c, Q16_16.sub(a,b) = ca,b=Q16_16Totality theorem for Q16.16 subtractiontools/lean/Semantics/Semantics/DynamicCanal.leanLean701,703,704,705LAYER_D_INVARIANTSalgebraic_bind
712703Q16_16_Mul_Total_TheoremDynamic Canal Theory∃c, Q16_16.mul(a,b) = ca,b=Q16_16Totality theorem for Q16.16 multiplicationtools/lean/Semantics/Semantics/DynamicCanal.leanLean701,702,704,705LAYER_D_INVARIANTSalgebraic_bind
713704Q16_16_Div_Total_TheoremDynamic Canal Theory∃c, Q16_16.div(a,b) = ca,b=Q16_16, b≠0Totality theorem for Q16.16 divisiontools/lean/Semantics/Semantics/DynamicCanal.leanLean701,702,703,705LAYER_D_INVARIANTSalgebraic_bind
714705DynamicCanalLambda_Total_TheoremDynamic Canal Theory∃c, λ(p,pressure) = cp=KernelParams, pressure=Q16_16Totality theorem for dynamic canal lambdatools/lean/Semantics/Semantics/DynamicCanal.leanLean692,701,702,703,704LAYER_D_INVARIANTScontrol_bind
715706Synaptic_Hotspot_DensityNeural Developmentρ(x,t) = ρ₀·exp(-(x-x₀)²/2σ²)·H(t-t_on)·H(t_off-t)x=position along dendrite, t=time, x₀=hotspot center, σ=width, t_on=onset, t_off=offset, H=HeavisideGaussian density function for adolescent synaptic hotspot formationdocs/neuroscience/SYNAPTIC_HOTSPOT_DYNAMICS.mdDocumented707,708LAYER_B_ROUTINGcontrol_bind
716707Adolescent_Formation_RateNeural Developmentdρ/dt = α·(1-ρ/ρ_max)·H(t-t_on) - β·ρ·H(t-t_off)α=formation rate, β=pruning rate, ρ_max=max density, t_on=adolescence onset, t_off=adolescence offsetDifferential equation for synapse formation vs pruning during adolescencedocs/neuroscience/SYNAPTIC_HOTSPOT_DYNAMICS.mdDocumented706,708,709LAYER_B_ROUTINGcontrol_bind
717708Pruning_Formation_BalanceNeural DevelopmentB(t) = ∫₀ᵗ (α·H(τ-t_on) - β·H(τ-t_off)) dτ / (α+β)Balance ratio between formation and pruning over timeQuantifies net synaptic change during developmental windowsdocs/neuroscience/SYNAPTIC_HOTSPOT_DYNAMICS.mdDocumented706,707,709LAYER_B_ROUTINGcontrol_bind
718709Mutation_Impact_ModelNeural Developmentα_mut = α·(1-δ·I[mutation])·γ_layerδ=impairment factor (0-1), I[mutation]=indicator for schizophrenia genes (Setd1a, Hivep2, Grin1), γ_layer=layer-specific coefficient (Layer 5=1.0)Gene mutation effects on adolescent synapse formation ratesdocs/neuroscience/SYNAPTIC_HOTSPOT_DYNAMICS.mdDocumented707,708,710LAYER_E_VERIFICATIONcontrol_bind
719710Layer_Specific_FormationNeural Developmentγ_layer = {L1:0.2, L2/3:0.5, L4:0.8, L5:1.0, L6:0.6}Cortical layer coefficients for hotspot formation propensityLayer 5 neurons show highest hotspot formation (control center)docs/neuroscience/SYNAPTIC_HOTSPOT_DYNAMICS.mdDocumented709LAYER_C_TOPOLOGYcontrol_bind
720711Local_Autonomy_WeightCephalopod Distributedw_local = γ · (1 - s_central)γ=autonomy coefficient (0.5-1.0), s_central=central signal strength [0,1]Local decision weight inversely proportional to central controldocs/neuroscience/CEPHALOPOD_DISTRIBUTED_NEURAL.mdDocumented712,713LAYER_B_ROUTINGcontrol_bind
721712Arm_ConsensusCephalopod Distributedconsensus = Σ_i (w_i · state_i) / Σ w_iw_i = w_local for arm i, state_i=local sensory stateWeighted consensus across semi-autonomous armsdocs/neuroscience/CEPHALOPOD_DISTRIBUTED_NEURAL.mdDocumented711,713LAYER_C_TOPOLOGYcontrol_bind
722713Distributed_Sensory_IntegrationCephalopod Distributedsensory_map = ⊕_j local_sensory_j⊕=XOR fusion, local_sensory_j=arm j sensory inputXOR-based fusion without central bottleneckdocs/neuroscience/CEPHALOPOD_DISTRIBUTED_NEURAL.mdDocumented711,712LAYER_C_TOPOLOGYcontrol_bind
723714Peripheral_Neuron_DensityCephalopod Distributedρ_peripheral = 0.67 · N_total67% of neurons in peripheral arms (vs central brain)Quantifies distributed neural architecturedocs/neuroscience/CEPHALOPOD_DISTRIBUTED_NEURAL.mdDocumented711,712,713LAYER_C_TOPOLOGYgeometric_bind
724715EI_Imbalance_AutismNeurodivergentξ = w_excite / w_inhibit; ξ_autism > ξ_neurotypicalξ=E/I ratio, w_excite=excitatory synaptic weight, w_inhibit=inhibitory synaptic weightExcitation/inhibition balance shifted toward excitation in autismdocs/neuroscience/NEURODIVERGENT_BRAIN_ARCHITECTURES.mdDocumented716,718LAYER_B_ROUTINGcontrol_bind
725716Local_Long_Range_Connectivity_AutismNeurodivergentκ = f_local / f_long; κ_autism > 1, κ_neurotypical ≈ 1κ=connectivity ratio, f_local=local connectivity density, f_long=long-range connectivity densityLocal over-connectivity with long-range under-connectivity in autismdocs/neuroscience/NEURODIVERGENT_BRAIN_ARCHITECTURES.mdDocumented715,718LAYER_C_TOPOLOGYgeometric_bind
726717Dopamine_Transport_DeficitNeurodivergentτ_clear = τ_normal / (1 - δ_adhd)τ_clear=dopamine clearance time, δ_adhd=deficit factor (0.2-0.4), τ_normal=neurotypical clearanceReduced dopamine transporter density in ADHD slows clearancedocs/neuroscience/NEURODIVERGENT_BRAIN_ARCHITECTURES.mdDocumented718,719LAYER_B_ROUTINGcontrol_bind
727718Sensory_Filter_Threshold_AutismNeurodivergentθ_autism = θ_neurotypical · (1 - σ_hyper)θ=sensory gating threshold, σ_hyper=hyper-sensitivity factor (0.3-0.6)Lower sensory gating threshold in autism causes hyper-sensitivitydocs/neuroscience/NEURODIVERGENT_BRAIN_ARCHITECTURES.mdDocumented715,716,717,719LAYER_B_ROUTINGcontrol_bind
728719Compensatory_Routing_WeightNeurodivergentw_comp = w_standard · (1 + λ_comp)w_comp=compensatory pathway weight, w_standard=standard pathway weight, λ_comp=compensation factor (0.2-0.8)Alternative neural pathway activation for standard functions in neurodivergent brainsdocs/neuroscience/NEURODIVERGENT_BRAIN_ARCHITECTURES.mdDocumented717,718LAYER_C_TOPOLOGYgeometric_bind
729720AdvancedBio_freeEnergySurprisalAdvancedBiotools/lean/Semantics/Semantics/Extensions/AdvancedBioDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
730721AdvancedBio_fisherDeltaFitnessAdvancedBiotools/lean/Semantics/Semantics/Extensions/AdvancedBioDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
731722AdvancedBio_neutralEvolutionRateAdvancedBiotools/lean/Semantics/Semantics/Extensions/AdvancedBioDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
732723AdvancedBio_NeuralFieldStateAdvancedBiotools/lean/Semantics/Semantics/Extensions/AdvancedBioDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
733724AdvancedBio_wilsonCowanUpdateAdvancedBiotools/lean/Semantics/Semantics/Extensions/AdvancedBioDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
734725AdvancedBio_remodelingErrorAdvancedBiotools/lean/Semantics/Semantics/Extensions/AdvancedBioDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
735726AuditoryMasking_upperMaskingSlopeAuditoryMaskingtools/lean/Semantics/Semantics/Extensions/AuditoryMaskingDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
736727AuditoryMasking_signalToMaskRatioAuditoryMaskingtools/lean/Semantics/Semantics/Extensions/AuditoryMaskingDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
737728AuditoryMasking_isSignalSalientAuditoryMaskingtools/lean/Semantics/Semantics/Extensions/AuditoryMaskingDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
738729AuditoryMasking_specificLoudnessAuditoryMaskingtools/lean/Semantics/Semantics/Extensions/AuditoryMaskingDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
739730AuditoryMasking_totalLoudnessAuditoryMaskingtools/lean/Semantics/Semantics/Extensions/AuditoryMaskingDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
740731BioPhotonics_fluenceRateUpdateBioPhotonicstools/lean/Semantics/Semantics/Extensions/BioPhotonicsDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
741732BioPhotonics_photonEmissionRateBioPhotonicstools/lean/Semantics/Semantics/Extensions/BioPhotonicsDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
742733BioPhotonics_lightIntensityAtDepthBioPhotonicstools/lean/Semantics/Semantics/Extensions/BioPhotonicsDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
743734BiologicalControl_biologicalHamiltonianBiologicalControltools/lean/Semantics/Semantics/Extensions/BiologicalControlDynamics.leanLeanLAYER_J_DYNAMICScontrol_bind
744735BiologicalControl_satisfyRequisiteVarietyBiologicalControltools/lean/Semantics/Semantics/Extensions/BiologicalControlDynamics.leanLeanLAYER_J_DYNAMICScontrol_bind
745736BiologicalControl_bellmanErrorBiologicalControltools/lean/Semantics/Semantics/Extensions/BiologicalControlDynamics.leanLeanLAYER_J_DYNAMICScontrol_bind
746737BiologicalControl_paretoEfficiencyBiologicalControltools/lean/Semantics/Semantics/Extensions/BiologicalControlDynamics.leanLeanLAYER_J_DYNAMICScontrol_bind
747738BiologicalExergy_exergyDestructionBiologicalExergytools/lean/Semantics/Semantics/Extensions/BiologicalExergyDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
748739BiologicalExergy_minimumEntropyProductionUpdateBiologicalExergytools/lean/Semantics/Semantics/Extensions/BiologicalExergyDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
749740BiologicalExergy_maximumEntropyProductionRateBiologicalExergytools/lean/Semantics/Semantics/Extensions/BiologicalExergyDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
750741BiologicalExergy_actualMetabolicWorkBiologicalExergytools/lean/Semantics/Semantics/Extensions/BiologicalExergyDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
751742BiologicalRegulation_fluxControlCoefficientBiologicalRegulationtools/lean/Semantics/Semantics/Extensions/BiologicalRegulationDynamics.leanLeanLAYER_J_DYNAMICScontrol_bind
752743BiologicalRegulation_isControlLawfulBiologicalRegulationtools/lean/Semantics/Semantics/Extensions/BiologicalRegulationDynamics.leanLeanLAYER_J_DYNAMICScontrol_bind
753744BiologicalRegulation_adaptationUpdateBiologicalRegulationtools/lean/Semantics/Semantics/Extensions/BiologicalRegulationDynamics.leanLeanLAYER_J_DYNAMICScontrol_bind
754745BiologicalRegulation_isAdaptedBiologicalRegulationtools/lean/Semantics/Semantics/Extensions/BiologicalRegulationDynamics.leanLeanLAYER_J_DYNAMICScontrol_bind
755746BiologicalRegulation_optimalRegulatoryModeBiologicalRegulationtools/lean/Semantics/Semantics/Extensions/BiologicalRegulationDynamics.leanLeanLAYER_J_DYNAMICScontrol_bind
756747CancerMetabolic_cancerOnsetProbCancerMetabolictools/lean/Semantics/Semantics/Extensions/CancerMetabolicDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
757748CancerMetabolic_elasticityCoefficientCancerMetabolictools/lean/Semantics/Semantics/Extensions/CancerMetabolicDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
758749CancerMetabolic_checkConnectivityTheoremCancerMetabolictools/lean/Semantics/Semantics/Extensions/CancerMetabolicDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
759750CancerMetabolic_priceSelectionTermCancerMetabolictools/lean/Semantics/Semantics/Extensions/CancerMetabolicDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
760751CardiacYield_averageBiomassWeightCardiacYieldtools/lean/Semantics/Semantics/Extensions/CardiacYieldDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
761752CardiacYield_totalBiomassYieldCardiacYieldtools/lean/Semantics/Semantics/Extensions/CardiacYieldDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
762753CardiacYield_gatingVariableUpdateCardiacYieldtools/lean/Semantics/Semantics/Extensions/CardiacYieldDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
763754CardiacYield_nobleSodiumCurrentCardiacYieldtools/lean/Semantics/Semantics/Extensions/CardiacYieldDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
764755CardiacYield_nobleK1ConductanceCardiacYieldtools/lean/Semantics/Semantics/Extensions/CardiacYieldDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
765756CellularSignaling_goldbeterKoshlandSwitchCellularSignalingtools/lean/Semantics/Semantics/Extensions/CellularSignalingDynamics.leanLeanLAYER_J_DYNAMICScontrol_bind
766757CellularSignaling_MitoticStateCellularSignalingtools/lean/Semantics/Semantics/Extensions/CellularSignalingDynamics.leanLeanLAYER_J_DYNAMICScontrol_bind
767758CellularSignaling_tysonStepCellularSignalingtools/lean/Semantics/Semantics/Extensions/CellularSignalingDynamics.leanLeanLAYER_J_DYNAMICScontrol_bind
768759CellularSignaling_molecularRepressionCellularSignalingtools/lean/Semantics/Semantics/Extensions/CellularSignalingDynamics.leanLeanLAYER_J_DYNAMICScontrol_bind
769760CellularSignaling_chemotacticFluxCellularSignalingtools/lean/Semantics/Semantics/Extensions/CellularSignalingDynamics.leanLeanLAYER_J_DYNAMICScontrol_bind
770761CognitiveAcoustic_integratedInformationPhiCognitiveAcoustictools/lean/Semantics/Semantics/Extensions/CognitiveAcousticDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
771762CognitiveAcoustic_gnwGatingCognitiveAcoustictools/lean/Semantics/Semantics/Extensions/CognitiveAcousticDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
772763CognitiveAcoustic_orchOrCollapseTimeCognitiveAcoustictools/lean/Semantics/Semantics/Extensions/CognitiveAcousticDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
773764CognitiveAcoustic_sonarRangeCognitiveAcoustictools/lean/Semantics/Semantics/Extensions/CognitiveAcousticDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
774765CognitiveAcoustic_gammatoneEnvelopeCognitiveAcoustictools/lean/Semantics/Semantics/Extensions/CognitiveAcousticDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
775766CognitiveAcoustic_xenobotReplicationProbCognitiveAcoustictools/lean/Semantics/Semantics/Extensions/CognitiveAcousticDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
776767CognitiveLearning_associativeStrengthUpdateCognitiveLearningtools/lean/Semantics/Semantics/Extensions/CognitiveLearningDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
777768CognitiveLearning_levySearchProbabilityCognitiveLearningtools/lean/Semantics/Semantics/Extensions/CognitiveLearningDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
778769CognitiveLearning_isCognitiveSwitchOptimalCognitiveLearningtools/lean/Semantics/Semantics/Extensions/CognitiveLearningDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
779770CognitiveLearning_memorySamplingProbCognitiveLearningtools/lean/Semantics/Semantics/Extensions/CognitiveLearningDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
780771ConstrainedEnergy_constrainedTEEConstrainedEnergytools/lean/Semantics/Semantics/Extensions/ConstrainedEnergyDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
781772ConstrainedEnergy_energyCompensationConstantConstrainedEnergytools/lean/Semantics/Semantics/Extensions/ConstrainedEnergyDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
782773ConstrainedEnergy_metabolicCeilingConstrainedEnergytools/lean/Semantics/Semantics/Extensions/ConstrainedEnergyDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
783774ConstrainedEnergy_metabolicScopeConstrainedEnergytools/lean/Semantics/Semantics/Extensions/ConstrainedEnergyDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
784775ConstrainedEnergy_maintenanceBudgetConstrainedEnergytools/lean/Semantics/Semantics/Extensions/ConstrainedEnergyDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
785776ConstructalMuscle_optimalBranchingRatioConstructalMuscletools/lean/Semantics/Semantics/Extensions/ConstructalMuscleDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
786777ConstructalMuscle_muscleShorteningVelocityConstructalMuscletools/lean/Semantics/Semantics/Extensions/ConstructalMuscleDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
787778ConstructalMuscle_totalMuscleForceConstructalMuscletools/lean/Semantics/Semantics/Extensions/ConstructalMuscleDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
788779ConstructalMuscle_surfaceVolumeRatioConstructalMuscletools/lean/Semantics/Semantics/Extensions/ConstructalMuscleDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
789780CorticalScaling_corticalNeuronScalingCorticalScalingtools/lean/Semantics/Semantics/Extensions/CorticalScalingDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
790781CorticalScaling_whiteMatterScalingCorticalScalingtools/lean/Semantics/Semantics/Extensions/CorticalScalingDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
791782CorticalScaling_isDendriticBranchLawfulCorticalScalingtools/lean/Semantics/Semantics/Extensions/CorticalScalingDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
792783CorticalScaling_synapsisPerPairInvariantCorticalScalingtools/lean/Semantics/Semantics/Extensions/CorticalScalingDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
793784EcologicalInformation_margalefRichnessEcologicalInformationtools/lean/Semantics/Semantics/Extensions/EcologicalInformationDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
794785EcologicalInformation_shannonDiversityEcologicalInformationtools/lean/Semantics/Semantics/Extensions/EcologicalInformationDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
795786EcologicalInformation_isSystemStableEcologicalInformationtools/lean/Semantics/Semantics/Extensions/EcologicalInformationDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
796787EcologicalInformation_informationSheddingEcologicalInformationtools/lean/Semantics/Semantics/Extensions/EcologicalInformationDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
797788EcologicalNetwork_andEcologicalNetworktools/lean/Semantics/Semantics/Extensions/EcologicalNetworkDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
798789EcologicalNetwork_redfieldCheckEcologicalNetworktools/lean/Semantics/Semantics/Extensions/EcologicalNetworkDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
799790EcologicalNetwork_hollingType1EcologicalNetworktools/lean/Semantics/Semantics/Extensions/EcologicalNetworkDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
800791EcologicalNetwork_hollingType2EcologicalNetworktools/lean/Semantics/Semantics/Extensions/EcologicalNetworkDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
801792EcologicalNetwork_hollingType3EcologicalNetworktools/lean/Semantics/Semantics/Extensions/EcologicalNetworkDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
802793EcologicalNetwork_networkConnectanceEcologicalNetworktools/lean/Semantics/Semantics/Extensions/EcologicalNetworkDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
803794EcologicalNetwork_taylorsVarianceEcologicalNetworktools/lean/Semantics/Semantics/Extensions/EcologicalNetworkDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
804795Epidemiological_basicReproductionNumberEpidemiologicaltools/lean/Semantics/Semantics/Extensions/EpidemiologicalDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
805796Epidemiological_herdImmunityThresholdEpidemiologicaltools/lean/Semantics/Semantics/Extensions/EpidemiologicalDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
806797Epidemiological_SIRStateEpidemiologicaltools/lean/Semantics/Semantics/Extensions/EpidemiologicalDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
807798Epidemiological_sirUpdateEpidemiologicaltools/lean/Semantics/Semantics/Extensions/EpidemiologicalDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
808799EpidemiologicalTrophic_reedFrostNewCasesEpidemiologicalTrophictools/lean/Semantics/Semantics/Extensions/EpidemiologicalTrophicDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
809800EpidemiologicalTrophic_trophicWaveUpdateEpidemiologicalTrophictools/lean/Semantics/Semantics/Extensions/EpidemiologicalTrophicDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
810801EpidemiologicalTrophic_trophicKineticsEpidemiologicalTrophictools/lean/Semantics/Semantics/Extensions/EpidemiologicalTrophicDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
811802EvolutionaryLandscape_frequencyChangeGradientEvolutionaryLandscapetools/lean/Semantics/Semantics/Extensions/EvolutionaryLandscapeDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
812803EvolutionaryLandscape_populationMeanFitnessEvolutionaryLandscapetools/lean/Semantics/Semantics/Extensions/EvolutionaryLandscapeDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
813804EvolutionaryLandscape_isDriftDominantEvolutionaryLandscapetools/lean/Semantics/Semantics/Extensions/EvolutionaryLandscapeDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
814805EvolutionaryLandscape_isAscendingPeakEvolutionaryLandscapetools/lean/Semantics/Semantics/Extensions/EvolutionaryLandscapeDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
815806EvolutionaryNetwork_inEvolutionaryNetworktools/lean/Semantics/Semantics/Extensions/EvolutionaryNetworkDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
816807EvolutionaryNetwork_hawkDoveMixedStrategyEvolutionaryNetworktools/lean/Semantics/Semantics/Extensions/EvolutionaryNetworkDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
817808EvolutionaryNetwork_isStrategyStableEvolutionaryNetworktools/lean/Semantics/Semantics/Extensions/EvolutionaryNetworkDynamics.leanLeanLAYER_J_DYNAMICSthermodynamic_bind
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23812392AMMR_WitnessRootAMMRtools/lean/Semantics/Semantics/AMMR.leanLeanLAYER_E_VERIFICATIONmetric_bind
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23972408AVMRInformation_shellEntropyAVMRInformationtools/lean/Semantics/Semantics/AVMRInformation.leanLeanLAYER_A_COMPRESSIONinformational_bind
23982409AVMRInformation_shellEntropyBoundAVMRInformationtools/lean/Semantics/Semantics/AVMRInformation.leanLeanLAYER_A_COMPRESSIONinformational_bind
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24002411AngrySphinxPolicy_enforceAngrySphinxGateAngrySphinxPolicytools/lean/Semantics/Semantics/AngrySphinxPolicy.leanLeanLAYER_L_APPLICATIONphysical_bind
24012412AtomicResolution_compressionContractsAtomicResolutionAtomicResolutiontools/lean/Semantics/Semantics/AtomicResolution.leanLeanLAYER_A_COMPRESSIONinformational_bind
24022413Atoms_SemanticFrustrationWaveAtomstools/lean/Semantics/Semantics/Atoms.leanLeanLAYER_K_SIGNALsignal_bind
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24052416BracketShellCount_ShellSystemBracketShellCounttools/lean/Semantics/Semantics/BracketShellCount.leanLeanLAYER_L_APPLICATIONphysical_bind
24062417AMMR_invariantOfAMMRtools/lean/Semantics/Semantics/AMMR.leanLeanLAYER_D_INVARIANTSstate_bind
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24142425ASCIIArtCompetition_evaluateSemanticSimilarityASCIIArtCompetitiontools/lean/Semantics/Semantics/ASCIIArtCompetition.leanLeanLAYER_M_LEAN_SEMANTICSlogic_bind
24152426ASCIIGen_ASCIIEncodingDataASCIIGentools/lean/Semantics/Semantics/ASCIIGen.leanLeanLAYER_I_ENCODINGinformational_bind
24162427ASCIIGen_analyzeASCIIEncodingASCIIGentools/lean/Semantics/Semantics/ASCIIGen.leanLeanLAYER_I_ENCODINGinformational_bind
24172428ASICTopology_WorkloadASICTopologytools/lean/Semantics/Semantics/ASICTopology.leanLeanLAYER_G_ENERGYthermodynamic_bind
24182429ASICTopology_projectWorkloadToTopologyASICTopologytools/lean/Semantics/Semantics/ASICTopology.leanLeanLAYER_G_ENERGYthermodynamic_bind
24192430AVMRInformation_shellEntropyAVMRInformationtools/lean/Semantics/Semantics/AVMRInformation.leanLeanLAYER_A_COMPRESSIONinformational_bind
24202431AVMRInformation_shellEntropyBoundAVMRInformationtools/lean/Semantics/Semantics/AVMRInformation.leanLeanLAYER_A_COMPRESSIONinformational_bind
24212432AdaptiveFabric_CMYKStateAdaptiveFabrictools/lean/Semantics/Semantics/AdaptiveFabric.leanLeanLAYER_F_CONTROLcontrol_bind
24222433AdaptiveFabric_FabricStateAdaptiveFabrictools/lean/Semantics/Semantics/AdaptiveFabric.leanLeanLAYER_F_CONTROLcontrol_bind
24232434Bandyopadhyay_Cycle_StringStarStringStarI_total = I_bulk + I_horizon + I_vacuumI_total=total information, I_bulk=kinetic matter, I_horizon=holographic surface, I_vacuum=ambient radiationUnitary information conservation across three mutually exclusive phasesString-Star-Manifold/Rupayan52-String-Star-Manifold-153b359/docs/THEORY.mdPending2437,2438,2439,2440,2441LAYER_D_INVARIANTSinformational_bind
24242435Hawking_Temperature_StringStarStringStarT_H = ℏc³/(8πGMk_B)T_H=Hawking temperature, ℏ=reduced Planck, c=speed of light, G=gravitational constant, M=mass, k_B=Boltzmann constantInverse relationship between mass and black hole temperatureString-Star-Manifold/Rupayan52-String-Star-Manifold-153b359/docs/THEORY.mdPending2436LAYER_G_ENERGYthermodynamic_bind
24252436Bekenstein_Hawking_Entropy_StringStarStringStarS = A/4S=entropy, A=surface areaHolographic principle: entropy proportional to horizon surface areaString-Star-Manifold/Rupayan52-String-Star-Manifold-153b359/docs/THEORY.mdPending2436,2439LAYER_G_ENERGYthermodynamic_bind
24262437Schwarzschild_Radius_StringStarStringStarr_s = 2GM/c²r_s=event horizon radius, G=gravitational constant, M=mass, c=speed of lightEvent horizon radius for non-rotating black holeString-Star-Manifold/Rupayan52-String-Star-Manifold-153b359/docs/THEORY.mdPending2436,2440LAYER_C_TOPOLOGYgeometric_bind
24272438Quadrupole_GW_Power_StringStarStringStarP = (32/5)(G⁴/c⁵)(m₁²m₂²(m₁+m₂))/r⁵P=gravitational wave power, G=gravitational constant, c=speed of light, m₁,m₂=masses, r=separationOrbital decay energy loss via gravitational radiationString-Star-Manifold/Rupayan52-String-Star-Manifold-153b359/docs/THEORY.mdPending2436LAYER_H_ALGEBRAphysical_bind
24282439Relativistic_Gamma_StringStarStringStarγ = 1/√(1 - v²/c²)γ=Lorentz factor, v=velocity, c=speed of lightTime dilation factor for relativistic velocitiesString-Star-Manifold/Rupayan52-String-Star-Manifold-153b359/engine/v17_1_True_Omega.pyPending2436LAYER_H_ALGEBRAphysical_bind
24292440AVMRInformation_shellEntropyAVMRInformationtools/lean/Semantics/Semantics/AVMRInformation.leanLeanLAYER_A_COMPRESSIONinformational_bind
24302441AVMRInformation_shellEntropyBoundAVMRInformationtools/lean/Semantics/Semantics/AVMRInformation.leanLeanLAYER_A_COMPRESSIONinformational_bind
24312442Scale_Space_Gaussian_SmoothingScaleSpaceL(x,t) = G(x,t) * f(x), G(x,t) = (1/2πt)exp(-|x|²/2t)L=scale-space representation, G=Gaussian kernel, f=input signal, t=scale parameterStructures emerge at appropriate scales via Gaussian smoothing; foundational for scale-space theoryEQUATION_FOREST_INDEX.mdPending2443,2444LAYER_C_TOPOLOGYgeometric_bind
24322443Wavelet_TransformScaleSpaceW(a,b) = (1/√|a|)∫ f(x)ψ*((x-b)/a)dxW=wavelet coefficient, a=scale, b=translation, ψ=mother waveletLocalized frequency analysis; structure emerges from coherence and distributed frequency across scaleEQUATION_FOREST_INDEX.mdPending2442,2445LAYER_C_TOPOLOGYgeometric_bind
24332444Renormalization_GroupScaleSpace∂g/∂l = β(g)g=coupling constant, l=logarithmic scale, β=beta functionPhysical behaviors evolve consistently across scales; universality classes emerge from fixed pointsEQUATION_FOREST_INDEX.mdPending2442,2446LAYER_D_INVARIANTSphysical_bind
24342445Fractal_Self_SimilarityScaleSpacef(λx) = λ^H f(x)f=fractal function, λ=scale factor, H=Hurst exponentRecursive patterns across scales; same structure at different magnificationsEQUATION_FOREST_INDEX.mdPending2443,2446LAYER_C_TOPOLOGYgeometric_bind
24352446Cellular_Automaton_Rule110ScaleSpaces_i(t+1) = f(s_{i-1}(t), s_i(t), s_{i+1}(t)) with Rule110 truth tables=cell state, f=update rule, i=position, t=timeMinimal computational complexity producing rich emergent behavior from simple deterministic rulesEQUATION_FOREST_INDEX.mdPending2445,2447LAYER_H_ALGEBRAcontrol_bind
24362447Turing_PatternsScaleSpace∂u/∂t = D_u∇²u + f(u,v), ∂v/∂t = D_v∇²v + g(u,v)u,v=morphogen concentrations, D=diffusion coefficients, f,g=reaction kineticsSpatial self-organization via diffusion-driven instability; resonant structures emerge from simple interactionsEQUATION_FOREST_INDEX.mdPending2446,2448LAYER_C_TOPOLOGYgeometric_bind
24372448Shannon_EntropyScaleSpaceH = -Σ p(x)log₂ p(x)p=probability distribution, H=entropy in bitsInformation content measurement; quantifies uncertainty and guides emergence of coherent patterns from chaosEQUATION_FOREST_INDEX.mdPending2449,2450LAYER_A_COMPRESSIONinformational_bind
24382449NegentropyScaleSpaceN = H_max - HH_max=maximum entropy, H=actual entropySpontaneous formation of structured, meaningful states from initial randomness; negative entropy as order measureEQUATION_FOREST_INDEX.mdPending2448,2450LAYER_A_COMPRESSIONinformational_bind
24392450Phase_TransitionScaleSpaceL = L₀ + A|t - t_c|^β, C = C₀ + B|t - t_c|^-αL=order parameter, C=specific heat, t=temperature, t_c=critical point, α,β=critical exponentsEmergence thresholds marking dramatic shifts in complexity and coherence; critical phenomena at phase boundariesEQUATION_FOREST_INDEX.mdPending2448,2451LAYER_G_ENERGYthermodynamic_bind
24402451Fourier_TransformScaleSpaceF(ω) = ∫ f(t)e^(-iωt)dtF=frequency domain, ω=angular frequency, f=time domain, t=timeDecomposes complexity into simpler wave-like patterns; standing waves as stable emergent formsEQUATION_FOREST_INDEX.mdPending2447,2452LAYER_C_TOPOLOGYgeometric_bind
24412452Twistor_TheoryScaleSpaceT = CP³ (complex projective 3-space) as space of null geodesicsT=twistor space, CP³=complex projective spaceEmergent dimensionality from relationships; structure not fixed but emergent from relationships and informational flowEQUATION_FOREST_INDEX.mdPending2451LAYER_C_TOPOLOGYgeometric_bind
24422453Vant_Hoff_Osmotic_PressureDesalinationΠ = i·c·R·TΠ=osmotic pressure (Pa), i=van't Hoff factor, c=molar concentration (mol/m³), R=gas constant (8.314 J/mol·K), T=temperature (K)Fundamental equation for osmotic pressure; determines minimum pressure required for reverse osmosisdesalination/osmotic_pressure.mdPending2454,2455LAYER_G_ENERGYthermodynamic_bind
24432454Reverse_Osmosis_EnergyDesalinationW_min = Π·V·(1/η_pump)W_min=minimum work (J), Π=osmotic pressure, V=volume, η_pump=pump efficiencyMinimum energy required to overcome osmotic pressure for water purificationdesalination/reverse_osmosis_energy.mdPending2453,2456LAYER_G_ENERGYthermodynamic_bind
24442455Specific_Energy_ConsumptionDesalinationSEC = (P·V)/(Q·η)SEC=specific energy (kWh/m³), P=pressure (Pa), V=volume (m³), Q=flow rate (m³/s), η=efficiencyStandard metric for desalination energy efficiency; benchmark ~3-4 kWh/m³ for seawater ROdesalination/sec_metrics.mdPending2454,2457LAYER_G_ENERGYthermodynamic_bind
24452456Membrane_PermeabilityDesalinationJ_w = A·(ΔP - Δπ)J_w=water flux (kg/m²·s), A=water permeability coefficient, ΔP=hydraulic pressure difference, Δπ=osmotic pressure differenceSolution-diffusion model for water transport through semi-permeable membranesdesalination/membrane_permeability.mdPending2453,2455,2458LAYER_C_TOPOLOGYphysical_bind
24462457Gibbs_Free_Energy_MixingDesalinationΔG_mix = nRT(x_A ln x_A + x_B ln x_B)ΔG_mix=Gibbs free energy of mixing, n=moles, x=mole fractionsThermodynamic cost of separating salt from water; lower bound for desalination energydesalination/thermodynamics.mdPending2455,2459LAYER_G_ENERGYthermodynamic_bind
24472458Concentration_PolarizationDesalinationC_m = C_b·exp(J_w/k)C_m=membrane surface concentration, C_b=bulk concentration, J_w=water flux, k=mass transfer coefficientSalt accumulation at membrane surface reduces effective driving force; critical for RO efficiencydesalination/concentration_polarization.mdPending2456,2460LAYER_C_TOPOLOGYphysical_bind
24482459Enthalpy_VaporizationDesalinationΔH_vap = 40.66 kJ/mol at 100°C (water)ΔH_vap=enthalpy of vaporization, varies with temperatureEnergy required for thermal desalination (distillation, multi-stage flash)desalination/thermal_distillation.mdPending2457,2461LAYER_G_ENERGYthermodynamic_bind
24492460Salt_RejectionDesalinationR = 1 - (C_p/C_f)R=rejection fraction, C_p=permeate concentration, C_f=feed concentrationMembrane selectivity metric; 99.7% rejection typical for seawater ROdesalination/membrane_selectivity.mdPending2458,2462LAYER_C_TOPOLOGYphysical_bind
24502461Recovery_RateDesalinationY = Q_p/Q_fY=recovery rate, Q_p=permeate flow, Q_f=feed flowFraction of feed water converted to product; 40-50% typical for seawater ROdesalination/system_design.mdPending2459,2463LAYER_G_ENERGYthermodynamic_bind
24512462Fouling_ResistanceDesalinationR_f = (R_t - R_m)/R_mR_f=fouling resistance, R_t=total resistance, R_m=membrane resistanceMembrane degradation over time due to scaling, biofouling, colloidal depositiondesalination/fouling_dynamics.mdPending2460,2464LAYER_C_TOPOLOGYphysical_bind
24522463Exergy_Analysis_DesalinationDesalinationEx = (H - H₀) - T₀(S - S₀)Ex=exergy, H=enthalpy, S=entropy, T₀=reference temperatureSecond-law efficiency analysis; identifies thermodynamic losses in desalination processesdesalination/exergy_analysis.mdPending2461,2454LAYER_G_ENERGYthermodynamic_bind
24532464Incompressible_Navier_StokesFluid Dynamics∂v/∂t + (v·∇)v = -∇P/ρ + ν∇²v + gv=velocity field, P=pressure, ρ=density, ν=kinematic viscosity, g=gravityMomentum conservation for incompressible viscous flow; fundamental equation of fluid dynamicsfluid_dynamics/navier_stokes.mdPending2465,2466,2467,2495LAYER_C_TOPOLOGYphysical_bind
24542465Incompressible_ContinuityFluid Dynamics∇·v = 0v=velocity fieldMass conservation for incompressible flow; density is constantfluid_dynamics/continuity.mdPending2464,2468LAYER_C_TOPOLOGYphysical_bind
24552466Bernoulli_Equation_IncompressibleFluid DynamicsP + ½ρv² + ρgh = constantP=pressure, ρ=density, v=velocity, g=gravity, h=elevationEnergy conservation along streamline for steady incompressible flow; pressure-velocity relationshipfluid_dynamics/bernoulli.mdPending2464,2469,2492LAYER_G_ENERGYthermodynamic_bind
24562467Poiseuille_FlowFluid DynamicsQ = πΔP r⁴ / (8μL)Q=volumetric flow rate, ΔP=pressure drop, r=pipe radius, μ=dynamic viscosity, L=lengthLaminar flow through circular pipe; pressure-driven flow model for membrane channelsfluid_dynamics/poiseuille.mdPending2464,2470,2503LAYER_C_TOPOLOGYphysical_bind
24572468Stokes_FlowFluid DynamicsF = 6πμRvF=drag force, μ=dynamic viscosity, R=particle radius, v=velocityLow Reynolds number flow (creeping flow); viscous forces dominate inertial forcesfluid_dynamics/stokes_flow.mdPending2465,2471,2501LAYER_C_TOPOLOGYphysical_bind
24582469Boundary_Layer_ThicknessFluid Dynamicsδ ≈ √(νx/U)δ=boundary layer thickness, ν=kinematic viscosity, x=distance from leading edge, U=free stream velocityGrowth of viscous boundary layer along flat plate; critical for membrane surface flowfluid_dynamics/boundary_layer.mdPending2466,2472,2498LAYER_C_TOPOLOGYphysical_bind
24592470Shear_Stress_ViscousFluid Dynamicsτ = μ(∂u/∂y)τ=shear stress, μ=dynamic viscosity, u=velocity, y=normal directionNewton's law of viscosity; viscous stress proportional to velocity gradientfluid_dynamics/shear_stress.mdPending2467,2473,2493LAYER_C_TOPOLOGYphysical_bind
24602471Reynolds_TransitionFluid DynamicsRe_crit ≈ 2300 (pipe), 5×10⁵ (flat plate)Re=ρvL/μ, Re_crit=critical Reynolds numberTransition from laminar to turbulent flow; determines flow regime in membrane systemsfluid_dynamics/reynolds_transition.mdPending2468,2474,2497LAYER_C_TOPOLOGYphysical_bind
24612472Friction_Factor_LaminarFluid Dynamicsf = 64/Ref=friction factor, Re=Reynolds numberDarcy-Weisbach friction factor for laminar pipe flow; pressure drop calculationfluid_dynamics/friction_factor.mdPending2469,2475,2507LAYER_C_TOPOLOGYphysical_bind
24622473Vorticity_EquationFluid Dynamics∂ω/∂t + (v·∇)ω = (ω·∇)v + ν∇²ωω=vorticity (∇×v), v=velocity, ν=kinematic viscosityEvolution of fluid rotation; vortex stretching and diffusion in incompressible flowfluid_dynamics/vorticity.mdPending2470,2476LAYER_C_TOPOLOGYphysical_bind
24632474Pressure_Drop_DarcyFluid DynamicsΔP = f(L/D)(ρv²/2)ΔP=pressure drop, f=friction factor, L=length, D=diameter, ρ=density, v=velocityPressure loss in pipe flow; energy dissipation due to frictionfluid_dynamics/darcy_weisbach.mdPending2471,2464LAYER_G_ENERGYthermodynamic_bind
24642475Laminar_Parabolic_ProfileFluid Dynamicsu(r) = u_max(1 - r²/R²)u(r)=velocity at radius r, u_max=centerline velocity, R=pipe radiusParabolic velocity profile for fully developed laminar pipe flow; maximum at center, zero at wallsfluid_dynamics/velocity_profile.mdPending2472,2467,2504LAYER_C_TOPOLOGYphysical_bind
24652476Stream_FunctionFluid Dynamicsψ = ∫ u dy - ∫ v dxψ=stream function, u,v=velocity componentsAutomatically satisfies incompressible continuity; streamlines follow ψ=constant contoursfluid_dynamics/stream_function.mdPending2473,2465LAYER_C_TOPOLOGYphysical_bind
24662477Kutta_Joukowski_TheoremFluid DynamicsL = ρ·V·ΓL=lift per unit span, ρ=fluid density, V=free stream velocity, Γ=circulationFundamental theorem connecting circulation to lift in 2D potential flowfluid_dynamics/kutta_joukowski.mdPending2478,2479,2481,2480LAYER_C_TOPOLOGYphysical_bind
24672478Circulation_DefinitionFluid DynamicsΓ = ∮ v·dlΓ=circulation, v=velocity, dl=infinitesimal path elementLine integral of velocity around closed curve; quantifies fluid rotationfluid_dynamics/circulation.mdPending2477,2480,2485LAYER_C_TOPOLOGYphysical_bind
24682479Magnus_ForceFluid DynamicsF_M = S·ρ·V·ΓF_M=Magnus force, S=spin parameter, ρ=density, V=velocity, Γ=circulationLift force on spinning cylinder/sphere due to circulation; perpendicular to velocity and spin axisfluid_dynamics/magnus_effect.mdPending2477,2481,2486LAYER_C_TOPOLOGYphysical_bind
24692480Lift_Coefficient_CirculationFluid DynamicsC_L = 2Γ/(V·c)C_L=lift coefficient, Γ=circulation, V=velocity, c=chord lengthRelates circulation to dimensionless lift coefficient for 2D airfoilsfluid_dynamics/lift_coefficient.mdPending2481,2482LAYER_C_TOPOLOGYphysical_bind
24702481Magnus_EffectFluid DynamicsF_M = ρ·V·Γ·SF_M=Magnus force, ρ=density, V=velocity, Γ=circulation, S=spin parameterForce on rotating object due to circulation; perpendicular to velocity and spin axisfluid_dynamics/magnus_effect.mdPending2479,2480LAYER_C_TOPOLOGYphysical_bind
24712482Circulation_LiftFluid DynamicsL = ρ·V·ΓL=lift, ρ=density, V=velocity, Γ=circulationLift force due to circulation; fundamental principle of aerodynamicsfluid_dynamics/circulation_lift.mdPending2480,2481LAYER_C_TOPOLOGYphysical_bind
24722483N_Space_CirculationManifold DynamicsΓ_n = ∮_∂Ω ωΓ_n=n-dimensional circulation, ω=(n-1)-form, ∂Ω=boundary of n-manifold ΩGeneralized Stokes theorem: circulation as integral of differential form over manifold boundary; generalizes 2D circulation to n-dimensionsmanifold_dynamics/n_space_circulation.mdPending2486,2487LAYER_C_TOPOLOGYgeometric_bind
24732484N_Space_Magnus_ForceManifold DynamicsF_M^n = ρ·(V × ω_n)F_M^n=n-dimensional Magnus force tensor, ρ=density, V=velocity vector, ω_n=n-dimensional circulation pseudovectorMagnus effect generalized to n-space: cross product of velocity with circulation pseudovector yields lift force in perpendicular hyperplanemanifold_dynamics/n_space_magnus.mdPending2485,2488LAYER_C_TOPOLOGYgeometric_bind
24742485FAMM_Frustration_TensorN-Space DynamicsΦ_μν = (Σ_thermal + Σ_steric + Σ_magnus)_μν / Σ_magnetic_μνΦ_μν=frustration tensor (rank-2), Σ=stress tensors (rank-2), μ,ν=manifold indicesFAMM frustration generalized to n-space as rank-2 tensor field on manifold; competing stress tensors create geometric frustration in n-dimensional assemblyn_space/famm_frustration_tensor.mdPending2485,2489LAYER_C_TOPOLOGYcontrol_bind
24752486Magnus_FAMM_CouplingN-Space Dynamics∂_t Φ_μν = -∇_μ F_M^ν + ∇_ν F_M^μ + λ·Φ_μν∂_t=time derivative, ∇=covariant derivative, F_M^ν=Magnus force vector, λ=coupling constantDynamical coupling between Magnus circulation and FAMM frustration tensor on n-manifold; Magnus force drives frustration evolution via covariant transportn_space/magnus_famm_coupling.mdPending2486,2487,2490LAYER_C_TOPOLOGYcontrol_bind
24762487N_Space_Circulation_MetricManifold Dynamicsg_μν Γ^μ Γ^ν = constg_μν=metric tensor, Γ^μ=circulation pseudovector componentsCirculation magnitude invariant under coordinate transformations on n-manifold; provides geometric constraint for Magnus-FAMM couplingmanifold_dynamics/circulation_metric.mdPending2485,2490LAYER_D_INVARIANTSstate_bind
24772488FAMM_Magnus_ReductionN-Space DynamicsΦ_eff = Φ_μν - α·(F_M^μ F_M_ν)/g_μνΦ_eff=effective frustration, α=reduction coefficient, F_M=Magnus force, g=metricMagnus force reduces effective frustration tensor when aligned with circulation; enables assembly at lower magnetic field in n-dimensional flown_space/famm_magnus_reduction.mdPending2487,2488,2491LAYER_G_ENERGYthermodynamic_bind
24782489N_Space_Flow_CirculationManifold DynamicsΓ^μ = ε^{μν₁...ν_{n-1}} ∂_ν₁ v_{ν₂} ... ∇_ν_{n-1} v_nΓ=circulation pseudovector, ε=Levi-Civita tensor, v=velocity, ∇=covariant derivativeCirculation generated by velocity gradients in n-dimensional flow; generalizes vorticity to n-space for Magnus effect calculationmanifold_dynamics/n_space_flow_circulation.mdPending2485,2490LAYER_C_TOPOLOGYgeometric_bind
24792492→2490Manifold_Bernoulli_EquationManifold DynamicsP + ½ρ g_μν v^μ v^ν + ρΦ = const (along streamlines)P=pressure scalar field, ρ=density, g_μν=metric tensor, v^μ=velocity vector field, Φ=potential scalar fieldBernoulli principle generalized to n-dimensional manifold: kinetic energy via metric contraction, potential as scalar field on manifold, conserved along integral curves of velocity fieldmanifold_dynamics/manifold_bernoulli.mdPending2488,2493,2468LAYER_G_ENERGYthermodynamic_bind
24802493→2491Manifold_Viscous_Stress_TensorManifold Dynamicsτ_μν = μ(∇_μ v_ν + ∇_ν v_μ - (2/3)g_μν ∇·v) + λ g_μν ∇·vτ_μν=viscous stress tensor (rank-2), μ=shear viscosity, λ=bulk viscosity, ∇=covariant derivative, g_μν=metric, v=velocityNewtonian viscosity generalized to n-manifold: stress tensor proportional to symmetric velocity gradient with covariant derivatives; includes bulk viscosity term for compressibilitymanifold_dynamics/viscous_stress_tensor.mdPending2492,2494,2500LAYER_C_TOPOLOGYphysical_bind
24812494→2492Manifold_Strain_Rate_TensorManifold Dynamicse_μν = ½(∇_μ v_ν + ∇_ν v_μ)e_μν=strain rate tensor (rank-2), ∇=covariant derivative, v=velocitySymmetric velocity gradient tensor on manifold; measures local deformation rate using covariant derivatives instead of partial derivativesmanifold_dynamics/strain_rate_tensor.mdPending2493,2495,2496LAYER_C_TOPOLOGYgeometric_bind
24822495→2493Manifold_Navier_StokesManifold Dynamicsρ(∂_t v^μ + v^ν ∇_ν v^μ) = -g^μν ∂_ν P + ∇_ν τ^μν + ρ f^μρ=density, v=velocity, ∇=covariant derivative, g^μν=inverse metric, P=pressure, τ=viscous stress tensor, f=external forceNavier-Stokes equation on n-manifold: material derivative with covariant transport, viscous stress tensor, and external forces; reduces to classical NS in Cartesian coordinatesmanifold_dynamics/manifold_navier_stokes.mdPending2493,2496,2466LAYER_C_TOPOLOGYphysical_bind
24832496→2494Manifold_Viscous_DissipationManifold DynamicsΦ_viscous = τ_μν e^μν = μ e_μν e^μν + λ (∇·v)²Φ_viscous=dissipation rate, τ=stress tensor, e=strain rate, μ=shear viscosity, λ=bulk viscosityViscous energy dissipation on manifold via contraction of stress and strain rate tensors; quantifies irreversible energy loss to heat in n-dimensional flowmanifold_dynamics/viscous_dissipation.mdPending2493,2497LAYER_G_ENERGYthermodynamic_bind
24842497→2495Manifold_Reynolds_NumberManifold DynamicsRe = ρ g_μν v^μ v^ν L / μRe=Reynolds number, ρ=density, g_μν=metric, v=velocity, L=characteristic length, μ=viscosityReynolds number generalized to manifold using metric contraction for kinetic energy; characterizes flow regime (laminar vs turbulent) on curved geometriesmanifold_dynamics/manifold_reynolds.mdPending2494,2498,2473LAYER_C_TOPOLOGYphysical_bind
24852498→2496Manifold_Boundary_LayerManifold Dynamicsδ ≈ √(ν x / g_μν v^μ v^ν)δ=boundary layer thickness, ν=kinematic viscosity, x=distance, g_μν=metric, v=velocityBoundary layer thickness on manifold: viscous effects scale with metric-contracted velocity; generalizes flat-plate Blasius solution to curved surfacesmanifold_dynamics/manifold_boundary_layer.mdPending2495,2499,2471LAYER_C_TOPOLOGYphysical_bind
24862499→2497Manifold_No_Slip_ConditionManifold Dynamicsv^μ|_∂Ω = 0 at boundary ∂Ωv^μ=velocity field, ∂Ω=manifold boundaryNo-slip boundary condition generalized: velocity vanishes on manifold boundary in all directions; essential for viscous flow on curved surfaces with boundariesmanifold_dynamics/manifold_no_slip.mdPending2496,2500LAYER_C_TOPOLOGYphysical_bind
24872500→2498Manifold_Viscosity_TensorManifold Dynamicsη_μνρσ = μ (g_μρ g_νσ + g_μσ g_νρ - (2/3) g_μν g_ρσ) + λ g_μν g_ρση_μνρσ=fourth-rank viscosity tensor, μ=shear viscosity, λ=bulk viscosity, g_μν=metricAnisotropic viscosity as fourth-rank tensor on manifold; generalizes isotropic Newtonian viscosity to curved geometries where viscosity may vary with directionmanifold_dynamics/viscosity_tensor.mdPending2497,2501LAYER_C_TOPOLOGYphysical_bind
24882501→2499Manifold_Stokes_FlowManifold Dynamics∇_ν τ^μν + g^μν ∂_ν P = 0 (low Re limit)∇=covariant derivative, τ=viscous stress tensor, g^μν=inverse metric, P=pressureStokes flow (creeping flow) on manifold: viscous forces dominate inertial forces; pressure and viscous stress balance via covariant derivatives for low Reynolds number flowmanifold_dynamics/manifold_stokes_flow.mdPending2498,2502,2470LAYER_C_TOPOLOGYphysical_bind
24892502→2500Manifold_Poiseuille_Flow_RateManifold DynamicsQ = (π R⁴ ΔP) / (8 μ L_g) · (1 + κ R²)Q=volumetric flow rate, R=channel radius, ΔP=pressure drop, μ=viscosity, L_g=geodesic length, κ=curvature correctionPoiseuille flow rate on curved manifold: Hagen-Poiseuille law with geodesic length and curvature correction for non-Euclidean channel geometrymanifold_dynamics/manifold_poisuelle_flow.mdPending2494,2503,2469LAYER_C_TOPOLOGYphysical_bind
24902503→2501Manifold_Parabolic_Profile_CurvedManifold Dynamicsv(r) = v_max [1 - (r/R)²] · (1 - κ r²/6)v(r)=velocity at radius r, v_max=centerline velocity, R=channel radius, κ=curvatureParabolic velocity profile on curved manifold: modified by curvature term that reduces velocity near walls in curved channelsmanifold_dynamics/curved_velocity_profile.mdPending2502,2504LAYER_C_TOPOLOGYphysical_bind
24912504→2502Manifold_Pressure_Drop_CurvedManifold DynamicsΔP = (8 μ L_g Q) / (π R⁴) · (1 + κ R²)ΔP=pressure drop, μ=viscosity, L_g=geodesic length, Q=flow rate, R=radius, κ=curvaturePressure drop along curved channel: increased resistance due to curvature; geodesic length replaces Euclidean lengthmanifold_dynamics/curved_pressure_drop.mdPending2502,2505LAYER_G_ENERGYthermodynamic_bind
24922505→2503Manifold_Shear_Stress_PoiseuilleManifold Dynamicsτ_w = (R ΔP) / (2 L_g) · (1 + κ R²/2)τ_w=wall shear stress, R=radius, ΔP=pressure drop, L_g=geodesic length, κ=curvatureWall shear stress in curved Poiseuille flow: curvature increases stress at channel wallsmanifold_dynamics/curved_shear_stress.mdPending2503,2506LAYER_C_TOPOLOGYphysical_bind
24932506→2504Manifold_Hydraulic_ResistanceManifold DynamicsR_h = (8 μ L_g) / (π R⁴) · (1 + κ R²)R_h=hydraulic resistance, μ=viscosity, L_g=geodesic length, R=radius, κ=curvatureHydraulic resistance on manifold: geometric factor includes curvature correction for non-straight channelsmanifold_dynamics/manifold_hydraulic_resistance.mdPending2504,2507LAYER_G_ENERGYthermodynamic_bind
24942507→2505Manifold_Friction_Factor_PoiseuilleManifold Dynamicsf = 64/Re · (1 + κ R²/4)f=friction factor, Re=Reynolds number, κ=curvature, R=radiusFriction factor for laminar flow in curved channels: curvature increases effective friction beyond classical 64/Remanifold_dynamics/curved_friction_factor.mdPending2505,2508LAYER_C_TOPOLOGYphysical_bind
24952511→2509Manifold_Poiseuille_Entrance_LengthManifold DynamicsL_e = 0.06 Re D · (1 + κ D)L_e=entrance length, Re=Reynolds number, D=diameter, κ=curvatureEntrance length for fully developed Poiseuille flow on manifold: curvature extends development length in curved channelsmanifold_dynamics/manifold_entrance_length.mdPending2509,2512LAYER_C_TOPOLOGYphysical_bind
24962610Acoustic_ImpedanceAcousticZ = |∇f| = √(Σ(∂f/∂xᵢ)²)Z=acoustic impedance, ∇f=gradient vector, xᵢ=coordinatesAcoustic impedance as gradient magnitude; high-gradient regions cause reflection/refraction; metric tensor on acoustic manifoldtools/lean/Semantics/Semantics/MorphicDSP.leanLean2609LAYER_C_TOPOLOGYgeometric_bind
24972611Acoustic_GeodesicAcousticγ(t) = argmin ∫|∇f(γ(t))|dtγ(t)=geodesic path, ∇f=gradient field, t=parameterGeodesic flow following gradient descent on acoustic manifold; sound rays as gradient trajectoriestools/lean/Semantics/Semantics/MorphicDSP.leanLean2609LAYER_C_TOPOLOGYgeometric_bind
24982612Acoustic_ResonanceAcousticωₙ ∝ ⟨|∇f|⟩ωₙ=resonant frequency, ⟨|∇f|⟩=average gradient magnitudeResonance eigenmode prediction via Laplacian eigenvalue approximation; standing waves as eigenfunctions on manifoldtools/lean/Semantics/Semantics/MorphicDSP.leanLean2609LAYER_C_TOPOLOGYgeometric_bind
24992613Gradient_InterferenceAcoustic∇f_total = ∇f₁ + ∇f₂∇f_total=combined gradient, ∇f₁,∇f₂=individual gradientsInterference pattern via gradient field superposition; constructive/destructive interference as gradient alignment/misalignmenttools/lean/Semantics/Semantics/MorphicDSP.leanLean2609LAYER_C_TOPOLOGYgeometric_bind
25002614Acoustic_Shannon_EntropyAcousticH₁ = -Σ p_b log₂ p_b (over |∇f|)H₁=Shannon entropy, p_b=gradient magnitude probabilityAcoustic Shannon entropy over gradient magnitudes; measures disorder in acoustic gradient fieldtools/lean/Semantics/Semantics/EntropyMeasures.leanLean2609LAYER_A_COMPRESSIONinformational_bind
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