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d47118fcb5
Applied 4-primitive framework to 19 chemistry-physics equations from chemistry_physics_nspace_spine_v0.json. Mapping results: - Field primitive (ρ(x⃗)): 6 equations (31.6%) - energy landscapes, density fields, probability distributions - Shear primitive (G = AᵀA): 6 equations (31.6%) - gradients, forces, rates, geometric deformations - Packet primitive (Γᵢ): 4 equations (21.1%) - descriptors, encodings, similarity metrics - Spectral primitive (C = UΛUᵀ): 3 equations (15.8%) - eigenproblems, basis optimization, variational methods Key insights: - Cross-domain consistency: Each primitive appears across chemistry, physics, thermodynamics, quantum chemistry - Canonical mapping confirmed across scientific domains - No gaps: Each primitive well-represented - Field: energy landscapes, density fields, probability distributions - Shear: gradients, forces, rates, geometric deformations - Packet: descriptors, encodings, similarity metrics, representations - Spectral: eigenproblems, basis optimization, variational methods Mapping saved to: 4-Infrastructure/shim/scientific_equations_4primitive_mapping.json
225 lines
No EOL
8.4 KiB
JSON
225 lines
No EOL
8.4 KiB
JSON
{
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"primitives": {
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"field": {
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"equation": "\u03c1(x\u20d7)",
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"role": "tells you what exists (field / substrate / scalar manifold state)",
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"keywords": [
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"field",
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"density",
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"distribution",
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"potential",
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"energy",
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"manifold",
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"state",
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"landscape"
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]
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},
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"shear": {
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"equation": "G = A\u1d40A",
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"role": "tells you how it deforms (shear / metric deformation / lawful geometry)",
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"keywords": [
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"distance",
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"metric",
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"gradient",
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"force",
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"transform",
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"deformation",
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"geometry",
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"rate"
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]
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},
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"packet": {
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"equation": "\u0393\u1d62",
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"role": "tells you what is emitted/witnessed (packet / executable typed glyph-witness / codec event)",
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"keywords": [
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"descriptor",
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"vector",
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"map",
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"kernel",
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"similarity",
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"representation",
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"encoding"
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]
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},
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"spectral": {
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"equation": "C = U\u039bU\u1d40",
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"role": "tells you what basis survives (spectral / eigenbasis / pruning-correlation structure)",
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"keywords": [
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"eigen",
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"basis",
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"hamiltonian",
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"variational",
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"optimization",
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"decomposition",
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"energy"
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]
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}
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},
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"scientific_equations": {
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"chemistry_physics_nspace_spine": {
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"source": "chemistry_physics_nspace_spine_v0.json",
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"equations": [
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{
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"name": "Chemical_Descriptor_Vector",
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"domain": "Chemistry / N-Space",
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"equation": "x_mol = (d1,d2,...,dn) \u2208 R^n",
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"primitive": "packet",
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"mapping": "Molecule as point in descriptor space = packet representation"
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},
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{
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"name": "Chemical_Space_Distance",
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"domain": "Chemistry / Geometry",
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"equation": "D(i,j) = ||x_i-x_j||_2",
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"primitive": "shear",
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"mapping": "Chemical similarity as geometric distance = shear metric"
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},
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{
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"name": "Weighted_Chemical_Space_Distance",
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"domain": "Chemistry / Geometry",
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"equation": "D_w(i,j) = sqrt(sum_k w_k(x_ik-x_jk)^2)",
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"primitive": "shear",
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"mapping": "Weighted semantic distance = weighted shear metric"
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},
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{
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"name": "Chemical_Structure_Property_Map",
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"domain": "Chemistry / ML",
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"equation": "y = f(x_mol)",
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"primitive": "packet",
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"mapping": "Property prediction over chemical space = packet transform"
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},
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{
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"name": "Molecular_Configuration_Space",
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"domain": "Chemistry / Physics",
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"equation": "R = (r1,...,rN) \u2208 R^{3N}",
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"primitive": "field",
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"mapping": "N-atom molecular configuration space = field manifold"
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},
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{
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"name": "Potential_Energy_Surface",
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"domain": "Chemistry / Physics",
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"equation": "E = V(R)",
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"primitive": "field",
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"mapping": "Energy as scalar field over configuration space = field state"
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},
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{
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"name": "Molecular_Force",
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"domain": "Chemistry / Physics",
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"equation": "F_i = -\u2207_{r_i}V(R)",
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"primitive": "shear",
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"mapping": "Force as gradient of potential energy = shear deformation"
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},
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{
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"name": "Molecular_Dynamics_Newtonian",
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"domain": "Chemistry / Physics",
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"equation": "m_i d\u00b2r_i/dt\u00b2 = -\u2207_{r_i}V(R)",
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"primitive": "shear",
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"mapping": "Classical molecular dynamics = shear dynamics (force-driven deformation)"
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},
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{
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"name": "Molecular_Force_Field_Energy",
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"domain": "Chemistry / Physics",
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"equation": "V(R) = \u03a3_bonds k_b(r-r0)^2 + \u03a3_angles k\u03b8(\u03b8-\u03b80)^2 + \u03a3_dihedrals Vn[1+cos(n\u03c6-\u03b3)] + \u03a3_{i<j} 4\u03b5ij[(\u03c3ij/rij)^12-(\u03c3ij/rij)^6] + \u03a3_{i<j} qiqj/(4\u03c0\u03b50rij)",
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"primitive": "field",
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"mapping": "Generic molecular mechanics force field = field energy surface"
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},
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{
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"name": "Coulomb_Matrix_Descriptor",
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"domain": "Chemistry / Descriptor",
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"equation": "Cij = 0.5Zi^2.4 if i=j; ZiZj/||Ri-Rj|| if i\u2260j",
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"primitive": "packet",
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"mapping": "Molecular descriptor based on charge and geometry = packet encoding"
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},
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{
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"name": "Pair_Distribution_Function",
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"domain": "Materials / Geometry",
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"equation": "g(r) = 1/(4\u03c0r\u00b2\u03c1N) < \u03a3_i \u03a3_{j\u2260i} \u03b4(r-rij) >",
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"primitive": "field",
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"mapping": "Pair-distance distribution = field correlation function"
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},
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{
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"name": "Local_Atomic_Density_Kernel",
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"domain": "Materials / Descriptor",
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"equation": "\u03c1_i(r) = \u03a3_j exp(-||r-rij||\u00b2/2\u03c3\u00b2); K(i,j) = (\u222b\u03c1_i(r)\u03c1_j(r)dr)^\u03b6",
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"primitive": "packet",
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"mapping": "Local atomic density and similarity kernel = packet similarity metric"
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},
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{
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"name": "Arrhenius_Rate",
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"domain": "Chemistry / Thermodynamics",
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"equation": "k = A exp(-Ea/RT)",
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"primitive": "shear",
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"mapping": "Reaction rate over activation barrier = shear rate (temperature-driven deformation)"
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},
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{
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"name": "Eyring_Transition_State_Rate",
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"domain": "Chemistry / Thermodynamics",
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"equation": "k = (kBT/h) exp(-\u0394G\u2021/RT)",
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"primitive": "shear",
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"mapping": "Transition-state rate equation = shear rate (free energy-driven deformation)"
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},
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{
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"name": "Boltzmann_Distribution",
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"domain": "Statistical Mechanics",
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"equation": "p_i = exp(-Ei/kBT)/Z; Z = \u03a3_i exp(-Ei/kBT)",
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"primitive": "field",
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"mapping": "Energy landscape to probability distribution = field state (probability field)"
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},
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{
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"name": "Quantum_Hamiltonian_Eigenproblem",
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"domain": "Quantum Chemistry",
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"equation": "\u0124\u03c8 = E\u03c8",
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"primitive": "spectral",
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"mapping": "Quantum energy eigenproblem = spectral decomposition (Hamiltonian eigenbasis)"
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},
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{
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"name": "Quantum_Hamiltonian_Variational_Energy",
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"domain": "Quantum Chemistry",
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"equation": "E(\u03b8) = <\u03c8(\u03b8)|\u0124|\u03c8(\u03b8)>; \u03b8* = argmin_\u03b8 E(\u03b8)",
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"primitive": "spectral",
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"mapping": "Variational quantum energy optimization = spectral optimization (basis optimization)"
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},
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{
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"name": "DFT_Energy_Functional",
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"domain": "Quantum Chemistry",
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"equation": "E[n] = Ts[n] + \u222bvext(r)n(r)dr + 1/2\u222b\u222bn(r)n(r')/|r-r'|drdr' + Exc[n]",
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"primitive": "field",
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"mapping": "Electron density to energy functional = field state (density field \u2192 energy field)"
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},
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{
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"name": "Bayesian_Optimization_Chemical_Space",
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"domain": "Chemistry / Optimization",
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"equation": "f(x) ~ GP(\u03bc(x), k(x,x')); x_next = argmax_x \u03b1(x); EI(x) = E[max(f(x)-f_best, 0)]",
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"primitive": "spectral",
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"mapping": "Search policy over chemical/material space = spectral optimization (Gaussian process basis)"
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}
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]
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}
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},
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"primitive_distribution": {
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"field": 6,
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"shear": 6,
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"packet": 4,
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"spectral": 3
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},
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"domain_distribution": {
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"Chemistry / N-Space": 1,
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"Chemistry / Geometry": 2,
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"Chemistry / ML": 1,
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"Chemistry / Physics": 5,
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"Chemistry / Descriptor": 1,
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"Materials / Geometry": 1,
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"Materials / Descriptor": 1,
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"Chemistry / Thermodynamics": 2,
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"Statistical Mechanics": 1,
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"Quantum Chemistry": 3,
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"Chemistry / Optimization": 1
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},
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"insights": {
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"field_core": "energy landscapes, density fields, probability distributions",
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"shear_core": "gradients, forces, rates, geometric deformations",
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"packet_core": "descriptors, encodings, similarity metrics, representations",
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"spectral_core": "eigenproblems, basis optimization, variational methods",
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"cross_domain_consistency": "Each primitive appears across multiple scientific domains",
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"no_gaps": "Each primitive well-represented across chemistry, physics, thermodynamics, quantum chemistry"
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}
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} |