{ "primitives": { "field": { "equation": "\u03c1(x\u20d7)", "role": "tells you what exists (field / substrate / scalar manifold state)", "keywords": [ "field", "density", "distribution", "potential", "energy", "manifold", "state", "landscape" ] }, "shear": { "equation": "G = A\u1d40A", "role": "tells you how it deforms (shear / metric deformation / lawful geometry)", "keywords": [ "distance", "metric", "gradient", "force", "transform", "deformation", "geometry", "rate" ] }, "packet": { "equation": "\u0393\u1d62", "role": "tells you what is emitted/witnessed (packet / executable typed glyph-witness / codec event)", "keywords": [ "descriptor", "vector", "map", "kernel", "similarity", "representation", "encoding" ] }, "spectral": { "equation": "C = U\u039bU\u1d40", "role": "tells you what basis survives (spectral / eigenbasis / pruning-correlation structure)", "keywords": [ "eigen", "basis", "hamiltonian", "variational", "optimization", "decomposition", "energy" ] } }, "scientific_equations": { "chemistry_physics_nspace_spine": { "source": "chemistry_physics_nspace_spine_v0.json", "equations": [ { "name": "Chemical_Descriptor_Vector", "domain": "Chemistry / N-Space", "equation": "x_mol = (d1,d2,...,dn) \u2208 R^n", "primitive": "packet", "mapping": "Molecule as point in descriptor space = packet representation" }, { "name": "Chemical_Space_Distance", "domain": "Chemistry / Geometry", "equation": "D(i,j) = ||x_i-x_j||_2", "primitive": "shear", "mapping": "Chemical similarity as geometric distance = shear metric" }, { "name": "Weighted_Chemical_Space_Distance", "domain": "Chemistry / Geometry", "equation": "D_w(i,j) = sqrt(sum_k w_k(x_ik-x_jk)^2)", "primitive": "shear", "mapping": "Weighted semantic distance = weighted shear metric" }, { "name": "Chemical_Structure_Property_Map", "domain": "Chemistry / ML", "equation": "y = f(x_mol)", "primitive": "packet", "mapping": "Property prediction over chemical space = packet transform" }, { "name": "Molecular_Configuration_Space", "domain": "Chemistry / Physics", "equation": "R = (r1,...,rN) \u2208 R^{3N}", "primitive": "field", "mapping": "N-atom molecular configuration space = field manifold" }, { "name": "Potential_Energy_Surface", "domain": "Chemistry / Physics", "equation": "E = V(R)", "primitive": "field", "mapping": "Energy as scalar field over configuration space = field state" }, { "name": "Molecular_Force", "domain": "Chemistry / Physics", "equation": "F_i = -\u2207_{r_i}V(R)", "primitive": "shear", "mapping": "Force as gradient of potential energy = shear deformation" }, { "name": "Molecular_Dynamics_Newtonian", "domain": "Chemistry / Physics", "equation": "m_i d\u00b2r_i/dt\u00b2 = -\u2207_{r_i}V(R)", "primitive": "shear", "mapping": "Classical molecular dynamics = shear dynamics (force-driven deformation)" }, { "name": "Molecular_Force_Field_Energy", "domain": "Chemistry / Physics", "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", "primitive": "field", "mapping": "Pair-distance distribution = field correlation function" }, { "name": "Local_Atomic_Density_Kernel", "domain": "Materials / Descriptor", "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", "primitive": "packet", "mapping": "Local atomic density and similarity kernel = packet similarity metric" }, { "name": "Arrhenius_Rate", "domain": "Chemistry / Thermodynamics", "equation": "k = A exp(-Ea/RT)", "primitive": "shear", "mapping": "Reaction rate over activation barrier = shear rate (temperature-driven deformation)" }, { "name": "Eyring_Transition_State_Rate", "domain": "Chemistry / Thermodynamics", "equation": "k = (kBT/h) exp(-\u0394G\u2021/RT)", "primitive": "shear", "mapping": "Transition-state rate equation = shear rate (free energy-driven deformation)" }, { "name": "Boltzmann_Distribution", "domain": "Statistical Mechanics", "equation": "p_i = exp(-Ei/kBT)/Z; Z = \u03a3_i exp(-Ei/kBT)", "primitive": "field", "mapping": "Energy landscape to probability distribution = field state (probability field)" }, { "name": "Quantum_Hamiltonian_Eigenproblem", "domain": "Quantum Chemistry", "equation": "\u0124\u03c8 = E\u03c8", "primitive": "spectral", "mapping": "Quantum energy eigenproblem = spectral decomposition (Hamiltonian eigenbasis)" }, { "name": "Quantum_Hamiltonian_Variational_Energy", "domain": "Quantum Chemistry", "equation": "E(\u03b8) = <\u03c8(\u03b8)|\u0124|\u03c8(\u03b8)>; \u03b8* = argmin_\u03b8 E(\u03b8)", "primitive": "spectral", "mapping": "Variational quantum energy optimization = spectral optimization (basis optimization)" }, { "name": "DFT_Energy_Functional", "domain": "Quantum Chemistry", "equation": "E[n] = Ts[n] + \u222bvext(r)n(r)dr + 1/2\u222b\u222bn(r)n(r')/|r-r'|drdr' + Exc[n]", "primitive": "field", "mapping": "Electron density to energy functional = field state (density field \u2192 energy field)" }, { "name": "Bayesian_Optimization_Chemical_Space", "domain": "Chemistry / Optimization", "equation": "f(x) ~ GP(\u03bc(x), k(x,x')); x_next = argmax_x \u03b1(x); EI(x) = E[max(f(x)-f_best, 0)]", "primitive": "spectral", "mapping": "Search policy over chemical/material space = spectral optimization (Gaussian process basis)" } ] } }, "primitive_distribution": { "field": 6, "shear": 6, "packet": 4, "spectral": 3 }, "domain_distribution": { "Chemistry / N-Space": 1, "Chemistry / Geometry": 2, "Chemistry / ML": 1, "Chemistry / Physics": 5, "Chemistry / Descriptor": 1, "Materials / Geometry": 1, "Materials / Descriptor": 1, "Chemistry / Thermodynamics": 2, "Statistical Mechanics": 1, "Quantum Chemistry": 3, "Chemistry / Optimization": 1 }, "insights": { "field_core": "energy landscapes, density fields, probability distributions", "shear_core": "gradients, forces, rates, geometric deformations", "packet_core": "descriptors, encodings, similarity metrics, representations", "spectral_core": "eigenproblems, basis optimization, variational methods", "cross_domain_consistency": "Each primitive appears across multiple scientific domains", "no_gaps": "Each primitive well-represented across chemistry, physics, thermodynamics, quantum chemistry" } }