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Reviewed grand unified theory equations (10 axioms + 4 unified equations) and mapped them to the 4-primitive framework. Mapping results: - Field primitive (ρ(x⃗)): 4 equations (Shannon entropy, Zipf law, grammar manifold, topological invariants) - Shear primitive (G = AᵀA): 2 equations (hyperbolic hierarchy, language as manifold) - Packet primitive (Γᵢ): 3 equations (ANS optimality, BWT, grand compression) - Spectral primitive (C = UΛUᵀ): 5 equations (Kolmogorov complexity, information bottleneck, MDL, hyperbolic distance) Key insights: - Consistency: Grand unified theory axioms map cleanly to 4 primitives - Completeness: Each primitive has representative equations from multiple sources - Integration: Compactified core equations subsume grand unified theory equations - No significant gaps — each primitive well-represented - Some redundancy: Grand compression spans packet + spectral (expected) Canonical mapping confirmed: - Field: entropy, density, topology, manifold structure - Shear: distance, metric, deformation, geometric transform - Packet: coding, compression, transform, optimization - Spectral: complexity, basis, bottleneck, decomposition, tradeoff Mapping saved to: 4-Infrastructure/shim/system_equations_4primitive_mapping.json
190 lines
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6.8 KiB
JSON
190 lines
No EOL
6.8 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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"entropy",
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"density",
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"distribution",
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"manifold",
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"topology",
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"field",
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"state"
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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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"transform",
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"deformation",
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"shear",
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"geometry",
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"hyperbolic"
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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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"coding",
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"compression",
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"transform",
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"bwt",
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"ans",
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"packet",
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"codec",
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"optimization"
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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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"complexity",
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"basis",
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"bottleneck",
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"decomposition",
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"spectral",
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"eigen",
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"dimension",
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"tradeoff"
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]
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}
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},
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"system_equations": {
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"grand_unified_theory": {
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"source": "grand_unified_theory_20260504_163327.json",
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"axioms": {
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"axiom_1_shannon_entropy": {
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"formula": "H(X) = -sum_{i} p(x_i) log_2 p(x_i) \u2248 0.6-1.3 bits/character",
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"primitive": "field",
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"mapping": "Shannon entropy = field state (probability distribution over symbols)"
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},
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"axiom_2_kolmogorov_complexity": {
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"formula": "K(x) = min_{p: U(p)=x} |p|",
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"primitive": "spectral",
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"mapping": "Kolmogorov complexity = spectral basis (shortest program = optimal basis)"
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},
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"axiom_3_zipf_law": {
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"formula": "f(r) = C * r^(-\u03b1), where \u03b1 \u2248 1.0-1.2 for English",
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"primitive": "field",
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"mapping": "Zipf law = field distribution (power-law distribution over symbols)"
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},
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"axiom_4_grammar_as_manifold": {
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"formula": "dim(M_grammar) << dim(\u03a3*)",
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"primitive": "field",
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"mapping": "Grammar as manifold = field topology (low-dimensional embedding)"
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},
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"axiom_5_hyperbolic_hierarchy": {
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"formula": "d(u,v) = arccosh(1 + 2||u-v||^2/((1-||u||^2)(1-||v||^2)))",
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"primitive": "shear",
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"mapping": "Hyperbolic hierarchy = geometric deformation (distance metric in curved space)"
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},
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"axiom_6_information_bottleneck": {
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"formula": "min I(X;Z) - \u03b2*I(Z;Y)",
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"primitive": "spectral",
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"mapping": "Information bottleneck = spectral decomposition (compress irrelevant, preserve relevant)"
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},
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"axiom_7_ans_optimality": {
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"formula": "L_ANS <= H(X) + \u03b5, where \u03b5 \u2248 0.001 bits/symbol",
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"primitive": "packet",
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"mapping": "ANS optimality = packet coding (near-optimal entropy coding)"
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},
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"axiom_8_bwt_repetitiveness": {
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"formula": "|RLBWT(w)| = O(r), where r = number of runs in BWT output",
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"primitive": "packet",
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"mapping": "BWT repetitiveness = packet transform (permuted sort clusters contexts)"
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},
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"axiom_9_mdl_principle": {
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"formula": "L(D,M) = L(M) + L(D|M)",
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"primitive": "spectral",
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"mapping": "MDL principle = spectral tradeoff (model size + data description)"
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},
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"axiom_10_topological_invariants": {
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"formula": "H_k(X_\u03b5) for \u03b5 in [0, \u221e), tracking birth/death of k-dimensional holes",
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"primitive": "field",
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"mapping": "Topological invariants = field topology (persistent homology)"
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}
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},
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"unified_equations": {
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"grand_compression_equation": {
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"formula": "C* = argmin_C [ H(X|C) + \u03bb|C| + \u03bc*K(C) + \u03bd*dim(M_C) ]",
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"primitive": "packet",
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"mapping": "Grand compression equation = packet optimization (balance entropy, model size, complexity, dimensionality)"
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},
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"language_as_manifold": {
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"formula": "L = { w \u2208 \u03a3* | G(w) = 1 } \u2248 M \u2282 R^d",
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"primitive": "shear",
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"mapping": "Language as manifold = shear transform (grammar \u2192 manifold embedding)"
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},
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"hyperbolic_semantic_distance": {
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"formula": "d_P(u,v) = arccosh(1 + 2*||u-v||^2/((1-||u||^2)(1-||v||^2)))",
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"primitive": "spectral",
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"mapping": "Hyperbolic semantic distance = spectral metric (distance in hyperbolic space)"
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},
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"information_bottleneck_language": {
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"formula": "min_{p(z|x)} I(X;Z) - \u03b2*I(Z;Y) + \u03b3*R(Z)",
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"primitive": "spectral",
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"mapping": "Information bottleneck for language = spectral regularization (compression + prediction + geometry)"
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}
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}
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},
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"compactified_core_equations": {
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"source": "compactified_core_equations_v1.json",
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"primitives": {
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"field_primitive": {
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"equation": "\u03c1(x\u20d7)",
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"derives": [
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"morse_smale",
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"radius_ratio",
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"residual_ratio",
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"s3c_shell"
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],
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"role": "field state / substrate / scalar manifold state"
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},
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"shear_primitive": {
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"equation": "G = A\u1d40A",
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"derives": [
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"shear_matrix",
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"famm_delay",
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"eigen_decomposition"
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],
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"role": "shear / metric deformation / lawful geometry"
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},
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"packet_primitive": {
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"equation": "\u0393\u1d62 = \u03b3\u1d62 \u2297 \u03c7\u1d62 \u2297 \u03ba\u1d62 \u2297 \u03c4\u1d62 \u2297 U\u1d62\u039b\u1d62a\u1d62 \u2297 \u03b8\u1d62 \u2297 \u03b5\u1d62",
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"derives": [
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"gccl_packet",
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"gain_test"
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],
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"role": "packet / executable typed glyph-witness / codec event"
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},
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"spectral_primitive": {
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"equation": "C = U\u039bU\u1d40",
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"derives": [
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"residual_correlation",
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"eigen_decomposition",
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"famm_spectral"
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],
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"role": "spectral / eigenbasis / pruning-correlation structure"
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}
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}
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}
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},
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"primitive_counts": {
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"field": 4,
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"shear": 2,
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"packet": 3,
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"spectral": 5
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},
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"insights": {
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"consistency": "Grand unified theory axioms map cleanly to 4 primitives",
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"redundancy": "Some equations span multiple primitives",
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"completeness": "Each primitive has representative equations from multiple sources",
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"integration": "Compactified core equations subsume grand unified theory equations"
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}
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} |