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972d6643e2
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
261 lines
12 KiB
Python
261 lines
12 KiB
Python
#!/usr/bin/env python3
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"""
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Map System Equations to 4-Primitive Framework
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==============================================
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Review all system equations and map them to the 4 primitives:
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- Field primitive (ρ(x⃗))
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- Shear primitive (G = AᵀA)
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- Packet primitive (Γᵢ)
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- Spectral primitive (C = UΛUᵀ)
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"""
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import json
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from pathlib import Path
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RESEARCH_STACK = Path("/home/allaun/Documents/Research Stack")
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# 4-primitive framework
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PRIMITIVES = {
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"field": {
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"equation": "ρ(x⃗)",
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"role": "tells you what exists (field / substrate / scalar manifold state)",
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"keywords": ["entropy", "density", "distribution", "manifold", "topology", "field", "state"]
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},
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"shear": {
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"equation": "G = AᵀA",
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"role": "tells you how it deforms (shear / metric deformation / lawful geometry)",
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"keywords": ["distance", "metric", "transform", "deformation", "shear", "geometry", "hyperbolic"]
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},
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"packet": {
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"equation": "Γᵢ",
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"role": "tells you what is emitted/witnessed (packet / executable typed glyph-witness / codec event)",
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"keywords": ["coding", "compression", "transform", "bwt", "ans", "packet", "codec", "optimization"]
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},
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"spectral": {
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"equation": "C = UΛUᵀ",
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"role": "tells you what basis survives (spectral / eigenbasis / pruning-correlation structure)",
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"keywords": ["complexity", "basis", "bottleneck", "decomposition", "spectral", "eigen", "dimension", "tradeoff"]
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}
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}
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# System equations from grand unified theory
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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) ≈ 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^(-α), where α ≈ 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(Σ*)",
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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) - β*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) + ε, where ε ≈ 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_ε) for ε in [0, ∞), 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) + λ|C| + μ*K(C) + ν*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 ∈ Σ* | G(w) = 1 } ≈ M ⊂ R^d",
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"primitive": "shear",
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"mapping": "Language as manifold = shear transform (grammar → 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) - β*I(Z;Y) + γ*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": "ρ(x⃗)",
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"derives": ["morse_smale", "radius_ratio", "residual_ratio", "s3c_shell"],
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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 = AᵀA",
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"derives": ["shear_matrix", "famm_delay", "eigen_decomposition"],
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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": "Γᵢ = γᵢ ⊗ χᵢ ⊗ κᵢ ⊗ τᵢ ⊗ UᵢΛᵢaᵢ ⊗ θᵢ ⊗ εᵢ",
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"derives": ["gccl_packet", "gain_test"],
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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 = UΛUᵀ",
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"derives": ["residual_correlation", "eigen_decomposition", "famm_spectral"],
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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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def analyze_mapping():
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print("=" * 70)
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print(" SYSTEM EQUATIONS → 4-PRIMITIVE FRAMEWORK MAPPING")
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print("=" * 70)
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print("\n4-PRIMITIVE FRAMEWORK:")
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for prim, data in PRIMITIVES.items():
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print(f"\n{prim.upper()}: {data['equation']}")
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print(f" Role: {data['role']}")
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print(f" Keywords: {', '.join(data['keywords'])}")
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print("\n" + "=" * 70)
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print(" GRAND UNIFIED THEORY EQUATIONS")
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print("=" * 70)
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gut = SYSTEM_EQUATIONS["grand_unified_theory"]
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print(f"\nSource: {gut['source']}")
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print(f"10 axioms, 4 unified equations")
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print("\nAXIOMS:")
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for ax_name, ax_data in gut["axioms"].items():
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prim = ax_data["primitive"].upper()
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print(f"\n{ax_name}:")
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print(f" Formula: {ax_data['formula']}")
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print(f" Primitive: {prim}")
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print(f" Mapping: {ax_data['mapping']}")
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print("\nUNIFIED EQUATIONS:")
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for eq_name, eq_data in gut["unified_equations"].items():
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prim = eq_data["primitive"].upper()
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print(f"\n{eq_name}:")
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print(f" Formula: {eq_data['formula']}")
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print(f" Primitive: {prim}")
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print(f" Mapping: {eq_data['mapping']}")
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print("\n" + "=" * 70)
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print(" PRIMITIVE DISTRIBUTION")
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print("=" * 70)
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primitive_counts = {"field": 0, "shear": 0, "packet": 0, "spectral": 0}
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for ax_data in gut["axioms"].values():
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primitive_counts[ax_data["primitive"]] += 1
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for eq_data in gut["unified_equations"].values():
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primitive_counts[eq_data["primitive"]] += 1
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print(f"\nField primitive (ρ(x⃗)): {primitive_counts['field']} equations")
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print(f"Shear primitive (G = AᵀA): {primitive_counts['shear']} equations")
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print(f"Packet primitive (Γᵢ): {primitive_counts['packet']} equations")
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print(f"Spectral primitive (C = UΛUᵀ): {primitive_counts['spectral']} equations")
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print("\n" + "=" * 70)
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print(" COMPACTIFIED CORE EQUATIONS")
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print("=" * 70)
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cce = SYSTEM_EQUATIONS["compactified_core_equations"]
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print(f"\nSource: {cce['source']}")
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for prim, data in cce["primitives"].items():
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print(f"\n{prim}:")
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print(f" Equation: {data['equation']}")
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print(f" Derives: {', '.join(data['derives'])}")
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print(f" Role: {data['role']}")
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print("\n" + "=" * 70)
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print(" INTEGRATION ANALYSIS")
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print("=" * 70)
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print("\nGrand unified theory equations map to:")
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print(f" - Field primitive: {primitive_counts['field']} equations (Shannon entropy, Zipf law, grammar manifold, topological invariants)")
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print(f" - Shear primitive: {primitive_counts['shear']} equations (hyperbolic hierarchy, language as manifold)")
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print(f" - Packet primitive: {primitive_counts['packet']} equations (ANS optimality, BWT, grand compression)")
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print(f" - Spectral primitive: {primitive_counts['spectral']} equations (Kolmogorov complexity, information bottleneck, MDL, hyperbolic distance)")
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print("\nCompactified core equations:")
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print(" - Field primitive: derives Morse-Smale, radius_ratio, residual_ratio, S3C shells")
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print(" - Shear primitive: derives shear_matrix, FAMM delays, eigen_decomposition")
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print(" - Packet primitive: derives GCCL packet, gain test")
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print(" - Spectral primitive: derives residual correlation, eigen_decomposition, FAMM spectral")
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print("\n" + "=" * 70)
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print(" KEY INSIGHTS")
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print("=" * 70)
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print("\n1. Consistency: Grand unified theory axioms map cleanly to 4 primitives")
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print("2. Redundancy: Some equations span multiple primitives (e.g., grand compression = packet + spectral)")
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print("3. Completeness: Each primitive has representative equations from multiple sources")
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print("4. Integration: Compactified core equations subsume grand unified theory equations")
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print("5. Canonical mapping:")
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print(" - Field: entropy, density, topology, manifold structure")
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print(" - Shear: distance, metric, deformation, geometric transform")
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print(" - Packet: coding, compression, transform, optimization")
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print(" - Spectral: complexity, basis, bottleneck, decomposition, tradeoff")
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# Save mapping
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output_file = RESEARCH_STACK / "4-Infrastructure/shim/system_equations_4primitive_mapping.json"
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with open(output_file, 'w') as f:
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json.dump({
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"primitives": PRIMITIVES,
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"system_equations": SYSTEM_EQUATIONS,
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"primitive_counts": primitive_counts,
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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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}, f, indent=2)
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print(f"\n✓ Mapping saved to: {output_file}")
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if __name__ == "__main__":
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analyze_mapping()
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