#!/usr/bin/env python3 """ Map System Equations to 4-Primitive Framework ============================================== Review all system equations and map them to the 4 primitives: - Field primitive (ρ(x⃗)) - Shear primitive (G = AᵀA) - Packet primitive (Γᵢ) - Spectral primitive (C = UΛUᵀ) """ import json from pathlib import Path RESEARCH_STACK = Path("/home/allaun/Documents/Research Stack") # 4-primitive framework PRIMITIVES = { "field": { "equation": "ρ(x⃗)", "role": "tells you what exists (field / substrate / scalar manifold state)", "keywords": ["entropy", "density", "distribution", "manifold", "topology", "field", "state"] }, "shear": { "equation": "G = AᵀA", "role": "tells you how it deforms (shear / metric deformation / lawful geometry)", "keywords": ["distance", "metric", "transform", "deformation", "shear", "geometry", "hyperbolic"] }, "packet": { "equation": "Γᵢ", "role": "tells you what is emitted/witnessed (packet / executable typed glyph-witness / codec event)", "keywords": ["coding", "compression", "transform", "bwt", "ans", "packet", "codec", "optimization"] }, "spectral": { "equation": "C = UΛUᵀ", "role": "tells you what basis survives (spectral / eigenbasis / pruning-correlation structure)", "keywords": ["complexity", "basis", "bottleneck", "decomposition", "spectral", "eigen", "dimension", "tradeoff"] } } # System equations from grand unified theory SYSTEM_EQUATIONS = { "grand_unified_theory": { "source": "grand_unified_theory_20260504_163327.json", "axioms": { "axiom_1_shannon_entropy": { "formula": "H(X) = -sum_{i} p(x_i) log_2 p(x_i) ≈ 0.6-1.3 bits/character", "primitive": "field", "mapping": "Shannon entropy = field state (probability distribution over symbols)" }, "axiom_2_kolmogorov_complexity": { "formula": "K(x) = min_{p: U(p)=x} |p|", "primitive": "spectral", "mapping": "Kolmogorov complexity = spectral basis (shortest program = optimal basis)" }, "axiom_3_zipf_law": { "formula": "f(r) = C * r^(-α), where α ≈ 1.0-1.2 for English", "primitive": "field", "mapping": "Zipf law = field distribution (power-law distribution over symbols)" }, "axiom_4_grammar_as_manifold": { "formula": "dim(M_grammar) << dim(Σ*)", "primitive": "field", "mapping": "Grammar as manifold = field topology (low-dimensional embedding)" }, "axiom_5_hyperbolic_hierarchy": { "formula": "d(u,v) = arccosh(1 + 2||u-v||^2/((1-||u||^2)(1-||v||^2)))", "primitive": "shear", "mapping": "Hyperbolic hierarchy = geometric deformation (distance metric in curved space)" }, "axiom_6_information_bottleneck": { "formula": "min I(X;Z) - β*I(Z;Y)", "primitive": "spectral", "mapping": "Information bottleneck = spectral decomposition (compress irrelevant, preserve relevant)" }, "axiom_7_ans_optimality": { "formula": "L_ANS <= H(X) + ε, where ε ≈ 0.001 bits/symbol", "primitive": "packet", "mapping": "ANS optimality = packet coding (near-optimal entropy coding)" }, "axiom_8_bwt_repetitiveness": { "formula": "|RLBWT(w)| = O(r), where r = number of runs in BWT output", "primitive": "packet", "mapping": "BWT repetitiveness = packet transform (permuted sort clusters contexts)" }, "axiom_9_mdl_principle": { "formula": "L(D,M) = L(M) + L(D|M)", "primitive": "spectral", "mapping": "MDL principle = spectral tradeoff (model size + data description)" }, "axiom_10_topological_invariants": { "formula": "H_k(X_ε) for ε in [0, ∞), tracking birth/death of k-dimensional holes", "primitive": "field", "mapping": "Topological invariants = field topology (persistent homology)" } }, "unified_equations": { "grand_compression_equation": { "formula": "C* = argmin_C [ H(X|C) + λ|C| + μ*K(C) + ν*dim(M_C) ]", "primitive": "packet", "mapping": "Grand compression equation = packet optimization (balance entropy, model size, complexity, dimensionality)" }, "language_as_manifold": { "formula": "L = { w ∈ Σ* | G(w) = 1 } ≈ M ⊂ R^d", "primitive": "shear", "mapping": "Language as manifold = shear transform (grammar → manifold embedding)" }, "hyperbolic_semantic_distance": { "formula": "d_P(u,v) = arccosh(1 + 2*||u-v||^2/((1-||u||^2)(1-||v||^2)))", "primitive": "spectral", "mapping": "Hyperbolic semantic distance = spectral metric (distance in hyperbolic space)" }, "information_bottleneck_language": { "formula": "min_{p(z|x)} I(X;Z) - β*I(Z;Y) + γ*R(Z)", "primitive": "spectral", "mapping": "Information bottleneck for language = spectral regularization (compression + prediction + geometry)" } } }, "compactified_core_equations": { "source": "compactified_core_equations_v1.json", "primitives": { "field_primitive": { "equation": "ρ(x⃗)", "derives": ["morse_smale", "radius_ratio", "residual_ratio", "s3c_shell"], "role": "field state / substrate / scalar manifold state" }, "shear_primitive": { "equation": "G = AᵀA", "derives": ["shear_matrix", "famm_delay", "eigen_decomposition"], "role": "shear / metric deformation / lawful geometry" }, "packet_primitive": { "equation": "Γᵢ = γᵢ ⊗ χᵢ ⊗ κᵢ ⊗ τᵢ ⊗ UᵢΛᵢaᵢ ⊗ θᵢ ⊗ εᵢ", "derives": ["gccl_packet", "gain_test"], "role": "packet / executable typed glyph-witness / codec event" }, "spectral_primitive": { "equation": "C = UΛUᵀ", "derives": ["residual_correlation", "eigen_decomposition", "famm_spectral"], "role": "spectral / eigenbasis / pruning-correlation structure" } } } } def analyze_mapping(): print("=" * 70) print(" SYSTEM EQUATIONS → 4-PRIMITIVE FRAMEWORK MAPPING") print("=" * 70) print("\n4-PRIMITIVE FRAMEWORK:") for prim, data in PRIMITIVES.items(): print(f"\n{prim.upper()}: {data['equation']}") print(f" Role: {data['role']}") print(f" Keywords: {', '.join(data['keywords'])}") print("\n" + "=" * 70) print(" GRAND UNIFIED THEORY EQUATIONS") print("=" * 70) gut = SYSTEM_EQUATIONS["grand_unified_theory"] print(f"\nSource: {gut['source']}") print(f"10 axioms, 4 unified equations") print("\nAXIOMS:") for ax_name, ax_data in gut["axioms"].items(): prim = ax_data["primitive"].upper() print(f"\n{ax_name}:") print(f" Formula: {ax_data['formula']}") print(f" Primitive: {prim}") print(f" Mapping: {ax_data['mapping']}") print("\nUNIFIED EQUATIONS:") for eq_name, eq_data in gut["unified_equations"].items(): prim = eq_data["primitive"].upper() print(f"\n{eq_name}:") print(f" Formula: {eq_data['formula']}") print(f" Primitive: {prim}") print(f" Mapping: {eq_data['mapping']}") print("\n" + "=" * 70) print(" PRIMITIVE DISTRIBUTION") print("=" * 70) primitive_counts = {"field": 0, "shear": 0, "packet": 0, "spectral": 0} for ax_data in gut["axioms"].values(): primitive_counts[ax_data["primitive"]] += 1 for eq_data in gut["unified_equations"].values(): primitive_counts[eq_data["primitive"]] += 1 print(f"\nField primitive (ρ(x⃗)): {primitive_counts['field']} equations") print(f"Shear primitive (G = AᵀA): {primitive_counts['shear']} equations") print(f"Packet primitive (Γᵢ): {primitive_counts['packet']} equations") print(f"Spectral primitive (C = UΛUᵀ): {primitive_counts['spectral']} equations") print("\n" + "=" * 70) print(" COMPACTIFIED CORE EQUATIONS") print("=" * 70) cce = SYSTEM_EQUATIONS["compactified_core_equations"] print(f"\nSource: {cce['source']}") for prim, data in cce["primitives"].items(): print(f"\n{prim}:") print(f" Equation: {data['equation']}") print(f" Derives: {', '.join(data['derives'])}") print(f" Role: {data['role']}") print("\n" + "=" * 70) print(" INTEGRATION ANALYSIS") print("=" * 70) print("\nGrand unified theory equations map to:") print(f" - Field primitive: {primitive_counts['field']} equations (Shannon entropy, Zipf law, grammar manifold, topological invariants)") print(f" - Shear primitive: {primitive_counts['shear']} equations (hyperbolic hierarchy, language as manifold)") print(f" - Packet primitive: {primitive_counts['packet']} equations (ANS optimality, BWT, grand compression)") print(f" - Spectral primitive: {primitive_counts['spectral']} equations (Kolmogorov complexity, information bottleneck, MDL, hyperbolic distance)") print("\nCompactified core equations:") print(" - Field primitive: derives Morse-Smale, radius_ratio, residual_ratio, S3C shells") print(" - Shear primitive: derives shear_matrix, FAMM delays, eigen_decomposition") print(" - Packet primitive: derives GCCL packet, gain test") print(" - Spectral primitive: derives residual correlation, eigen_decomposition, FAMM spectral") print("\n" + "=" * 70) print(" KEY INSIGHTS") print("=" * 70) print("\n1. Consistency: Grand unified theory axioms map cleanly to 4 primitives") print("2. Redundancy: Some equations span multiple primitives (e.g., grand compression = packet + spectral)") print("3. Completeness: Each primitive has representative equations from multiple sources") print("4. Integration: Compactified core equations subsume grand unified theory equations") print("5. Canonical mapping:") print(" - Field: entropy, density, topology, manifold structure") print(" - Shear: distance, metric, deformation, geometric transform") print(" - Packet: coding, compression, transform, optimization") print(" - Spectral: complexity, basis, bottleneck, decomposition, tradeoff") # Save mapping output_file = RESEARCH_STACK / "4-Infrastructure/shim/system_equations_4primitive_mapping.json" with open(output_file, 'w') as f: json.dump({ "primitives": PRIMITIVES, "system_equations": SYSTEM_EQUATIONS, "primitive_counts": primitive_counts, "insights": { "consistency": "Grand unified theory axioms map cleanly to 4 primitives", "redundancy": "Some equations span multiple primitives", "completeness": "Each primitive has representative equations from multiple sources", "integration": "Compactified core equations subsume grand unified theory equations" } }, f, indent=2) print(f"\n✓ Mapping saved to: {output_file}") if __name__ == "__main__": analyze_mapping()