{
  "primitives": {
    "field": {
      "equation": "\u03c1(x\u20d7)",
      "role": "tells you what exists (field / substrate / scalar manifold state)",
      "keywords": [
        "entropy",
        "density",
        "distribution",
        "manifold",
        "topology",
        "field",
        "state"
      ]
    },
    "shear": {
      "equation": "G = A\u1d40A",
      "role": "tells you how it deforms (shear / metric deformation / lawful geometry)",
      "keywords": [
        "distance",
        "metric",
        "transform",
        "deformation",
        "shear",
        "geometry",
        "hyperbolic"
      ]
    },
    "packet": {
      "equation": "\u0393\u1d62",
      "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 = U\u039bU\u1d40",
      "role": "tells you what basis survives (spectral / eigenbasis / pruning-correlation structure)",
      "keywords": [
        "complexity",
        "basis",
        "bottleneck",
        "decomposition",
        "spectral",
        "eigen",
        "dimension",
        "tradeoff"
      ]
    }
  },
  "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) \u2248 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^(-\u03b1), where \u03b1 \u2248 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(\u03a3*)",
          "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) - \u03b2*I(Z;Y)",
          "primitive": "spectral",
          "mapping": "Information bottleneck = spectral decomposition (compress irrelevant, preserve relevant)"
        },
        "axiom_7_ans_optimality": {
          "formula": "L_ANS <= H(X) + \u03b5, where \u03b5 \u2248 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_\u03b5) for \u03b5 in [0, \u221e), 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) + \u03bb|C| + \u03bc*K(C) + \u03bd*dim(M_C) ]",
          "primitive": "packet",
          "mapping": "Grand compression equation = packet optimization (balance entropy, model size, complexity, dimensionality)"
        },
        "language_as_manifold": {
          "formula": "L = { w \u2208 \u03a3* | G(w) = 1 } \u2248 M \u2282 R^d",
          "primitive": "shear",
          "mapping": "Language as manifold = shear transform (grammar \u2192 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) - \u03b2*I(Z;Y) + \u03b3*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": "\u03c1(x\u20d7)",
          "derives": [
            "morse_smale",
            "radius_ratio",
            "residual_ratio",
            "s3c_shell"
          ],
          "role": "field state / substrate / scalar manifold state"
        },
        "shear_primitive": {
          "equation": "G = A\u1d40A",
          "derives": [
            "shear_matrix",
            "famm_delay",
            "eigen_decomposition"
          ],
          "role": "shear / metric deformation / lawful geometry"
        },
        "packet_primitive": {
          "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",
          "derives": [
            "gccl_packet",
            "gain_test"
          ],
          "role": "packet / executable typed glyph-witness / codec event"
        },
        "spectral_primitive": {
          "equation": "C = U\u039bU\u1d40",
          "derives": [
            "residual_correlation",
            "eigen_decomposition",
            "famm_spectral"
          ],
          "role": "spectral / eigenbasis / pruning-correlation structure"
        }
      }
    }
  },
  "primitive_counts": {
    "field": 4,
    "shear": 2,
    "packet": 3,
    "spectral": 5
  },
  "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"
  }
}