[
  {
    "concept_name": "Fundamental Theorem of Algebra",
    "synthesis": {
      "concept_type": "theorem",
      "formal_statement": "Every non-constant polynomial with complex coefficients has at least one complex root",
      "ontological_structures": [
        {"pattern": "Δ", "evidence": "PRIMARY — distinguishes polynomials that have roots from algebraic completeness of ℂ", "primary": true},
        {"pattern": "⇄", "evidence": "Relates polynomial degree to number of roots"},
        {"pattern": "⟳", "evidence": "Minimal — static existence statement"}
      ],
      "dimension_hints": "D=1 (foundational) — pure algebra, no spatial structure",
      "attribute_dominant": "Δ",
      "complexity": "foundational (1)",
      "elimination_test": "Remove Δ (existence/completeness distinction) → theorem vanishes, it IS about algebraic closure. Remove ⇄ (degree-root relation) → existence still proven. Remove ⟳ → already static. Δ is essential.",
      "related": ["Complex analysis", "Algebraic closure", "Polynomial roots", "Gauss"]
    }
  },
  {
    "concept_name": "Brouwer Fixed Point Theorem",
    "synthesis": {
      "concept_type": "theorem",
      "formal_statement": "Every continuous function from a compact convex set to itself has at least one fixed point",
      "ontological_structures": [
        {"pattern": "⇄", "evidence": "PRIMARY — theorem IS the relation f(x)=x, correspondence between function and point", "primary": true},
        {"pattern": "Δ", "evidence": "Distinguishes fixed from non-fixed points"},
        {"pattern": "⟳", "evidence": "Minimal — static existence, no dynamics"}
      ],
      "dimension_hints": "D=1 (foundational) — topological meta-principle, dimension-independent",
      "attribute_dominant": "⇄",
      "complexity": "foundational (1)",
      "elimination_test": "Remove ⇄ (f(x)=x relation) → theorem vanishes, no fixed point concept. Remove Δ (fixed/non-fixed distinction) → f(x)=x relation still defines the point. Remove ⟳ → already static. ⇄ is essential.",
      "related": ["Topology", "Fixed points", "Game theory", "Economics equilibria"]
    }
  },
  {
    "concept_name": "Bayes' Theorem",
    "synthesis": {
      "concept_type": "theorem",
      "formal_statement": "P(A|B) = P(B|A)P(A)/P(B) — relates conditional probabilities inversely",
      "ontological_structures": [
        {"pattern": "⇄", "evidence": "PRIMARY — theorem IS the bidirectional relation: P(A|B) ↔ P(B|A)", "primary": true},
        {"pattern": "Δ", "evidence": "Distinguishes prior from posterior probability"},
        {"pattern": "⟳", "evidence": "Update process from prior to posterior via evidence"}
      ],
      "dimension_hints": "D=1 (foundational) — pure probability theory, no spatial structure",
      "attribute_dominant": "⇄",
      "complexity": "foundational (1)",
      "elimination_test": "Remove ⇄ (inverse conditional relation) → theorem vanishes, it IS the inversion formula. Remove Δ (prior/posterior distinction) → inversion still holds. Remove ⟳ (update) → static formula still valid. ⇄ is essential.",
      "related": ["Probability theory", "Statistical inference", "Machine learning", "Bayesian statistics"]
    }
  },
  {
    "concept_name": "Peano Axioms",
    "synthesis": {
      "concept_type": "axioms",
      "formal_statement": "Five axioms defining natural numbers: 0 exists, successor function, induction principle",
      "ontological_structures": [
        {"pattern": "Δ", "evidence": "PRIMARY — axioms DEFINE what natural numbers are, establish their distinction", "primary": true},
        {"pattern": "⟳", "evidence": "Successor operation generates sequence recursively"},
        {"pattern": "⇄", "evidence": "Induction relates base case to general case"}
      ],
      "dimension_hints": "D=1 (foundational) — axiomatic foundation of arithmetic",
      "attribute_dominant": "Δ",
      "complexity": "foundational (1)",
      "elimination_test": "Remove Δ (definitional boundary of ℕ) → axioms lose meaning, they ARE the definition. Remove ⟳ (successor) → static set, not constructive ℕ. Remove ⇄ (induction) → weaker system but definition survives. Δ is essential.",
      "related": ["Natural numbers", "Mathematical induction", "Arithmetic", "Foundations of mathematics"]
    }
  },
  {
    "concept_name": "Cauchy-Riemann Equations",
    "synthesis": {
      "concept_type": "equations",
      "formal_statement": "∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x — necessary and sufficient for complex differentiability",
      "ontological_structures": [
        {"pattern": "⇄", "evidence": "PRIMARY — equations ARE the equivalence: holomorphic ↔ satisfies CR", "primary": true},
        {"pattern": "Δ", "evidence": "Distinguishes holomorphic from merely differentiable"},
        {"pattern": "⟳", "evidence": "Minimal — constraint on derivatives, not evolution"}
      ],
      "dimension_hints": "D=2 (linear/planar) — operates on complex plane ℂ ≅ ℝ²",
      "attribute_dominant": "⇄",
      "complexity": "foundational (1)",
      "elimination_test": "Remove ⇄ (holomorphic↔CR equivalence) → equations lose significance. Remove Δ → equivalence still holds. Remove ⟳ → already static. ⇄ is essential.",
      "related": ["Complex analysis", "Holomorphic functions", "Conformal mapping", "Analytic continuation"]
    }
  },
  {
    "concept_name": "Green's Theorem",
    "synthesis": {
      "concept_type": "theorem",
      "formal_statement": "∮_C (L dx + M dy) = ∬_R (∂M/∂x - ∂L/∂y) dA — relates line integral to double integral",
      "ontological_structures": [
        {"pattern": "⇄", "evidence": "PRIMARY — theorem IS the equivalence: boundary integral ↔ region integral", "primary": true},
        {"pattern": "Δ", "evidence": "Distinguishes boundary from interior"},
        {"pattern": "⟳", "evidence": "Minimal — static integral relation"}
      ],
      "dimension_hints": "D=2 (planar) — operates on 2D regions in plane",
      "attribute_dominant": "⇄",
      "complexity": "foundational (1)",
      "elimination_test": "Remove ⇄ (boundary↔interior relation) → theorem vanishes. Remove Δ → relation still holds. Remove ⟳ → already static. ⇄ is essential.",
      "related": ["Vector calculus", "Stokes' theorem", "Fundamental theorem of calculus", "Divergence theorem"]
    }
  },
  {
    "concept_name": "Stokes' Theorem",
    "synthesis": {
      "concept_type": "theorem",
      "formal_statement": "∫_∂Ω ω = ∫_Ω dω — generalized theorem relating boundary to interior via exterior derivative on n-manifolds",
      "ontological_structures": [
        {"pattern": "⇄", "evidence": "PRIMARY — theorem IS the relation: boundary ↔ interior via differential forms", "primary": true},
        {"pattern": "Δ", "evidence": "Distinguishes manifold boundary from interior"},
        {"pattern": "⟳", "evidence": "Minimal — static integral identity"}
      ],
      "dimension_hints": "D=1 (foundational) — meta-principle valid on arbitrary n-dimensional manifolds, dimension-independent",
      "attribute_dominant": "⇄",
      "complexity": "synthetic (3)",
      "elimination_test": "Remove ⇄ (boundary↔interior correspondence) → theorem vanishes. Remove Δ → correspondence still holds. Remove ⟳ → already static. ⇄ is essential. NOTE: X=3 because unifies differential geometry + topology + analysis.",
      "related": ["Differential geometry", "Exterior calculus", "Differential forms", "Green's theorem", "Divergence theorem"]
    }
  },
  {
    "concept_name": "Divergence Theorem",
    "synthesis": {
      "concept_type": "theorem",
      "formal_statement": "∬_∂V F·n dS = ∭_V ∇·F dV — relates surface flux to volume divergence",
      "ontological_structures": [
        {"pattern": "⇄", "evidence": "PRIMARY — theorem IS the relation: surface flux ↔ volume source", "primary": true},
        {"pattern": "Δ", "evidence": "Distinguishes boundary from interior volume"},
        {"pattern": "⟳", "evidence": "Flux represents flow, but theorem itself is static"}
      ],
      "dimension_hints": "D=3 (volumetric) — operates on 3D regions",
      "attribute_dominant": "⇄",
      "complexity": "foundational (1)",
      "elimination_test": "Remove ⇄ (flux↔divergence relation) → theorem vanishes. Remove Δ → relation still holds. Remove ⟳ → already static integral identity. ⇄ is essential.",
      "related": ["Vector calculus", "Gauss's law", "Fluid dynamics", "Electromagnetism"]
    }
  },
  {
    "concept_name": "Fundamental Theorem of Arithmetic",
    "synthesis": {
      "concept_type": "theorem",
      "formal_statement": "Every integer > 1 has unique prime factorization (up to order)",
      "ontological_structures": [
        {"pattern": "Δ", "evidence": "PRIMARY — establishes primes as fundamental distinct building blocks", "primary": true},
        {"pattern": "⇄", "evidence": "Relates composite numbers to prime factors"},
        {"pattern": "⟳", "evidence": "Factorization is a recursive decomposition process"}
      ],
      "dimension_hints": "D=1 (foundational) — number theory, no spatial structure",
      "attribute_dominant": "Δ",
      "complexity": "recursive (2)",
      "elimination_test": "Remove Δ (primes as atomic units) → uniqueness loses meaning. Remove ⇄ → distinction still holds. Remove ⟳ (recursive factorization) → uniqueness remains but process lost. Δ is essential. NOTE: X=2 because factorization is self-similar recursive decomposition.",
      "related": ["Number theory", "Prime numbers", "Unique factorization domains", "Euclidean algorithm"]
    }
  },
  {
    "concept_name": "Tarski's Undefinability Theorem",
    "synthesis": {
      "concept_type": "theorem",
      "formal_statement": "Arithmetic truth cannot be defined within arithmetic itself",
      "ontological_structures": [
        {"pattern": "Δ", "evidence": "PRIMARY — distinguishes truth from provability, definable from undefinable", "primary": true},
        {"pattern": "⇄", "evidence": "Self-referential relation: system attempting to define its own truth"},
        {"pattern": "⟳", "evidence": "Minimal — logical constraint, not temporal"}
      ],
      "dimension_hints": "D=1 (foundational) — meta-mathematical, logical hierarchy",
      "attribute_dominant": "Δ",
      "complexity": "recursive (2)",
      "elimination_test": "Remove Δ (truth/provability distinction) → theorem vanishes. Remove ⇄ (self-reference) → you can still have undefinability via other means. Remove ⟳ → already static. Δ is essential. NOTE: X=2 because meta-level self-reference.",
      "related": ["Mathematical logic", "Gödel incompleteness", "Semantic paradoxes", "Hierarchy of languages"]
    }
  }
]