[
  {
    "concept_name": "Turing Completeness",
    "synthesis": {
      "concept_type": "property",
      "formal_statement": "A computational system is Turing-complete if it can simulate any Turing machine",
      "ontological_structures": [
        {"pattern": "Δ", "evidence": "PRIMARY — property IS the distinction: systems that can vs cannot compute all computable functions", "primary": true},
        {"pattern": "⇄", "evidence": "Relates different computational models via equivalence"},
        {"pattern": "⟳", "evidence": "Minimal — describes capability, not execution process"}
      ],
      "dimension_hints": "D=1 (foundational) — meta-property of computational systems",
      "attribute_dominant": "Δ",
      "complexity": "foundational (1)",
      "elimination_test": "Remove Δ (the classification itself) → property vanishes, it IS about distinguishing universal from non-universal systems. Remove ⇄ (equivalence between models) → distinction remains. Remove ⟳ → already static property. Δ is essential.",
      "related": ["Computability theory", "Lambda calculus", "Church-Turing thesis", "Universal computation"]
    }
  },
  {
    "concept_name": "Shannon's Information Entropy",
    "synthesis": {
      "concept_type": "measure",
      "formal_statement": "H(X) = -Σ p(x) log p(x) — quantifies expected information content of a message",
      "ontological_structures": [
        {"pattern": "⇄", "evidence": "PRIMARY — entropy IS the relation: probability distribution ↔ information content", "primary": true},
        {"pattern": "Δ", "evidence": "Distinguishes high-entropy (uncertain) from low-entropy (predictable) sources"},
        {"pattern": "⟳", "evidence": "Minimal — static measure, not transmission process"}
      ],
      "dimension_hints": "D=1 (foundational) — meta-measure of information",
      "attribute_dominant": "⇄",
      "complexity": "foundational (1)",
      "elimination_test": "Remove ⇄ (probability↔information relation) → entropy vanishes, it IS the quantification formula. Remove Δ (high vs low entropy) → consequence of measure. Remove ⟳ → already static. ⇄ is essential.",
      "related": ["Information theory", "Compression", "Channel capacity", "Thermodynamic entropy"]
    }
  },
  {
    "concept_name": "Boolean Algebra",
    "synthesis": {
      "concept_type": "algebraic structure",
      "formal_statement": "Algebra with operations AND, OR, NOT satisfying laws: idempotence, commutativity, distributivity, De Morgan",
      "ontological_structures": [
        {"pattern": "⇄", "evidence": "PRIMARY — algebra IS the relational structure: operations ↔ laws ↔ truth values", "primary": true},
        {"pattern": "Δ", "evidence": "Distinguishes true from false (binary logic)"},
        {"pattern": "⟳", "evidence": "Minimal — static algebraic structure"}
      ],
      "dimension_hints": "D=1 (foundational) — basis of digital logic",
      "attribute_dominant": "⇄",
      "complexity": "foundational (1)",
      "elimination_test": "Remove ⇄ (operations and their laws) → algebra vanishes, it IS the relational structure. Remove Δ (true/false distinction) → presupposed by operations. Remove ⟳ → already static. ⇄ is essential.",
      "related": ["Digital logic", "Circuit design", "Propositional logic", "Set theory"]
    }
  },
  {
    "concept_name": "Gödel's Incompleteness Theorems",
    "synthesis": {
      "concept_type": "theorem",
      "formal_statement": "Any consistent formal system sufficient for arithmetic contains true statements that cannot be proven within the system",
      "ontological_structures": [
        {"pattern": "Δ", "evidence": "PRIMARY — theorems ARE about the distinction: provable vs true, complete vs incomplete", "primary": true},
        {"pattern": "⇄", "evidence": "Relates consistency, completeness, and expressiveness"},
        {"pattern": "⟳", "evidence": "Self-reference creates recursive structure"}
      ],
      "dimension_hints": "D=1 (foundational) — meta-mathematical limit",
      "attribute_dominant": "Δ",
      "complexity": "recursive (2)",
      "elimination_test": "Remove Δ (provable/true distinction) → theorems vanish, they ARE about this gap. Remove ⇄ (relations between properties) → important but distinction is essence. Remove ⟳ (self-reference) → mechanism of proof, not content. Δ is essential.",
      "related": ["Mathematical logic", "Formal systems", "Metamathematics", "Undecidability"]
    }
  },
  {
    "concept_name": "CAP Theorem",
    "synthesis": {
      "concept_type": "theorem",
      "formal_statement": "Distributed systems cannot simultaneously guarantee Consistency, Availability, and Partition tolerance — pick at most 2",
      "ontological_structures": [
        {"pattern": "Δ", "evidence": "PRIMARY — theorem IS the impossibility distinction: cannot achieve all three properties", "primary": true},
        {"pattern": "⇄", "evidence": "Relates three system properties in trade-off"},
        {"pattern": "⟳", "evidence": "Minimal — describes system classification, not runtime dynamics"}
      ],
      "dimension_hints": "D=1 (foundational) — meta-classification of distributed system trade-offs, not spatial",
      "attribute_dominant": "Δ",
      "complexity": "foundational (1)",
      "elimination_test": "Remove Δ (impossibility of CAP simultaneously) → theorem vanishes, it IS about the tripartition classification. Remove ⇄ (trade-off between properties) → consequence of impossibility. Remove ⟳ → already static classification. Δ is essential.",
      "oscillation_notes": "Initially mapped as constraint T7+P8 (structural limit), but CAP is domain-specific (distributed systems only), not universal like Heisenberg/Gödel. Single-machine databases have CAP 'for free'. Corrected to node Σ₁₁₁₊, clusters with Turing Completeness as impossibility/classification result in CS.",
      "related": ["Distributed systems", "Database theory", "Network partitions", "System design"]
    }
  },
  {
    "concept_name": "Big-O Notation",
    "synthesis": {
      "concept_type": "notation",
      "formal_statement": "f(n) = O(g(n)) — f grows at most as fast as g asymptotically",
      "ontological_structures": [
        {"pattern": "⇄", "evidence": "PRIMARY — notation IS the relation: algorithm complexity ↔ asymptotic bound", "primary": true},
        {"pattern": "Δ", "evidence": "Distinguishes complexity classes (O(1), O(n), O(n²), etc.)"},
        {"pattern": "⟳", "evidence": "Minimal — describes growth, not execution"}
      ],
      "dimension_hints": "D=1 (foundational) — meta-measure of algorithmic complexity",
      "attribute_dominant": "⇄",
      "complexity": "foundational (1)",
      "elimination_test": "Remove ⇄ (asymptotic comparison) → notation vanishes, it IS the relational statement f≤cg. Remove Δ (complexity classes) → consequence of comparison. Remove ⟳ → already static notation. ⇄ is essential.",
      "oscillation_notes": "Notation vs theorem: Big-O is a formal convention (like Leibniz notation for derivatives), not an ontological assertion. Mapping is correct but has different weight — it's a tool, not a claim about the world.",
      "related": ["Algorithm analysis", "Complexity theory", "Asymptotic analysis", "Performance"]
    }
  },
  {
    "concept_name": "Hash Function Collision Resistance",
    "synthesis": {
      "concept_type": "property",
      "formal_statement": "Computationally infeasible to find two distinct inputs x ≠ y such that h(x) = h(y)",
      "ontological_structures": [
        {"pattern": "Δ", "evidence": "PRIMARY — property IS the distinction between secure (collision-resistant) and insecure hash functions", "primary": true},
        {"pattern": "⇄", "evidence": "Relates input space to output space via compression"},
        {"pattern": "⟳", "evidence": "Minimal — property of function, not hashing process"}
      ],
      "dimension_hints": "D=1 (foundational) — meta-property of security classification, not spatial mapping",
      "attribute_dominant": "Δ",
      "complexity": "foundational (1)",
      "elimination_test": "Remove Δ (secure/insecure distinction) → property vanishes, it IS about classification of hash functions. Remove ⇄ (input↔output mapping) → computational mechanism, not ontological content. Remove ⟳ → already static property. Δ is essential.",
      "oscillation_notes": "Originally D=2 (input→output space mapping), but ontological content is not spatial — it's a security property classification. Corrected to D=1 foundational. Clusters with Turing Completeness (computational boundary distinctions).",
      "related": ["Cryptography", "Hash functions", "Data structures", "Digital signatures"]
    }
  },
  {
    "concept_name": "Lambda Calculus",
    "synthesis": {
      "concept_type": "formal system",
      "formal_statement": "Formal system for expressing computation via function abstraction (λx.M) and application (M N)",
      "ontological_structures": [
        {"pattern": "⇄", "evidence": "PRIMARY — calculus IS the relational system: abstraction ↔ application ↔ reduction rules", "primary": true},
        {"pattern": "⟳", "evidence": "β-reduction is a rewriting process"},
        {"pattern": "Δ", "evidence": "Distinguishes bound from free variables"}
      ],
      "dimension_hints": "D=1 (foundational) — basis of functional programming",
      "attribute_dominant": "⇄",
      "complexity": "foundational (1)",
      "elimination_test": "Remove ⇄ (abstraction/application relation) → calculus vanishes, it IS the computational structure. Remove ⟳ (reduction process) → mechanism of evaluation, but structure remains. Remove Δ → variable distinction presupposed. ⇄ is essential.",
      "related": ["Functional programming", "Type theory", "Church-Turing thesis", "Programming languages"]
    }
  },
  {
    "concept_name": "Dijkstra's Algorithm",
    "synthesis": {
      "concept_type": "algorithm",
      "formal_statement": "Finds shortest path in weighted graph by iteratively selecting minimum-distance unvisited node",
      "ontological_structures": [
        {"pattern": "⟳", "evidence": "PRIMARY — algorithm IS the iterative process: select → update → repeat until complete", "primary": true},
        {"pattern": "⇄", "evidence": "Relates nodes, edges, distances in optimization"},
        {"pattern": "Δ", "evidence": "Distinguishes visited from unvisited nodes"}
      ],
      "dimension_hints": "D=2 (planar) — operates on graph structure (network topology)",
      "attribute_dominant": "⟳",
      "complexity": "foundational (1)",
      "elimination_test": "Remove ⟳ (iterative selection process) → algorithm vanishes, it IS the step-by-step procedure. Remove ⇄ (distance relations) → data but process remains essence. Remove Δ (visited/unvisited) → mechanism. ⟳ is essential.",
      "related": ["Graph theory", "Shortest path", "Optimization", "Network routing"]
    }
  }
]