[
  {
    "concept": "P vs NP",
    "coordinate": "Σ112₊",
    "category": "node",
    "status": "OPEN",
    "confidence": 0.88,
    "D": 1, "A": 1, "X": 2, "P": "+",
    "archetype": "segmentation",
    "attribute_dominant": "Δ",
    "millennium_status": "OPEN",
    "note": "Already mapped in Batch 3. Included for completeness."
  },
  {
    "concept": "Riemann Hypothesis",
    "coordinate": "Σ212₋",
    "category": "node",
    "status": "OPEN",
    "confidence": 0.85,
    "D": 2, "A": 1, "X": 2, "P": "-",
    "archetype": "aggregation",
    "attribute_dominant": "Δ",
    "elimination_test": "Remove Δ → no distinction between zeros and non-zeros, between critical line and rest of plane → concept destroyed",
    "dimension_rationale": "Zeros form pattern in complex plane (literal 2D). Critical line Re(s)=1/2 is planar structure",
    "complexity_rationale": "X=2: ζ encodes info about primes, which define ζ — numbers speak about themselves through the zeta function",
    "polarity_rationale": "P=-: zeros AGGREGATE on critical line (contraction/concentration). If false → fragmentation (P=+)",
    "millennium_status": "OPEN",
    "tensions": "P± borderline — aggregation vs fragmentation depends on truth of conjecture. NEW CELL.",
    "frontier_note": "Humanity cannot yet prove whether zeros aggregate on critical line"
  },
  {
    "concept": "Hodge Conjecture",
    "coordinate": "Σ123₊",
    "category": "node",
    "status": "OPEN",
    "confidence": 0.86,
    "D": 1, "A": 2, "X": 3, "P": "+",
    "archetype": "thread",
    "attribute_dominant": "⇄",
    "elimination_test": "Remove ⇄ → correspondence cohomology↔subvarieties destroyed → concept gone",
    "dimension_rationale": "Foundational identification/correspondence between structures (D=1)",
    "complexity_rationale": "X=3: genuine synthesis of algebraic topology + algebraic geometry + complex analysis",
    "millennium_status": "OPEN",
    "tensions": "Same cell as Stokes (solved) and BSD (open) → Σ₁₂₃₊ = cell of great mathematical bridges",
    "frontier_note": "Synthetic threading between mathematical frameworks is partially understood (Stokes) but not complete"
  },
  {
    "concept": "Yang-Mills Existence and Mass Gap",
    "coordinate": "Σ411₊",
    "category": "node",
    "status": "OPEN",
    "confidence": 0.82,
    "D": 4, "A": 1, "X": 1, "P": "+",
    "archetype": "instant",
    "attribute_dominant": "Δ",
    "elimination_test": "Remove Δ → no distinction vacuum/first excitation → mass gap vanishes → concept destroyed",
    "dimension_rationale": "QFT is inherently temporal; excitation requires dynamical content",
    "complexity_rationale": "X=1: defines existence and spectral property of gauge theory. QFT components derive from same framework → not X=3",
    "millennium_status": "OPEN",
    "tensions": "Archetype 'instant' resonates weakly with mass gap. Gap is more barrier-like (T7-adjacent). Lowest confidence in batch.",
    "frontier_note": "Vacuum/excitation distinction not rigorously proven — lacuna in temporal distinction zone"
  },
  {
    "concept": "Navier-Stokes Existence and Smoothness",
    "coordinate": "Σ412₊",
    "category": "node",
    "status": "OPEN",
    "confidence": 0.84,
    "D": 4, "A": 1, "X": 2, "P": "+",
    "archetype": "scansion",
    "attribute_dominant": "Δ",
    "elimination_test": "Remove Δ → no smooth/singular distinction → problem vanishes. Oscillation Δ/⟳ resolved: essence=Δ (distinction), mechanism=⟳ (temporal evolution)",
    "dimension_rationale": "Question is explicitly temporal: do solutions survive for ALL TIME or blow up in finite time?",
    "complexity_rationale": "X=2: asks whether equations 'behave well' — meta-question about the nature of their own solutions",
    "millennium_status": "OPEN",
    "tensions": "Δ/⟳ oscillation resolved via §3.2 Step 2, margin: tight. NEW CELL.",
    "frontier_note": "Whether temporal evolution preserves smooth/singular distinction — lacuna in temporal recursive distinction"
  },
  {
    "concept": "Birch and Swinnerton-Dyer Conjecture",
    "coordinate": "Σ123₊",
    "category": "node",
    "status": "OPEN",
    "confidence": 0.87,
    "D": 1, "A": 2, "X": 3, "P": "+",
    "archetype": "thread",
    "attribute_dominant": "⇄",
    "elimination_test": "Remove ⇄ → correspondence rank↔order_of_zero destroyed → concept gone",
    "dimension_rationale": "Foundational identification between arithmetic rank and analytic order (D=1)",
    "complexity_rationale": "X=3: genuine synthesis of algebraic geometry + complex analysis + number theory",
    "millennium_status": "OPEN",
    "tensions": "Same cell as Hodge → two Millennium Problems clustering in synthetic-thread zone. Strong cluster validation.",
    "frontier_note": "Arithmetic↔analytic bridge remains open. Same frontier cell as Hodge."
  },
  {
    "concept": "Poincaré Conjecture",
    "coordinate": "Σ322₋",
    "category": "node",
    "status": "SOLVED",
    "confidence": 0.87,
    "D": 3, "A": 2, "X": 2, "P": "-",
    "archetype": "foundation",
    "attribute_dominant": "⇄",
    "elimination_test": "Remove ⇄ → equivalence simply-connected↔S³ gone → concept destroyed. Oscillation Δ/⇄ resolved: presupposes Δ, asserts ⇄",
    "dimension_rationale": "Literally about 3-manifolds — volumetric topological content (D=3)",
    "complexity_rationale": "X=2: topology of topology — classifies the space of possible spaces (meta-level)",
    "polarity_rationale": "P=-: REDUCES possibilities (simply connected 3-manifold MUST be S³, cannot be anything else)",
    "millennium_status": "SOLVED (Perelman 2003)",
    "tensions": "D=3 for theorem content, but proof (Ricci flow) is D=4. Mapped theorem, not proof. NEW CELL.",
    "frontier_note": "CLOSED frontier. Only solved Millennium Problem. Only P=- in batch."
  }
]