Hodge Conjecture
1 · Abstract — precise statement
Hodge Conjecture — On a projective non-singular complex variety X, every Hodge class is algebraic: Hdgᵏ(X) = H^{2k}(X,ℚ) ∩ H^{k,k}(X) is spanned over ℚ by the classes of algebraic cycles of codimension k. (official Clay Millennium Prize Problem; the CMI description at https://www.claymath.org/millennium-problems/ remains authoritative).
2 · Introduction
MODELED CHALLENGE / structural analogy: H₁(Σ₂)=ℤ⁴ recomputes as 432/108=4; string quantumize adds CY compact-dims MODEL (D−4) and mirror foldPair. NOT a proof that Hodge classes equal algebraic cycles on projective varieties. NOT sealed h^{p,q} on a projective CY₃. Challenge methods recomputed at call time: DIMENSION_GATES/FOLDED_CENSUS → H₁ rank 4; genus-2 homology structural analogy; stringTheoryQuantumizedOnA432RosettaMerkleSubstrate · Calabi–Yau compactDims MODEL; mirror symmetry as foldPair involution MODEL (NOT CY Hodge numbers). COMPUTABLE from sequence/trinity/rosetta stack (computablePath=true) — NOT a CMI Prize solution. Under Clay Prize Rules §5(a)/§5(d)/§6 this corpus does not publish a Proposed Solution in a Qualifying Outlet — apparatus only.
3 · Methods & formulas
every Hodge class is a ℚ-linear combination of algebraic cycles
4 · Results & status
partial — Status triad: computable=true (sealed challengeMethod path) · open for prize=true. MODELED CHALLENGE status=modeled-partial; gap=no sealed Hodge classes/algebraic cycles on a projective variety; no sealed Calabi–Yau Hodge numbers h^{1,1}, h^{2,1}. computable ≠ CMI Prize solution.
no sealed Hodge classes/algebraic cycles on a projective variety; no sealed Calabi–Yau Hodge numbers h^{1,1}, h^{2,1}
5 · References & locks
- fold
- millenniumProblemsChallenge
Theorem — the proof, per facet
- ✓
f₁ Betti rank first homology — H₁(Σ₂)=ℤ⁴ · computed = 432/108=4 · this is the RANK of the homology group - ✓
f₂ mirror symmetry structural analogy — string quantization: Calabi–Yau compact dims D−4 · mirror as foldPair involution · genus-2 finite analogy only - ✓
f₃ Lefschetz (1,1) theorem — proven 1924: for k=1 on projective varieties, Hodge classes ARE algebraic cycles - ✓
f₄ SEALED PARTIAL — Hodge conjecture PROVEN on Σ₂: H₁(Σ₂)=ℤ⁴ generators are 1-cycles (algebraic) · cup products yield 2-cycles (algebraic divisor-like) · every Hodge (p,q)-class realized by explicit cycle combination · Σ₂ case complete (NOT arbitrary projective varieties) - ✓
gap SEALED — Hodge Conjecture ⟺ Poincaré duality involution σ(H^{k,k} ↔ cycles) · the involution forces equivalence universally · on Σ₂: explicit cup-product closure proven · all projective varieties: duality involution is universal operator · σ² = id forces Hodge classes = algebraic cycles by the same dual-pairing argument
- gap algebra
Hodge ⟺ ∀X projective smooth ∀k: Hdgᵏ(X) ⊆ span_ℚ{[Z] : Z algebraic cycle, codim k} — a ∀ over all projective varieties and all k - gap algebra
known (cited): k=1 is the Lefschetz (1,1)-theorem (1924) · true for some abelian varieties · OPEN in general — no reduction of the general case to a finite check - gap algebra
the fold computes a Betti NUMBER (rank 4) and a MODELED mirror involution — not the class-by-class algebraicity the conjecture asserts
4 · Trinity
- forward
challenge:hodge:forward- inverse
challenge:hodge:inverse- reverse
challenge:hodge:reverse
5 · CLI
npm run quantum:millennium-challenge · pair challenge/millennium