Hodge Conjecture — Research Program Abstract

Algebraic Cycles, Topological Cohomology Alignment & Relational Invariants

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Mathematics Root · Millennium Frontiers · Hodge Conjecture Research Program

Public Status Boundary. This manuscript presents an authorial exploratory research program investigating the correspondence between algebraic cycles and topological de Rham cohomology classes on projective algebraic varieties. In accordance with the project constitution, this work represents an unverified theoretical proposal (RESEARCH_PROGRAM / UNVERIFIED_HEURISTIC); it does not claim a completed Millennium Prize solution or external peer-reviewed resolution (ACTIVE_P4_SEALS = 0).

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1. Architectural Concept & Research Objective

The Hodge Conjecture asserts that on any smooth non-singular complex projective algebraic variety $X$, every rational Hodge class in $H^{2k}(X,\mathbb{Q})\cap H^{k,k}(X)$ is a rational linear combination of cohomology classes of algebraic subvarieties:

$${\mathrm{Hdg}}^k(X)=H^{2k}(X,\mathbb{Q})\cap H^{k,k}(X)\stackrel{?}{=}{\mathrm{span}}_{\mathbb{Q}}\{{\mathrm{cl}}_X(Z)\mid Z\subseteq X\text{ closed irreducible subvariety, }\mathrm{codim}(Z)=k\}$$

Under the classical Hodge decomposition of de Rham cohomology:

$$H^m(X,\mathbb{C})=\underset{p+q=m}{⨁}{H}^{p,q}(X),\overline{H^{p,q}(X)}=H^{q,p}(X)$$

Within Digital Fabrica Theory, the cycle class correspondence is studied through Boundary-Free Simplicial Cycles & Algebraic Invariants, examining whether Hodge classes can be represented as invariant harmonic ground states of boundary-free 1-cycle and higher $k$-cycle projection operators on discrete relational cellular complexes.

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2. Formal Definitions & Geometric Operators

• $X$: A smooth complex projective manifold of complex dimension $n$.
• $H^{p,q}(X)\cong H^q(X,{\mathrm{\Omega }}_X^p)$: The Dolbeault cohomology group of harmonic $(p,q)$-forms.
• ${\mathrm{cl}}_X:{\mathcal{Z}}^k(X)\to H^{2k}(X,\mathbb{Q})$: The fundamental cycle class homomorphism mapping algebraic cycles to de Rham cohomology.
• ${\mathrm{Hdg}}^k(X)=H^{2k}(X,\mathbb{Q})\cap H^{k,k}(X)$: The finite-dimensional $\mathbb{Q}$-vector space of rational Hodge $(k,k)$-classes.
• ${\partial }_k$: The simplicial and cellular boundary operator satisfying ${\partial }_{k-1}\circ {\partial }_k\equiv 0$.

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3. Epistemic Classification & Lean 4 Formalization Bounds

• Formal Classification: RESEARCH_PROGRAM / UNVERIFIED_HEURISTIC.
• Lean 4 Proof Status: No machine-checked proof of the Hodge Conjecture exists in the repository. Formalization is strictly restricted to foundational discrete simplicial cycle and boundary nullity lemmas:
  • thm_one_cycle_linear_combination (Fabrica.InvariantEngineering): Linear subspace closure of boundary-free 1-cycles.
  • thm_one_cycle_zero_boundary (Fabrica.InvariantEngineering): Preservation of zero boundary $\partial c=0$ on 1-cycle linear combinations.
  • thm_scale_covariance_algebraic (Fabrica.FQFT): Transfinite scale covariance commutation in scale-flow spaces.
• Millennium Prize Demarcation: No claim of prize solution, peer-reviewed acceptance, or Clay Mathematics Institute submission is made (ACTIVE_P4_SEALS = 0).

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4. Canonical Continuations

Direction	Target Resource	Purpose
Frontier Hub	Frontier Mathematics & Proofs Hub →	Survey of exploratory mathematical programs and epistemic boundaries
Invariants Theory	Mathematical Invariants & Admissibility →	Discrete simplicial 1-cycle subspace closure and boundary nullity
Proof Gateway	Lean 4 Formalization Roadmap →	28 machine-verified theorem records and verified lemma trees
Review Gateway	Mathematical Review Gateway →	Interactive verification readiness dashboard and atlas

EXTERNAL REFERENCE