% Paper 25: Hodge Conjecture — TOE canon, application series. Léon Fernando Vlegels, CC BY 4.0.
% Status: outside the load-bearing spine; conditional translation of a Clay problem into the framework's geometry. Registry items are scope markers.
% Registry: P025_1_c, P025_2_c, P025_3_c (scope) — ledger: Paper 40 and addenda/verify/tbs_registry.json.
% Verifier: verify_P025.py (historical series; includes expected-fail checks that document recorded gaps).

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\newtheorem{theorem}{Theorem}[section]
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\title{The Hodge Conjecture \\[0.5em]
\large Structure, Reductions, and the Geometric Support Principle}

\author{L\'{e}on Fernando Vlegels \\
\small Independent Researcher}
\date{2025-12}

\begin{document}
\maketitle
\tableofcontents
\newpage

\begin{abstract}
We analyze the structure of the Hodge Conjecture, reducing it to a 
single statement: the Geometric Support Principle (GSP), which asserts 
that every primitive rational Hodge class pairs non-trivially with some 
algebraic cycle. We prove that GSP for middle-dimensional classes implies 
the full conjecture via Lefschetz decomposition and Hard Lefschetz duality. 
For non-middle dimensions, we establish GSP using Lefschetz (1,1) and 
duality. For middle dimensions, we show that GSP is equivalent to the 
algebraic classes spanning the primitive Hodge classes, which is the essential 
content of Hodge. We prove GSP (and hence Hodge) for varieties of 
dimension $\leq 3$, and identify the cone-theoretic obstruction that 
must be overcome for a general proof.
\end{abstract}

\paragraph{Scope.} This paper is part of the corpus's application series. It translates the Hodge Conjecture into the framework's geometry and derives conditional results inside that translation. It does not claim a solution at the standard of the Clay Mathematical Institute: the translation dictionary itself is among the registered open items, and the corpus's load-bearing results do not depend on this paper. The registry (Paper 40) records the specific defects.

\section{Introduction}

\begin{conjecture}[Hodge]
On a smooth complex projective variety $X$, every rational $(p,p)$-class 
is a $\mathbb{Q}$-linear combination of algebraic cycle classes:
\begin{equation}
H^{p,p}(X) \cap H^{2p}(X, \mathbb{Q}) = \text{Alg}^p(X)_{\mathbb{Q}}
\end{equation}
\end{conjecture}

We approach this through a systematic reduction to a single principle.

\section{The Geometric Support Principle}

\begin{definition}
A class $\gamma \in H^{p,p}(X, \mathbb{Q})$ has \textbf{geometric support} 
if there exists an algebraic cycle $Z$ of codimension $n-p$ such that:
\begin{equation}
\langle \gamma, [Z] \rangle = \int_X \gamma \wedge [Z] \neq 0
\end{equation}
\end{definition}

\begin{principle*}[Geometric Support Principle (GSP)]
Every non-zero primitive rational Hodge class has geometric support.
\end{principle*}

\begin{theorem}[Main Reduction]
\label{thm:main}
The following are equivalent:
\begin{enumerate}
\item The Hodge Conjecture
\item GSP for all smooth projective varieties
\item GSP for middle-dimensional classes on even-dimensional varieties
\end{enumerate}
\end{theorem}

The proof occupies Sections 4--5.

\section{Background Results}

\subsection{The Lefschetz Theorems}

Let $X$ be smooth projective of dimension $n$, with $L \in H^{1,1}(X,\mathbb{Z})$ 
the class of a hyperplane section.

\begin{theorem}[Lefschetz (1,1)]
$H^{1,1}(X) \cap H^2(X, \mathbb{Z}) = c_1(\text{Pic}(X))$.
\end{theorem}

Every integral $(1,1)$-class is the first Chern class of a line bundle, 
hence algebraic.

\begin{theorem}[Hard Lefschetz]
For $0 \leq k \leq n$, the map $L^k: H^{n-k}(X) \to H^{n+k}(X)$ is an 
isomorphism.
\end{theorem}

\begin{definition}
The \textbf{primitive cohomology}:
\begin{equation}
H^m_{\text{prim}}(X) = \ker(L^{n-m+1}: H^m(X) \to H^{2n-m+2}(X))
\end{equation}
\end{definition}

Every class decomposes uniquely as $\alpha = \sum_{j \geq 0} L^j \alpha_j$ 
with primitive $\alpha_j$.

\subsection{Hodge-Riemann Bilinear Relations}

\begin{theorem}[Hodge-Riemann]
The bilinear form on $H^{p,p}_{\text{prim}}(X, \mathbb{R})$:
\begin{equation}
Q(\alpha, \beta) = (-1)^p \int_X \alpha \wedge \beta \wedge L^{n-2p}
\end{equation}
is positive definite.
\end{theorem}

This is the fundamental positivity constraint from projective geometry.

\section{Reduction to Primitive Classes}

\begin{proposition}
\label{prop:lefschetz_alg}
If $\beta \in \text{Alg}^k(X)_{\mathbb{Q}}$, then $L^j \cdot \beta \in \text{Alg}^{k+j}(X)_{\mathbb{Q}}$.
\end{proposition}

\begin{proof}
$L = [H]$ for hyperplane $H$. Then $L^j \cdot [Z] = [H^j \cap Z]$.
\end{proof}

\begin{theorem}[Reduction to Primitives]
\label{thm:primitive_reduction}
The Hodge Conjecture holds iff every primitive Hodge class is algebraic.
\end{theorem}

\begin{proof}
($\Rightarrow$) Trivial.

($\Leftarrow$) Any Hodge class $\alpha$ decomposes as $\alpha = \sum_j L^j \alpha_j$ 
with primitive Hodge classes $\alpha_j$. If each $\alpha_j$ is algebraic, 
Proposition \ref{prop:lefschetz_alg} implies $\alpha$ is algebraic.
\end{proof}

\section{GSP Implies Hodge}

\subsection{The Orthogonal Complement Argument}

Let $V = H^{p,p}_{\text{prim}}(X, \mathbb{Q})$ and $W = \text{Alg}^p_{\text{prim}}(X)_{\mathbb{Q}}$.

Define the intersection pairing:
\begin{equation}
\langle \cdot, \cdot \rangle: H^{p,p}(X,\mathbb{Q}) \times H^{n-p,n-p}(X,\mathbb{Q}) \to \mathbb{Q}
\end{equation}

\begin{definition}
The \textbf{algebraic orthogonal}:
\begin{equation}
W^\perp = \{\gamma \in V : \langle \gamma, [Z] \rangle = 0 \text{ for all } Z \in \text{Alg}^{n-p}(X)\}
\end{equation}
\end{definition}

\begin{theorem}
\label{thm:gsp_implies_hodge}
If GSP holds, then $W^\perp = 0$, hence $W = V$ (i.e., Hodge holds).
\end{theorem}

\begin{proof}
If $\gamma \in W^\perp$ with $\gamma \neq 0$, then $\gamma$ has no 
geometric support, contradicting GSP.
\end{proof}

\subsection{Non-Middle Dimensions}

For $p \neq n/2$, either $p < n/2$ or $p > n/2$.

\begin{theorem}
\label{thm:non_middle}
GSP holds for $p \neq n/2$.
\end{theorem}

\begin{proof}
\textbf{Case 1}: $p > n/2$ (equivalently, $n - p < p$).

By induction, assume Hodge holds for codimension $n - p$ (base case:
codimension 1 is Lefschetz (1,1)).

Then $H^{n-p,n-p}(X,\mathbb{Q}) = \text{Alg}^{n-p}(X)_{\mathbb{Q}}$.

For $\gamma \in H^{p,p}_{\text{prim}}(X,\mathbb{Q})$, $\gamma \neq 0$:

By Poincar\'e duality, $\gamma$ pairs non-trivially with some class in
$H^{n-p,n-p}(X,\mathbb{Q})$. That class is algebraic by the induction
hypothesis.

Therefore $\gamma$ has geometric support.

\textbf{Case 2}: $p < n/2$ (equivalently, $n - p > p$).

Hard Lefschetz: $L^{n-2p}: H^{p,p}(X) \xrightarrow{\sim} H^{n-p,n-p}(X)$.

For primitive $\gamma$, Hodge-Riemann positivity implies $L^{n-2p}\gamma \neq 0$.

Setting $p' := n-p$, we have $p' > n/2$ (since $p < n/2$), so Case 1 applies to $L^{n-2p}\gamma \in H^{p',p'}(X)$.
By Case 1, $L^{n-2p}\gamma$ has geometric support: it pairs non-trivially with some algebraic cycle $Z$ of codimension $p' = n-p$, hence of complementary codimension $p$.

But:
\begin{equation}
\langle L^{n-2p}\gamma, [Z] \rangle = \langle \gamma, L^{n-2p}[Z] \rangle
\end{equation}

and $L^{n-2p}[Z] = [H^{n-2p} \cap Z]$ is algebraic of codimension $n-p$.

Therefore $\gamma$ pairs non-trivially with an algebraic cycle.
\end{proof}

\subsection{The Middle Dimension}

For $n = 2m$ even and $p = m$:

\begin{theorem}
\label{thm:middle_equiv}
The following are equivalent:
\begin{enumerate}
\item GSP for codimension $m$ on $2m$-dimensional varieties
\item $\text{Alg}^m_{\text{prim}}(X)_{\mathbb{Q}} = H^{m,m}_{\text{prim}}(X, \mathbb{Q})$
\item The Hodge Conjecture
\end{enumerate}
\end{theorem}

\begin{proof}
(1) $\Rightarrow$ (2): Theorem \ref{thm:gsp_implies_hodge}.

(2) $\Rightarrow$ (3): Theorem \ref{thm:primitive_reduction} + Theorem \ref{thm:non_middle}.

(3) $\Rightarrow$ (1): If all Hodge classes are algebraic, every Hodge 
class pairs non-trivially with itself (by Hodge-Riemann), hence with an 
algebraic class.
\end{proof}

\begin{remark}[TBS]\label{rmk:tbs:P025_1_c}
The proof of Theorem~\ref{thm:non_middle} (Case~1, $p > n/2$) is circular. The induction step assumes ``Hodge holds for codimension $n-p$,'' meaning it assumes $H^{n-p,n-p}(X,\mathbb{Q}) = \operatorname{Alg}^{n-p}(X)_{\mathbb{Q}}$ for the specific value $n - p < p$. But this is exactly the Hodge Conjecture at codimension $n-p$, which is an open problem for $n - p \geq 2$. The induction base case (codimension 1, Lefschetz (1,1)) is valid; the induction step for $p > n/2$ and $n - p \geq 2$ uses the conjecture as a hypothesis. The theorem as stated (``GSP holds for $p \neq n/2$'') therefore is not proven for $p > n/2$ with $n - p \geq 2$, i.e., for $n \geq 4$ and $\lceil n/2 \rceil < p \leq n-2$.
\end{remark}

This completes the proof of Theorem \ref{thm:main}.

\section{Proven Cases}

\subsection{Surfaces ($n = 2$)}

\begin{theorem}
The Hodge Conjecture holds for smooth projective surfaces.
\end{theorem}

\begin{proof}
For $n = 2$:
\begin{itemize}
\item $p = 1$: Lefschetz (1,1)
\item $p = 2$: $H^{2,2}(X) \cap H^4(X,\mathbb{Q}) = \mathbb{Q} \cdot [\text{pt}]$
\end{itemize}
\end{proof}

\subsection{Threefolds ($n = 3$)}

\begin{theorem}
The Hodge Conjecture holds for smooth projective threefolds.
\end{theorem}

\begin{proof}
For $n = 3$:
\begin{itemize}
\item $p = 1$: Lefschetz (1,1)
\item $p = 2$: Theorem \ref{thm:non_middle} with $n-p = 1$ (use Lefschetz (1,1))
\item $p = 3$: $\mathbb{Q} \cdot [\text{pt}]$
\end{itemize}
\end{proof}

\subsection{First Open Case: Fourfolds}

For $n = 4$, the first non-trivial case is $p = 2$ (middle dimension).

We need: every $\gamma \in H^{2,2}_{\text{prim}}(X, \mathbb{Q})$ is algebraic.

This is where the conjecture becomes genuinely difficult.

\section{The Cone-Theoretic Obstruction}

\subsection{The Effective Cone}

\begin{definition}
The \textbf{effective cone}:
\begin{equation}
\text{Eff}^p(X) = \left\{ \sum_i a_i [Z_i] : a_i \geq 0, Z_i \text{ subvariety of codim } p \right\} \subset H^{p,p}(X, \mathbb{R})
\end{equation}
\end{definition}

This is a closed convex cone.

\begin{proposition}
$\text{Alg}^p(X)_{\mathbb{R}} = \text{Eff}^p(X) - \text{Eff}^p(X)$ (differences of effective classes).
\end{proposition}

\subsection{The Dual Cone}

\begin{definition}
For the middle dimension ($p = n/2$), the \textbf{dual effective cone}:
\begin{equation}
\text{Eff}^{n/2}(X)^\vee = \{\gamma \in H^{n/2,n/2}(X,\mathbb{R}) : \langle \gamma, \eta \rangle \geq 0 \text{ for all } \eta \in \text{Eff}^{n/2}(X)\}
\end{equation}
\end{definition}

\begin{proposition}
\label{prop:dual_span}
$\text{Eff}^{n/2}(X)^\vee - \text{Eff}^{n/2}(X)^\vee$ spans $\text{Alg}^{n/2}(X)_{\mathbb{R}}^\perp$
under the intersection pairing.
\end{proposition}

\begin{remark}[TBS]\label{rmk:tbs:P025_2_c}
Proposition~\ref{prop:dual_span} is stated without proof. The claimed identity $(\operatorname{Eff}^\vee - \operatorname{Eff}^\vee) = \operatorname{Alg}^\perp$ (under the intersection pairing) is a non-trivial duality statement that would require at minimum: (i) showing $\operatorname{Eff}^\vee$ is full-dimensional in $H^{n/2,n/2}(X,\mathbb{R})$, which requires the effective cone to have non-empty interior; (ii) showing the duality pairing is non-degenerate on the relevant subspaces; and (iii) a finite-dimensionality argument for the span. None of these steps is provided. For the effective cone to have non-empty interior requires the existence of sufficiently many algebraic cycles, which is again a statement in the direction of the Hodge Conjecture.
\end{remark}

\subsection{The Obstruction}

\begin{theorem}[Cone-Theoretic Characterization]
GSP for middle dimension is equivalent to:
\begin{equation}
H^{n/2,n/2}_{\text{prim}}(X, \mathbb{Q}) \cap \text{Int}(\text{Eff}^\vee \cup -\text{Eff}^\vee) \subset \text{Alg}^{n/2}(X)_{\mathbb{Q}}
\end{equation}
where $\text{Int}$ denotes interior with respect to Hodge-Riemann metric.
\end{theorem}

\begin{proof}
A primitive Hodge class $\gamma$ with $Q(\gamma,\gamma) > 0$ lies in the 
interior of $\text{Eff}^\vee \cup -\text{Eff}^\vee$ (one sign or the other, 
depending on whether it pairs positively or negatively with effective classes).

If this interior meets $H^{n/2,n/2}_{\text{prim}}(X,\mathbb{Q})$ only in 
algebraic classes, then GSP holds.

Conversely, if GSP holds, every rational class in this interior pairs 
non-trivially with some effective class, hence lies in the span of 
algebraic classes.
\end{proof}

\subsection{The Missing Ingredient}

The obstruction to proving GSP is:

\begin{center}
\textit{Can a rational Hodge class with positive Hodge-Riemann norm lie 
outside the span of algebraic classes while remaining in the dual 
effective cone?}
\end{center}

Algebraically, this is possible (the cones don't determine the rational structure).

Geometrically, the constraint is that $X$ is projective and algebraic 
classes come from actual subvarieties.

\textbf{The Hodge Conjecture asserts}: The geometric constraints force 
the answer to be NO.

\section{The Three-Layer Interpretation}

\subsection{Layer Structure}

From $\alpha^{-1} = 4\pi^3 + \pi^2 + \pi$:

\begin{center}
\begin{tabular}{ccc}
\toprule
\textbf{Layer} & \textbf{Object} & \textbf{Role} \\
\midrule
Bulk & $H^*(X, \mathbb{C})$ & All cohomology \\
Boundary & $H^{p,p}(X) \cap H^{2p}(X, \mathbb{Q})$ & Hodge classes \\
Edge & $\text{Alg}^p(X)_{\mathbb{Q}}$ & Algebraic classes \\
\bottomrule
\end{tabular}
\end{center}

\subsection{Why Boundary Should Equal Edge}

The Hodge-Riemann positivity provides the key constraint:

For $\gamma \in H^{p,p}_{\text{prim}}(X, \mathbb{Q})$ with $\gamma \neq 0$:
\begin{equation}
Q(\gamma, \gamma) > 0
\end{equation}

This is a \textbf{positive energy condition}. The class $\gamma$ carries 
non-trivial "geometric weight."

In the three-layer framework: Boundary classes have positive energy. 
This energy must be "grounded" in Edge (algebraic support).

A Hodge class with positive energy but no algebraic support would be 
"ungrounded": a cohomological configuration with no geometric realization.

The projective constraint (Chow's theorem: analytic = algebraic) suggests
such ungrounded configurations cannot exist.

\begin{remark}[TBS]\label{rmk:tbs:P025_3_c}
The appeal to Chow's theorem (``analytic = algebraic'') to justify that ``ungrounded'' Hodge classes cannot exist conflates two distinct settings. Chow's theorem states that every closed analytic \emph{subvariety} of a projective space is algebraic, i.e., defined by polynomial equations. However, a rational Hodge class $\gamma \in H^{p,p}(X,\mathbb{Q})$ is not an analytic subvariety; it is a cohomology class. The existence of an algebraic cycle representing $\gamma$ (the content of the Hodge Conjecture) is a far stronger statement than Chow's theorem, and is not implied by it. Chow's theorem operates at the level of subsets of projective space; the Hodge Conjecture operates at the level of cohomology classes. The ``energy grounding'' argument (Hodge-Riemann positivity $\Rightarrow$ algebraic support) is a heuristic intuition, not a mathematical argument.
\end{remark}

\section{Conclusion}

We have established:

\begin{enumerate}
\item \textbf{Reduction}: Hodge $\Leftrightarrow$ GSP $\Leftrightarrow$ 
middle-dimensional algebraic spanning

\item \textbf{Non-middle dimensions}: GSP holds by Lefschetz (1,1) + 
Hard Lefschetz duality

\item \textbf{Low dimensions}: Hodge holds for $\dim X \leq 3$

\item \textbf{Obstruction}: Middle-dimensional GSP requires rational 
Hodge classes with positive norm to lie in the algebraic span, a 
constraint between rational structure and cone geometry

\item \textbf{Interpretation}: Hodge-Riemann positivity (positive energy) 
should force algebraic support (geometric grounding)
\end{enumerate}

The Hodge Conjecture reduces to proving that on a projective variety, 
the effective cone and rational structure interact so that every positive-norm 
rational Hodge class has algebraic support.

This is the geometric content of "Boundary = Edge."

\appendix

\section{Proof of Lefschetz (1,1)}

For completeness, we sketch the proof.

The exponential sequence:
\begin{equation}
0 \to \mathbb{Z} \to \mathcal{O}_X \xrightarrow{\exp} \mathcal{O}_X^* \to 0
\end{equation}

induces:
\begin{equation}
H^1(X, \mathcal{O}_X^*) \xrightarrow{c_1} H^2(X, \mathbb{Z}) \to H^2(X, \mathcal{O}_X)
\end{equation}

The image of $c_1$ consists of classes mapping to 0 in $H^2(X, \mathcal{O}_X) = H^{0,2}(X)$.

These are exactly $H^2(X, \mathbb{Z}) \cap H^{1,1}(X)$.

Since $H^1(X, \mathcal{O}_X^*) = \text{Pic}(X)$ (line bundles), every integral 
$(1,1)$-class is $c_1(L)$ for some line bundle, hence algebraic.

\section{The Standard Conjectures}

Grothendieck's Standard Conjectures would imply Hodge.

\textbf{Conjecture B} (Lefschetz type): The Lefschetz isomorphism is induced 
by an algebraic correspondence.

\textbf{Conjecture C} (K\"unneth type): The K\"unneth components of the diagonal 
are algebraic.

\textbf{Conjecture D} (Hodge type): Numerical and homological equivalence 
coincide.

Deligne proved B in characteristic 0. The full standard conjectures remain open.

\section{Known Special Cases}

\begin{center}
\begin{tabular}{cc}
\toprule
\textbf{Variety Type} & \textbf{Status} \\
\midrule
Surfaces & Proven \\
Threefolds & Proven \\
Abelian varieties (dim $\leq 4$) & Proven \\
Products of curves & Proven \\
Fermat hypersurfaces & Proven (many cases) \\
General type, $\dim \geq 4$ & Open \\
\bottomrule
\end{tabular}
\end{center}


\begin{thebibliography}{99}

\bibitem{Paper1}
L.~F.~Vlegels,
\textit{The Perfect Stable Sphere: Deriving $\alpha^{-1}$ from $(B^4, S^3)$ Geometry},
This volume (2025).

\bibitem{Paper3}
L.~F.~Vlegels,
\textit{Mathematical Foundations of Geometric Fundamental Physics},
This volume (2025).

\bibitem{Paper4}
L.~F.~Vlegels,
\textit{The Three-Layer Ontology of Physical Reality},
This volume (2025).

\bibitem{Paper11}
L.~F.~Vlegels,
\textit{Geometric First Principles: A Unified Framework for Fundamental Constants},
This volume (2025).

\end{thebibliography}


\section*{Status (2026-06)}
Outside the corpus's load-bearing spine: a conditional translation of the Hodge Conjecture into the framework's geometry, not a resolution. The induction over non-middle dimensions is circular and the application of Chow's theorem does not hold as written (registry C-I-09, C-I-10, C-II-10); algebraic representability of rational Hodge classes --- the full content of the conjecture --- is not established. The framework offers a geometric organising picture, not a proof. The corpus claims no resolution of the Clay problem (disposition: registered-not-claimed, Addendum~400).

\end{document}
