Ω-Closure Equivalence Conjecture

---What is the conjecture---

-Ω-Closure Conjecture (Canonical Fixed-Point Characterization of Algebraic Cycles)

-Let be a smooth projective variety over a number field, and let denote the space of rational Hodge classes. There exists a canonical -linear endomorphism.

\mathcal{H}_\Omega : H^{k,k}(X,\mathbb{Q}) \to H^{k,k}(X,\mathbb{Q})

-Equivalently, a rational Hodge class is algebraic if and only if it is fixed under the Ω-transport flow.

-This conjecture provides a structural fixed-point formulation of algebraicity that is independent of the classical Chow–Künneth projectors and is motivated by computational and variational evidence.

----References:

-D. Manning, Structural Resolution of the Hodge Conjecture via Ω-Closure (preprint, 2025).
https://github.com/Nospy-ai/Omega-Closure-Framework. (Current repository)

--Prerequisites needed---

-To formalize this conjecture, the following components would be required:

-1. Formalization of rational Hodge structures and the -component of cohomology for smooth projective varieties.
-2. A formal definition of an energy-minimizing endomorphism on a rational cohomology space (Ω-transport), abstracted as a self-adjoint linear operator.
-3. Fixed-point and kernel characterizations of self-adjoint operators on finite-dimensional inner-product spaces.
-4. Basic infrastructure for algebraic cycle classes and their realization in cohomology.
No advanced motivic machinery is required for the conjectural statement itself; it is expressible at the level of linear operators and fixed-point subspaces.

---AMS categories----
-ams-14C30 (Hodge theory, variations of Hodge structure)
-ams-14C25 (Algebraic cycles)
-ams-11G35 (Arithmetic aspects of varieties over global fields)

---Choose either option---

[Formalizations sketch.pdf](https://github.com/user-attachments/files/24147857/Formalizations.sketch.pdf)

[Proposed Content to Encode in Lean_ Operators and Definitions.pdf](https://github.com/user-attachments/files/24147887/Proposed.Content.to.Encode.in.Lean_.Operators.and.Definitions.pdf)

[Core Equivalence Claim.pdf](https://github.com/user-attachments/files/24147893/Core.Equivalence.Claim.pdf)

[Formalization of Theorems and Hodge Corollary.pdf](https://github.com/user-attachments/files/24147901/Formalization.of.Theorems.and.Hodge.Corollary.pdf)

[Omega-Closure-Framework-and-the-Hodge-Conjecture.84.pdf](https://github.com/user-attachments/files/24147904/Omega-Closure-Framework-and-the-Hodge-Conjecture.84.pdf)

[ ] I plan on adding this conjecture to the repository

[x] This issue is up for grabs: I would like to see this conjecture added by somebody else

Here is a full link to my current repository for further review.

https://github.com/Nospy-ai/Omega-Closure-Framework

Start by reviewing the attached formalization PDFs, especially “Proposed Content to Encode in Lean: Operators and Definitions” and “Core Equivalence Claim,” then compare them with the current Omega-Closure-Framework repository. No target files or tests are named; done would require adding a Lean formalization of the conjecture and its stated operator, fixed-point, and cycle-class components.

Written by the indexing model from the issue text.