# AI provenance: Dense Algebraic Loci of Hodge Classes

Authors: Deep Bhattacharjee and Ushashi Bhattacharya. No AI system is an author or contributor.

## How the work was produced

Deep Bhattacharjee carried out the constructions on (F2) and (F3'), the search for a closure of the Hodge conjecture, the Lean, Julia, C, C++, Python and shell verifications with AI assistance from Claude (Anthropic) in Claude Code. Claude drafted most of the text and figures under his direction. Both authors approved the final version.

## How the results are checked

Every hand proof was read against the cited literature. Every finite quantity is recomputed by programs in at least two languages; the combinatorial and arithmetic ones are proved in Lean 4 (62 theorems, 52 axiom-free, 2 on propext and Quot.sound, 8 on propext). Machine checks found and corrected four slips before release (a Hasse invariant at 2, a Picard number of a twist family, the dimension of a dihedral Hecke algebra, the sign of the square of the Fricke isogeny). No proved statement rests on floating point; the enumerations and linear-programming bounds that do are labelled as computations.

## What is not claimed

The Hodge conjecture is not proved. It is shown equivalent to (F2) and (F3'), both open; each construction stops at a step the paper names.

## Labels given on the Hexagon form

- Text of the contribution: a mix of human-written and AI-generated (to be confirmed by the authors).
- Human understanding at submission: some parts, but not all, checked and understood by a human (to be confirmed by the authors).

## Where everything is

Release v2.7.0: https://github.com/creelie/fermat-hodge-/releases/tag/v2.7.0.