# Hodge–Tate proof dependency adapter

Family 032: The rational Hodge conjecture for CM abelian varieties. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Connect the claimed rational Hodge results for CM abelian varieties with their finite-field Tate and standard-conjecture consequences.

## Applicability and commercial boundary

CM, abelian-variety and coefficient hypotheses matter. A separate withdrawn Weil-class proof must not automatically invalidate this family or be ignored if actually used.

## Initial business decision

Research reference or conditional symbolic/verification engineering. No recurring buyer need, practical implementation or profitability has been demonstrated. Defer a standalone commercial product until a constructive example and a measurable workflow improvement exist. These records also identify unsupported inferences that an evidence platform could flag.

## Next verification action

Inspect the manuscript dependency chain and map any withdrawal to a specific proof and version.

## Evidence scope

The repository catalog statement was individually read. Main-paper construction, effectiveness and selected formal-scope comparison remain queued for this record. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[The rational Hodge conjecture for CM abelian varieties](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf).