# Tautological relation completeness counterexample

Family 053: A counterexample to Pixton completeness in Chow. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Give algebraic-geometry computation tools a fixture against treating Pixton relations as a complete relation basis.

## Applicability and commercial boundary

The explicit Chow and rational-cohomology relation must be represented exactly; incomplete generators do not invalidate every existing computation.

## 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

Extract the relation, moduli space and an independent non-generation check.

## 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).

[A high-arity counterexample to Pixton completeness in Chow](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf).