# Overgroup equation solvability contract

Family 256: Nonsingular systems of equations over arbitrary groups. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Prevent group equation solvers from reporting a solution in the original group when the theorem guarantees a solution only in an overgroup.

## Applicability and commercial boundary

Full row rank of the rational exponent-sum matrix is required; an embedding in a quotient does not give a cheap witness inside the coefficient group.

## 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 one explicit equation and construct the ambient extension separately.

## 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).

[Nonsingular systems of equations over arbitrary groups](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026/nonsingular-systems-over-arbitrary-groups.pdf).