# Decidable word-problem embedding contract

Family 250: Boone–Higman embeddings with higher finiteness. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Record the claimed equivalence between decidable word problem and embedding into a finitely presented simple group, with finiteness-property companions.

## Applicability and commercial boundary

Embedding existence does not transfer an efficient word-problem algorithm or bound presentation growth.

## 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

Read constructive embedding costs and exact source/target finiteness assumptions.

## 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).

[Finite algebraic envelopes and the Boone–Higman conjecture](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026/paper.pdf).