# Finite algebra representation solver completeness contract

Family 206: Finite lattice representation and undecidability. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Symbolic solver teams can record undecidability of whether an input finite lattice admits the specified finite-algebra or subgroup-interval representation.

## Applicability and commercial boundary

Finite input size alone does not guarantee a universal terminating decider when potential witnesses are unbounded. Restricted classes and particular verified examples remain useful.

## Initial business decision

Research reference or conditional symbolic/verification engineering. No practical implementation, recurring buyer need or profitability has been demonstrated; a weak business bridge is explicitly deferred.

## Next verification action

Add representation class, witness budget and unknown-result semantics to an evidence adapter.

## Evidence scope

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

[Finite congruence lattices: characterization and undecidability](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026/paper.pdf).