# Symmetry-preserving query language expressiveness audit

Family 243: Separating choiceless counting from polynomial time and witnessed choice. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Database and logic-language researchers can compare choiceless counting with witnessed-choice features on unordered structures.

## Applicability and commercial boundary

The separation is for the specified full formalism; a polynomial-time F3 linear system can still be solved using conventional algorithms. This is not a general data query performance impossibility.

## 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

Map the finite-structure language features and construct a small illustrative linear system.

## 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).

[Choiceless polynomial time with counting does not capture polynomial time](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Choiceless-polynomial-time-with-counting-does-not-capture-polynomial-time-September-23-2026/paper.pdf).