# Measured-action cost comparison reference

Family 259: A group without fixed price. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Distinguish costs of free probability-measure-preserving actions of a finitely generated group.

## Applicability and commercial boundary

Measured group-theoretic cost is unrelated to monetary cost; infinite actions and asymptotic extensions cannot be sold as business optimization.

## 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

Map action assumptions and a finite teaching illustration.

## 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 group without fixed price](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-group-without-fixed-price-October-5-2026/unequal-costs-rank-100-amalgam.pdf).