# Elementary-basis witness and algebra audit

Family 169, source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. Selected-source addendum, 9 October 2026 Australia/Brisbane. All earlier editions are preserved.

## Current decision

Investigate a small exact CAS/reference integration; demand unvalidated. No buyer interview, paid pilot or profitability evidence has been obtained.

## Reviewed finding and application boundary

The witness-only interface includes an expansion proof for every color count. The formal selected table is noncomputably chosen from a proved finite history criterion. The paper supplies a separate terminating packet procedure and a path-three exact expansion, with no polynomial running-time or unimodality guarantee. Abstract Hessenberg module consequences do not choose a geometric basis or canonical isomorphism.

## Evidence extent

- [01-introduction.tex](../../../../../source/openai-math/preprints/Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026/build/sections/01-introduction.tex): entire 312-line introduction. Natural unit interval graph elementary positivity, individual witness and terminating finite procedure; no time efficiency or elementary unimodality claim.
- [06-witness.tex](../../../../../source/openai-math/preprints/Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026/build/sections/06-witness.tex): lines 1–130 and 560–629. Finite-cone/root packet setup and selected history/algorithm/consequence passage; middle proof not reviewed.

[Source metadata](source.json), [individual definition observations](../../definition-hole-review-queue-2026-10-09-v3.json), [manuscript ledger](../../../../checkpoints/manuscript-ledger-2026-10-09-v4.json). Selected definition/body and challenge comparison found no intended-model mismatch; imported dependency semantics, complete proof, independent kernel and executable correspondence are not verified. Intentional challenge placeholders do not establish source proof failure.

## Next action

Develop finite controls where they answer a concrete modeling question, inspect quantitative construction constants separately, and compare business usefulness with an existing expert workflow. Preserve unknown hypotheses and source claims explicitly.