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Positive-basis algebra and witness audit

Decision: Prototype bounded CAS integration; low commercial confidence. Edition: 9 October 2026 Australia/Brisbane. This is a new catalog hypothesis, not an established market or a claim of commercial novelty.

Research finding

Family 169 claims a nonnegative q-polynomial elementary expansion of the chromatic quasisymmetric function of every natural unit interval graph. Its source witness assigns a partition to each eligible nondescent permutation, with an expansion proof for every number of colors. The selected Comparator configuration has no theorem_names, but its returned dependent witness type carries this proof obligation. The formal construction chooses a finite successful table; the paper gives a separate deterministic terminating history/packet procedure. No polynomial running-time guarantee or elementary-unimodality claim is supplied.

Problem and buyer

An algebra research group, CAS maintainer or specialist software team can compare formulas yet lack an easily replayed record of the graph ordering, q-ascent convention, basis conversion and proposed witness labels. A convention error or omitted eligible permutation can produce plausible coefficients. The initial buyer hypothesis is a team repeatedly validating exact algebra computations. Research budgets and existing free tools may make this a small integration service rather than a recurring commercial product.

What the finding could enable

The constructive result could ultimately supply a positive-expansion API, individual permutation witness explorer or independently checked symbolic certificate adapter. The implemented reference is narrower: enumerate a small graph's proper colorings, convert the exact finite polynomial into the elementary basis, reconstruct every coefficient and compare an optional supplied witness. These operations are conventional algebra, not an implementation of the new packet construction. They make that future implementation testable on known examples.

Technical and commercial limits

chromatic_basis_reference.py accepts a zero-indexed extensive nondecreasing h on at most six vertices and uses exactly n colors. It caps two million enumerated/probed/arithmetic work units; exhaustion returns unknown. The reference checks the complete degree-n polynomial in those n variables. It does not produce an all-r formal proof, a uniform source witness, a geometric basis or a polynomial-time algorithm. Sage was consulted through documentation; no Sage runtime comparison was executed.

The formal raw table encoding has C(2n-1,n-1)^(n^n) possible tables for positive n: 10^27 at n=3 and 35^256 at n=4. This is an encoding-space count, not an observed running time, a required exhaustive enumeration or a lower bound on another construction. The paper's finite packet procedure is different. Its abstract Hessenberg representation consequence does not specify a preferred isomorphism, geometric basis or cup-product structure.

Minimal architecture

Graph/order/convention manifest -> exact bounded polynomial reference -> elementary-basis coefficient report -> optional permutation/partition witness checker -> hash-pinned source and finite evidence export. A future packet backend should retain the same small reference and return explicit unavailable/unknown states. A CAS adapter should reuse the CAS arithmetic rather than build a separate algebra system.

Existing alternatives and differentiation

Sage already exposes chromatic quasisymmetric functions and symmetric-function basis conversions. A standalone coefficient calculator is therefore a weak product. The proposed difference is a reusable review record that pins conventions, validates a supplied witness and attaches source scope, finite limits and reproducibility metadata. This difference has not been tested with a maintainer. Sage graph API, Sage symmetric/quasisymmetric tutorial.

Monetization hypothesis

Test an AUD 1,500–5,000 exact-computation review or adapter only if an existing team has recurring work. At an assumed AUD 3,000 fee and 12 hours at AUD 180/hour, AUD 840 remains before sales, support and overhead; 17 hours exceed the fee. These prices and labor assumptions are invented planning values, not quotes. Shared evidence infrastructure must not be counted again as a second customer. A free research integration may be the correct outcome if budgets or repeat usage are absent.

Validation experiment

106 finite checks cover empty/complete graph formulas through six vertices, every eligible h through four vertices, exact full-polynomial reconstruction, permutation mass identities, the paper's path-three expansion and Sage's documented path-four monomial example. The path-three report is q e_(2,1)+(1+q+q^2)e_(3). A supplied four-permutation witness passes; changing one partition fails, and omitting one is incomplete. This is finite technical evidence; it does not validate a source-wide theorem, general recall or demand.

The next usefulness test is whether a maintainer resolves a concrete convention/witness error faster with the record than with existing CAS output. Prepare the comparison locally before any explicitly authorized outreach.

Conditions to reject or defer

Reject a standalone paid calculator if Sage already answers the whole workflow. Defer a production positive-witness backend until the packet construction has a selected executable implementation and practical bounds. Reject recurring pricing if every graph review needs bespoke expert work, exact size caps exclude real workloads or a witness report changes no research decision.

Next concrete action

Replay the supplied path-three witness, inspect the saved result, then specify a Sage interchange record and one deliberately incorrect convention control. Continue reading the middle packet/history proof separately. No buyer, deployment or paid pilot has been engaged.

Pinned source evidence