Generator-invariant local efficacy for phylogenetic invariants

Research-attempt provenance

Definitions and mathematical statement

Shared notation.

Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

A phylogenetic model assigns a probability distribution to observed leaf states on a tree using hidden internal states and stochastic transition matrices along edges. A phylogenetic invariant is a polynomial vanishing on that model. Flattening groups tensor indices into blocks to produce a matrix or lower-order tensor; rank constraints can then give polynomial minors. The exact constraints depend on the mutation model and state cardinality.

Record scope and qualifications.

The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. No tree model, data law, significance measure, or comparison criterion is given. The usefulness question remains a methodological prompt rather than an optimization claim. The supplied catalogue marks this record partially solved; its present research status was not independently checked. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim.

Statement recorded in the supplied source.

Problem 32. "Finding good invariants": Do some of the invariants have more of a statis-tical/biological significance than others? [S2]

Source status.

The supplied catalogue labels this source record partially solved; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness.

Short English statement

The source asks which phylogenetic invariants have stronger statistical or biological usefulness.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20001208 (AIM-COMPUTATION-0046). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 AIM Workshop Problem Lists, source linked by catalogue record AIM-COMPUTATION-0046. https://aimath.org/WWN/compalgstat/compalgstat.pdf.

Preparation and review.

The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 20001208 and code AIM-COMPUTATION-0046 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Literature and context check

The original AIM Computational Algebraic Statistics workshop PDF was consulted. The preserved extract does not specify every model, parameterization or performance criterion needed by its broad question. The following declared model is an actual test of one route, not a replacement target or a claim of novelty for elementary statistical or algebraic facts.

Research attempt: Raw invariant magnitude is not an intrinsic efficacy criterion

If a polynomial $f$ vanishes on a model, then $cf$ for any nonzero constant $c$ and $f^2$ have the same zero set. A ranking by $|f(\widehat p)|$ alone can therefore be changed arbitrarily by rescaling a generator. For a concrete independence model on a binary pair, take $f=p_{00}p_{11}-p_{01}p_{10}$ and the family \[ p_{00}=p_{11}=1/4+\delta,\quad p_{01}=p_{10}=1/4-\delta,\qquad |\delta|<1/4. \] Direct multiplication gives $f=\delta$ and $f^2=\delta^2$. The derivative of $f^2$ at the independent point is zero, whereas that of $f$ along this family is one. Thus applying the same first-order linear noise calibration to both generators is invalid. This does not make $f^2$ intrinsically useless: a test using a correctly calibrated squared statistic can retain the information in $|f|$.

An exact finite-sample alternative in this particular family avoids asymptotic claims. Encode observations as spins $Y_1,Y_2$, and set $Z=Y_1Y_2\in\{-1,1\}$. Its expectation is $4\delta$ and variance $1-16\delta^2$. For $N$ independent observations, $\widehat r=N^{-1}\sum Z_i$ has mean $4\delta$ and variance $(1-16\delta^2)/N$. Under the independence point $\delta=0$, $\Pr(|\widehat r|\ge t)\le1/(Nt^2)$ by Markov on its square. Under $|4\delta|>t$, failure to exceed that threshold has probability at most $1/[N(|4\delta|-t)^2]$ by the centered variance bound.

These calculations associate usefulness with a specific alternative direction, sample law, threshold and noise scale. Multiplying $\widehat r$ and its threshold by the same nonzero magnitude changes neither decision nor these bounds. They do not establish that the determinant invariant is best for every phylogenetic alternative, and they do not attach biological meaning to a polynomial merely because it belongs to a generating set.

Verification and adversarial audit

The determinant, spin expectation and variance were computed exactly. The strict interior range guarantees valid probabilities. Squaring an invariant was distinguished from squaring a statistically calibrated test, avoiding a false universal loss-of-power claim.

Audit corrections.

Generator rescaling and squaring were tested explicitly; neither raw magnitude nor a zero derivative was promoted to a universal biological ranking.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The source supplies no model, biological alternative or loss function. The first gap is defining those targets and comparing appropriately normalized statistics, especially at singular points where first-order calibration vanishes.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://aimath.org/WWN/compalgstat/compalgstat.pdf (consulted 2026-10-11).

Model and contributor disclosure

OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration, repository-format review, and a model-assisted mathematical audit of the written arguments; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.