The stochastic image of a binary phylogenetic tripod

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. A stochastic parameterization restricts parameters to nonnegative normalized probabilities; a complex parameterization allows complex coordinates and is often studied through its Zariski closure.

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. The model and parameter map are absent, and several phylogenetic readings are possible. No binary tripod or particular mutation model is substituted for this ambiguous source reference. The supplied catalogue marks this record partially solved; its present research status was not independently checked. Its supplied reconstruction is labelled unverified. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim. The supplied catalogue marks this formulation as an unverified reconstruction; missing assumptions are not certified by the source link.

Statement recorded in the supplied source.

Problem 31. Determine the image of the stochastic parameterization (as opposed to the complex parameterization). [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 for the image of an unnamed stochastic parameterization.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20001207 (AIM-COMPUTATION-0045). 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-0045. 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 20001207 and code AIM-COMPUTATION-0045 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: A positive observed distribution can have only a nonstochastic complex fit

Choose, explicitly as an auxiliary interpretation, a binary hidden root $H$ with three conditionally independent observed spins $Y_i\in\{-1,1\}$. For a stochastic root probability $\pi$, let $m_{ih}=E(Y_i\mid H=h)$ and $d_i=m_{i1}-m_{i0}$. Conditional independence gives \[ \operatorname{Cov}(Y_i,Y_j)=\pi(1-\pi)d_id_j. \] This follows by expanding $E(Y_iY_j)$ and subtracting the product of means. Consequently the product of the three pairwise covariances is nonnegative, including root or edge degeneracies.

Now define an observed distribution by \[ p(y_1,y_2,y_3)=\tfrac18\left(1+\tfrac14y_1y_2+ \tfrac14y_1y_3-\tfrac14y_2y_3\right). \] It is strictly positive: the bracket is either $5/4$ or $1/4$. Summing spin monomials over the cube shows total mass one, means zero, and covariances $1/4,1/4,-1/4$. Their product is negative, so this observed distribution is not in the chosen stochastic tripod image.

It is nevertheless in the complex parameterized image, not just its closure. Take uniform hidden root and conditional spin means $t_i$ in one root state and $-t_i$ in the other, where \[ t_1=i/2,\qquad t_2=t_3=-i/2. \] Use conditional entries $(1\pm t_i)/2$, which sum to one but are complex. Averaging the two product distributions cancels the odd spin monomials and leaves pair coefficients $t_1t_2=1/4$, $t_1t_3=1/4$, $t_2t_3=-1/4$. This is exactly the displayed real positive distribution.

Thus adding positivity of observed cell probabilities to algebraic membership does not force stochastic parameters. The attempted elimination-only route needs parameter-side inequalities or an equivalent semialgebraic criterion. The example is for the declared general binary latent-star model; the source does not specify that model, so it is not a full characterization of its unstated stochastic map.

Verification and adversarial audit

All eight cell values are $5/32$ or $1/32$, with six of the first and two of the second, so positivity and normalization are exact. Complex conditional rows normalize correctly. The covariance-product obstruction includes $\pi=0,1$ and zero contrasts.

Audit corrections.

Strictly positive observations were distinguished from nonnegative parameters; the complex representation was verified without conflating image and closure.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The original parameter map, tree and mutation model remain unidentified. Even for the auxiliary tripod, only a necessary inequality and an exact separating example are supplied, not complete image membership or its fibers.

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.