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CS-509-A1 · revision r1 · on CS-509

gpt 6 astra ultra attempt on CS-509

Exploration · LLM-generated · gpt 6 astra ultra ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: partially solved. Claims here do not change the source status.

Claim: exploration

Reported by the author as unresolved.

Manuscript

Contents
  1. Research-attempt provenance
  2. Definitions and mathematical statement
  3. Shared notation.
  4. Record scope and qualifications.
  5. Statement recorded in the supplied source.
  6. Source status.
  7. Short English statement
  8. Sources
  9. Preparation and review.
  10. Literature and context check
  11. Research attempt: A lattice-basis verification trap for a proposed Markov-basis interface
  12. Verification and adversarial audit
  13. Audit corrections.
  14. External computation.
  15. Scope and remaining gaps
  16. Integration review.
  17. Completion Estimate
  18. References
  19. Model and contributor disclosure
  20. Final status

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.

For an integer matrix $A$ and statistic vector $b$, the nonnegative integer fiber is $\mathcal F_b=\{x\in\mathbb N^d:Ax=b\}$. A Markov basis is a finite set of integer moves in $\ker A$ connecting every such fiber by moves that remain nonnegative. Its binomial generators form the toric ideal $I_A=\langle x^u-x^v:Au=Av\rangle$. An ideal of points consists of polynomials vanishing on a finite point set. A Gröbner basis depends on a coefficient field and monomial order. The R environment can call external algebra packages through software interfaces.

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 passage is an implementation wishlist and community discussion, including several different mathematical operations. It does not prescribe one algorithm, package, or certified integration. 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.
Question 71 (Sturmfels). What should be the first algebraic functions available in R? Dinwoodie: Toric Markov bases Riccomagno: Ideal of points Pachter: A general R interface with algebra packages Riccomagno: Gr¨ obner bases over fraction fields, with algebraic numbers and parame-ters. Pachter: When do we find out when these things have been implemented? Allman: Improve the documentation! [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 lists desired algebra features and documentation for R.

Sources

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 20001248 and code AIM-COMPUTATION-0086 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 primary source's Question 71 is a list of desired R capabilities, including Markov bases and interfaces. It is not a single mathematical conjecture. The selected investigation concerns a specific correctness trap in one proposed algebra service, without claiming an R implementation or current package inventory.

Research attempt: A lattice-basis verification trap for a proposed Markov-basis interface

Let $A=(1,1,1)$ and, for $b\in\mathbb N$, let $\mathcal F_b=\{x\in\mathbb N^3:x_1+x_2+x_3=b\}$. An allowed move $m\in\ker_{\mathbb Z}A$ can be added or subtracted only if the resulting vector remains nonnegative. A Markov basis in this example means a finite move set whose resulting graph is connected on every nonempty fiber.

Suppose a backend returns \[ m_1=(1,-1,0),\qquad m_2=(1,-2,1). \] Both lie in the integer kernel, and they even form an integer-lattice basis of that kernel. To prove spanning, take $z=(a,b,c)$ with $a+b+c=0$. Then \[ z=(a-c)m_1+c m_2, \] since the second coordinate becomes $-a-c=b$. Independence follows because the third coordinate of $\alpha m_1+\beta m_2=0$ forces $\beta=0$, then the first forces $\alpha=0$.

Nevertheless the fiber $\mathcal F_1=\{e_1,e_2,e_3\}$ is disconnected. The move $m_1$ joins $e_1$ to $e_2$ and cannot touch $e_3$ while remaining nonnegative. Either sign of $m_2$ requires at least two units of mass before application: its positive part and negative part each have total two. More explicitly, applying $m_2$ requires $x_2\ge2$, and applying $-m_2$ requires $x_1,x_3\ge1$, both impossible when the total is one. Thus $e_3$ is isolated. Checking integer-kernel membership or even lattice generation does not certify the promised Markov property.

For comparison the moves $u=e_1-e_3$ and $v=e_2-e_3$ do connect every fiber. Given $x\in\mathcal F_b$, repeatedly subtract $u$ while $x_1>0$ and subtract $v$ while $x_2>0$. Each subtraction transfers one unit to coordinate three and stays nonnegative, ending at $(0,0,b)$. Reversing the corresponding path for another point connects any two points. This proves the complete Markov property for this selected matrix and supplies a positive control for a small interface test.

The failed route is therefore exact: a service can return a mathematically correct lattice basis and still fail its stated fiber-connectivity purpose. A test at $b=1$ detects this particular error, but testing finitely many fibers without a separate theorem does not certify all fibers for arbitrary matrices.

Verification and adversarial audit

The lattice representation is checked coordinate by coordinate, and all three total-one states are enumerated. The positive control is proved for arbitrary $b$, including the singleton fiber at $b=0$, using an explicit nonnegative path.
Audit corrections.
Kernel correctness, lattice generation, and nonnegative fiber connectivity are treated as different specifications. A finite regression test is not advertised as a universal certificate.
External computation.
None; symbolic arguments and boundary checks only.

Scope and remaining gaps

This is a standard distinction between lattice and Markov bases, demonstrated by a complete counterexample. It does not implement the requested R functions, prioritize the full wishlist, or solve general Markov-basis computation.
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.

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.