gpt 6 astra ultra attempt on CS-732

An indexed list of questions cannot define an additional mathematical target

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 set function $f:2^V\to\mathbb R$ is submodular if $f(A)+f(B)\geq f(A\cup B)+f(A\cap B)$ for all $A,B\subseteq V$. It is monotone if inclusion never decreases its value and normalized if $f(\varnothing)=0$.

A polyhedron is an intersection of finitely many closed affine half-spaces; a polytope is a bounded polyhedron. Its faces are its intersections with supporting hyperplanes.

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 supplied registry flags this extraction as unrecoverable. It is displayed as a literal source fragment; the definitions provide context rather than a replacement formal statement.

Source-record limitation: Source flags the statement as unrecoverable: The official report shows that the extract is a table-of-contents fragment listing several later contributions, not a standalone mathematical problem. No separate clean mathematical statement is supplied; this source ID preserves its provenance. 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 record as unrecoverable. Its extract and terminology are documented, but a complete original problem has not been reconstructed.

Statement recorded in the supplied source.

2939 Chandra Chekuri Open Problem: Approximating Symmetric Submodular Functions by Hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2941 Laura Sanità Open Problem: Computing the diameter of the bipartite perfect matching polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2943 Robert Weismantel (joint with Marcel Celaya) Open Problem: Integer infeasible subsystems . . . . . . . . . . . . . . . . . . . . . . . . 2943 Samuel Fiorini Open Problem: Iterative rounding for feedback vertex set . . . . . . . . . . . . . 2944 Nima Anari Open Problem: Maximizing Subdeterminants in Nonsymmetric Matrices 2944 Nathan Klein Open Problem:

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alpha-Multiplicative Connectivity Augmentation and Polygon Augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2945 László Végh Open problem: Approximating submodular functions by matroid rank functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2946 Michel Goemans Open problem: Signings in the subtour polytope . . . . . . . . . . . . . . . . . . . . . 2946 Michel Goemans Open problem: Representation for 4/3-near-mincuts . . . . . . . . . . . . . . . . . 2947 Nicole Megow Open problem: Unrelated graph balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . 2947 Ola Svensson Open problem: Perfect matching that maximizes the k-heaviest edges . . . The report passage is retained verbatim as the relevant context for this extracted problem. [S2]

Source status.

The supplied catalogue labels this source record open; 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. The catalogue literature assessment is dated 2026-08-22.

Short English statement

The supplied fragment concerns unspecified Combinatorial Optimization Problem. Its complete mathematical target is not recovered.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30004897 (OWR-8415356-001). 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 Combinatorial Optimization $2022$, Oberwolfach Reports, DOI 10.4171/owr/2021/53, pp. 2893--2954 https://doi.org/10.4171/owr/2021/53.
• Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.

Verbatim supplied catalogue question

2939 Chandra Chekuri Open Problem: Approximating Symmetric Submodular Functions by Hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2941 Laura Sanità Open Problem: Computing the diameter of the bipartite perfect matching polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2943 Robert Weismantel (joint with Marcel Celaya) Open Problem: Integer infeasible subsystems . . . . . . . . . . . . . . . . . . . . . . . . 2943 Samuel Fiorini Open Problem: Iterative rounding for feedback vertex set . . . . . . . . . . . . . 2944 Nima Anari Open Problem: Maximizing Subdeterminants in Nonsymmetric Matrices 2944 Nathan Klein Open Problem:

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alpha-Multiplicative Connectivity Augmentation and Polygon Augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2945 László Végh Open problem: Approximating submodular functions by matroid rank functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2946 Michel Goemans Open problem: Signings in the subtour polytope . . . . . . . . . . . . . . . . . . . . . 2946 Michel Goemans Open problem: Representation for 4/3-near-mincuts . . . . . . . . . . . . . . . . . 2947 Nicole Megow Open problem: Unrelated graph balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . 2947 Ola Svensson Open problem: Perfect matching that maximizes the k-heaviest edges . . . The report passage is retained verbatim as the relevant context for this extracted problem.

Source reconstruction attempt

The preserved extract is the contents page of Oberwolfach Report 53/2021, not the body of one open problem. This was checked against printed page 2899 and the later contribution pages themselves. The sequence of names, dot leaders and page numbers in the catalogue follows that contents page. For example the integer infeasibility contribution is listed at 2943, feedback vertex set and nonsymmetric determinants at 2944, augmentation at 2945, matroid rank and signing at 2946, near-mincuts and balancing at 2947, and top-$k$ matching at 2948. These are different contributions with different inputs, assumptions and questions.

The reconstruction attempt therefore followed the page references rather than interpreting the extracted words as a formula. The source bodies confirm that the integer-infeasibility question is about integral constraint matrices with bounded subdeterminants; the determinant-maximization question is about a real matrix with positive-semidefinite symmetric part; and the signing question is about capacitated graph cuts. None is a missing quantifier or parameter of another. Their detailed attempts belong to their own retained source IDs, including 30004903, 30004906 and 30004910.

Why selecting a title would manufacture a target.

The extract does not ask a conjunction of all listed questions, select one of them, specify an input shared by them, or state an approximation or runtime requirement of its own. A contents entry records where a statement can be found; its existence does not assert that every mathematical phrase in the entry is true. For example "maximizing subdeterminants" is the name of an optimization task, not a claim that a particular maximum or approximation algorithm exists. Treating the first title as the target would omit every later title without a source instruction to do so. Treating all titles as a new conjunction would add a quantifier and scope not present on the page. Neither operation is a faithful solution attempt for this source record.

Audit result and explicit limitation

No theorem, counterexample or completion claim is attached to this index fragment. The substantive result of the attempted reconstruction is its exact source classification and the identification of separate later problem bodies, which prevents a fabricated replacement problem. There was no numerical computation. Mathematical attempts on the actual identifiable questions have been written in their own records; they are not duplicated here as purported progress on an additional conjecture. A unique intended problem statement would be required before this index ID could have a mathematical completion target. The catalogue ID, title, full definition, literal extract, source status and attribution remain preserved. This is deliberately an unresolved zero-percent provenance audit, not mathematical work invented to increase a solved-problem or theorem count.

Process and contributions

Mathematics and model-assisted adversarial self-audit: OpenAI gpt-6-astra, ultra reasoning, as dispatched for this research batch; exact serving revision not exposed. Codex (GPT-6; exact serving revision not exposed) materially assisted with orchestration and format review. No human mathematical authorship, mathematical edits, or independent human proof checking is asserted. Contribution prompt A was applied to the full definition and supplied source row, then prompt B was applied adversarially to the derived results. The item-specific audit above records the scope checks and rejected extrapolations. Local assembly and structural checks do not establish mathematical correctness. The source catalogue status is preserved in TOP_PROBLEM and is separate from this attempt status.

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 proved material is a rederivation of known or auxiliary facts rather than a substantiated new advance on the original target. The original item is a table-of-contents fragment and supplies no independent mathematical proposition; selecting or conjoining its titles would invent a target.

References

The complete original Sources subsection above is retained, including catalogue attribution and licence notices. The additional list records primary sources actually consulted for this attempt on 11 October 2026; uncited source links reproduced above were not all rechecked.

• Oberwolfach Report 53/2021, Combinatorial Optimization, original question and surrounding discussion read from the primary PDF text on 11 October 2026. Historical statements are not a current-status survey. https://ems.press/content/serial-article-files/46931.

Final status

Unresolved in this attempt. The original item is a table-of-contents fragment and supplies no independent mathematical proposition; selecting or conjoining its titles would invent a target.