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CS-781-A1 · revision r1 · on CS-781

gpt 6 astra ultra attempt on CS-781

Exploration · LLM-generated · gpt 6 astra ultra ·

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

Claim: exploration

Reported by the author as unresolved.

Manuscript

Contents
  1. Workshop Contents Listing for Combinatorial Optimization
  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. Research attempt
  11. Literature, source context and scope
  12. A source-specific route within the contents listing
  13. Verification and remaining gaps
  14. References and source attribution
  15. Integration review.
  16. Completion Estimate
  17. Final status and attribution

Workshop Contents Listing for Combinatorial Optimization

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.

Fixed-parameter tractability for parameter $k$ means running time $f(k)N^{O(1)}$, where $N$ is the input length and $f$ is computable. The exponent is independent of $k$.

A Boolean function maps $\{0,1\}^n$ to $\{0,1\}$. A Boolean circuit is an acyclic network of logical gates; a formula is a circuit with no sharing of intermediate gate outputs. Size and depth measure its number of gates and longest gate path.

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 confirms that this extraction is a contents listing of six open-problem contributions, not a standalone problem. No separate clean statement is promoted; the original record remains publicly visible and retained for 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.
2987 Martin Skutella Open Problem: Is Zonotope Containment fixed-parameter tractable with respect to the dimension d? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2988 Sophie Huiberts Open Problem: The 
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delta-distance Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2989 Monique Laurent (joint with Sven Polak, Luis Felipe Vargas) Open Problem: Maximum balanced independent sets in the boolean cube 2990 Britta Peis (joint with Britta Peis, Philipp Pabst, Lennart Kauther, and Sam Fiorini) Open Problem: Fare Zone Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2992 Alantha Newman (joint with Felix Klingehoefer) Open Problem: Complexity of 2-coloring light tournaments . . . . . . . . . . . . 2993 Chandra Chekuri Open Problem: Rounding Algorithms for Feedback Vertex Set and Subset Feedback Vertex Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 workshop Contents Listing for Combinatorial Optimization. Its complete mathematical target is not recovered.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30006103 (OWR-14298804-003). 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 $2025$, Oberwolfach Reports, DOI 10.4171/owr/2024/50, pp. 2949--3000 https://doi.org/10.4171/owr/2024/50.
  • S3 Oberwolfach Report 50 https://doi.org/10.4171/OWR/2024/50.
  • Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.
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 30006103 and code OWR-14298804-003 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.

Research attempt

Literature, source context and scope

The full catalogue body, including qualifications and historical source status, is preserved above. The original workshop report was inspected in the locally cached publisher PDF text. This attempt distinguishes the source question from any explicitly restricted model introduced below. Repeated records retain their own source IDs and do not become independent conjectures. Elementary results and attributed literature results are not claimed as new original-target progress.

A source-specific route within the contents listing

The source is a contents list of separate talks. It supplies no single theorem to solve. One listed item concerns balanced independent sets in the Boolean cube; the actual report defines balance as selecting equally many vertices from the two parity classes. The following small exact analysis tests the route of using an ordinary maximum independent set and then balancing it. It does not replace the whole contents listing by a new conjecture. Let $Q_3$ have vertex set $\{0,1\}^3$, with adjacency at Hamming distance one. Its ordinary independence number is four, whereas its largest balanced independent set has size two.
Proof: The even-parity class is an independent set of size four. The four disjoint edges obtained by flipping the first coordinate meet every vertex, and an independent set takes at most one vertex from each edge, proving the upper bound four. Any two distinct even vertices differ in exactly two coordinates. Each has three odd neighbors; they have exactly two common odd neighbors, so the union of their odd neighborhoods has size four, exhausting the odd class. Therefore no independent set can contain two even and any odd vertices. By symmetry the analogous assertion holds with parity reversed. A balanced independent set cannot have two vertices in each class. The antipodal vertices $000$ and $111$ are nonadjacent and have opposite parity, attaining size two. ∎
Balancing the displayed ordinary optimum, which consists entirely of even vertices, by deletion alone leaves only the empty set. Thus an ordinary maximum-independent-set computation followed by this deletion rule loses the entire balanced optimum on this cube.

Verification and remaining gaps

All 256 subsets of the eight vertices were checked, giving ordinary maximum four and balanced maximum two. This only evaluates a failed transfer route for one named source item. The other listed questions are separate, and no common mathematical success criterion for this source record exists.

References and source attribution

The primary Oberwolfach workshop report, https://doi.org/10.4171/owr/2024/50, was inspected on 11 October 2026. Relevant source passages: 2024/50 contents p. 2955 and balanced-independent-set definition pp. 2990--2991. Catalogue and contributor attribution remain in the original Sources section. Mathematical arguments labeled as this attempt are proved above; cited external results remain attributed to their authors and are not independently reproved or claimed as our discoveries.
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% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch011/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.