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CS-788-A1 · revision r1 · on CS-788

gpt 6 astra ultra attempt on CS-788

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. Duplicate Fiorini Feedback-Vertex-Set Question
  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. Attempted inference from an integrality-gap bound
  13. Verification and remaining gaps
  14. References and source attribution
  15. Integration review.
  16. Completion Estimate
  17. Final status and attribution

Duplicate Fiorini Feedback-Vertex-Set Question

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 feedback vertex set is a set of graph vertices whose removal leaves no cycles. A fractional relaxation replaces selected/unselected decisions by nonnegative numerical variables and valid linear constraints. An extreme point cannot be written as a proper convex combination of two other feasible points.

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 commentary on the still-open Fiorini question represented canonically by OWR-14298804-016. No separate clean statement is promoted; the original record remains publicly visible and retained for provenance.

Related catalogue records: 30004905. Separate source IDs are preserved for their 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.
The paper also showed that a previous LP relaxation suggested by Chekuri and Madan [4] also has integrality gap of 2 and this resolved an open problem. In particular, the original question of Fiorini is not yet resolved. 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 duplicate Fiorini Feedback-Vertex-Set Question. Its complete mathematical target is not recovered.

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 30006114 and code OWR-14298804-017 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. Related canonical identities: 30004905. 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.

Attempted inference from an integrality-gap bound

This fragment emphasizes that a gap-two theorem for an LP did not settle Fiorini's extreme-point question. We test the tempting general inference that a covering relaxation with integrality gap at most two must have a coordinate at least one half at every extreme point. The following abstract covering polyhedron disproves that inference; it is explicitly not asserted to be an FVS strong-density polyhedron. Let $q=(1/3,1/3,1/3,1/3)$ and let $\mathcal I$ be the six incidence vectors of two-element subsets of $\{1,2,3,4\}$. Set \[ Q=\operatorname{conv}(\mathcal I\cup\{q\})+\mathbb R_{\ge0}^4. \]
Proposition: The point $q$ is an extreme point with every coordinate below $1/2$, while for every nonnegative cost vector the minimum integral cost over $Q$ is at most $3/2$ times the minimum fractional cost.
Proof: The all-ones objective has unique minimizer $q$: its value there is $4/3$, while every vector in $\mathcal I$ has value two and a nonzero nonnegative added vector increases the value. A unique minimizer is extreme, because a nontrivial convex decomposition into feasible points would make both endpoints minimizers. Every integral vector of $Q$ has at least two coordinates at least one. Indeed, if its convex representation assigns positive weight to $q$, all four coordinates are positive, hence are at least one when integral. If that weight is zero, the coordinate sum before the nonnegative addition is two, and a vector with only one positive integral coordinate cannot dominate a convex combination of two-element incidence vectors, each of whose coordinates is at most one. Therefore the integral optimum is $I=c_{(1)}+c_{(2)}$, the sum of the two smallest costs, attained by the corresponding incidence vector. The fractional optimum is $\min\{I,\tfrac13\sum_i c_i\}$. Since $c_{(1)}+c_{(2)}\le\tfrac12\sum_i c_i$, in either case $I$ is at most $3/2$ times that optimum. This remains true if the optimum is zero, by the same inequality. ∎
Thus the numerical integrality-gap assertion by itself cannot supply the desired extreme-point coordinate. Specific structure of the strong-density inequalities is essential; the constructed $Q$ is only an obstruction to the logical shortcut.

Verification and remaining gaps

For equal unit costs, fractional and integral values are $4/3$ and two. The bound was additionally checked on all 625 cost vectors in $\{0,1,2,3,4\}^4$. No graph realizing $Q$ as the named relaxation is asserted, and no counterexample to Fiorini's actual conjecture is obtained.

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 pp. 2993--2994, gap theorem versus original Fiorini question. 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.

https://chekuri.cs.illinois.edu/papers/fvs-polyhedral-mpa.pdf. Chandrasekaran, Chekuri, Fiorini, Kulkarni and Weltge, Polyhedral aspects of feedback vertex set and pseudoforest deletion set. Opened primary 48-page paper; equation $1$, the strong-density polyhedron, and Conjecture 1 on printed pages 3--4 were inspected. Its integrality-gap results are distinct from the unresolved extreme-point assertion.

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.