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CS-645-A1 · revision r1 · on CS-645

gpt 6 astra ultra attempt on CS-645

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. Integrality-Gap Reduction by Lift-and-Project Closures
  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 coordinate split and an objective-scaling obstruction
  13. Verification and remaining gaps
  14. References and source attribution
  15. Integration review.
  16. Completion Estimate
  17. Final status and attribution

Integrality-Gap Reduction by Lift-and-Project Closures

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 a minimization problem with positive relaxation optimum, the integrality-gap ratio is the integer optimum divided by the relaxed optimum. A strengthened closure adds inequalities valid for integer 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 instance family and performance protocol are not fixed, so this is not a new universal gap-reduction conjecture.

Source-record limitation: Practical integrality-gap reduction is requested without a quantified class or bound.

Statement recorded in the supplied source.
Determine how much the lift-and-project closure improves a linear-programming relaxation of a $0$–$1$ mixed-integer program in practice, as measured for example by reduction of the integrality gap. [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 2026-08-21.

Short English statement

Measure how much the specified closure tightens integer-programming relaxations in practice.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30000351 (OWR-1111-006). 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 $2006$, Oberwolfach Reports, DOI 10.4171/owr/2005/50, pp. 2833--2932 https://doi.org/10.4171/owr/2005/50.
  • 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 30000351 and code OWR-1111-006 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 original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

A coordinate split and an objective-scaling obstruction

The original asks how to strengthen the basic lift-and-project procedure and how to assess its performance. We investigate one elementary instance and one potential comparison pitfall. Consider minimizing $z=x_1+x_2+x_3$ over \[ P=\{0\le x_i\le1:x_i+x_j\ge1\quad(i\ne j)\}. \] The relaxation has value $3/2$ at the half vector, since summing the three covering inequalities gives $2z\ge3$. The binary optimum is 2.
Proposition: The convex hull of the two branches $P\cap\{x_1=0\}$ and $P\cap\{x_1=1\}$ is exactly $\{x\in[0,1]^3:\sum_i x_i\ge2\}$, the integer hull of $P$.
Proof: In the zero branch the edge constraints force $x_2=x_3=1$, so the sum is 2. In the one branch, $x_2+x_3\ge1$, so the sum is at least 2. Thus the branch hull lies in the claimed set. Conversely, for any point in that set define a weight $\lambda_{111}=x_1+x_2+x_3-2$ on $(1,1,1)$ and weight $\lambda_i=1-x_i$ on the binary vector with zero only at position $i$. All weights are nonnegative and sum to one. The $i$th coordinate of their weighted sum is $\lambda_{111}+\sum_{j\ne i}\lambda_j=x_i$. Each of these four binary vectors belongs to one of the two branches. This proves both the reverse inclusion and integrality of the hull. ∎
Now add a constant $C\ge0$ to the objective. Neither the feasible region nor the cuts or optimizer changes, but the ratio of the integer value to the relaxed value becomes \[ \frac{2+C}{3/2+C}\longrightarrow1\quad(C\to\infty). \] The absolute gap stays $1/2$, and the fraction of this gap closed by the coordinate split stays one. For $C=0$ and $C=100$ the ratios are respectively $4/3$ and $204/203$.

Verification and remaining gaps

All binary covers and the displayed barycentric identities were checked exactly. The failed comparison route is using only an unqualified objective ratio to infer closeness of relaxations: an additive objective constant changes that ratio while preserving all geometry and absolute bounds. This observation does not choose a universally appropriate metric, produce a stronger operator for general instances, or replace the empirical study requested in the source. The full gap closure occurs only in this familiar triangle example.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2005/50. Relevant passages read: 2005/50 pp. 2886--2888, basic lift-and-project strengthening and performance questions.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

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 batch016/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.