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CS-646 · computer science · mathematics · Optimization

Higher-Row Cuts from Cylindrical Simplices

Status Open · unsolvedmath.comLLM status Open unverified

Statement

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 valid cutting inequality holds for all mixed-integer feasible points while excluding some relaxation points. Split cuts use disjunctions $\pi^Tx\leq a$ or $\pi^Tx\geq a+1$ with integral $\pi,a$; multirow cuts use several rows jointly.

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 cylindrical-simplex construction and meaning of finite generation are source-specific. The practical-effectiveness clause supplies no numerical benchmark or theorem bound.

Source-record limitation: The two-row inequality and cylindrical-simplex construction referred to by the question are not supplied.

Statement recorded in the supplied source.
For mixed-integer cutting inequalities derived from cylindrical simplices, determine whether the two-row inequality is generally not finitely generated by mixed-integer Gomory cuts or split cuts; whether useful, easily computed $k$-row inequalities arise from $k$-simplices for $k>2$; and whether these cuts are computationally effective alone or with Gomory mixed-integer cuts. [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

Resolve the generation, higher-dimensional construction, and computational-use questions for the stated simplex-derived cutting inequalities.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30000352 (OWR-1111-008). 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 30000352 and code OWR-1111-008 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.

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

CS-646-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-646

Higher-Row Cuts from Cylindrical Simplices

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 valid cutting inequality holds for all mixed-integer feasible points while excluding some relaxation points. Split cuts use disjunctions $\pi^Tx\leq a$ or $\pi^Tx\geq a+1$ with integral $\pi,a$; multirow cuts use several rows jointly.

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 cylindrical-simplex construction and meaning of finite generation are source-specific. The practical-effectiveness clause supplies no numerical benchmark or theorem bound.

Source-record limitation: The two-row inequality and cylindrical-simplex construction referred to by the question are not supplied.

Statement recorded in the supplied source.
For mixed-integer cutting inequalities derived from cylindrical simplices, determine whether the two-row inequality is generally not finitely generated by mixed-integer Gomory cuts or split cuts; whether useful, easily computed $k$-row inequalities arise from $k$-simplices for $k>2$; and whether these cuts are computationally effective alone or with Gomory mixed-integer cuts. [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

Resolve the generation, higher-dimensional construction, and computational-use questions for the stated simplex-derived cutting inequalities.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30000352 (OWR-1111-008). 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 30000352 and code OWR-1111-008 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 computable cut from one explicit lattice-free simplex

The report discusses two-row corner polyhedra, lattice-free cuts and higher-dimensional extensions. We work in a specified $k$-row corner model: $x=f+\sum_{j=1}^m r_js_j\in\mathbb Z^k$, $s_j\ge0$, where $k\ge2$, $f=(1/2,\ldots,1/2)$ and $r_j\in\mathbb R^k$. Let \[ L=\{x\in\mathbb R^k:x_i\ge0,\ \sum_i x_i\le k\},\qquad \psi(r)=\max\left\{\max_i(-2r_i),\frac2k\sum_i r_i\right\}. \]
Proposition: The simplex $L$ has no integer point in its interior, contains $f$ in its interior, and gives the valid corner inequality $\sum_j\psi(r_j)s_j\ge1$.
Proof: An integer point in the interior would have every coordinate at least 1 and sum strictly less than $k$, which is impossible. The coordinates of $f$ are positive and its sum is $k/2<k$. In translated coordinates $r=x-f$, the strict interior inequalities are $-2r_i<1$ for every $i$ and $(2/k)\sum_i r_i<1$. Each associated linear form $\ell$ satisfies $\ell(r_j)\le\psi(r_j)$. If $s_j\ge0$ and $\sum_j\psi(r_j)s_j<1$, linearity gives \[ \ell\left(\sum_jr_js_j\right)=\sum_j\ell(r_j)s_j\le\sum_j\psi(r_j)s_j<1 \] for every facet form. Then $f+\sum_jr_js_j$ lies in the interior of $L$, contradicting its being integral. This proves validity. Also $\psi(r)\ge0$: if some coordinate of $r$ is nonpositive the corresponding $-2r_i$ is nonnegative; otherwise the sum form is positive. Thus the expression has the expected nonnegative gauge coefficients, including $\psi(0)=0$. ∎
For the positive coordinate rays $r_j=e_j$, $j=1,\ldots,k$, the coefficient is $2/k$ and the cut is $\sum_js_j\ge k/2$. It is tight at $s_j=1/2$, where $x=(1,\ldots,1)$ is integral. Computing all coefficients takes $O(km)$ arithmetic operations when the rays are explicitly given.

Verification and remaining gaps

The coefficients and tight point were checked for $k=2,3,4$, and a finite set of half-integer nonnegative $s$ values on coordinate rays was enumerated. This is a complete standard lattice-free-cut calculation for one chosen simplex. It does not prove that these cuts generate the whole corner polyhedron, identify maximal useful lattice-free sets, or establish practical effectiveness for the source's examples. The attempted step from one efficiently computable valid family to a complete finite description has no justification here; the source itself discusses obstructions to such completeness.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2005/50. Relevant passages read: 2005/50 pp. 2912--2914, two-row corner polyhedra, lattice-free sets and higher-row 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.

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