Quotients and Presentations of L-Convex Objects

Quotients and Presentations of L-Convex Objects

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 function is $L$-convex under the source's discrete-convexity convention, which combines translation behavior with a lattice submodularity condition. Flow presentations encode its feasible set or epigraph through network constraints.

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 discrete domain, translation convention, flow presentations, and allowed quotients are not defined in the supplied question.

Source-record limitation: The two flow presentations, domain, and proposed quotient operations are not defined by the supplied extraction. The primary Oberwolfach Report 2025/12, p. 532, describes two flow presentations of alcoved polytopes using support numbers and facet volumes. The supplied reconstructed question instead names $L$-convex sets and functions. This is a disagreement about the mathematical objects, not merely missing definitions; no equivalence or corrected target is inferred, and the supplied statement is preserved.

Statement recorded in the supplied source.

Describe the correspondence between the two flow-based presentations of $L$-convex sets and functions. Define natural quotient operations for $L$-convex sets and $L$-convex functions, and establish their basic structural properties. [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

Relate the two stated flow descriptions and develop quotient operations for the specified discrete-convex objects.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30006215 (OWR-14299089-001). 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 Mini-Workshop: Alcoved Polytopes in Physics and Optimization $2025$, Oberwolfach Reports, DOI 10.4171/owr/2025/12, pp. 531--558 https://doi.org/10.4171/owr/2025/12.
• 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 30006215 and code OWR-14299089-001 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.

There is a source mismatch in the preserved catalogue body. The report's introduction asks separately about two flow presentations of alcoved polytopes, by support numbers and facet volumes, and about quotients of $L$-convex sets and functions. Its later detailed quotient talk concerns $M$-convex objects. These statements cannot silently be merged into an asserted correspondence of two $L$-convex flow presentations. The following is a deliberately restricted test of projection as one possible operation; it is not a proposed definition of the requested quotient.

Coordinate elimination for integer difference constraints

Fix $n\ge2$, integer constants $c_{ij}$ for $i\ne j$, and \[ S=\{x\in\mathbb Z^n:x_i-x_j\le c_{ij}\text{ for all }i\ne j\}. \] This class has the usual set-level $L$-convex closure properties: translation by integer multiples of the all-one vector, and componentwise minimum and maximum. Translation preserves differences. If $x_i\le x_j+c$ and $y_i\le y_j+c$, monotonicity of minimum and maximum gives respectively $\min(x_i,y_i)\le\min(x_j,y_j)+c$ and the corresponding maximum inequality. Applying this to every constraint proves the assertion, including an empty set.

Failure of arbitrary linear images

Even the $L$-convex set $\mathbb Z^2$ need not remain so under an arbitrary integral linear map. Its image under $(x,y)\mapsto(x,y,x+y)$ contains $(1,0,1)$ and $(0,1,1)$, but not their componentwise minimum $(0,0,1)$. Hence arbitrary image-taking does not supply an operation preserving these set axioms. Coordinate projection succeeds because its fibers are integer intervals governed by explicit compatible bounds.

Verification and remaining gaps

Finite bounded enumerations test the elimination formula, including a contradictory two-edge cycle, and the exact image obstruction. These standard difference-constraint facts do not address function quotients, a universal quotient property, or the separate support-number/facet-volume correspondence for alcoved polytopes. No repair to the immutable catalogue wording has been made.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2025/12. Relevant passages read: OWR 2025/12, printed page532 introduction and pages548--549 quotient talk.

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, read and reviewed the written mathematical arguments, independently reran the finite checks, 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 batch026/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.