Claimed Collaborative

Counterexamples to Zak's modified integer round-down conjecture for the skiving stock problem

A self-published paper claims that the additive integrality gap of the skiving stock (bin covering) problem's pattern LP relaxations is unbounded, refuting Zak's 2003 conjecture that it always stays below two, by transferring OpenAI's unrefereed bin-packing construction; it states that the proof was generated by Claude Fable 5.1 and GPT-5.6 Sol Pro.

Lab: Independent
Model: Claude Fable 5.1 and GPT-5.6 Sol Pro
Field: Computer science
Date: 2026-10-07
Human collaborators: Eric Winnington
Problem posed: 2003 · open 23 yrs

Sources

Original work

GitHubUnbounded integrality gaps and additive hardness for the skiving stock problem (PDF)↗

What was found

The skiving stock problem, called bin covering in the algorithms literature, asks for the most disjoint groups of items whose sizes each reach a threshold. Zak conjectured in 2003 that the gap between the standard pattern LP relaxation and the integer optimum is always less than two, the covering counterpart of the modified integer round-up property for cutting stock. The paper maps OpenAI's bin-packing instances of 24 September 2026, which have unbounded configuration-LP gaps, to covering instances by the complement $b_i = 2 - a_i$ with threshold 9, and proves that saturation of the packing LP carries both LP values across while a covering deficit $d$ becomes a packing excess of at most $\lceil 3d/2 \rceil$ . It concludes that the gaps of both the ordinary and the stronger proper relaxation are unbounded, and that distinguishing instances that cover $B$ bins from those that cannot cover $B - c$ is NP-hard for every fixed $c$ .

Novelty check

The paper cites Zak (2003) and the later literature on round-down properties, including Martinovic and Scheithauer's transfer between round-up and round-down failures under integral volume, and distinguishes its cardinality-saturated transfer from those without claiming novelty for the complement identity. vibemathed lists the problem as a candidate with no earlier resolution, and no earlier refutation was found in the registry. The source construction is part of the OpenAI release carried by the registry's collection entry.

Caveats and known objections

Self-published on GitHub and unrefereed, and it inherits everything about its source: OpenAI's bin-packing result is an unrefereed preprint from an unreleased model, and the paper says plainly that it neither rebuilt that result's Lean formalization nor ran its comparator check, and that its own transfer is not formalized. Nothing here is machine-checked end to end, and no reader outside the author is on record. The disclosure is short: one line states that the proof was generated by a combination of Claude Fable 5.1 and GPT-5.6 Sol Pro, and a closing note says ChatGPT assisted with drafting and organizing the literature. The author's own role is not described, so autonomy is graded collaborative, the weaker of two defensible readings: the paper's line supports ai-led, and vibemathed labels the contribution AI co-developed.

Also recorded at

vibemathedzak-s-modified-integer-round-down-conjecture

Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.

Nobody outside the lab has checked this yet.

Reading the primary source closely enough to say whether it supports the claim counts as a check, and you are credited on the entry.

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Entry history (1 event)

2026-10-20AddedEntered the registry graded Claimed and Collaborative.

Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.

Graded claimed for verification and collaborative for autonomy. What these mean.

Cite this entry

Plain text

whataifound.org. (2026). Counterexamples to Zak's modified integer round-down conjecture for the skiving stock problem. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-10-07-skiving-stock-round-down

BibTeX

@misc{whataifound-independent-2026-down,
  title        = {Counterexamples to Zak's modified integer round-down conjecture for the skiving stock problem},
  author       = {{whataifound.org}},
  year         = {2026},
  howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
  note         = {Result by Independent. Verification: Claimed. Autonomy: Collaborative.},
  url          = {https://whataifound.org/finding/2026-10-07-skiving-stock-round-down}
}

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