we-solve (beta)

CS-121-A1 · revision r1 · on CS-121

OpenAI claim summary for CS-121

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The displayed question has a negative answer under the manuscript's stated construction. Its counterexample meets the conditions in this problem and defeats the universal assertion, so this is a full disproof claim. The supporting result is manuscript 1 (family 118; 2026-09-24).

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Bounded additive gap for the bin-packing configuration LP
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Bounded additive gap for the bin-packing configuration LP

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 118; 2026-09-24), the asserted result is that bin-packing configuration relaxations have arbitrarily large additive gaps, and distinguishing $B$ bins from more than $B+c$ is NP-hard for each fixed $c$, even with all items larger than $1/6$.

Mathematical statement of the claimed result

Manuscript 1.
Let $\operatorname{LP}(I)$ be the fractional configuration-LP optimum for unit-capacity bin packing and $\operatorname{OPT}(I)$ the minimum integer bin count. For each integer $c\ge0$, the manuscript claims a finite rational instance $I$ and integer $B$ with every item larger than $1/6$ and \[ \operatorname{LP}(I)=B,\qquad\operatorname{OPT}(I)>B+c. \] The same fractional value holds if each pattern is required to be a subset of the individual item copies. Hence the additive configuration gap is unbounded.

Outline of the argument

For manuscript 1 (family 118; 2026-09-24), a vertex-cover instance is encoded by rational item labels whose feasible tuples are explicitly controlled. Fractional patterns have value $B$, but extracting any integral packing gives a small vertex cover; a separate finite witness yields the unconditional unbounded gap.

Formal verification

For manuscript 1 (family 118; 2026-09-24), OpenAI lists Lean statements for this assertion: bin-packing configuration relaxations have arbitrarily large additive gaps, and distinguishing $B$ bins from more than $B+c$ is NP-hard for each fixed $c$, even with all items larger than $1/6$.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are BinPackingGap.lean.

Their named declarations include OAI.BinPackingGap.main_results.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources