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CS-115 · computer science · mathematics · Computational complexity

$d$-to-$1$ Games Conjecture

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For every integer $d\geq2$ and every $s>0$, does there exist an alphabet size $k$ such that it is NP-hard to distinguish satisfiable $d$-to-$1$ Label-Cover instances from instances of value at most $s$?

Formulation references: [S1], [S2].

Short English statement

For every integer $d\geq2$ and every $s>0$, does there exist an alphabet size $k$ such that it is NP-hard to distinguish satisfiable $d$-to-$1$ Label-Cover instances from instances of value at most $s$?

Sources

Catalogue entry

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

Contributions 1

CS-115-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-115

Perfect-completeness hardness is established for projections with exactly two preimages. The statement here includes every integer fiber size $d\ge2$, and no theorem or reduction for all those sizes is supplied.

\$d\$-to-\$1\$ Games Conjecture

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 105; 2026-09-23), the asserted result is that exact-two projection games are NP-hard to distinguish between fully satisfiable instances and arbitrarily small fixed value, with a fixed alphabet.

Mathematical statement of the claimed result

Manuscript 1.
For every fixed rational $\delta\in(0,1)$, the claim gives $q(\delta)\ge2$ and a deterministic polynomial reduction from $3$SAT to explicit unweighted $2$-to-$1$ projection games with left alphabet $[2q]$, right alphabet $[q]$, and exactly two preimages per right label, satisfying \[ \varphi\text{ satisfiable}\Rightarrow\operatorname{val}(G)=1, \qquad\varphi\text{ unsatisfiable}\Rightarrow\operatorname{val}(G)\le\delta. \] The nonempty edge-occurrence list and every projection table are output explicitly. An accompanying graph consequence gives absolute $C,k_0$ such that for each fixed $k\ge k_0$, distinguishing a $k$-colorable graph from one for which every $k$-coloring properly colors at most $1-1/k+C\log k/k^2$ of its edges is NP-hard. The Boolean Not-Two predicate accepts zero, one or three true literals; the further claimed hardness gap is satisfiable versus value at most $5/8+\varepsilon$ for every fixed $0<\varepsilon<3/8$. Thus NP has three nonadaptive query PCPs with perfect completeness and soundness $5/8+\varepsilon$, using $O(\log n)$ random bits.

Outline of the argument

For manuscript 1 (family 105; 2026-09-23), matrix slices organize a tree of partial function tables. Joint pointwise evaluations retain perfect completeness, while hidden-child sampling and posterior control decode inconsistent answers without increasing the two-element projection fibers.

Formal verification

For manuscript 1 (family 105; 2026-09-23), OpenAI lists Lean statements for this assertion: exact-two projection games are NP-hard to distinguish between fully satisfiable instances and arbitrarily small fixed value, with a fixed alphabet.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are PerfectCompleteness.lean.

Their named declarations include OAI.PerfectCompleteness.Theorem11.exists_reduction.

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

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