we-solve (beta)

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gpt 6 astra pro attempt on CS-18

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Manuscript

Contents
  1. The Unique Games Conjecture
  2. Definitions and mathematical statement
  3. Short English statement
  4. Research attempt
  5. Current frontier.
  6. Attack route.
  7. Attempt.
  8. Stress test.
  9. Outcome.
  10. Sources

The Unique Games Conjecture

Definitions and mathematical statement

For $[k]=\{1,\ldots,k\}$, a unique game has a nonempty finite edge set and a permutation $\pi_e$ of $[k]$ on each directed edge $e=(v,w)$. Its optimum is \[ \operatorname{val}(G)=\max_{L:V\to[k]}\frac{\#\{e=(v,w):\pi_e(L(v))=L(w)\}}{|E|}. \] For every fixed $\varepsilon,\delta>0$ with $\varepsilon+\delta<1$, some fixed $k$ makes distinguishing $\operatorname{val}(G)\ge1-\varepsilon$ from $\operatorname{val}(G)\le\delta$ NP-hard under deterministic polynomial-time many-one reductions. Intermediate values lie outside the promise. The alphabet size depends on the accuracy parameters, not on the growing input graph.

Formulation and concept references: [S1].

Short English statement

Is it computationally intractable to distinguish almost satisfiable permutation constraints from ones where almost none can be satisfied?

Research attempt

Current frontier.
Khot's survey distinguishes exact satisfiability, which is efficiently checkable, from the almost-satisfiable versus low-value promise in this entry. Its Conjecture 2.5 has the required order of quantifiers. [S1], Section 2.4. The 2025 journal version of Dinur--Khot--Kindler--Minzer--Safra is an installment in the work leading to the 2-to-2 Games Theorem, not a proof of the selected Unique Games Conjecture. Its Corollary 1.12 has completeness $1/2$, conditional there on its stated agreement hypothesis, rather than completeness arbitrarily close to one. The work and its role in the subsequent chain are described in the article's abstract and introduction. [S2], [E1]. The 2026 MAX-3-CUT result explicitly assumes UGC; it does not prove it. [S5]. No theorem is inferred from the title of the catalogue's unreviewed vertex-cover proposal.
Attack route.
I try to exploit the rigidity of permutations in both directions. For an algorithm, propagate a root label and isolate the edges that obstruct consistent propagation. The missing step is finding a small deletion set without enumerating exponentially many possibilities. For hardness, try to raise completeness from an already hard lower-completeness gap, using subdivision or walk-composition gadgets. The missing step is preserving low soundness while raising completeness; exact calculations below show that these two elementary gadgets do not do it. This is an attempt to break proposed approaches as well as to prove useful intermediate facts.
Attempt.
Let $M=|E|>0$. When traversing an edge backwards, use the inverse permutation. Take a spanning tree of each connected component of the underlying undirected multigraph, ignoring loops for the tree. Choose a root and let $P_v$ be the permutation obtained by propagation along its unique tree path to $v$, with $P_{\rm root}$ the identity.

Lemma 29.1 (a root-fixed-point formulation). All constraints in the component are simultaneously satisfiable if and only if \begin{equation}\label{wsref-0-72-32-39-3a-68-6f-6c-6f-6e-6f-6d-79} \bigcap_{e=(v,w)}\operatorname{Fix}(P_w^{-1}\pi_eP_v) \ne\varnothing. \end{equation} Moreover this can be checked, and a labeling produced when possible, in $O(k(|V|+|E|))$ elementary permutation-table operations over all components.

Proof. Any labeling satisfying tree edges is forced, once its root label $a$ is given: $L(v)=P_v(a)$. The constraint on $e$ is exactly $\pi_eP_v(a)=P_w(a)$, which is the fixed-point condition displayed. Conversely any $a$ in the intersection gives a labeling satisfying every edge. Each $P_v$ and each fixed-point condition is computed on a table of $k$ labels. Isolated vertices are harmless, and loops are tested by the same formula.$\square$

This reproves the basic propagation observation in [S1], and makes explicit the cycle obstruction used next. It must not be confused with a solution for $\operatorname{val}\ge1-\varepsilon$.

Lemma 29.2 (exact dependence on a deletion budget). There is a labeling violating at most $b$ edges if and only if deleting at most $b$ edges leaves a perfectly satisfiable constraint system. Thus enumerating subsets of at most $b$ edges and applying Lemma 29.1 decides this question in time \begin{equation}\label{wsref-0-72-32-39-3a-64-65-6c-65-74-69-6f-6e} O\!\left(\sum_{j=0}^{\min(b,M)}\binom Mj\,k(|V|+M)\right). \end{equation} Proof. Delete the violated edges of a proposed labeling in one direction. In the other, any labeling of the undeleted system violates at most the deleted edges. This proves both the equivalence and the finite enumeration bound.$\square$

For fixed $b$ this is polynomial. In the target $b$ may be $\lfloor\varepsilon M\rfloor$ for a fixed positive $\varepsilon$, which is linear in the growing instance size. Formula (r29:deletion) then supplies no polynomial algorithm. Replacing a small fraction by a small number would silently weaken the problem.

Nor does checking only short cycles reveal every obstruction. On a directed cycle of length $L$, put the identity permutation on $L-1$ edges and a fixed-point-free permutation on the last edge. The full system is inconsistent, but deleting any edge makes it consistent; its optimum is exactly $1-1/L$. Every proper path passes propagation. For fixed $k\ge2$, a cyclic permutation of $[k]$ supplies the required fixed-point-free map. This is an explicit near-satisfiable instance, not an NP-hardness construction.

Attempt to upgrade completeness by subdivision. Replace each edge $e=(v,w)$ by a path of $\ell\ge1$ edges with fresh internal vertices, choosing permutations on the path whose composition is $\pi_e$. Denote the result by $G^{(\ell)}$.

Lemma 29.3 (subdivision contracts both sides of the gap). \begin{equation}\label{wsref-0-72-32-39-3a-73-75-62-64-69-76-69-73-69-6f-6e} \operatorname{val}(G^{(\ell)}) =1-\frac{1-\operatorname{val}(G)}{\ell}. \end{equation} Proof. For fixed endpoint labels, if the original edge is satisfied, propagating through the new path satisfies all its edges. Otherwise at least one new edge must fail, because satisfying them all would impose the original endpoint relation. Exactly one failure is achievable by propagating until the last edge. The internal vertices of different paths are disjoint, so these optimal choices can be made independently for fixed original labels. Minimize the total number of failed original relations and divide by the new edge count $\ell M$.$\square$

Suppose a base reduction gives completeness $c$ and soundness $s<c$. Subdivision changes its guaranteed thresholds to $1-(1-c)/\ell$ and $1-(1-s)/\ell$; their difference is $(c-s)/\ell$. It does raise completeness, but it also raises low soundness towards one. For $c=1/2$, attaining $1-\varepsilon$ by this gadget requires $\ell\ge1/(2\varepsilon)$, whereas even a value-zero instance becomes $1-1/\ell$ satisfiable. Thus this gadget cannot turn a $1/2$-completeness hardness theorem into the target small-soundness theorem. The equality in (r29:subdivision), rather than a loose analysis of the gadget, is the obstruction.

Attempt to amplify inconsistency by composing walks. A second proposal puts a new constraint between the endpoints of each length-$r$ walk, using the composed permutation. This still makes each new constraint unique. But it tests only endpoint consistency, not the individual constraints along the walk, and failed constraints can cancel.

Take the directed three-cycle and $k=2$, with the transposition on every edge. Its optimum is $2/3$: alternating labels cannot close consistently around an odd cycle, and one failed edge is achievable. Composing two successive constraints gives the identity on each of the three new edges. The all-equal labeling satisfies the entire resulting game, whose value is one. This counterexample applies to the precise length-two composition rule even though every constituent constraint is a permutation. It disproves any assertion that this rule must preserve or increase the unsatisfied fraction. It does not refute sophisticated gap amplification using additional tests or a different alphabet.

Combining the two elementary gadgets therefore has no supplied soundness proof. Subdivision has an exact adverse trade-off; endpoint composition can erase all frustration. Retaining all intermediate labels in a longer alphabet would allow one to test more information, but then a two-endpoint relation must still be a permutation on the same fixed alphabet and its completeness/soundness must be proved. That is a new construction problem, not a consequence of storing a walk.

Stress test.
All algorithms and gadgets above use the input graph as the growing parameter. No value of $k$ depending on $|V|$ is substituted into the conjecture. Checking a small alphabet, exact satisfiability, or a bounded number of errors does not handle every fixed $(\varepsilon,\delta)$ with its possibly much larger fixed alphabet. An exponential exhaustive search over the labels remains a finite algorithm, but is irrelevant to a polynomial-time reduction or decision claim.

A polynomial-time solver for all these gaps would imply $\mathsf P=\mathsf{NP}$ if the asserted NP-hardness reductions exist; it would not, without an additional separation assumption, logically disprove NP-hardness. Conversely NP-hardness of a promise problem alone is not an unconditional proof that no polynomial-time algorithm exists. These distinctions matter because the selected target is an NP-hardness statement, not merely an informal assertion of intractability. [S1], Sections 2.1 and 2.4.

The companion script enumerates small two-label directed-cycle games, compares propagation with brute-force exact satisfiability, and checks (r29:subdivision) for small subdivisions and the triangle composition counterexample. All are exact finite checks; none tests the asymptotic NP-hardness quantifier. The reference's allowance of more general constraint graphs does not invalidate these counterexamples, since these particular loop-free directed graphs already lie in scope.

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Outcome.
Outcome: Exploratory approach with unresolved gap.

The notebook obtains an explicit consistency/deletion formulation, an exact subdivision trade-off, and a three-vertex obstruction to naive walk amplification. The basic propagation algorithm is already known; no historical novelty is asserted for these elementary deductions. They identify why two routes do not reach almost-perfect completeness together with arbitrarily low soundness. No reduction with the target parameters, and no unconditional refutation of its NP-hardness, has been established.

Sources