Polymorphism Characterization of Constraint Approximation Thresholds

Research-attempt provenance

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 constraint satisfaction problem assigns labels from a finite alphabet to variables so as to satisfy specified relations on tuples. The value of an instance is the maximum fraction of satisfied constraints; a gap problem distinguishes two separated ranges of this value.

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. Source-record limitation: The supplied assessment identifies this as an equivalent formulation of Unique Games; existing canonical records 43 and 30001374 cover that target.

The supplied assessment identifies this as an equivalent formulation of Unique Games, rather than an independent new target.

Related catalogue records: 43, 30001374. Separate source IDs are preserved for their provenance.

Statement recorded in the supplied source.

Is the approximation threshold of every constraint-satisfaction problem characterized by the existence of approximate non-dictator polymorphisms? [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 2026-08-21.

Short English statement

The source asks about polymorphism Characterization of Constraint Approximation Thresholds.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30001376 (OWR-4135-007). 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 Complexity Theory $2010$, Oberwolfach Reports, DOI 10.4171/owr/2009/52, pp. 2787--2850 https://doi.org/10.4171/owr/2009/52.
• S3 Khot unique-games paper https://doi.org/10.1109/SFCS.2002.1181954.
• 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 30001376 and code OWR-4135-007 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. Related canonical identities: 43, 30001374. 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.

Literature and context check

The primary report on printed pages 2821--2823 defines approximate polymorphisms through a distribution over coordinatewise distributional functions, and a pseudorandomness requirement on each function in its support under every product marginal. It also states a characterization conditional on the Unique Games Conjecture. That conditional source theorem is not an unconditional solution. This attempt tests concrete candidate operations against the actual approximation and low-influence requirements.

Research attempt: Exact polymorphisms versus approximate quality and influence

A non-dictator exact operation can destroy approximate quality

Boolean majority of three inputs preserves every 2-SAT clause when all three input assignments satisfy it. To prove this, if the output falsified a clause with two literals, at least two input assignments would falsify its first literal and at least two would falsify its second. These two subsets of a three-element set intersect, so one input assignment would falsify both literals, a contradiction. Negated literals obey the same majority rule, and unit clauses are immediate.

Nevertheless majority is not an $\alpha$-approximate polymorphism for every positive $\alpha$ in the source's all-quality-level definition, even for positive unit clauses. Take the three constraints $x_1,x_2,x_3$ and input assignments $100,010,001$. Each satisfies fraction $c=1/3$, but coordinatewise majority produces $000$ of value zero. The requirement of expected value at least $\alpha c$ fails for every $\alpha>0$. This is a concrete obstruction to the route "an exact non-dictator polymorphism automatically gives a nontrivial approximation ratio." It does not rule out other approximate operations.

A baseline that genuinely satisfies the distributional requirement

For a Boolean CSP in which independent fair bits satisfy each allowed constraint with probability at least $\rho$, the constant distributional function which outputs the fair-bit distribution, ignoring all its inputs, has zero influence under every input distribution. The source samples its output independently at different variables, so its expected objective is at least $\rho$ on every instance. Since every input quality $c$ is at most one, this is at least $\rho c$. It is therefore a $\rho$-approximate polymorphism satisfying every positive influence tolerance. For Max-Cut with edges between distinct vertices, independent fair endpoints differ with probability $1/2$, giving the explicit ratio $1/2$. The distinct-variable restriction avoids silently treating a loop constraint as having positive random satisfaction probability.

Averaging dictators does not meet the source quantifiers

Another proposed operation is to choose an input index uniformly once and return that entire assignment. As a distribution over deterministic coordinatewise functions this preserves expected quality at least $c$ and so is 1-approximate. However every function in the distribution's support is itself a dictator. Under uniform binary inputs, its chosen coordinate has fixed nonzero influence: with the conditional-variance convention for the Boolean indicator it is $1/4$ (with resampling-disagreement convention it is $1/2$). The source requires all influences to be less than an arbitrarily prescribed tolerance for every supported function, not only after averaging over the random choice of dictator. Hence this construction cannot make the required influence tend to zero. Sampling a fresh dictator separately at every variable is a different operation with different correlations and cannot inherit the whole-assignment quality argument.

Verification and adversarial audit

The majority counterexample uses a language containing unit clauses, which is a legitimate restricted 2-SAT language; all three input quality bounds hold exactly. The fair-bit operation uses the report's independent variable-level output sampling. The dictator discussion distinguishes outer randomness selecting a function from inner independent sampling at each variable, and states the influence normalization used for its numerical value.

Audit corrections.

Retained the all-input-quality requirement and per-supported-function influence quantifier; did not equate syntactic non-dictatorship with arbitrarily low influence.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

No universal approximation-threshold characterization is established or refuted. The source's UGC-dependent theorem is acknowledged, while the attempts identify concrete failures of two shortcuts and prove a valid low-influence random-assignment baseline in an explicit language class.

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% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://ems.press/content/serial-article-files/46253 (consulted 2026-10-11).

Model and contributor disclosure

The parent Codex GPT-6 model has reviewed the written mathematical arguments at their declared scopes. This is model-assisted review, not independent human verification. OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration and repository-format review; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.