we-solve (beta)

CS-773 · computer science · mathematics · Logic and computability · Optimization

Abelian Embeddings and Gap-CSP Integrality Gaps

Status Open · unsolvedmath.comLLM status Open unverified

Statement

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 predicate $P$ is a finite relation on label tuples, and a gap-CSP distinguishes instances whose best satisfiable fraction is above one threshold from those below another. An Abelian embedding maps labels into an Abelian group so that satisfying tuples obey the specified additive relation.

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. The exact embedding relation, nontriviality condition, alphabet, and threshold conventions are not supplied; both displayed gaps are retained.

Source-record limitation: The Abelian embedding relation and nontriviality convention needed for the conjectural criterion are not supplied.

Statement recorded in the supplied source.
Can the complexity of $\operatorname{gap-CSP}\text{-}P[1,s]$ differ from that of $\operatorname{gap-CSP}\text{-}P[1-\varepsilon,s]$ only when $P$ admits a nontrivial Abelian embedding? [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 2026-08-22.

Short English statement

Determine whether the stated complexity change near perfect satisfiability requires a nontrivial additive embedding.

Sources

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 30005917 and code OWR-14298371-009 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. 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.

Catalogue entry

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

Contributions 1

CS-773-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-773

Abelian Embeddings and Gap-CSP Integrality Gaps

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 predicate $P$ is a finite relation on label tuples, and a gap-CSP distinguishes instances whose best satisfiable fraction is above one threshold from those below another. An Abelian embedding maps labels into an Abelian group so that satisfying tuples obey the specified additive relation.

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. The exact embedding relation, nontriviality condition, alphabet, and threshold conventions are not supplied; both displayed gaps are retained.

Source-record limitation: The Abelian embedding relation and nontriviality convention needed for the conjectural criterion are not supplied.

Statement recorded in the supplied source.
Can the complexity of $\operatorname{gap-CSP}\text{-}P[1,s]$ differ from that of $\operatorname{gap-CSP}\text{-}P[1-\varepsilon,s]$ only when $P$ admits a nontrivial Abelian embedding? [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 2026-08-22.

Short English statement

Determine whether the stated complexity change near perfect satisfiability requires a nontrivial additive embedding.

Sources

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 30005917 and code OWR-14298371-009 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. 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.

Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

Checking embeddings and the loss from keeping only their equations

The primary report supplies the definition omitted from the short extract: a nontrivial Abelian embedding of a predicate $P$ consists of maps $\sigma_i:\Sigma_i\to A$ into an Abelian group, not all constant, whose sum is zero on every satisfying tuple. Its precise conjecture is further qualified through integrality-gap structure. We test what the embedding equations alone can and cannot decide, without replacing that conjecture by a blanket tractability claim.
Proposition: The even-parity predicate on three bits has a nontrivial Abelian embedding. The three-bit OR predicate has none. Nevertheless a collection of predicates all sharing a nontrivial Abelian embedding can have an unsatisfiable instance whose associated Abelian equations are jointly satisfiable.
Proof: For even parity, use $A=\mathbb F_2$ and $\sigma_i(x)=x$. For OR, fix a coordinate $i$ and set some other coordinate to one. The tuples with $x_i=0$ and $x_i=1$ both satisfy OR. Subtracting their embedding equations gives $\sigma_i(0)=\sigma_i(1)$. Repeating for each $i$ makes every map constant, contradicting nontriviality, in every Abelian group. For the last claim take the alphabet $\Sigma=\{0,1\}^2$ and, for each $s\in\{0,1\}^3$, define \[ P_s((a_1,b_1),(a_2,b_2),(a_3,b_3))=1 \] exactly when $a_1+a_2+a_3=0$ over $\mathbb F_2$ and at least one $b_i\ne s_i$. Every $P_s$ has the common nonconstant embedding $\sigma_i(a,b)=a$. On three variables impose all eight predicates $P_s$. Setting all $a_i=0$ satisfies every embedding equation. Yet each choice of the three $b_i$ fails the constraint indexed by $s=(b_1,b_2,b_3)$, so the full instance is unsatisfiable. If the parity of the $a_i$ is even it satisfies exactly seven of the eight constraints; if odd it satisfies none. Thus its exact optimum is $7/8$. ∎
This gives a concrete failure of the route "solve the embedding equations and read off a satisfying assignment." Those equations retain only a necessary condition on each support. The source's proposed hybrid method also uses a semidefinite relaxation and rounding; the example neither analyzes nor contradicts that combined method. Conversely the lack of an embedding for OR by itself is not a proof of any approximation threshold.

Verification and remaining gaps

All eight bit tuples were checked for the OR cancellation witnesses, and all $4^3$ assignments were enumerated for the eight-predicate instance, confirming optimum $7/8$ and simultaneous linear feasibility. The even-parity embedding was checked directly. The 2026 primary survey abstract was consulted for continuing inverse-theorem context, but no general theorem from it is claimed proved here. The source's exact integrality-gap conjecture, approximation dichotomy and higher-order correlation inverse problems remain outside this finite obstruction.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2024/27. Relevant passages read: 2024/27 pp.1547--1549, actual Abelian-embedding definition, integrality-gap footnote and hybrid algorithm.

https://arxiv.org/abs/2602.22183. Read primary 2026 survey abstract on general k-wise correlations and applications; no full inverse theorem or broad dichotomy is claimed checked.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

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% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch, read and reviewed its written mathematical arguments, and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch025/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.

Go to contribution →