Constraint-Satisfaction Table Caption

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 specified domain to variables so that all stated relations hold.

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 supplied registry flags this extraction as unrecoverable. It is displayed as a literal source fragment; the definitions provide context rather than a replacement formal statement.

The intended regularity of the solution and the table context are missing.

Source-record limitation: Source flags the statement as unrecoverable: The official report confirms that the extraction is a table caption summarizing a taxonomy, not a standalone open mathematical problem. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim. The supplied catalogue marks this record as unrecoverable. Its extract and terminology are documented, but a complete original problem has not been reconstructed.

Statement recorded in the supplied source.

Does a CSP have a well-behaved solution? The report passage is retained verbatim as the relevant context for this extracted problem. [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

The fragment asks about an unspecified well-behaved solution of a constraint problem.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30005023 (OWR-9790359-004). 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 Mini-Workshop: Descriptive Combinatorics, LOCAL Algorithms and Random Processes $2023$, Oberwolfach Reports, DOI 10.4171/owr/2022/8, pp. 429--455 https://doi.org/10.4171/owr/2022/8.
• 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 30005023 and code OWR-9790359-004 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.

Literature and context check

I read the original table and its surrounding text on pages 443--444. The extracted sentence is literally the caption "Does a CSP have a well-behaved solution?" for a table distinguishing continuous, Borel and measurable solutions under different local-lemma hypotheses. No one missing theorem is recoverable from the caption alone. The following auxiliary compactness proof makes explicit why ordinary existence is weaker than the table's regularity questions; it does not replace the damaged target with a new conjecture.

Research attempt: Countable CSP compactness and the failure of a fixed local test

A complete countable finite-domain compactness argument.

Let variables be $v_1,v_2,\ldots$, with a fixed nonempty finite domain $D$, and let constraints $C_1,C_2,\ldots$ each involve finitely many variables. Assume every finite set of constraints is satisfiable. For each $n$, let $T_n$ be the set of assignments to $v_1,\ldots,v_n$ that extend to an assignment satisfying $C_1,\ldots,C_n$; this extension may freely assign all other variables. Each $T_n$ is nonempty and finite. Restricting a member of $T_{n+1}$ to its first $n$ entries gives a member of $T_n$. These levels form a finitely branching tree with a node at every depth. Starting at its root, select a child having descendants at arbitrarily large depths. Such a child exists because there are finitely many children: if each had a finite depth bound, their maximum would bound the parent's descendants. Fix an ordering of $D$ and choose the first qualifying child at each step. This constructs an infinite branch and a full assignment $a$. For a particular $C_i$, choose $n$ at least $i$ and at least all variable indices occurring in $C_i$. The first $n$ branch entries lie in $T_n$, so extend to a solution of $C_i$; since its variables are already fixed, $a$ itself satisfies $C_i$. Hence all constraints are simultaneously satisfied.

The branch rule requires determining whether a node has arbitrarily deep extensions. That is an infinitary property, not an algorithm of bounded local radius and not a continuity or Borel-selection proof on an uncountable space of instances. Compactness alone has not supplied any of the adjectives in the table caption.

A fixed-radius satisfiability test cannot replace compactness.

Take proper two-coloring constraints on an odd cycle of length $L>2r+1$, where $r\geq1$. Every radius-$r$ induced neighborhood is a path and has a proper two-coloring, obtained by alternating colors along its vertices. The entire cycle is not two-colorable: following its $L$ edges flips a color an odd number of times and returns the opposite color to the starting vertex. Thus passing all radius-$r$ neighborhood satisfiability checks does not imply a global solution. This does not contradict the compactness theorem, whose hypothesis quantifies over every finite set of constraints, including all constraints of this finite cycle. The discrepancy is exactly between independently chosen local witnesses and one compatible assignment on their overlaps.

Verification and adversarial audit

The finite domain is nonempty and every constraint has finite scope. The tree levels use extendability to the first $n$ constraints, so a constraint involving a high-index variable is not accidentally checked before its variables exist. The cycle length makes each stated neighborhood a path; the case $r=0$ can also be handled by isolated-vertex neighborhoods but is not needed. Neither example is presented as a counterexample to a stated row of the source table.

Audit corrections.

Read and retained the actual table-caption provenance. Distinguished finite satisfiability, fixed-radius tests and regular selection, and stated the finite-domain hypotheses needed by the branch construction.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The original record is an unrecoverable table caption and remains so. This provides an actual compactness argument and a locality counterexample for a clearly declared auxiliary CSP model, with no claimed resolution of the table's Borel, measurable or continuous local-lemma theorems and open entries.

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/46944 (consulted 2026-10-11).

Model and contributor disclosure

Codex GPT-6 completed written-proof review of this attempt at its declared scope; this is model-assisted review, not independent human proof checking. 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.