Ramsey number R(5,5)

Qualify proof-aware canonical cubes or symmetry-first encoding on a representative hard raw-K43 tranche before production migration.

First action: Generate two shallow deterministic cube levels from one exact raw-K43 encoding; canonicalize prefixes; measure orbit reduction, tail skew, total proof volume, and checked coverage under the locked solver/checker portfolio.

Stop or redirect when: Stop on any coverage or checker disagreement, or unless held-out total CPU falls by at least 20 percent after canonicalization and proof assembly costs.

• Canonical counterexample-guided exact repair within one retained boundary.
  Enumerate at most 64 trial-1 models, block each verified known boundary assignment, and require dual checking plus canonical novelty.

• complete known-class boundary blocking
  Freeze a 30-vertex core, quotient the twelve residual labels by row-lex order, block eight validated 426-bit supplied-class assignments, and verify UNSAT with DRAT
  Reopen only if: Restore writable lab-state submission, resume the bound cold checkpoint, and obtain a final audit covering all 656 hosts with physical-CNF equality and s VERIFIED.
• symmetry-normalized exact boundary repair
  Remove the non-invariant Hamming counter, impose lexicographically nondecreasing 30-bit destroyed-to-core rows, exhaustively validate tied-row coverage, and permit one source-blocked SAT model.
  Reopen only if: Provide a proved class-level block or canonical augmentation, or demonstrate a source, mapping, comparator, or checker defect.
• canonical counterexample-guided exact repair
  Parse the immutable boundary CNF once and append one 426-literal clause blocking each verified supplied-corpus primary assignment.
  Reopen only if: A sound class-level block, exhaustively validated symmetry quotient, or demonstrated source/mapping/checker defect.
• proof-core compression for structured Ramsey exclusions
  Strip symmetry and cardinality auxiliaries from the burden-zero formulas, deletion-minimize raw K5-clause cores, canonicalize their signed incidence hypergraphs, and derive parameterized obstruction templates across the 20 second distances.
• novel order-42 basin discovery
  Freeze an induced order-30 graph, solve its 426 boundary edges exactly with minimum Hamming distance 60, then independently validate and canonicalize the first model.
  Reopen only if: Install replayable complete-boundary model blocking with a predeclared model budget.
• exact subgraph-count feature reconstruction
  Use authenticated edge-extremal catalogues to derive exact triangle, pointed-neighbourhood, and induced-feature bounds, then reconstruct the published rational I2/I3 system before seeking one new inequality.
• coding-theory distance distributions and Terwilliger moment bounds
  Freeze each degree histogram, impose degree-pair totals and the union of edge/nonedge row-distance supports, then test ordinary binary Krawtchouk inequalities. An explicit central-distance primal template replaced millions of numerical LP solves.
  Reopen only if: Retry the same uncoloured stage only after a demonstrated artifact or mathematical defect. Treat coloured edge marginals or adjacency-aware identities as a separately justified mechanism.
• certificate-carrying cube-and-conquer
  Partition the frozen normalized formula on four declared cross-part primary variables, independently verify the cover and physical leaf reconstruction, and require a checked model or DRAT/LRAT certificate per resolved leaf.
  Reopen only if: A materially different proof-logging solver or encoding, justified branching/budget evidence, or a demonstrated parent, cover, or checker defect.
• symmetry-normalized certificate-carrying SAT
  Represent every complement/relabel orbit by fixing N(0)={1,...,20}, independently audit the resulting 41 units, and run one bounded proof-logging SAT experiment.
  Reopen only if: A checked complete cube cover, materially different encoding or proof-logging solver, justified larger resources, or a demonstrated normalization-packet defect.
• novel order-42 basin discovery
  Encode every order-42 Ramsey constraint and require each degree to be 20 or 21; validate the encoding on exhaustive small controls, then run bounded proof-logging SAT.
  Reopen only if: Change the exact search configuration using the proved fixed-neighborhood normalization, a materially different encoding or solver, or justified additional resources.
• certificate-carrying two-orbit SAT
  Assign one relaxation variable to every raw five-set/color origin, impose independently derived prefix and suffix sequential at-most-one constraints, solve both encodings, and verify DRAT, LRAT, specialization, and full-graph semantics.
  Reopen only if: Reopen this strategy only after a concrete defect in its retained provenance, multiplicity, encoding, mapping, proof, or coverage evidence.
• linkage-tree gene-pool optimal mixing for gray-box Ramsey MaxSAT
  Maintain a diverse population of exact low-burden colorings. Rebuild a linkage tree each generation from elite-population mutual information, active-K5 co-occurrence, or a hybrid of both. Traverse its edge subsets from small to large, copy one subset from a donor, and accept only moves that pass the exact incremental burden evaluator and the archive's diversity rule. Compare learned, gray-box, hybrid, univariate, and uniform-crossover masks rather than assuming the visible K5 factorization is the best linkage model.

• Promote the result as fully cold-validated now.
  Fresh semantic replay covers only 7 of 656 hosts.
  Reopen only if: Complete all 656 host replays and final cold formula/proof audit
• A ninth or otherwise unblocked completion exists in the represented record-21 frozen-core row-lex boundary family.
  Verified UNSAT
  Reopen only if: A demonstrated mismatch in CNF generation, proof checking, block semantics, or symmetry coverage
• Use artifacts/known-class-embedding-prefix49-audit/provenance.json as proof that historical runtime imports were frozen.
  Recorded Git commits do not reveal temporary uncommitted worktree changes.
  Reopen only if: none; use the corrected v2 report.
• Accept stored NetworkX summaries alone as offline proof of complete mapping streams.
  Full VF2 mapping streams are not retained.
  Reopen only if: Retain complete VF2 sidecars or cold-regenerate every required stream.
• Treat the 49-host prefix as a supplied-class block or exhaustive classification.
  Only 49 of 656 supplied profiles were processed, and the supplied 656 are historically non-exhaustive.
  Reopen only if: Complete a locked 656-host census and separately justify any claimed corpus completeness.
• Continue lab-ramsey-r55-872a7c3ee855 as proof-carrying production.
  Imported modules are not explicit hashed inputs, historical dirty state is unavailable, and the aggregate 30-second gate is not enforced.
  Reopen only if: Do not reopen this job; replace it with a hash-locked fail-closed job.
• Accept stored NetworkX summaries alone as offline proof of the second complete mapping stream.
  The full NetworkX streams are not retained in host artifacts.
  Reopen only if: Retain complete second-path sidecars or cold-regenerate every positive stream.
• Treat the reviewed 33-host prefix as a complete supplied-class block.
  Only 33 of 656 supplied host representations have been processed.
  Reopen only if: Complete and cold-audit all 656 canonical hosts.
• Continue the same quotient formula with an arbitrary larger individual-normal-form cutoff.
  The quotient is not a class block, and the first post-source-block model is another supplied class.
  Reopen only if: A proved class-level block or canonical augmentation, an artifact defect, or a predeclared experiment that directly meets a field-progress gate.
• Continue the identical labelled-blocking loop with a larger arbitrary cutoff.
  The declared 64 vectors yielded no novel class and collapsed to eight S_12 normal forms; labelled blocks cannot exclude an isomorphism class.
  Reopen only if: Install a proved class-level block or sound symmetry quotient, or demonstrate an artifact defect.