Exact covering number C(15,6,3)

A corrected bounded audit found no VeriPB/CakePB executable or source-name candidate in PATH and five declared roots, and the installed GitHub connector exposed no repository for either checker. An independent traversal reproduced the result, preserved the epoch-128 calibration hashes, and rejected five receipt mutations. No target formula or cube ran; 30 <= C(15,6,3) <= 31 is unchanged.

CaDiCaL 1.7.3's previously missed --lratveripb path emitted a deterministic 129-byte VeriPB 2.0 proof for a transparent two-clause contradiction. Independent exact semantics, four proof mutations, a satisfiable base mismatch, two mode controls, and a nine-file rerun passed. VeriPB and CakePB remain unavailable, so this is emitter-only infrastructure progress; no target cube ran and 30 <= C(15,6,3) <= 31 is unchanged.

The exact overlap-one two-subset Venn projection for labelled H=3K5 is feasible. A producer found a 30-block type vector over 18 types; a separate all-5005-block reconstruction verified all 29 rows, rejected five mutations, and the producer reran byte-identically. This closes only one automorphism orbit of an aggregate relaxation and leaves 30 <= C(15,6,3) <= 31 unchanged.

Exact one-subset intersection-histogram enumeration found that all 124 degree/cut-compatible (s,H(S)) pairs survive. Independent bitset DP, all-32768-subset identity checks, five mutations, and a byte-identical producer rerun passed. The result closes only this H(S)-aggregate route; C(15,6,3) remains between 30 and 31.

Global block-pair intersection moments give two exact identities and the necessary interval 15+ceil(Q/3)<=R<=floor(115+7Q/6). Independent reconstruction and four mutation controls passed. All 50 saved marginal witnesses pass with wide margins, so the lemma is retained but unstructured sampling is redirected. The covering range remains 30 through 31.

The eleven-implication encoding exactly realizes the 55 canonical third blocks in the fixed-r=2 branch and is materially smaller than the failed selector DNF. Independent validation passed. Nevertheless, both seed-paired CaDiCaL conflict-rate ratios were below one, all runs were UNKNOWN, and no witness or proof was produced; the maintained range is unchanged.

An exact two-anchor stabilizer quotient reduced the distinguished third-block domain from 5005 labelled blocks to 55 representatives throughout the complete fixed-r=2 branch. Independent reconstruction passed. The 55-selector DNF realization failed the frozen two-seed CaDiCaL gate and produced no witness or proof.

The global incidence-balance LRAT discriminator failed its 16 MiB gate. Independent checks confirmed the formula and rejected the incomplete prefix. No legitimate covering case was eliminated, so 30 <= C(15,6,3) <= 31 remains unchanged.

The depth-two first-hit discriminator passed exactly: 18933 forced-representative coordinates collapse to 250 colored unordered-pair orbits with counts 14,43,30,92,71, and a separate composition-table checker covers all 2003001 labelled pairs per profile. The predeclared empirical depth-four projection is 76434, but comparison with validated global frontiers shows the owner map is ordinary symmetry bookkeeping, not additional completion-case elimination. No complete link, cover, or UNSAT certificate was produced, so 30 <= C(15,6,3) <= 31 remains unchanged.

The PB proof route was held because its pinned producer/replayers are absent and revision-conflicted. A solver-free redirect established and independently checked 2,4,3,6,5 stabilizer orbits for the five forced root-link profiles. Whole-orbit first-hit ownership reduces aggregate residual selector coordinates from 40020 to 18933. No link tail or global cover was classified, so 30 <= C(15,6,3) <= 31 remains unchanged.

The proposed joint local-admissibility/triple-excess shared-support route was falsified at preflight. For H=3K5 its natural formulation has 43260 variables and 430535 nonzeros, and exact incidence tightness proves that its 30-support threshold is already the original fixed-H 30-cover problem. Independent reconstruction and mutation controls passed; no solver search was run and 30 <= C(15,6,3) <= 31 is unchanged.

A universal per-block four-moment lemma was implemented as an exact lift filter. It retained only 169 and 167 of 5005 candidate blocks for the two named epoch-114 H/e controls and found required triples with zero eligible support, proving those two stored excess vectors do not lift. Independent DP reconstruction, a cyclic boundary control, four mutations, and a byte-identical rerun passed. Neither complete H nor any global cover case was excluded, so 30 <= C(15,6,3) <= 31 is unchanged.

The frozen unseeded sequential-unary root-link selector search failed its known-SAT five-second qualification. Forced SAT and independent model validation passed, but the unforced arm timed out. No unknown type or global case was classified, so 30 <= C(15,6,3) <= 31 is unchanged.

All five flat Z3 root-link feasibility runs timed out after 20 seconds each, including a type with a source-verified witness. The independent audit validated the five target vectors, the archived positive control, and three mutations. No link type was newly classified and 30 <= C(15,6,3) <= 31 is unchanged.

The high-three-bit owner passed for all 18316 orbit-0 and 7798 orbit-5 canonical profiles. Exact vectors are [2231,2342,2283,2251,2273,2323,2311,2302] and [933,962,993,973,995,983,1030,929]; maximum 2342. Independent finite-coverage validation and five mutations passed, and the producer reran byte-identically. Combined with prior orbit-1 evidence, the 62437-profile count-level frontier now has 82 restart chunks with maximum 3682. This eliminates no profile and leaves 30 <= C(15,6,3) <= 31 unchanged.

A direct double count forces every putative cover's triple-excess vector to be a capped triangle decomposition of 3*K15+4*H. Exact independently checked decompositions exist for 3K5, 2C15, and 48 deterministic sampled weighted 4-regular H. The relaxation eliminated no graph, so unstructured sampling is closed and 30 <= C(15,6,3) <= 31 remains unchanged.

The high-bit parent-rich stress pilot passed for all 21182 canonical depth-four internal-orbit-1 profiles. Eight chunks contain [2664,2679,2637,2561,2655,2636,2706,2644] records. Independent validation reconstructed all 80851 deletion-parent incidences over 1236 parent keys, checked membership in the frozen 1775-key ledger, verified orbit weights and exact union, reproduced clean/resume and checker receipts byte-for-byte, and rejected seven corruptions. This is scoped ownership infrastructure only; no profile was eliminated and 30 <= C(15,6,3) <= 31 is unchanged.

The count-only high-bit preflight passed. Exact loads are [2664,2679,2637,2561,2655,2636,2706,2644], totaling 21182 canonical profiles from 78278 raw tables. Independent key-level reconstruction, restart identity, rerun identity, the frozen low-bit control, and six mutations passed. This qualifies only the next parent-rich orbit-1 pilot; C(15,6,3) remains between 30 and 31.

The genuine r=2 exact-degree-12 incidence formula was reconstructed exactly and tested with qualified ASCII-DRAT production. The corrected matched-work run hit the exact 8 MiB proof cap, and its incomplete prefix was independently rejected. No SAT witness or complete UNSAT certificate was produced, so 30 <= C(15,6,3) <= 31 is unchanged.

The eight-way low-bit FNV subchunk hypothesis was falsified. C++ bucket 0 contained 5204 profiles and triggered the frozen cap; independent exhaustive Python enumeration found exact loads [5204,0,5558,0,5246,0,5174,0], totaling 21182. This rules out one partition only. Parent fields, restart identity, the full ledger, and C(15,6,3) remain unresolved.

The full-ledger chunk-size hypothesis was falsified. Python Burnside DP and standalone C++ enumeration agree on every raw and canonical count across all 61 internal orbits, totaling 62437. Chunks 0, 1, and 5 contain 18316, 21182, and 7798 profiles, so the 5000-record cap is invalid. Full-checker preflight passed, but no missing chunk was materialized and C(15,6,3) remains between 30 and 31.

A generic pair-surplus depth-four chunk producer passed a clean-versus-resume gate on internal orbits 16 and 38. It produced 2550 profiles; independent Burnside and parent checks passed; chunks and manifests were byte-identical; semantic mutations, scope drift, and baseline divergence were rejected. This validates segmented ownership infrastructure only, so 30 <= C(15,6,3) <= 31 remains unchanged.

The complete frozen type-(8,4^13) depth-four layer was scanned with exact individual-degree capacities. All 167 blockers replay independently, including three not found by the prior aggregate filter. The result is a narrow local reduction only; its 7.3959% rejection rate fails the 90% route gate, and 30 <= C(15,6,3) <= 31 remains unchanged.

A complete 4.08% internal-edge slice of the exact depth-four pair-surplus profile layer was materialized: 17337 raw S11 tables give 2548 canonical profiles representing 18667722645 labelled prefixes, and every deletion parent lies among 296 keys in the frozen depth-three ledger. A different Burnside/polynomial checker, seven mutations, a byte-identical rerun, and a cap boundary control passed. This is structural infrastructure only; 30 <= C(15,6,3) <= 31 is unchanged.

The complete four-row degree-feasible pair-surplus prefix universe has exactly 1292182981632 labelled prefixes, 1341414 S11 multiset states, and 62437 S4 x S11 profiles. Independent fixed-point validation, six mutations, and a byte-identical producer rerun passed. This is structural progress only; 30 <= C(15,6,3) <= 31 is unchanged.

Binary DRAT qualified on the known q0-m0 UNSAT control and cut its proof by 52.378979%, with fresh conversion and dual replay. The frozen rank-one target nevertheless hit exactly 16 MiB and returned UNKNOWN; an independent checker reconstructed the formula and rejected the prefix. No link, exclusion, or covering bound resulted.

A hash-owned root-link completion formula was built and independently reconstructed. Degree conservation proved its exact-eight totalizer redundant, reducing variables by 21.16% and clauses by 28.83%. A verifier-incompatible padded-header defect was found and repaired. The canonical ASCII-DRAT run still hit 16 MiB and its prefix was explicitly rejected, so no link, exclusion, or covering bound resulted.

The complete degree-feasible three-row pair-surplus prefix universe contains exactly 2942341831 labelled prefixes, 9001 S12 multiset states, and 1775 S3 x S12 profiles. Every one of the 82 depth-two profiles occurs as a deletion parent. Independent Burnside/DP validation, deterministic reruns, seven mutations, and cap boundaries passed. This is structural progress only; 30 <= C(15,6,3) <= 31 is unchanged.

Exact nauty weight-coloured gadget codes and an independent weighted GraphMatcher partition agreed on all 210 unordered-root instances of H=2*C15 and its one-edge weight mutation. They produced 7 and 56 orbits, a weight-blind control reverted the mutation to 7, 13 corruptions were rejected, and both receipts reran byte-identically. This validates bounded ownership infrastructure only; 30 <= C(15,6,3) <= 31 is unchanged.

Exact nauty coloured-gadget codes and an independent weighted GraphMatcher partition agreed on all 210 unordered-root instances of the saved 3K5 and circulant surplus graphs. They yielded 2 and 7 root-pair orbits, selected one whole minimum-code orbit on each graph, rejected nine mutations, and reran byte-identically. The first checker failure exposed and repaired a hash-only receipt weakness. This is bounded ownership infrastructure only; 30 <= C(15,6,3) <= 31 is unchanged.

The retained epoch-94 type-2 local UNSAT proof replayed exactly in the current environment, including independent formula reconstruction, byte-identical LRAT conversion, dual intact acceptance, and eleven rejected mutations. This is a replication control only: no new cylinder was excluded and 30 <= C(15,6,3) <= 31 remains unchanged.

The forced pair-surplus multigraph has an exact independently checked two-row prefix layer: 3,489,526 labelled prefixes reduce to 126 ordered-root and 82 unordered-root orbits, each with one parent owner. This supplies a globally relevant ownership prefix but no complete H catalogue, cover, exclusion, or changed bound.

The fixed circulant-H maximum-q branch returned UNKNOWN and changed no covering bound. Independently, the complete fixed-H root catalogue was computed twice: 5005 labelled blocks form 185 orbits, exactly 70 of which satisfy q>=5. Combined with the saved anchor lemma, those 70 orbits are existence-complete for this one H.

The bounded MiniSat alternative-solver hypothesis failed: the exact H=3K5 CNF remained UNKNOWN after 5.04317 CPU seconds. Independent parsing and four fail-closed controls passed, but no cover, proof, local exclusion, or global bound was produced.

The exact H=3K5 local cylinder was compiled and audited. Fresh pair counters caused a proof-cap failure. Reusing inherited thresholds reduced the formula from 43153 variables and 203823 clauses to 33178 variables and 156603 clauses, but the reduced proof run also hit the cap and a 60-second no-proof run remained UNKNOWN. No covering bound changed.

The minimum domain-separated SHA-256 node of the validated type-(6,6,4^12) depth-four frontier was certified to have no 30-block completion. CaDiCaL returned UNSAT in 6.403 seconds; the 13.73 MB DRAT and 9.82 MB LRAT passed independent byte-identical CNF reconstruction, fresh conversion, dual replay, and eleven mutations. This is one local exclusion only, so the maintained range remains 30 <= C(15,6,3) <= 31.

The pending independent replay passed and validates the complete type-(6,6,4^12) partial root-link frontier through depth four with counts 1,3,30,381,9429. No root profile or 30-block cover was decided, so 30 <= C(15,6,3) <= 31 remains unchanged.

Canonical-parent and packed global-closure generators independently emitted identical claimed (6,6,4^12) root-link frontiers through depth four, with counts 1,3,30,381,9429. A structural checker passed, but the predeclared exhaustive third replay could not be queued, so no completeness promotion or covering bound change is claimed.

The one-direction triple-support variant was proved projection-equivalent and independently reconstructed. It removed 13,650 raw clauses and substantially reduced active clauses, but slightly increased active variables, failing the predeclared joint gate. No search ran and 30 <= C(15,6,3) <= 31 remains unchanged.

The exact r=0 point stabilizer supports sound simultaneous row and residual-column lex ordering. The 12-comparator CNF was independently reconstructed and attacked successfully, but one-round preprocessing made both active dimensions larger. No solver search ran, and the exact range remains 30 to 31.

A complete proof package certifies that q0-m0 has no 30-block completion. The DRAT is 8757790 bytes and the LRAT is 4537565 bytes; independent reconstruction, two full LRAT replayers, fresh DRAT reconversion, and nine controls passed. This is one local prefix exclusion only, so C(15,6,3) remains open with range 30 to 31.

Native external LRAT failed the q0-m0 8-MiB gate twice with an identical capped prefix. The no-proof arm reported UNSAT after 9310 conflicts, but no complete proof exists. Independent reconstruction and rejection checks passed. The global range remains 30 <= C(15,6,3) <= 31.

The 32-key sample was frozen and independently reconstructed. Lab registration failed before any job existed, so the full tranche was not run interactively. The first frozen leaf was run twice locally and deterministically hit the exact 8-MiB DRAT cap before SAT/UNSAT. The maintained range remains 30 <= C(15,6,3) <= 31.

A clean-room global orbit-closure implementation exactly reproduced the complete reported type-0 frontier through depth four: counts 1,2,14,128,2258, all 2403 records, all 2402 canonical parents, and both frontier digests matched. A separate checker and fresh rerun passed. The result is scoped verification infrastructure and leaves 30 <= C(15,6,3) <= 31 unchanged.

The canonical-header proof bridge passed independently on q3. Fresh CaDiCaL DRAT was verified and converted by drat-trim; the LRAT was accepted by CakeLPR; independent reconstruction and reconversion matched exactly; seven controls failed closed or detected mutation. This certifies one named cylinder only and leaves 30 <= C(15,6,3) <= 31 unchanged.

The canonical-header proof bridge passed independently on q0 and q1. Fresh CaDiCaL DRATs were verified and converted by drat-trim, the LRATs were accepted by CakeLPR, independent reconstruction and reconversion matched exactly, and six controls failed closed. This repairs a verification-infrastructure blocker on two named cylinders but does not change 30 <= C(15,6,3) <= 31.

Four complete canonical root-prefix cylinders were built and independently reconstructed. Three are UNSAT with CakeLPR-accepted LRATs, one is UNKNOWN, and the dual-replay scale gate failed because lrat-check segfaulted. The maintained range remains 30 <= C(15,6,3) <= 31.

The C3 quotient and all four exact-count CNFs passed independent reconstruction. Four seed-0 CaDiCaL 1.7.3 runs at 20 seconds each all returned UNKNOWN. No cover, UNSAT proof, restricted exclusion or global bound change resulted.

The exact PB calibration and its independent semantic controls passed, but the proof toolchain gate failed closed. No proof was emitted or replayed, no legitimate covering branch was tested, and the maintained range remains 30 <= C(15,6,3) <= 31.

The solver-free cross-artifact dominance gate redirected. Independent reconstruction found 249 unique fixed-first owners and four valid terminal fixed-link predicates, but their raw bases differ. Consequently none of 996 pairs licensed implication, no live owner was eliminated, and the maintained range remains 30 <= C(15,6,3) <= 31.

Pinned DRAT-to-LRAT transport was independently qualified on one already known local UNSAT cylinder. Fresh 13270-byte DRAT converted to a dual-replayed 185277-byte LRAT, but this was 1.4114 times the 131272-byte direct LRAT. Four final semantic mutations were rejected. No cover or new exclusion was produced, so 30 <= C(15,6,3) <= 31 remains unchanged.

Canonical ownership passed as arithmetic on the exact retained objects but failed its research hypothesis. The fixed-link group is order two; the only complete graph orbit was already known. Four certified partial cylinders generate six labelled images, overlap, and cover no declared complete frontier. Exact dominance leaves three maximal labelled cylinders in two orbits. No covering bound changed.

A fresh clean-room audit reconstructed all six 33162-variable, 156392-clause incidence formulas and verified their exhaustive overlapping case cover. Six sequential seed-0 five-second CaDiCaL runs all returned UNKNOWN, totaling 104678 conflicts in 29.95 process seconds. No witness or UNSAT certificate was produced, so 30 <= C(15,6,3) <= 31 remains unchanged.

The external-only profile-129 run returned UNKNOWN. Its 1697671-byte LRAT prefix was rejected by both proof replayers. Throughput was 123.9669 conflicts/process-second and proof growth was 2829.4517 bytes/conflict, failing both frozen scale gates. No covering bound or profile state changed.

The genuine r=5 formula returned UNKNOWN. Its incomplete 1908453-byte LRAT was dual-rejected, all semantic controls passed, and no covering bound changed.

A deterministic 11-row fixed-link equality cylinder was certified UNSAT. Its 7433-variable, 31251-clause CNF and 131272-byte LRAT passed clean-room reconstruction, dual intact replay, dual final-line-deletion rejection, four repaired-hash semantic mutations, and a byte-identical checker rerun. Its sibling is UNKNOWN. The global range remains 30 <= C(15,6,3) <= 31.

A deterministic 22-row fixed-link equality cylinder was certified UNSAT. Its 8177-variable, 34190-clause CNF and 462268-byte LRAT passed independent reconstruction, dual intact replay, dual final-line-deletion rejection, and four content-mutation controls. The 23-row sibling is UNKNOWN. The global range remains 30 <= C(15,6,3) <= 31.

Both automorphism-stable halves of the epoch-71 pair-row conditioning are independently certified UNSAT. The result generalizes a two-profile exclusion to two explicit cylinders containing at least six checked admissible simple profiles, but remains confined to one fixed root link and does not change the global range.

A 12,821-variable, 52,445-clause fixed-link exact-pair formula was certified UNSAT. Its proof excludes the selected graph and one distinct automorphic image, but no whole link or global cover family. The maintained range remains 30 <= C(15,6,3) <= 31.

The exact profile-129 S9 normalization and 5.364% CNF compression were independently validated. Nevertheless, all four matched runs were UNKNOWN, conflict throughput and proof-efficiency gates both failed, and both replay tools rejected every incomplete LRAT prefix. No covering bound or profile state changed.

Profile-129 was correctly materialized and independently checked, but the frozen CaDiCaL run returned UNKNOWN. Both replay tools rejected its incomplete LRAT prefix. No covering bound or profile state changed.

One conditional multiplicity-eight neighborhood type was excluded. Its exact CNF is UNSAT with a dual-replayed 9.56 MB LRAT, and a materially different DFS exhausted an even weaker repeated-block formulation. The maintained covering range remains 30 <= C(15,6,3) <= 31.

The exact CaDiCaL 1.7.3 -P1 proof-compression mechanism failed. Fresh -P0 reproduced the retained complete LRAT exactly. -P1 crossed the 16 MiB cap before termination and was worse on both proof bytes and process time. No cover or new exclusion was produced.

The matched support-three method gate failed. No cover or exhaustive exclusion was produced, so the maintained range remains 30 <= C(15,6,3) <= 31.

All six retained incidence CNFs were independently reconstructed clause-for-clause. The audit checked 938352 clauses and explicit S6 x S9 maps for all 5004 second blocks. Independent and mutation checks passed. This validates a reusable exhaustive encoding but produces neither a cover nor an UNSAT certificate, so the maintained range remains 30 <= C(15,6,3) <= 31.

A sound two-cut profile lemma and independently checked implementation eliminated 11 of 256 sampled one-cut survivors. The observed 4.296875% rate failed the predeclared 26-node scale gate, so the unchanged full scan is closed. The exact covering range remains 30 <= C(15,6,3) <= 31.

The exact 16-row no-proof r=2 frontier was structurally valid but found no SAT model and achieved only 0.6956225517262626x the unsplit conflict rate. Fourteen raw UNSAT statuses lack proofs and carry no mathematical force. The covering range remains 30 <= C(15,6,3) <= 31.

A sound residual cut-capacity filter independently excluded 164 of 2258 recorded type-0 depth-4 root-link nodes. This is genuine local progress but only 7.26 percent, below the frozen 10 percent scale gate. Internal-subset capacity excluded nothing, no cover or global UNSAT certificate was obtained, and the exact range remains 30 <= C(15,6,3) <= 31.

An exact, independently checked 16-cube partition of the legitimate r=2 incidence formula was produced. Five fresh proof-producing leaves were sampled before cube 4 exceeded the 1 MiB LRAT cap. All five results were UNKNOWN and both replayers rejected every prefix. The control was correctly skipped, no case was excluded, and 30 <= C(15,6,3) <= 31 remains unchanged.

The six-syntax CNF/LRAT pipeline passed under exact resource bounds and independent checks. All leaves were intentionally inconsistent degree controls, so no legitimate cover was excluded and the maintained range is unchanged.

The retained base's 89400 cardinality clauses and all 105 pair threshold maps were reconstructed exactly. A counter-sharing E5 projection would contain 50400 variables and 282393 clauses. Its clause gate passes, but its variable gate fails by 1355 variables, so no formula or solver run was authorized. The maintained covering range is unchanged.

A hash-bound, one-process proof calibration of the raw owner-H6 r=2 CNF returned UNKNOWN. The proof prefix stayed below all caps but was explicitly rejected by lrat-check and CakeLPR. Positive, final-line-deletion, independent-replay, and three receipt-mutation controls passed. No mathematical case was eliminated and 30 <= C(15,6,3) <= 31 remains unchanged.

The stale r=5 owned-H6 lead was rejected because exact degree 12 makes r>=3 impossible. Three sound owner-H6 formulas were compiled and independently checked. r=1 and r=2 passed the frozen 5% preprocessing gate; r=2 was selected. No solver search ran and the exact covering range did not change.

H6 was built and independently verified as the exact fixed-first q>=6 branch. It compresses seven exact-q cases into one CNF and passed the bounded H6/q6 rate and RSS gate. All four searches were UNKNOWN, and every partial proof was rejected twice. The exact range remains 30<=C(15,6,3)<=31.

E5 was implemented and independently verified as the exact fixed-first q=5 equality branch. Four proof-producing runs were UNKNOWN, all partial proofs were rejected by two fresh replayers, and E5 failed its throughput gate by a wide margin. The exact range remains 30 <= C(15,6,3) <= 31.

A human-checkable excess-moment lemma and two deterministic implementations establish that every hypothetical 30-cover admits a q>=5 normalized anchor. This removes 21 of 249 necessary arithmetic profile types for global existence search. It supplies no cover or global exclusion, so the exact range remains 30 to 31.

The first residual block in every fixed-first lex-normalized putative 30-cover omits points 0 and 1, reducing that coordinate from 5004 to 1716 types. Explicit units failed the 5% preprocessing gate. The maintained range remains 30 <= C(15,6,3) <= 31.

A local lex-degree lemma was proved and independently checked: for one fixed root link, the first residual block must omit points 1 and 2. This reduces its canonical first-column frontier from 2211 to 672 cubes. A greatest-boundary cube was dual-replayed UNSAT, while a surviving least cube remained UNKNOWN. The global covering range is unchanged.

The fixed source link has exact automorphism group generated by (4 5). Its action yields 1419 fixed and 792 paired orbits on the 3003 residual blocks, so 2211 representative-first cubes are existence-complete under the actual CNF order. The reduction is local and only 1.3582x. No SAT or UNSAT terminal result was produced, and the covering range remains unchanged.

The pass did not produce a valid research result: RuntimeError: Codex gpt-5.6-sol failed rc=0: was rejected. Python 3.12.3 generated and independently reconstructed the CNF. CaDiCaL 1.7.3 performed the bounded search and emitted LRAT. Fresh GCC builds of retained lrat-check and CakeLPR sources rejected the incomplete proof. No CAS, proof assistant, cloud lab, external proof service, or human validator produced terminal evidence.\"\n}\n```"}} {"type":"item.completed","item":{"id":"item_1","type":"todo_list","items":[{"text":"Audit workspace, retained route artifacts, pinned solvers, and authoritative status","completed":true},{"text":"Predeclare the cheapest proof-producing calibration and its fail-closed gates","completed":true},{"text":"Run through the reproducibility harness and independently attack the result","completed":true},{"text":"Write immutable evidence/checkpoint artifacts and report exact scoped outcome","completed":true}]}} {"type":"turn.completed","usage":{"input_tokens":4489104,"cached_input_tokens":4323840,"output_tokens":41998,"reasoning_output_tokens":10553}}

A proof-compatible (8,4^13) root-profile incidence cube was implemented and independently reconstructed. The corrected 20-second run was UNKNOWN. The route failed its scale-up gate, but the epoch found and corrected a zero-padded DIMACS/LRAT replay defect and normalized nondeterministic verifier failure outputs into reproducible semantic receipts. The exact range remains 30 <= C(15,6,3) <= 31.

The pass did not produce a valid research result: RuntimeError: Codex gpt-5.6-sol failed rc=0: ent harness. Web search checked the Covering Repository context, Gordon-Kuperberg-Patashnik, and arXiv:2607.23766. Exploratory Z3 4.13.0, CaDiCaL 1.7.3, and HiGHS 1.2.0 probes were timeout/unknown and are not evidence. No CAS, proof assistant, cloud lab, external proof service, or human validator produced a terminal certificate.\"\n}\n```"}} {"type":"item.completed","item":{"id":"item_1","type":"todo_list","items":[{"text":"Audit workspace, delegate memos, prior artifacts, and official/current status","completed":true},{"text":"Predeclare and implement the cheapest bounded discriminator with independent checks","completed":true},{"text":"Run the experiment through the reproducibility harness and attack the result","completed":true},{"text":"Write immutable evidence/checkpoint artifacts and report the exact scoped outcome","completed":true}]}} {"type":"turn.completed","usage":{"input_tokens":16116197,"cached_input_tokens":15840768,"output_tokens":64087,"reasoning_output_tokens":15889}}

All nine conditional pair-star incidence leaves were built and independently reconstructed. Their preprocessing sizes were only about 1.3x smaller than the control, so every strict 2x gate failed and no solver search ran. The nine-template quotient remains valid infrastructure, but this unchanged composite should not be scaled.

A reusable-totalizer fixed-second profile formula was independently reconstructed and reduced clauses from 186668 to 164858, but all four five-second proof-producing runs were UNKNOWN. The geometric-mean conflict-rate ratio was only 1.011798 and proof bytes per conflict worsened to 1.022287, so both continuation gates failed. Both lrat-check and CakeLPR rejected every incomplete trace. The exact range remains 30 <= C(15,6,3) <= 31.

The integrated exact C5 two-orbit tail passed its 5x operational gate by a wide margin: 98827 versus 575 four-orbit prefixes under matched 100000-node arms. A separate implementation validated the index semantics and initial decisions. No witness was found, every root remained capped, and the exact range stays 30 <= C(15,6,3) <= 31.

A complete fixed-first exactly-once 249-profile manifest was built and independently checked. It retains one previously dual-replayed UNSAT profile and 248 unresolved profiles. A matched rank-2 calibration returned four UNKNOWN results and failed both continuation gates. The exact range remains 30 <= C(15,6,3) <= 31.

The retained rank-1 LRAT passed fresh dual replay. The new rank-2 profile was encoded exactly and independently reconstructed, but its 60-second solve returned UNKNOWN. Both proof checkers rejected the incomplete 19.5 MB LRAT prefix. No profile or bound was eliminated.

The independently ranked smallest unique-minimum-owner profile was encoded exactly and certified UNSAT. This excludes profile (0,1,25,0,0,3) but leaves 248 necessary profiles and the exact covering number unresolved. The audit also found and corrected a status-line parser failure without altering the raw solver receipt.

An exact two-orbit C5 tail index passed its memory, build, correctness, positive-control, independent-checking, and local speed gates. None of 256 calibration prefixes had a completing pair. No cover, C5 exclusion, global exclusion, or improved bound was obtained.

The exact C5 one-orbit tail index passed its efficiency and independent-soundness gates. No cover was found, all eight cases remained capped, and no covering bound changed.

The exact r=0,q=3 cell compiled at the predicted dimensions and passed independent structural and semantic checks. Its bounded proof-producing run returned UNKNOWN; two fresh replayers rejected the incomplete LRAT prefix. No profile, cell, or covering number was resolved.

A complete 13-way internal pair-excess partition was derived, compiled, and independently checked. Exact profile enumeration further reduced 278256 weak profiles to 249 necessary profiles in 30 live cells and proved nine (r,q) cells empty. The q=0 and q=12 proof-producing runs were both UNKNOWN, so C(15,6,3) remains between 30 and 31.

Corrected a delegate's 28-comparator count to 27, built and independently checked the 165674-clause strict residual formula, and ran the frozen six-run discriminator. All runs were UNKNOWN, the throughput gate failed, and both proof replayers rejected every incomplete prefix. No covering bound changed.

An exact, independently checked ten-profile frontier and selected profile CNF were produced. The selected profile is a tiny exact cell of the normalized simple r=2 candidate universe, but its 60-second proof-producing run returned UNKNOWN. Both proof replayers rejected the incomplete prefix. The exact range remains 30 <= C(15,6,3) <= 31.

A new fixed-block incidence lemma eliminates minimum-intersection branches r=3,4,5 and shows that r=2 has exactly ten simple profiles. The complete normalized r=2 CNF was independently reconstructed, but its bounded proof-producing run returned UNKNOWN. No cover or UNSAT branch proof was obtained, so the exact range remains 30 <= C(15,6,3) <= 31.

A complete disjoint six-way minimum-intersection frontier was proved and independently checked. It repairs the overlap in the prior second-block normalization and soundly removes lower-intersection candidates within five branches. No cover was found and no UNSAT branch was proved, so the exact range remains 30 <= C(15,6,3) <= 31.

The independent CakeLPR gate passed and a separately written checker reproduced it. This removes the proof-verifier blocker but proves no new covering bound; the exact range remains 30 <= C(15,6,3) <= 31.

The pair-excess prune was independently validated and rejected 29035 candidate extensions before depth six, but its direct 21-tuple implementation was 19.49 times slower than the matched baseline. Both arms hit all node caps, emitted no witness, and prove no exclusion. The exact range remains 30 <= C(15,6,3) <= 31.

An exact residual-normalizer reduction replaced 167 first-block choices by eight cases inside the C5-invariant family. A one-million-node DFS found no witness but was cap-limited in every case. Independent reconstruction and four mutation controls passed. The exact range remains 30 <= C(15,6,3) <= 31.

The retained epoch-27 and epoch-29 CNFs, LRATs, mutants, and legacy checker binaries were hash-audited. Both legacy binaries are byte-identical and derive from the same lrat-check.c source. CakeLPR was absent, so the predeclared gate returned BLOCKED and no proof replay or global search occurred. An independent checker and three mutation controls validated the fail-closed decision. The exact range remains 30 <= C(15,6,3) <= 31.

The impossible seed-6 r=5 leaf was replaced by a correctly scoped r=4 experiment. Twelve normalizations and 378 fixed-slot placements were independently reconstructed. The represented 4,956 assignments contain no cover. One maximal 126-assignment leaf was proved UNSAT by a replayed 15.9 MB LRAT. This is local proof-pipeline progress only; the exact range remains 30 <= C(15,6,3) <= 31.

The fixed-point-free C5 quotient and sparse feasibility model were independently validated. The clean ten-second MILP run timed out without a witness or useful telemetry. Thus the exact range remains 30 <= C(15,6,3) <= 31, and even the C5 symmetry class remains unresolved.

The exact value remains 30 <= C(15,6,3) <= 31. Epoch 27 repaired the proof-toolchain blocker: a full r=5 incidence calibration emitted a 2745-line LRAT proof that independent replay accepted and a truncation control rejected. A first padded-header run was invalidated after exposing octal parsing. No global cube was eliminated.

All exact-degree three-block cover repairs of the fixed seed-6 U=2 anchor were excluded. Of 4,060 deletion triples, 3,224 are cover-impossible by a sound union test. Two independent encodings agree on the remaining 836 cases, 8,484,420 canonical replacement triples, 8,414,065 valid operations, minimum scored deficit 2, and zero covers. The global range remains 30 <= C(15,6,3) <= 31.

The fixed seed-6 U=2 anchor has no disjoint block pair. All 9,882 q=3 two-block operations are identities or complement-equivalent to independently checked q=1/q=2 operations. Direct independent rescoring found 9,696 valid nonidentity q=3 families, minimum deficit 2, and zero covers. Combined with earlier receipts, no exact-degree two-block replacement improves this anchor. The global covering number remains open.

Twelve fresh noncyclic degree-regular starts produced an independently validated U=2 30-block family outside both known U=5 anchor classes. No cover was found. Complete independent q=1 and q=2 shells around this anchor also contained no cover or improving family. The exact covering number remains open.

The selected prefix has exactly 233 independently reproduced residual-excess orbits. Canonical augmentation recorded 10001 validated nodes but breached its cap during active depth 1 after reaching only profile 67. The present route is held; C(15,6,3) remains between 30 and 31.

The forced multiplicity-four four-block prefix has exactly nine independently reproduced S13 orbits and 61311250 valid labelled unordered instances. This corrects a factor-six error in all six delegate [1,1,1] orbit sizes. The result is only a prefix classification; C(15,6,3) remains between 30 and 31.

A hash-bound sparse PB formulation and independent checker were completed. They exactly compress the frozen leaf from 38727 to 1998 variables, but the single predeclared Z3 run timed out. No root-link node was eliminated and the covering range remains 30 through 31.

A hash-bound 28-node completion sample was prepared and independently reconstructed. The first exact-completion leaf remained UNKNOWN after 60 seconds, so the predeclared stop rule fired. Zero canonical nodes were eliminated and the covering range remains 30 through 31. The non-replayable 446902508-byte LRAT prefix was hash-recorded and removed; all decisive inputs, metadata, logs, and receipts remain.

A deterministic canonical root-link pilot recorded and independently validated 10001 canonical-parent nodes within the first forced degree type, exceeding the route cap in 85.374 seconds. The exact covering range remains unchanged.

The maintained range remains 30 <= C(15,6,3) <= 31. The tie anchor's complete degree-preserving radius-at-most-three shell contains 244,205 admissible operations with minimum deficit five and no cover. A materially different checker and replay/corruption controls passed. The result is strictly local.

The exact covering number remains open at 30 <= C(15,6,3) <= 31. Exact enumeration closes the degree-preserving radius-at-most-three repair shell around one fixed U=5 source anchor: 4,060 deletion triples reduce soundly to 33, yielding 294,624 admissible operations with minimum deficit five and no cover. A materially different checker and replay/mutation controls passed. No conclusion extends beyond this anchor.

Complement equivalence reduced 20,475 raw q=3 operations to 1,600 disjoint-pair operations. Exact enumeration produced 788 unique labelled families with combined minimum deficit 13 and no cover. Independent reconstruction passed. Composed with prior q<=2 receipts, this closes all one-step q<=3 and radius-two degree-preserving replacements from the two U=5 anchors, but does not change the global range 30 <= C(15,6,3) <= 31.

A complete q=1/q=2 audit from the unique epoch-14 U=5 tie found 21369 raw operations and 16291 unique nonidentity labelled families. Their minimum deficit is 5, uniquely attained by the original source. Composition with prior checked receipts shows that the source-containing U<=5 q<=2 component has exactly two vertices. No cover, improved state, or global covering-number bound was obtained.

A complete q=2 audit of the frozen degree-regular U=5 state found 15174 raw choices, 15146 legal operations, and 11626 unique nonidentity neighbor families. Their minimum deficit was 5, attained once. No cover or improved state was found; the global exact value remains open.

A deterministic 24-cell matched pilot completed 2.4 million valid proposals. Exact degree preservation passed all three route gates and produced a 30-block degree-regular state missing five triples, but no cover. Two independent exhaustive audits established that this state has no improving legal q=1 coupled swap. The exact covering range remains 30 to 31.

The proposed twelve-way canonical two-row split was rejected. Five root-row types are sound; twelve classes partition pointed endpoint states; materialized pointed prefixes form exactly 126 stabilizer orbits. An explicit 4-regular multigraph reaches two endpoint classes. The exact covering range remains 30<=C(15,6,3)<=31.

The residual missed-triple capacity predicate U_y<=10(12-r_y) was proved redundant after full root-link pair coverage. A complete 2^24 Boolean audit and independent SMT checker validated the implication. Two preliminary CaDiCaL runs were UNKNOWN and have no mathematical force. The exact range remains 30<=C(15,6,3)<=31.

The proposed root-link residual-budget filter was proved redundant for all five forced degree types. Exhaustive and independent audits passed, but no cover, exclusion, or improved covering bound was obtained.

The six incidence branches are structurally complete and independently checked, but they failed both predeclared 2x throughput gates. All solver cells were UNKNOWN. The exact covering range is unchanged. A proved residual-excess budget now strengthens the redirected canonical root-link route.

The frozen three-seed incidence-versus-selector discriminator passed all five gates. Incidence was substantially faster by solver-counter throughput and used about one tenth the memory. Every run was UNKNOWN, so the maintained range remains 30 <= C(15,6,3) <= 31.

A complete incidence-matrix SAT representation was generated and independently audited. It is substantially smaller than the selector baseline and showed higher search throughput at seed 0. Both solver results were UNKNOWN, so the exact range remains 30 to 31.

A reproducible four-cell benchmark tested one explicit rooted pair-excess profile. Independent audits passed, all solver calls returned UNKNOWN, and exact conditioning worsened conflicts and decisions in both encodings. The exact covering-number range remains 30 to 31. Initial relative-path harness defects failed before evidence and were repaired; only the final hash-bound run and checker receipt support the result.

Two independent exhaustive scans prove that the fixed archival 31-cover has no three-delete/two-insert repair. They agree on 4,495 deletions, 770 degree-feasible cases, 754 capacity-feasible cases, 18,313 insertion pairs, and zero repairs. The exact covering number remains unresolved at 30 <= C(15,6,3) <= 31.

The exact covering number remains unresolved. Two independent exhaustive implementations prove that the fixed archival 31-cover has no two-delete/one-insert repair: the pruned implementation checked 825,825 cases after safely eliminating 300 deletion pairs, and the independent implementation checked all 2,327,325 cases. The result is local and does not alter 30 <= C(15,6,3) <= 31.

The exact covering number remains unresolved. A complete six-case second-block normalization was proved and exhaustively checked, giving a safe 834/29 raw branch-representation reduction. The prescribed five-seed runner and independent checker passed a harness-only smoke audit, including correction of two fail-closed reporting defects. The scientific 60-second control was not run because checkpointed-lab registration was blocked before a job ID existed.

The exact value remains open. Four correctly offset fixed-block SAT formulas all timed out with UNKNOWN status. Pair bounds passed the literal two-encoding conflict gate, but formula growth and the weaker totalizer decision signal require multi-seed confirmation before scale-up.

The authoritative 30-31 status and historical method chain were audited. A promoted Terra cyclic enumerator was reproduced by a separate set-based implementation, exhaustively excluding the regular Z_15-invariant 30-block family and proving its maximum triple coverage is 440. Conditional pair- and triple-excess lemmas were derived. No unrestricted SAT search or exact-value claim was made.