2026-07-21 00:18 UTCgpt-5.6-sol · high

Authenticate Springer Supplementary Data 4 and exhaust every one-edge flip of its published two-conflict K43 matrix.

Progress

The two authenticated 3812-byte Springer bodies agree at SHA-256 c2429869f6fa47ab7388134b580b014efae01e6f0e474f5bab2233afb1ef6990. The seed has exactly two all-zero K5s, [6,12,17,36,42] and [6,12,31,36,42], and no all-one K5. All 903 single-edge flips were exhausted. The global radius-1 minimum burden is 2, attained by flips (6,12) and (36,42). Among the 741 flips avoiding vertices {6,12,36,42}, the minimum is 3, attained by seven recorded edges. No witness or global Ramsey bound was obtained.

Strategy and discriminator

exact destroy-and-repair around a published near-solution

Hash-gate two direct publisher retrievals, independently parse the raw matrix in Python and C, and compare complete monochromatic-K5 identity ledgers for all 903 one-edge flips.

Hypothesis: No single-edge flip of the authenticated publisher seed is a Ramsey witness, and every flip avoiding all four shared conflict vertices has at least three monochromatic K5s.

Test: Exhaust the lexicographically ordered 903 edges under two materially different exact encodings and compare every violation identity.

Rationale

Checker A scans every 5-subset and derives exact flip deltas; Checker B reparses the raw bytes in C and recursively enumerates graph and complement K5s afresh after every flip. Their complete ledgers agree. Mutation tests, malformed-input rejection, 41 full rescans, canonical deletion checks, and a UBSan replay passed.

Claims requiring scrutiny

For the frozen publisher matrix, every one-edge flip has at least two monochromatic K5s.
For the 741 flips whose edge avoids {6,12,36,42}, the exact minimum burden is three.
Deleting 6 or 42 yields the supplied-656 class represented by the complement of record 42; deleting 12 or 36 yields the complement of record 256.

Evidence and scope

artifacts/publisher_seed_radius1_report.json; SHA-256 0db3cca75888b70850b60e1b891d8e08349ab0bec6196cd8f4c528603fa04fe9; complete 903-entry identity ledger
.proof-experiments/20260721-000810-1f70a4; return code 0; 45.243 seconds; experiment JSON SHA-256 3ce459a1a214f1c86dcc9351e4cabbae961a210785cab8e576c78c2f0dc8f98f
.proof-experiments/20260721-001630-5af604; UBSan full-ledger replay passed; experiment JSON SHA-256 2d22200322a85d0874f75253e86be41eda5500c1bfc7a57f1f4ea8f648a07032
checkers/publisher_radius1_a.py; SHA-256 f34ea148c36ab74b0c11784f9b4d333dc915ef0a6bd3d70c7ef9c2aa03f5a535
checkers/publisher_radius1_b.c; SHA-256 b1bfd2a51511ae0f3e6caccbaa29235cf65df2f8a3cae7ccdfce934af1f5914a
records/delegate-provenance-epoch6.json; hash-linked Sol/Terra role disclosure

Computational experiments

.proof-experiments/20260720-235158-0cfef1 and 20260720-235208-a7e84e: byte-identical publisher retrievals
.proof-experiments/20260721-000810-1f70a4: decisive 903-flip experiment; global minimum 2; strict 741-flip minimum 3
.proof-experiments/20260721-001555-dfd5af: combined ASan/UBSan attempt aborted under the harness address-space limit and was not counted as validation
.proof-experiments/20260721-001630-5af604: repaired UBSan-only audit reproduced all 903 identities

Independent checker

Checker A independently parses publisher whitespace in Python and scans C(43,5) subsets with exact one-edge deltas. Checker B parses fixed raw byte positions in C and performs recursive bitset clique enumeration in the graph and complement separately for every flip.

Contribution gate

not_requested

No structured gate reasons were recorded in this legacy attempt; see the adjudication ledger.

Original model outcome: progress
Public classification: progress

Cross-domain transfers tested

Violation-hypergraph support -> an edit avoiding all four shared conflict vertices changes every relevant deletion remainder -> the resulting 741-flip slice has exact minimum burden 3.
Isolated adversarial reconstruction -> different encodings should expose bookkeeping errors -> complete-ledger comparison caught an initial Python delta defect that omitted unaffected seed conflicts; the defective result was discarded.

Established facts

The authenticated publisher seed has exactly two all-zero K5s and no all-one K5.
artifacts/publisher_seed_radius1_report.json · Publisher body SHA-256 c2429869f6fa47ab7388134b580b014efae01e6f0e474f5bab2233afb1ef6990 · computed
Every one of its 903 one-edge flips has burden at least 2.
Complete dual-checked flip ledger in artifacts/publisher_seed_radius1_report.json · Hamming distance exactly one from the frozen seed · computed
The 741 flips avoiding {6,12,36,42} have minimum burden 3.
Complete strict-family ledger and seven full minimizer identities · Exactly the recorded avoid-intersection predicate · computed

Ruled out in this epoch

Repair the authenticated publisher seed using exactly one edge flip.
All 903 one-edge flips · The exact global minimum burden remains 2. · artifacts/publisher_seed_radius1_report.json · Demonstrate a source-byte, parser, or checker defect against the retained controls; larger radii are a different route.
Escape all four known deletion remainders using one edge that avoids their deleted vertices.
All 741 edges disjoint from {6,12,36,42} · The exact minimum burden is 3. · avoid_all_shared_conflict_vertices family in the retained report · Use at least two coordinated edits or a materially different seed.

Open leads

Exhaust the complete radius-2 neighborhood and its deletion-remainder-change slice.
Only 407253 unordered pairs remain, small enough for an explicit score certificate and independent replay. · Implement lexicographically ordered C and Python radius-2 evaluators, retain every pair score and every minimizer identity, and stop at the first mismatch. · high · open

Continuation checkpoint

Objective: Exhaust and independently verify the publisher seed's complete radius-2 edge neighborhood.

First action: Run the cold replay command in CHECKPOINT.md, then enumerate the 407253 lexicographically ordered unordered edge pairs under separate C and Python implementations.

Stop condition: Stop on any source, parser, ordering, score, or identity mismatch; otherwise stop after every pair is retained and agreed. A zero must pass both complete checkers.

Next moves

Cold-replay the exact radius-1 command in CHECKPOINT.md.
Exhaust all C(903,2)=407253 unordered two-edge flips under independent C recursive-bitset and Python two-flip subset-delta encodings.
Separately classify radius-2 edit sets that change every deletion remainder: for each v in {6,12,36,42}, at least one edited edge is not incident to v.

Citations

Tool disclosure

GPT-5.6 Sol was principal investigator. Two precomputed GPT-5.6 Terra delegates supplied advisory literature-strategy and verification memos; Sol hash-verified them and independently reran all decisive evidence. Deterministic tools: Python 3.12.3, C/GCC 13.3.0, nauty labelg, curl 8.5.0/OpenSSL 3.0.13, SHA-256, jq, UBSan, an attempted ASan run limited by RLIMIT_AS, Git diagnostics, and the computational-researcher experiment harness. No SAT solver, CAS, proof assistant, or new subagent was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration: 1636.7s
Review state: not a result claim
Attempt ID: ramsey-r55-20260721-001847-bbfd27

Human review ledger

No human review recorded.