2026-07-21 12:33 UTCgpt-5.6-sol · high

Exact all-profile test of the uncoloured union-support adjacency-row Delsarte pair relaxation at n=43,44,45.

Progress

The hypothesis was falsified over the complete declared domain. There are 6,992,920, 953,580, and 106,076 handshake-parity profiles at orders 43, 44, and 45. The published extremal-excess interval removes none. Every profile satisfies the uncoloured pair LP by assigning distance 22 to even degree sums and distance 21 at n=43 or 23 at n=44,45 to odd degree sums. This changes no Ramsey bound.

Strategy and discriminator

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.

Hypothesis: The uncoloured union-support adjacency-row Delsarte pair LP excludes at least one degree histogram at n=43,44,45 surviving published degree and extremal-excess interval restrictions.

Test: Construct or refute an exact feasible pair-distance distribution for every admissible histogram; accept a cut only with exact arithmetic and independent reconstruction.

Rationale

Every selected distance lies in the exact union support for every allowed degree pair. For every possible number of even-degree vertices and every Krawtchouk index, the resulting integer Delsarte sum is nonnegative. The minima at n=43,44,45 are respectively 1,0,1.

Claims requiring scrutiny

The uncoloured union-support adjacency-row Delsarte pair LP cuts zero of the 8,052,576 handshake-parity degree histograms in the Ramsey degree bands at orders 43 through 45.
The 656 supplied order-42 controls yield 194 degree histograms and off-diagonal row distances 12 through 28 under byte-identical Python and C ledgers.
Across those controls, 12,144 ordered edge pairs attain 13 common neighbours and 12,144 ordered nonedge pairs attain 13 common nonneighbours.

Evidence and scope

artifacts/rowpair_v1/rowpair_v1_report.json; SHA-256 ef8171abecd9e53ff4915c1e17b594f167829669a02e6fb42e7d0c77ddf14ce3
artifacts/rowpair_v1/rowpair_v1_cold_audit.json; SHA-256 35c1135b3b26f68a2281e496cd060029177c3ede53271705988b2858b1aa952d
artifacts/rowpair_v1/control_ledger_python.tsv and control_ledger_c.tsv; common SHA-256 48df90ead7bfec093a33e14dad2906eb070413c1d4c2cd9d99e6e8ced3c1f9cc
docs/rowpair-v1.md contains the exact formulation and replay instructions.
records/research-state.json and CHECKPOINT.md contain the durable route disposition.

Computational experiments

.proof-experiments/20260721-121817-c060be: failed closed; complement control exposed an endpoint-excluding common-nonneighbour off-by-two bug in the new ledger.
.proof-experiments/20260721-122138-bd5745: corrected exact gate passed in 29.588 seconds with 42,136 KiB peak child RSS.
.proof-experiments/20260721-122217-a1e4eb: independent cold audit passed in 5.800 seconds with 54,484 KiB peak child RSS.

Independent checker

checkers/rowpair_certificate.py uses a three-term integer Krawtchouk recurrence instead of the producer's binomial sum, dynamic programming instead of recursive profile enumeration, exhausts all 33,867 labelled graphs through order 6, and rejects seven mutations. A separately written C parser/bitset ledger is byte-identical to the Python ledger.

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

Binary-code Delsarte bounds -> adjacency rows might exclude scalar Ramsey profiles -> the uncoloured union-support formulation instead admits a universal central-distance template and cuts none.
Published Ramsey excess identities -> edge-extremal intervals might sharply reduce the profile set -> every per-degree interval straddles zero, so the interval-level baseline retains all profiles.

Established facts

Every declared degree histogram at n=43,44,45 has the explicit central-distance feasible point in the uncoloured pair relaxation.
Exact support and Krawtchouk tables in artifacts/rowpair_v1/rowpair_v1_report.json, independently reconstructed by the cold checker. · The uncoloured union-support pair LP only; no edge-colour marginals, adjacency symmetry, triple moments, or graph-existence conclusion. · computed
The two independent control-ledger implementations agree exactly on all supplied graphs and complements.
The retained TSV files are byte-identical with SHA-256 48df90ead7bfec093a33e14dad2906eb070413c1d4c2cd9d99e6e8ced3c1f9cc. · The authenticated supplied 656 controls, not an exhaustive census. · computed

Ruled out in this epoch

Exclude an order-43, 44, or 45 degree histogram using only the union of edge/nonedge row-distance supports and ordinary Delsarte inequalities.
All 8,052,576 degree histograms satisfying the exact degree bands, vertex count, and handshake parity. · Every profile has an explicit feasible central-distance distribution. · Main report and independent cold audit. · Demonstrate a source, degree-domain, support, profile-count, or Krawtchouk-certificate defect. A coloured adjacency-aware model is a different route.

Open leads

Reconstruct a stronger order-45 I2/I3 scalar baseline from authenticated edge-extremal (4,5,n) data.
The edge-extremal interval is vacuous, while the published subgraph-count LP identifies triangle and refined neighbourhood features as the established next strengthening. · Fetch and hash the ANU archive index and r45extreme.tar.gz, inventory their schemas, and map every available feature to I2/I3 before generating profiles. · high · open

Continuation checkpoint

Objective: Determine whether authenticated edge-extremal archives contain enough triangle and pointed features to reconstruct the published I2/I3 order-45 baseline.

First action: Fetch and hash https://users.cecs.anu.edu.au/~bdm/data/ramsey.html and r45extreme.tar.gz, inventory their files, and map available fields to equations I2 and I3 in sources/rowpair/subgraph-counting-r55.pdf.

Stop condition: Stop if provenance fails, required I3 features are absent, or the archive only reproduces the already recorded edge extrema.

Next moves

Authenticate and inventory the public r45extreme archives.
Determine whether they expose the triangle and pointed-neighbourhood features needed to reconstruct the published I2/I3 scalar baseline for order 45.
Do not construct a custom Terwilliger SDP or enumerate the 8,052,576 profiles under the vacuous baseline.

Citations

Tool disclosure

GPT-5.6 Sol acted as principal investigator. Supplied GPT-5.6 Terra literature-strategy and experiment-verification memos were advisory and promoted with provenance; no new subagent was spawned. Deterministic tools were Python 3.12.3, GCC 13.3.0 with UndefinedBehaviorSanitizer, exact Python integers, SHA-256, jq, curl, pdftotext, and the computational-researcher experiment harness. The first run failed closed on an off-by-two ledger bug; only the corrected gate and cold audit support the result. No CAS, numerical LP solver, SAT solver, proof assistant, external publication, or system-level change was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration: 1981.0s
Review state: not a result claim
Attempt ID: ramsey-r55-20260721-123303-2aa131

Human review ledger

No human review recorded.