2026-08-09 21:11 UTCgpt-5.6-sol · high

Transferred the established pair-excess budget into a sound paired C5-quotient DFS and measured its pruning value and runtime cost.

No Progress

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.

Research-policy redirect

Evidence receipt creation failed; durable progress is withheld.

Strategy and discriminator

C5-invariant exact-cover DFS with pair-excess pruning

Maintain 21 pair-orbit multiplicities and reject a partial selection when its aggregate excess above multiplicity four exceeds six.

Hypothesis: Under matched sound 100000-node C5 DFS arms, the pair-excess constraint rejects at least one candidate before terminal depth six.

Test: Compare baseline and pair-enabled DFS arms at 100000 nodes each, then independently reconstruct the C5 pair quotient, threshold six, and prune-depth aggregates.

Rationale

The predeclared pruning signal passed, the quotient arithmetic and aggregates were independently reconstructed, and mutations failed closed. However, no root case was exhausted and the severe runtime regression blocks unchanged scale-up.

Claims requiring scrutiny

In the fixed C5 quotient, an extendable partial selection cannot have aggregate pair excess above four exceeding six.
The completed pair-enabled arm recorded 1685425 candidate rejections, of which 29035 occurred at depth at most five.
At equal 100000-node caps, the pair-enabled arm took 26.8641763204 seconds and the baseline took 1.3783247829 seconds.
No 30-block witness or mathematical exclusion was obtained.

Evidence and scope

python3 scripts/c5_pair_excess_pilot_v1.py --output artifacts/epoch32-20260809/c5_pair_excess_pilot_100k_receipt.json --node-limit-per-arm 100000 --time-limit-per-arm 30
python3 checkers/check_c5_pair_excess_pilot_v1.py --input artifacts/epoch32-20260809/c5_pair_excess_pilot_100k_receipt.json --output artifacts/epoch32-20260809/c5_pair_excess_checker_receipt.json
python3 checkers/test_c5_pair_excess_pilot_fail_closed_v1.py --input artifacts/epoch32-20260809/c5_pair_excess_pilot_100k_receipt.json --checker checkers/check_c5_pair_excess_pilot_v1.py --output artifacts/epoch32-20260809/c5_pair_excess_fail_closed_receipt.json
artifacts/epoch32-20260809/hash_manifest.json binds 17 decisive files with zero observed mismatches

Computational experiments

.proof-experiments/20260809-210541-effcc3 — both 100000-node arms were cap-limited; 29035 shallow pair rejections and no witness.
.proof-experiments/20260809-210631-88600c — independently reconstructed dimensions, quotient threshold, and prune aggregates.
.proof-experiments/20260809-210639-a96d74 — rejected threshold, count, and forged-witness mutations.

Independent checker

checkers/check_c5_pair_excess_pilot_v1.py independently reconstructs the pair quotient and any positive witness, but deliberately does not replay DFS transitions or validate a negative search claim.

Contribution gate

not_requested

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

Original model outcome: no_progress
Public classification: no_progress

Cross-domain transfers tested

Global loopless 4-regular pair excess -> quotient excess budget six under free C5 pair orbits -> 29035 observed nonterminal candidate rejections, but an unoptimized vector implementation lost 19.49x runtime.

Established facts

The C5 action partitions the 105 labelled pairs into 21 free orbits of size five.
Independent frozenset partition reconstruction in c5_pair_excess_checker_receipt.json · The fixed action with three disjoint five-cycles · computed
Every C5 block-orbit pair profile has total mass 15.
Independent reconstruction for all 1001 block orbits · All free C5 orbits of 6-subsets · computed
An extendable partial six-orbit selection has aggregate quotient pair excess at most six.
Every final pair multiplicity is at least four; quotient mass is 6*15=90 and baseline mass is 21*4=84; partial counts are monotone · Partial C5-orbit selections extendable to a C5-invariant 30-block cover · proved
The direct pair-enabled pilot was 19.4905 times slower than its matched baseline.
26.8641763204/1.3783247829 seconds in the hash-bound receipt · The recorded 100000-node paired Python experiment · computed

Ruled out in this epoch

Scale the current 21-tuple pair-excess kernel unchanged.
The current Python C5 DFS implementation and branching order · It produced a 19.49-fold runtime regression while exhausting no root case. · c5_pair_excess_pilot_100k_receipt.json and independent checker receipt · A packed incremental implementation must match or exceed baseline useful frontier progress per wall second.
Treat the bounded C5 miss as an exclusion.
All eight root cases in this epoch · Every case hit its node cap, and DFS transitions were not independently replayed. · Both arms report all_cases_exhausted=false and mathematical_exclusion_validated=false. · Complete every root case and independently replay the entire transition frontier.

Open leads

Exact one-orbit tail lookup
It directly attacks the established final-orbit failure concentration using profile and coverage indexing. · Index block orbits by cycle profile and, at one remaining slot, test exact residual profile plus coverage-mask containment. · high · open
CakeLPR certificate-integrity gate
It remains the prerequisite for a promotable global negative certificate. · Rerun the retained two-proof/two-mutant gate only after official sources exist locally with all pinned hashes. · normal · open
Packed pair-excess delta kernel
The mathematical prune is active, but candidate-level tuple scans dominate runtime. · Precompute packed contribution words and compare useful frontier progress under a five-second matched wall-time test. · low · open

Continuation checkpoint

Objective: Determine whether exact one-orbit completion indexing materially eliminates the C5 DFS late-failure bottleneck.

First action: Implement a profile-keyed map from residual cycle degree to unused block orbits and replace depth-five branching with exact mask-containment lookup.

Stop condition: Promote only a directly checked witness; hold or redirect if matched wall-time progress does not improve or if any completeness audit fails.

Next moves

Implement a residual-profile-indexed one-orbit tail lookup for depth-five states.
Run matched fixed-node and fixed-wall controls against the repaired baseline.
Revisit pair excess only with packed incremental evaluation and a predeclared wall-time continuation gate.
Keep global LRAT scale-up blocked until hash-pinned CakeLPR passes the two-proof/two-mutant qualification gate.

Citations

Tool disclosure

GPT-5.6 Sol was the principal investigator and designed, implemented, executed, and audited the discriminator. GPT-5.6 Terra delegates supplied advisory prior-art and verification memos; their agreement was not validation, and all relied-on facts were independently reconstructed in the main workspace. Python 3.12.3 performed exact enumeration, bitset DFS, hashing, and mutation controls. No SAT solver, CAS, proof assistant, cloud lab, or external proof service ran. Official-source retrieval failed at DNS, and no source or hash was fabricated.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration: 916.4s
Review state: evidence receipt failure; not durable progress
Attempt ID: covering-c1563-20260809-211108-ede0ad

Human review ledger

No human review recorded.