2026-08-10 06:41 UTCgpt-5.6-sol · high

Integrated the independently audited exact two-orbit C5 completion index into the selected-aware constructive DFS and compared it with the exact one-orbit tail under matched 100000-node, eight-root balanced arms.

No Progress

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.

Research-policy redirect

Evidence receipt creation failed; durable progress is withheld.

Strategy and discriminator

C5-invariant orbit-level constructive search

At two remaining block orbits, intersect a combined-degree-profile pair bitset with one pair-coverer bitset for every uncovered triple orbit, then reject pairs incident with selected orbits.

Hypothesis: Integrated exact two-orbit completion tests at least five times as many distinct reached four-orbit prefixes under the matched 100000-node balanced C5 DFS budget, or exhausts the same frontier with at least five times fewer nodes.

Test: Run one-orbit and two-orbit tail arms with identical roots, branching, memoization, feasibility pruning, 100000-node totals, and 12500-node per-root caps; compare memo-surviving four-orbit prefixes and directly check any witness.

Rationale

The deterministic matched count and independent reconstruction establish a reusable search-kernel improvement. They do not establish frontier completeness, and bounded absence of a witness cannot change either covering bound.

Claims requiring scrutiny

The pair index represents exactly 500500 unordered distinct C5 block-orbit pairs and has semantic SHA-256 1d5aafab7a2ac7d66f4a3a45d66b40b4b97873d2b3d61e1c7f47a6d812166746 in both implementations.
At 100000 nodes per arm, the one-tail baseline tested 575 four-orbit prefixes and the two-tail challenger tested 98827, a factor of 171.87304347826088.
The independent checker reproduced 256 initial exact tail answers, receipt arithmetic, a positive fixture, and three rejection controls.
No 30-block witness or mathematical exclusion was obtained.

Evidence and scope

Primary command recorded by .proof-experiments/20260810-063055-fab237 returned LIMITED_NO_WITNESS and operational gate PASS.
Independent command recorded by .proof-experiments/20260810-063151-f113b3 returned PASS.
sha256sum -c artifacts/epoch45-20260810/SHA256SUMS returned OK for every retained source, receipt, report, and experiment record.

Computational experiments

.proof-experiments/20260810-063055-fab237: matched 100000-node arms; 171.873x prefix gain, no witness, all cases capped
.proof-experiments/20260810-063151-f113b3: independent 500500-pair reconstruction, 256 query replays, and four controls passed

Independent checker

checkers/check_c5_integrated_two_tail_v1.py is a separately written frozenset implementation that does not import the primary search; it reconstructs both semantic hashes and naively replays the first 16 decisions in every root case and arm.

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

Exact meet-in-the-middle completion indexing -> prediction that moving the tail boundary from depth five to depth four would improve whole-search state coverage by at least 5x -> observed 171.873x fixed-node and 20.234x derived wall-rate gains.

Established facts

The integrated two-tail arm tested 98827 memo-surviving four-orbit states under the exact recorded protocol.
Primary receipt SHA-256 0b6d33b5bdcd2ae18a3f2419ec7c4efd99419ff5c7a376d27c1c06c8507e004e and independent receipt-arithmetic PASS · Fixed-point-free C5 family, eight roots, 100000-node arm · computed
The independent reconstruction of all 500500 pair semantics has the same stream hash as the primary index.
artifacts/epoch45-20260810/c5_integrated_two_tail_checker_receipt.json · All unordered distinct pairs of the 1001 C5 block orbits · computed

Ruled out in this epoch

Scale the unchanged integrated C5 search by merely increasing its arbitrary cutoff.
Current quotient, roots, branching, memoization, profile pruning, and pair-tail mechanism · A second capped constructive pass found no witness; a larger cutoff alone would yield neither a certified negative result nor a new structural contribution. · artifacts/epoch45-20260810/c5_integrated_two_tail_receipt.json · A witness-bearing heuristic, sound new bulk prune, independently audited complete C5 decomposition, or materially different search mechanism
Scale native-PB cubes immediately.
The currently retained OPB leaf and local toolchain · No independently pinned native-PB emitter/replayer pair or hash-bound global ownership manifest is available. · Sol read-only tool audit and artifacts/epoch45-20260810/terra_pb_toolchain_audit.md · A pinned emitter and independent replayer pass intact and truncated controls, and a complete cube-ownership manifest is independently checked

Open leads

Compact fixed-second reusable-counter CNF/LRAT leaf
It combines the stronger fixed-second symmetry with the reusable profile-counter representation and uses the already dual-calibrated proof pipeline. · Compile profile (1,0,24,1,0,3), truth-check the delta, then run the frozen two-seed five-second comparison. · high · open
Proof-producing global profile decomposition
The 249-profile manifest supplies an exact ownership frontier; terminal replayable leaves could settle 30 if all profiles close. · Use the compact fixed-second calibration to decide whether profile leaves can become certificate-sized. · normal · open
Witness-bearing C5 heuristic reopening
The tail kernel is fast and reusable, but deterministic prefix order currently yields only empty tails. · Only reopen after defining a materially different prefix ordering or invariant with a predeclared positive-hit signal. · low · open

Continuation checkpoint

Objective: Determine whether compact reusable counters can make a fixed-second rank-2 proof leaf materially cheaper without sacrificing proof replay.

First action: Start from the retained fixed-second r=0 representative CNF and compile full unary counters over residual columns 2..29 for profile (1,0,24,1,0,3).

Stop condition: End or redirect on semantic drift, symmetry-coverage failure, incomplete-proof acceptance, or failure to improve both conflict rate and LRAT bytes per conflict under the frozen matched test.

Next moves

Build the compact fixed-second r=0 rank-2 leaf using reusable full unary counters over columns 2 through 29.
Independently truth-check the counter and assumption delta and verify fixed-second symmetry coverage.
Run the frozen two-seed five-second CaDiCaL/LRAT comparison and require material improvement in both conflict rate and proof bytes per conflict.
Replay every terminal proof with lrat-check and CakeLPR; reject incomplete prefixes explicitly.

Citations

Tool disclosure

GPT-5.6 Sol was principal investigator. GPT-5.6 Terra delegates supplied advisory challenger/prior-art and experiment-verification memos, promoted with provenance; delegate agreement was not validation. Deterministic work used CPython 3.12.3, integer bitsets, frozensets, SHA-256, and the Proof Factory experiment harness. Web search and direct source reading checked status and prior art. No SAT solver, CAS, proof assistant, cloud lab, external proof service, or human validator was used this epoch.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration: 1341.7s
Review state: evidence receipt failure; not durable progress
Attempt ID: covering-c1563-20260810-064145-82c1a9

Human review ledger

No human review recorded.