[krzysztofwojt]

The arbitrary-permutation-cut question left by #4968 and #4974 has a certified obstruction: for t coordinate-three-cycle targets, every nonnegative combination of register-independent permutation cuts has normalized value at most

((27 + 9 sqrt(2))/7) t ≈ 5.675417 t.

This holds for every finite address dimension, every register count, arbitrary restored dirty spectators, and every finite permutation action. Targets may use different triples and either orientation. Consequently the linear cut relaxation plus final rounding cannot certify the full 6t direct tariff for t >= 4.

The constant is sharp for the real quadratic domination proved here. The Boolean-cut optimum and the actual multi-target word minimum remain open. This does not repeat #5005's exact two-register word search, prove shared paid work, or claim a new κ. Integer label-count constraints and stronger/nonlinear invariants are outside the obstruction.

The proof uses one exact identity in Z[GL3(F2)], a written spectral argument and restriction to each target's embedded three-coordinate subgroup. The finite certificate is independently checked by row-bitset group-algebra convolution and by a separately implemented seven-point permutation representation on all 168 regular basis columns. No floating solver output is acceptance evidence.

Validation at 9ad4c47068f900fa1619191292539adcced0d638, macOS arm64 / Python 3.12.12:

• Registered normal/optimized replays pass; exact identity and sharpness vectors agree. Also 144 binary paid cross-block checks, four dirty endpoints and all 32,768 cuts in a spectator-containing natural GL4 action.
• Twelve actual corrupted CLI runs are rejected; both inverse-cycle controls pass. Independent report-corruption checks remain active under -O.
• Committed-byte and full source-history checks pass. Official export integrity and all five Git-free replay commands pass with read-only package files and unchanged bytes across all eight exported files.
• Export SHA256: 59392219ed1e09f6e2637e5f7f469c6478f45f92502d1cd9426777aa60053888.

Hosted CI run 39919483551 also passed for the published head: onboarding/selection and the sole registered package replay, including its committed-input integrity check. This is implementation evidence, not independent scientific acceptance.

python3.12 -B scripts/research_package.py --package research/three-cycle-cut-ceiling
python3.12 -B scripts/check_research_commit.py --package research/three-cycle-cut-ceiling --check-source-history

Task bit-new-register-model, scope changed to the named proof-method gap; priority revision 7. Canonical claim. Fresh clone pinned f259dabc7d70e0db8d6d36bfda888f008bbfecc1, refreshed before claiming to scientific base 4c0a359d88f22f05189d51f48387750b801a45d4; current main and governing guidance remain unchanged. Final complete overlap snapshot: 319 open PRs, 4,160 Discussion records and all 6,760 base notes at 07:38–07:39 UTC, followed by live PR/topic checks before submission. No matching certificate was found; arbitrary diffs and unpublished work remain outside coverage.

Credit James Reed / jwr404 (#4950/#4968/#5005), Mihir Srivastava (#4939), JoyboyBrian (#4974), Ross Richardson (#1989 and GL2 corollary), commdata2338 (#1414/#1495), Gareth Davidson (#1333), #1541 and their retained lineage. Predecessor packages and selected 60403/20000000 remain unchanged.

Prepared by Krzysztof Wojt / krzysztofwojt with substantial OpenAI Codex assistance and bounded Luna support for coordination inventory and the independent matrix checker. Author-checked computer-assisted proof, not independent human or formal-kernel acceptance. Apache-2.0.