research: independently audit five-b UT4 gap coverage

[josecarlosgonzalezroque-design]

Change and mathematical scope

PR #2828 truncates its opposite-orientation five-b gap tables at three calls per outer label and four calls in total. This package supplies an explicit coverage certificate for those caps, including pure-dirty calls whose active projection is zero.

A separate 2+3 meet-in-the-middle implementation reconstructs all 5,664 compatible active profile pairs. Rank demand in the other gaps proves the 3/4 caps. Literal 3×3 matrix calculations reconstruct both orientations' bounded operator tables: 81,935 states representing 2,798,629 words per orientation. The optional comparison matches the complete profile set and every table entry/multiplicity against #2828 at 8a1000c.

Only research/ut4-five-b-coverage-audit/ is added. This is a scoped coverage/operator audit. It does not independently certify the remaining dirty-prefix filters, final whole-word rejection, inherited arbitrary-workspace compression, or a native supplier/new κ.

Task and overlap

Card: bit-new-register-model, changed to a complementary audit because the pending #2828 already supplies the requested construction-bound work. TASK.json records the selection, scope change and bottleneck connection.

Started from main b5e86d8; refreshed and rebased onto canonical main 23ef28c at 2026-10-16 01:18:45 UTC. Relevant instructions and scientific inputs were unchanged. Live PR searches found #2828 as the audit object; Discussion coverage remains incomplete and no advance claim was posted.

Validation

Python 3.12.14 on Windows; standard library only. The independent replay imports no predecessor code.

python3.12 -B scripts/research_package.py --package research/ut4-five-b-coverage-audit
python3.12 -B research/ut4-five-b-coverage-audit/compare_reference.py REFERENCE_ROOT

The second command is optional and requires the pinned reference export.

• Registered replay passed in 71.954 seconds; metadata and source pins passed.
• All 5,664 profiles passed direct degree-1–3 subsequence expansion of the fourteen retained quotient equations.
• Controls detect a changed middle coupling, unjustified smaller caps, lost pure-dirty contributions and reversed contraction order.
• The optional comparison matched the exact profile set and gap tables while preserving every original census output.
• The author's original verifier and controls also passed. Those reruns are not independent implementations.

The README documents exact-source export instructions, canonical digests and unrun obligations. No selected supplier cold replay or frontier change is claimed.

Credit

The full exclusion belongs to steph4n-gh's #2828, building on Gareth Davidson (#2027/#2314/#2343), Nick Petalas (#2175), ToikY (#1662/#1680), CinnaDavis (#1541), and Aaron Wang's profile method (#1690). No predecessor code is vendored. New work is Apache-2.0.

Prepared with substantial OpenAI Codex assistance; no independent human review or subagents.

Open next questions

Independently reconstruct the remaining dirty-prefix equations and final AC/ACB/BAC rejection for all fifteen-call five-b splits. Start from the then-current pinned #2828 head and reuse these profile/gap digests. This would audit the remaining computational implication; the inherited workspace-compression proof needs separate review. No computational blocker is known. See the README and scoped handoff.