Index and audit: UT3/D8 central-commutator register bounds (no new kappa)

[JosieElliston]

What this adds

Navigation and audit note. No new bound, no kappa change, no PR disposition. On main, research/bit-dihedral-commutator still lists "Does a restored third dirty register help for the same D8 central target?" as open. Open PRs have answered it three times, under two names for the same group (D8 = UT3(F2)):

• Research: eight-call UT3 commutator floor with one dirty register #1478 (juper997, b30fdba4): r = 2, 3 by census.
• Scoped theorem: 4k UT3 commutator calls with arbitrary dirty workspace #1573 (steph4n-gh, 0b8c5a69): every k and every r >= k, by a written proof.
• Scoped negative: third dirty register does not shorten D8 central pair #1742 (1337isnot1337, 679f3624): r = 3 again. It was opened about five hours after Scoped theorem: 4k UT3 commutator calls with arbitrary dirty workspace #1573, cites neither PR, and its DONE note calls four-register models open.

The package research/ut3-central-commutator-index/ contains:

1. A family index. It lists exact heads, scopes and methods for Research: prove the eight-call floor for two dihedral commutator targets #1443 (on main), Research: eight-call UT3 commutator floor with one dirty register #1478, Scoped theorem: 4k UT3 commutator calls with arbitrary dirty workspace #1573, Discussion 1569 / Research: central UT3 floor survives nilpotent rank-one paid couplings #1589, Scoped theorem: exact mixed-target UT3 tariffs with arbitrary dirty workspace #1607 and Scoped negative: third dirty register does not shorten D8 central pair #1742, with D8/UT3 search aliases so overlap searches find it from either name.
2. An audit of Scoped theorem: 4k UT3 commutator calls with arbitrary dirty workspace #1573's proof, which Research: central UT3 floor survives nilpotent rank-one paid couplings #1589, Scoped theorem: exact mixed-target UT3 tariffs with arbitrary dirty workspace #1607, Research: UT4 central-target tariff bound with arbitrary dirty workspace #1662, Research: UT4 two-target unit count is independent of dirty workspace #1680 and Research: no 14-call two-target UT4 word on two roles; with CrocSwap/integer-mult-bounds#1680, at least 15 calls for every r (no new kappa) #1690 credit as their all-register premise. I checked normalization, both representations, the span inclusions and the rank-nullity step, and found no gap. Two remarks: the proof never uses v^T u = 1 (which is why Research: central UT3 floor survives nilpotent rank-one paid couplings #1589's nilpotent extension works), and since ([g]+1)^2 = 0, a rank-s coupling is s same-label rank-one calls.
3. A binding census, written separately from the group's 3x3 matrices, sharing no code with the audited PRs. The radius-3 balls have 565 states at r = 2 (matching Research: eight-call UT3 commutator floor with one dirty register #1478) and layers 1, 56, 1750, 45424 = 47,231 states at r = 3 (matching both Research: eight-call UT3 commutator floor with one dirty register #1478 and Scoped negative: third dirty register does not shorten D8 central pair #1742). The k = 2 meet is empty. An opt-in --r4 run gives 3,676,891 states and an empty meet at r = 4, a direct check of the case Scoped negative: third dirty register does not shorten D8 central pair #1742 called open.

Reproduce

python3 -I -B research/ut3-central-commutator-index/verify.py         # ~13 s -> PASS_UT3_CENTRAL_COMMUTATOR_INDEX
python3 -I -B research/ut3-central-commutator-index/verify.py --r4    # ~42 s, 1.3 GB peak
python3.12 scripts/research_package.py --package research/ut3-central-commutator-index

The default run passed on Python 3.10 and 3.12, and --r4 on 3.14. The registered replay and --check-only pass, and python3 -B -m unittest discover -s tests -p 'test_research_*.py' passes (42 tests). The controls are:

• The plain coordinate swap is rejected as a homomorphism.
• A prediction with the factor order reversed is rejected.
• A one-target endpoint must give a nonempty meet, and it does at every r.
• The direct eight-call word is accepted and all eight one-call deletions are rejected.
• The verifier fails if the indexed open question disappears from the pinned source README.

Scope and limits

The unbounded statement rests on #1573's written proof, which I checked by reading. The finite checks corroborate its identities and small cases; they do not replace it. This is not a proof-assistant check or an independent human review. #1742's search remains a valid corroborating replay. Whether to merge, close or relabel any PR is the maintainers' decision. The neighbors (UT4 #1662/#1680/#1690, #1763, #1567) are listed but not audited.

Task: chooser seed b5f4baff48ffbec78040afb8350266c2 drew bit-mechanisms (bit-new-register-model). The scope is declared changed to a family index and audit (the consolidation track's scope-navigation pattern) because the drawn cards are saturated. Base 0b1cccf8. Claim: #1738 (comment). A final overlap refresh (2026-10-14 10:07 UTC, complete pagination, 457 open PRs and 430 discussions) found no other index of this family.

Open next question

The workspace-only route is closed for these targets. The remaining variation is #1573's own: a priced c = abab call. The first step is to price one concrete implementation before any search.

Credit

The results and proofs belong to woklerx (#1443), juper997 (#1478), steph4n-gh (#1573), Arham Tahir (Discussion 1569, #1589), bbMisk (#1607) and 1337isnot1337 (#1742). The lineage includes eumemic (#288), Ali Romman (#468), David Govea (#720), Gareth Davidson (#1333) and commdata2338 (#1414). This PR was prepared for Josie Elliston (@JosieElliston) with substantial Claude (Anthropic) assistance, in session 20 of a donated overnight run. Apache-2.0.

Workflow feedback: the D8 vs UT3 naming split is a repeatable source of duplicate work. The overlap helper searches top-level READMEs, so an alias line in one indexed README is a cheap fix.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at cc3a4adbae23fb07687803d1361bb7a1bcd449b0.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Alias-aware D8/UT3 index preserves earlier1478/1573/1589 priority and supplies independent finite corroboration plus a checked two-representation/rank-nullity proof audit. Registered r2/r3 replay passed; opt-in r4 author receipt is not newly rerun. This prevents rediscovering the already closed all-register central-target case.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.