[adamgall]

Result

For every even rotation period n, the formal F2[D_{2n}] register model with paid reflections only x,y, arbitrary binary rank-one idempotents, free scalar mixing, and any finite restored dirty workspace has the sharp separate bills for heterogeneous diagonal targets r^k_i x^e_i:

• x calls: sum d_n(k_i+e_i)
• y calls: sum d_n(k_i)

Here r=xy, e_i in {0,1}, and d_n is cyclic distance to zero. Coordinate words simultaneously attain both bounds, hence all nonnegative per-letter tariffs are exact.

The uniform lower-bound proof uses two n-address permutation representations with offsets c=n/2 and c=n/2-1, plus subadditive balanced-cut ranks. The offsets matter for n=2 mod 4; n=2,6,10 controls are included. Arbitrary scalar projectors contribute exactly their rank across the active cut, so this is not merely a coordinate-call argument or a finite-search extrapolation.

What is new

This closes the explicitly open finite-torsion question in #1911 (712593f8): its infinite-dihedral Laurent invariant does not descend through r^n=1. The new finite cut invariant does. In particular #1821's D16 central pair retains its 16-call minimum with arbitrarily many dirty roles, and the theorem covers all even periods and heterogeneous endpoints.

The all-register D8 central-target special case was already proved by steph4n-gh in #1573; it is credited, not claimed anew. Arham Tahir's #1919 rank-bounded D8 invocation tariff is distinct and also credited. #1792's stale-handoff warning was checked, along with local/merged #1443 and live all-state overlap.

Evidence and reproduction

Only research/finite-dihedral-cut-bound/ is added: written proof, standard-library verifier, deterministic receipt, full chooser selection, TASK/SOURCES/VERIFY metadata, and actual validation record.

uv run --python 3.12 --no-project python -I -B research/finite-dihedral-cut-bound/verify.py
uv run --python 3.12 --no-project python scripts/research_package.py --package research/finite-dihedral-cut-bound

Both direct and registered optimized replay passed on macOS arm64 / CPython 3.12.12:

• 1,027 complete formal forward/inverse dirty fixtures; 134,864 faithful-regular basis columns.
• 9,420 complete short mixed words, 47,872 representation products, and 1,088 target cut ranks.
• 3,720 rank-one call checks over all 155 projectors through four roles; all 174 scalar invertible maps through three roles; 1,088 additional arbitrary-idempotent rank checks.
• 9,249 omitted-call controls and 13 semantic/model controls reject; a corrupted receipt also rejects under -O.
• A nonzero invertible dirty shear hidden by both proof representations is detected by full formal and faithful-regular checks, guarding against represented-endpoint false positives.

The universal theorem is the written proof, not inferred from finite checks. No native supplier, selected cold replay, hosted CI, Lean, or independent human proof review is claimed.

Provenance and limits

• Exact base: 0b1cccf86e2f5132ed286a26ef0a681a495740ea.
• Independently sampled seed: 63537661f5ee8acaa486b70f82726911; track bit-mechanisms; emitted card bit-new-register-model; target-specific relaxation method.
• Pre-computation claim.
• Final shared overlap snapshot covered 584 open PRs and 565 discussion records, fully paginated; exact live predecessor heads and all-state dihedral search checked. This is not a novelty guarantee.

No new kappa or selected-supplier change. Odd periods, arbitrary free address maps, nonlinear invocation tariffs, and physical/compiler/frame/hosting/precision/tape admission remain outside scope. The lower bound grants scalar mixing free only as an optimistic formal relaxation; the attaining word needs no clean state, extra workspace, COPY, or uncharged restoration.

Prepared for Adam Gall with substantial OpenAI Codex GPT-6 Astra assistance in proof, implementation and writing. Apache-2.0; predecessor credit retained.

[Mister-Meeseeks]

Batch review at 5938ce3e91bbb8b5c2227af7976d444b2fbac7d4.

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

Reviewed scope: Two finite address-cut representations prove sharp separate reflection tariffs for all even rotation periods and arbitrary dirty roles, resolving1911s finite-torsion question.1027 full dirty fixtures,9420 mixed words and faithful regular checks pass. Odd periods and actual supplier realization remain open.

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.