Research: sharp odd-dihedral joint tariffs with dirty workspace

[RossRichardson]

For odd finite-dihedral rotation periods, the two separate reflection bounds miss one joint call for each longest reflection. This package proves the missing joint constraint for every odd period and every finite dirty-register workspace, completing the odd-period question left by Adam Gall's PR1941.

For heterogeneous formal targets r^k_i x^e_i, let A=sum d_n(k_i+e_i), B=sum d_n(k_i), and let t count the longest reflections. Every permitted word obeys X>=A, Y>=B, X+Y>=A+B+t; all pairs (A+j,B+t-j) are attained. The exact nonnegative tariff is alpha*A+beta*B+t*min(alpha,beta). For example, two longest D10 reflections require ten calls even with arbitrary restored dirty workspace; separate cuts and parity alone permit eight.

The new proof uses a half of the faithful regular action whose crossing rank equals each target's shortest alternating-word length. It credits and extends PR1941's separate cuts. This is a written all-parameter theorem in the explicit formal F2[D_(2n)] reflection-only, rank-one-idempotent model with free scalar mixing. No new kappa, selected supplier, physical compiler/hosting/frame/precision/tape admission, or Lean multiplication theorem is claimed.

Only research/odd-dihedral-joint-tariff/ is added. It contains the proof, full formal and separately implemented regular-action checks, independent specialist checker, failure controls, fixed receipt, exact source pins, task selection and limitations. No predecessor implementation is imported or frozen input changed. Credit Adam Gall #1941/#1911, commdata2338 #1414, Cinna Davis #1541/#1870/#1953, Ross Richardson #1766, Aaron Wang #1763, and the retained register-model lineage; their known special cases are not reclaimed as newly open.

Validation on macOS arm64 / CPython 3.12.12:

• Registered replay PASS: 510 complete single-target cases, all 155 projectors through four roles across five odd periods, 3,770 complete short words, 392 forward/inverse dirty pair words, and 84 larger mixed tuples. All 2,478 omitted-call controls reject.
• Separate Codex exact-verification specialist found no blocking proof/code defect; its no-import checker covers all 5,200 single targets for odd periods 3–101 and 216 full-matrix conjugations.
• Six package tests pass, including actual detector-removal sensitivity, source/receipt corruption, missing-evidence refusal, and isolated optimized read-only relocation. CPython 3.14.6 replay agrees; selected-record integrity and whitespace pass.
• Hosted CI and full selected cold replay are separate and are not claimed passed.

Reproduce: python3.12 -I -B scripts/research_package.py --package research/odd-dihedral-joint-tariff.

Exact contribution head: 691e6eb9417384d1820de4fdc167fc3e9797b75e. Research base: 0b1cccf86e2f5132ed286a26ef0a681a495740ea; main advanced to ecc75c79c22d909fd42aa68437eb77c4c1361835 during the run, with selected record, guidance, pinned inputs and package runner byte-identical. Final overlap refresh covered 535 open PRs and 592 Discussion records with complete pagination and no warnings; the precise scope is coordinated in Discussion1955. No novelty guarantee is inferred from searches.

Prepared for Ross Richardson with substantial OpenAI Codex assistance, including bit-supplier derivation and separate exact-verification review. This is AI-assisted checking, not independent human or formal-kernel review. Apache-2.0. The next useful mechanism must change the paid alphabet or supply a fully priced physical interface; another search in this same formal model cannot beat the direct bill.

[Mister-Meeseeks]

Batch review at 691e6eb9417384d1820de4fdc167fc3e9797b75e.

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

Reviewed scope: Odd-dihedral cut proof gives simultaneous per-letter and total tariff inequalities for arbitrary dirty roles, and matching formal words attain the weighted optimum. Exact quotient/cut corroboration and independent controls passed. Uniformity comes from the written argument, not bounded enumeration; no rational-frame or physical translation.

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.