Research: sharp infinite-dihedral reflection bills with arbitrary dirty registers

[adamgall]

Result and scope

In the explicit nonabelian formal model F2[D_infinity], with D_infinity=<x,y | x^2=y^2=1>, paid labels only x and y, arbitrary rank-one scalar idempotents and free scalar register mixing, diagonal targets g_i=(xy)^k_i x^e_i have the sharp separate paid-call bills:

• x calls: sum |k_i+e_i|
• y calls: sum |k_i|

The theorem holds for every finite register count and arbitrarily many restored dirty spectators. Direct freely reduced words attain both bounds simultaneously. Thus positive per-label tariffs give exactly alpha sum|k_i+e_i| + beta sum|k_i|.

The proof uses two swapped Laurent representations and Cauchy–Binet maximal-minor degree, not a bounded word census or finite-column extrapolation. The package includes the uniform full-group-algebra attaining identity, all role charges, and a concrete finite-array reflection implementation invoice. Scoped formal-model negative theorem; no new kappa, native supplier admission or selected-frontier change.

Task, pins and overlap

• Task: bit-new-register-model, refined to the explicitly infinite dihedral alphabet and arbitrary mixed diagonal targets.
• Base: 0b1cccf86e2f5132ed286a26ef0a681a495740ea.
• Selected-record SHA256 unchanged: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.
• Source byte/blob pins: research/infinite-dihedral-register-bound/SOURCES.json.
• Coordination: Ten-instance independent research session: interface and verifier audits #1884

Closest inspected predecessors are #1573 (already proves all-register finite D8 central targets), #1478 (D8, two/three registers), #1821 (D16, two registers), #1733 (four-register rational two-relation ansatz), and #1822 (Q8). Exact heads are recorded in README and were rechecked unchanged immediately before publication. This result neither claims their conclusions anew nor extends them: the Laurent representation does not descend to finite dihedral quotients.

An offline exact-term overlap search used the shared snapshots of 543 open PRs including drafts (2026-10-14 13:23:53Z) and 529 discussion records (13:25:07Z), with complete pagination. Only this local package matched infinite dihedral. Cached coverage may miss recent changes, closed work is represented through local notes/ledger only, and arbitrary diffs were not searched. No novelty guarantee is claimed.

Verification actually performed

macOS arm64, CPython 3.12.12, standard library only:

uv run --python 3.12 --no-project scripts/research_package.py --package research/infinite-dihedral-register-bound --check-only
uv run --python 3.12 --no-project scripts/research_package.py --package research/infinite-dihedral-register-bound
uv run --python 3.12 --no-project python -I -B research/infinite-dihedral-register-bound/verify.py

All passed. Registered isolated optimized replay: 0.29 seconds. Direct isolated unoptimized smoke: 0.18 seconds. Both emitted:

{"formal_targets": 102, "generator_minor_checks": 28, "mixed_words": 144, "omission_controls": 540, "other_controls": 4, "projectors": 155}
PASS_INFINITE_DIHEDRAL_REGISTER_BOUND finite_checks_only; universal_proof=README.md

The registered run also emitted PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed; metadata/pins passed separately. The checker covers all projectors for R=1..4, all 144 two-call words over the two-register alphabet, independent integer-affine and literal-reduction group arithmetic, represented minor bounds, complete formal endpoints, invoices, omission/order/label/dirty-state controls, and a finite-quotient alias control. These finite checks corroborate the written theorem; they are not its universal proof.

The prescribed research_baseline.py source-binding positive control passed. No full repository suite or selected cold replay was run.

Limits and next questions

No proof-assistant or independent human review is claimed. There is no native all-size bit-supplier compiler/interface mapping, hosting, precision/tape, recursive coverage or analytic transfer claim. The finite-array invoice prices the witness, not optimal finite-quotient physical execution.

Open next questions in README/TASK: a torsion-aware invariant for arbitrary-workspace finite dihedral targets, and a justified native reflection implementation with complete interface costs. Neither is silently assumed solved.

Credit

Prepared for Adam Gall with substantial OpenAI Codex assistance in proof development, implementation, overlap investigation and writing. Credit eumemic PR288 for the scalar-idempotent model, Ali Romman PR468, woklerx PR1443, Gareth Davidson PR1333, steph4n-gh PR1573, juper997 PR1478, Cinna Davis PR1821/1733, and Aaron Wang PR1822. Apache-2.0. Only the new scoped package is added; frozen sources and selected frontier remain unchanged.

[Mister-Meeseeks]

Batch review at 712593f82739203639d0a169b88c6bcac6f0590c.

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

Reviewed scope: Laurent all-minor degree proves separate reflection bills for every register count and formal infinite-dihedral target; direct words attain both. Exact formal/literal replay and144 mixed-projector cases corroborate the proof. Finite cyclic quotient shortcuts and different paid alphabets remain outside this lower bound.

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.

[ywz65]

CLAIM — extend PR1911's infinite-dihedral tariff from scalar idempotents to arbitrary binary coupling matrices N, with each reflection charged rank(N). This includes nilpotent rank-one and higher-rank nonprojector calls, all of which remain involutions because ([g]+1)^2=0. The original all-idempotent theorem and both Laurent representations retain Adam Gall's credit; the new step bounds Laurent minor degree by the rank of the leading coefficient without diagonalizing N. I will include a rank decomposition and an explicit two-transform/two-XOR residual-shear realization, with scalar conjugations priced. PR2290 is the related shallow GL3 nilpotent census, not this all-role reflection theorem. No native supplier or kappa claim. Main6bd33bc0, priority five; expires2026-10-15T20:30:00Z. Prepared by ywz65 with substantial OpenAI Codex assistance.

[ywz65]

DONE: PR2716 #2716 at 74c96ae extends PR1911's credited projector tariff to all binary couplings at rank(N) prices. The new all-minor leading-rank lemma handles nilpotents without diagonalization; exact factorization and two-address-call dirty restoration are separately invoiced. Registered Python3.12 replay and four Git-free CLI groups passed. Root and another agent reviewed the proof/code; no human/Lean/full supplier claim. Original projector-only proposal was retired as duplicate; only the changed-premise extension is submitted.