DONE: sharp reflection rounding radii (#5122; baseline CI failure)

[krzysztofwojt]

DONE / 2026-10-23 10:00 UTC / priority revision 7 / claim released

PR #5122, exact head dab8cc5917fbc76564367a973bd6cd1ed5a22ee3, resolves #5079 Open next question 1. For A_1(x,y)=(x+y,-y), the exact uniform infinity-norm rounding radius is floor(2m/3)/m on the complete refined grid and 2/3 on R^2. The proof gives the full integer-reflection-family formula, odd-grid corrections, explicit witnesses and one compatible attaining rule.

Normal/optimized registered checks, independent enumeration of 44,268 residues and 2,017,390 candidate outputs, larger attaining fixtures and eight rejecting controls pass. Exact publication-byte/source-history, official export integrity and read-only Git-free optimized replay pass. The hosted run failed in existing planning tests before this package ran; a clean checkout of canonical baseline 20a1e31bc6009b9b3f4d084bbaadc6350e34fcba, without this contribution, reproduces all eight failures. The separate existing repair is #5123. No hosted package pass is claimed.

The mathematical scope is complete and this claim is released. Evidence is author-checked with AI assistance, not human/formal acceptance, a physical caller or new kappa. Reader reachability and larger finite groups remain open; neither is reserved. A future hosted replay needs the repository planning repair, not a weaker mathematical check or altered frozen pins.

Credit Nick Petalas #5079, Sricharan Vagvala #1631, woklerx #1892, Sagar Thakrar #1537, Ross Richardson #1287 and the retained frame/dirty-echo lineage. Krzysztof Wojt with substantial OpenAI Codex assistance and bounded Luna support; Apache-2.0. The selected frontier is unchanged and no PR was merged or closed by this session.

Original advance claim, completed and released

CLAIM / 2026-10-23 09:24 UTC / priority revision 7

Question: resolve Open next question 1 of Nick Petalas's PR #5079: the least uniform infinity-norm error of one integer-valued rounder commuting with the nonorthogonal reflection A(x,y)=(x+y,-y). Determine the optimum on each complete refined grid (1/m)Z^2, m>=2, and on R^2, with an attaining globally compatible rule. If the same argument closes, state the exact dependence for A_k(x,y)=(x+k*y,-y); do not expand to arbitrary finite groups or a physical caller.

Base: fresh canonical main 81e9bd69e4a5b00ba9c713a1620e36d576d5522e; predecessor #5079 at 43f558c1b8fc14fb0ed33a4512fbc976beea6b0f. The finite-group existence theorem and stabilizer-compatible construction are cited premises, not replay targets. Sricharan Vagvala #1631, woklerx #1892, Sagar Thakrar #1537, Ross Richardson #1287 and the retained frame/dirty-echo lineage retain credit.

Method: minimize the full two-point orbit error, retain fixed-axis constraints, and prove matching global upper/lower bounds. Use an independent exact lattice enumeration on bounded refinements and literal equivariance/attainment tests to corroborate the written argument. Stop within two hours at a proved sharp formula or a precise obstruction; no larger numerical sweep without a proof target.

Value and overlap: #5079 explicitly requests this optimization and disclaims it. The complete overlap snapshot covers 318 open PRs (09:17Z), 4,229 Discussion records (09:18Z), and 6,824 main-tree notes. The latest PR bodies and Discussion updates were rechecked immediately before this claim, plus all-state rounding searches and recent krzysztofwojt topics. No same-question claim was found in this coverage. The account's Pauli, four-event association, dirty-fanout, permutation-cut, intact-region and address-twist work is excluded. Arbitrary diffs and unpublished work remain outside coverage.

Expected artifact: one focused proof and standalone exact verifier in research/reflection-rounding-radius/, or a concrete handoff. Current evidence is LOCAL-ONLY planning. This is a side theorem about mathematical rounding, not a #4100 caller discharge, paid implementation, supplier, new kappa or selected-frontier change. Independent human and formal proof-kernel review are unclaimed.

Claim expires: 2026-10-23T11:25:00Z.

Prepared by Krzysztof Wojt with substantial OpenAI Codex assistance and bounded Luna support. Apache-2.0.