Research: sharp inverse-square rounding bound for cyclic scans

[cbethin]

PR628 bounds periodic-scan rounding by 6(B+1)u/beta^3 and requires u <= beta^2/2. This mathematical refinement proves 4(B+1)u/beta^2 under u <= beta/2, uniformly over every cycle length, either direction, and arbitrary exact complex coefficients with modulus at most 1-beta.

The proof keeps a shared suffix-product factor in the accumulated rounding error and closure denominator. A one-entry dyadic family in the unchanged original scan has error at least u/(4*beta^2). Thus the inverse-square dependence is sharp for this algorithm and rounding rule. No defect in PR628's stated bound is alleged.

The resulting factor-cascade execution guard saves at least g fractional bits when beta=2^-g. At radix 9 and 128 target bits, the sufficient guard changes from 4,790 to 4,324 bits. Coefficient and phase preparation remain charged. The asymptotic O(p+r^4) precision/preparation bounds remain unchanged; this is neither a measured runtime speedup nor a new multiplication exponent. Integration into PR628 is not activated.

Reproduce with Python 3.11+ and the standard library:

python3 -B research/cyclic-scan-sharp-bound/verify.py

Expected: PASS_SHARP_CYCLIC_SCAN_BOUND.

Evidence includes 5,024 exact comparisons with a separately implemented finite-geometric inverse, 24 residual certificates through length 2,048, 64 exact sharp-family instances, 384 exact cascade recurrences, 276 inherited-budget comparisons, and seven mathematical/admission controls. The new integer-grid implementation agrees exactly with the original varying scan in 3,362 cases, including 482 constant-scan matches. Another 1,686 cases exercise the expanded precision admission. Agreement between scan implementations is not counted as independent validation of their shared algorithm; the direct inverse, residuals, symbolic family and written proof supply distinct checks.

Local Python 3.13 and isolated optimized Python 3.14 replays pass. A minimal offline package in a path with spaces passes. Source drift, changed expected evidence and output-overwrite controls fail closed. The selected-record inventory and staged whitespace checks pass. Full make verify, supplier cold replay, Linux, remote CI, the factorizer/full kernel, and proof-assistant validation are not claimed. There is no independent human review.

Only research/cyclic-scan-sharp-bound/ is added. Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. PR628 source: 0768f97d06c623b8211e651543c66135796b62bf, checked against Git and still current at the pre-submission inspection. OS seed 68 assigns the mathematical-audit track. No pinned coordination issue exists; the local claim records the scope. The full overlap review covers 581 open PRs through #762. The immediate publication check also inspects #763, a separate compact-bit repair audit; main remains at the pinned base.

Credit Chafik Boukhalfa / chafreaky for PR628's scan, original analysis and factor cascade. Joshua Snider / jsnider3's PR692 already applies the standard finite-geometric inverse to weighted stride cycles and the bilinear dual; that oracle method is not claimed new here. Its proof retains the cubic-margin scan bound. PR663/PR673 and PR744/PR758 concern distinct dirty-restoration and echo questions. Original notices are preserved, and no code from PR692 is consumed.

Prepared for Charles Bethin with substantial OpenAI Codex assistance in derivation, implementation, testing and writing. Apache-2.0. No literature-priority claim is made. The selected frontier, upstream sources and inherited full-multiplication obligations are unchanged.

[Mister-Meeseeks]

100-PR batch review at 983d66f0f2e48ee500ec7eee2da9244c001c5064.

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

Reviewed scope: Cyclic scan rounding bound improves to inverse-square margin under stated exact-coefficient assumptions; proof and matching dyadic family, factor preparation remains inherited.

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.