[aliromman]

Proves and implements a fixed-point direct solver for every real dyadic cyclic band matrix with positive diagonal and row-dominance gap delta>0. The gap may shrink with size. The required fractional guards are logarithmic in matrix size and inverse gap. This extends the constant-smallness periodic interface and the fixed385-row direct consumer to a quantified input contract. It does not change the complete kappa calculation or the selected frontier.

All-size proof, exact evidence, integer solver, and source pins.

Natural-order cyclic fill uses at most 4NW^2 factor products and at most (4W+1)N retained factor entries. Each complex RHS solve uses at most 8NW products and 4N component roundings/divisions. Products and Schur sums are exact integers; each factor entry is rounded once. The proof supplies an explicit factor residual, both triangular solve errors, and a uniform integer range for all allowed inputs. These are arithmetic counts, not a complete native or physical-tape tariff.

For the 4,097-row width-four fixture with gap 1/4096, 60 fractional bits suffice for the 2^-20 solution target. The proved signed width is 151 bits; the observed peak is 132. Factor preparation uses 261,836 products and each complex solve uses 130,960 products. Exact original-matrix residuals certify the target. This finite fixture corroborates the written all-size theorem; it is not a Gaussian production-family claim.

The nearest predecessor is Dugongue's PR #4554, specifically its constant-gap ordinary-band proof and periodic Woodbury reduction. Ryanson Jonathan's independent check retains that scope. The earlier batched principal-window note uses noncyclic factors. PR #5260 proves the fixed385-row consumer, with a matrix-specific certificate. The present result adds direct cyclic fill, one-round factor residuals and explicit variable-gap precision. Classical diagonal dominance, Wilkinson's error analysis and Higham's exposition retain their credit. No new general stability theorem or global literature priority is claimed.

Base: canonical main ff3cc58, priority revision7. Every declared source pin is an ancestor of this publication. Predecessor code is not imported or edited. Source identities retain the OpenAI/math and Harvey–van der Hoeven lineage documented in the source notice. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.

With CPython 3.12 and the standard library:

python3.12 -B scripts/research_package.py --package research/cyclic-band-gap-solver

Expected: PASS_CYCLIC_BAND_GAP in normal and optimized modes, followed by PASS_REGISTERED_RESEARCH_PACKAGE.

Both registered modes pass. A separately written pivoted rational solver checks 72 complex right-hand sides on 36 small cases and nine matrix/RHS scale pairs. The replay also checks 129 symbolic cyclic patterns, eight controls and exact residuals at 257, 1,025 and 4,097 rows. Four source mutations fail in both modes before receipt checks: omitted wrap entries, wrong update sign, wrong divisor and omitted row-gap validation. Source pins and committed bytes pass. Both read-only Git-free export replays pass; all nine files remain unchanged. Canonical main and scoped overlap were refreshed. Hosted CI passed both planning and the focused replay at this exact head.

Limitations: the matrix must meet the stated real dyadic contract. Matrix formation, source approximation, final output-grid rounding, physical movement, native relations and the outer invoice remain caller obligations. The full-history wrapper was not run in the shallow checkout; direct source ancestry passed. No selected cold replay, Lean, Docker, independent human review or proof-kernel acceptance is claimed.