# Verification and scope

The written proof establishes the strict bound e<=lc+l+4c-1 for every suspension with c>0. The resulting exact maximum B(n)=ceil(n^2/4+5n/2)+1 for n>=13 combines that upper bound with Swanepoel's credited lower construction. Version 1.1 also proves the one-exception bound B(n), the two-exception bound B(n)+3 for cores with c>=8, and hence B(n)<=f_3(n)<=B(n)+3 for sufficiently large n using Swanepoel's structural theorem.

The finite-circle maximum principle, cyclotomic trace lemma, sine-product classification, axis predecessor dynamics and equality contradiction were checked in detail by the originating researcher. The extension audit covers paired heights with one root, extreme-fiber radius counts, every small-deficit case, integer rounding, the exceptional reflected pair and the extremal radius-change argument. No known mathematical gap remains in these stated bounds. Independent review and proof-assistant verification have not occurred.

Both normal and optimized standard-library Python runs agree on 169675 exact diagnostic checks. These include 144008 cyclotomic rational-angle cases, root orders through 1024, finite abelian-group configurations, algebraic circle constructions, axis dynamics and floor/ceiling optimization. The calculations use exact integers, polynomial remainders and Fractions, not floating-point tolerance.

Normal and optimized extension runs agree on 3955936 exact diagnostics. They cover 1875965 bounded integer deficit profiles for each exception-count bound, 703 eligible free-plus-torsion axis models, 3241 closed torsion fibers, exact rational circle intersections, parity identities and retained negative controls. The general inequalities are proved algebraically and geometrically, not extrapolated from these tests.

The original attempted one-exception bound failed to be strict at l=5,c=8; that failed estimate remains archived. The extension improves the deficit-one estimate and proves the full one-exception upper bound, but equality profiles (a,b,d)=(2,0,0),(0,0,2) remain geometrically unresolved. Two reflected exceptions with D>=2 also remain unresolved. The frozen exceptional-set question is not closed and no source-wide solution increment is justified.

The original report and both preprint/published source versions were retained. Ten complete relevant PDF pages were visually inspected for the suspension theorem, and three additional complete arXiv pages for the structural theorem and exceptional-point estimates. The published source contains the one-unit suspension gap and the unrestricted upper bound B(n)+11. No later resolution was established by the bounded searches, but bibliographic novelty and absolute priority are not certified.

This preprint is AI-assisted, self-audited and unrefereed. Its metadata must not imply independent review, formal verification, a DOI assignment or an already completed public release.