# Verification report

Manuscript: B-Spline Laws and Nonreal Zeros for Weighted-Cycle Internal DLA
Author: Alper Ferudun, Mercury Software GmbH. Version 1.0, 29 September 2026.

The general proof is the matching of a finite harmonic-exit recurrence
with the classical Cox--de Boor recurrence, followed by a justified
penultimate-state cycle lift. The four-cycle region is characterized
in both directions, with all reversible fibers. The five-cycle witness
has primitive polynomial coefficients (1025,10575,13944,4960) and cubic
discriminant -38803806734400. Its transition probabilities and
normalization are explicit. The n=3,4 argument proves minimal cycle size.

Fresh isolated Python 3.9.6 and Python 3.13.5 runs passed 23,967 assertions
each, with identical reports except interpreter version:

- 1,712 exact cycle cases, including general directed probabilities.
- 963 cases independently compared with finite absorbing-chain linear systems.
- 108 nonperiodic line environments.
- 81 explicitly reconstructed four-cycle inverse fibers.
- 18 independently enumerated descent-major-index probability rows.
- Direct cubic discriminant, Sylvester determinant and Sturm root count.
- Four rejected false extensions/normalization mutations.

All computations use exact rational or integer arithmetic, not stochastic
simulation or floating-point root estimates. The three main formula
implementations are compared with a separate finite-state absorption
method. This is an independent computational route, NOT an independent
human/model review. The finite sample does not prove the general theorem.
Scripts, source hashes and full reports are included in the source archive.

The originating researcher audited the full written proof and inspected
all final PDF page images. The final build receipt records page count,
compiler results, warnings, visual checks and PDF hash. There is no
independent refereeing or proof-assistant verification.

Scope: a complete one-dimensional theorem and counterexample, not a
resolution of every higher-dimensional or circulant question in the AIM
source. The q-Eulerian law, harmonic-scale method and B-spline identities
are credited prior material. Possible folklore and differently indexed
prior art are not excluded; absolute priority is not certified.

No DOI or public release is asserted by this local verification report.