A proof of Mathar's recurrence for OEIS A026124 with exact boundary terms

This note proves Mathar's recurrence for OEIS A026124 from its nonnegative-walk definition, for every n >= 5 with a(0) = a(1) = 0. It provides explicit polynomial certificates and the exact exceptional residuals 8, -16, 8 at n = 2, 3, 4, together with executable integer-arithmetic checks. The all-index result follows from the proof; finite term comparisons test the implementations. The differential operator specializes Richard J. Mathar's published 2021 general formula. The generating function and the conjectured recurrence are credited to their prior sources. First-proof priority has not been established, and this deposit does not assert OEIS approval or independent human peer review. Byungwoong Yoo selected the research target, defined its scope, and directed the investigation. ChatGPT was used for symbolic derivation, proof drafting, implementation, and computational checks. The package records the verification performed and the remaining external review. The note and original project documentation are offered under CC BY 4.0 to the extent applicable rights are held; original code is MIT-licensed. Third-party materials retain their own terms and attribution.