Certified positivity of the odd Toeplitz complement at the linear threshold a=3N

Version 1.2 of this research compendium gives a reproducible computer-assisted positivity certificate for an explicit 55-dimensional symmetric matrix D_M arising from odd sine modes 5, 7, ..., 113 at the single linear boundary a=3N. For every integer N>=653, with M=N+2, it certifies negative off-diagonal entries and positive weighted row sums for a fixed positive rational vector. Weighted strict diagonal dominance then implies D_M is positive definite. The finite range 655<=M<=10000 is treated by direct matrix balls and the infinite tail by exact identities, Cauchy bounds, and Euler-Maclaurin/Boole remainder control. This record is a preprint candidate and has not been peer reviewed. It does not prove the even sector, the band 3N<=a<=4N, positivity at a fixed scale, or the Riemann hypothesis. Independent mathematical review and clean-room reproduction remain necessary. Licensing: the article, LaTeX source, and project-authored documentation are licensed under Creative Commons Attribution 4.0 International (CC BY 4.0). Original Python and shell certificate code is licensed under the MIT License. External runtime dependencies are not bundled and remain under their own licenses. The reserved DOI 10.5281/zenodo.22133134 will be registered only when this draft is published.