Computational Certificates for Boolean Rank Nine for Polygons with 35 to 40 Vertices

This record contains computational certificates and reproducibility materials accompanying the manuscript “Boolean Rank Nine for Polygons with 35 to 40 Vertices” by Zhu Dake, Qiuzhen College, Tsinghua University. The materials support the computer-assisted exclusion of coverings by eight rectangles for polygon slack matrix supports of orders 35 through 40. Together with previously published nine-rectangle coverings, this yields Boolean rank nine throughout this range. Two archives are provided: 1. eight-rectangle-threshold-certificate.zipThe main certificate archive contains fifteen CNF instances and their saved DRUP refutations, a reverse-unit-propagation checker with C# source code, formula generators, verification scripts, an eight-rectangle positive control at order 34, published nine-rectangle covering witnesses for orders 35 through 55, reproduction instructions, and a SHA-256 file manifest. 2. verification-35-55-supplement.zipThis archive contains supplementary verification scripts, fresh checking logs, and reports, including additional propagation-based checks for two proof instances. The equality of Boolean rank nine for orders 41 through 55 is a previously established result of Goucha, Gouveia, and Silva. The new lower-bound contribution of the accompanying manuscript concerns orders 35 through 40. Instructions and software requirements are provided in the README files within the archives. These materials document computational checks; no third-party audit or proof-assistant formalization is claimed.