Reproducible exhaustive searches for slope-distinct grid sets and reflection-invariant cyclic packings

We provide executable finite-enumeration arguments for two small extremal problems. Exact searches establish that a subset of the ten by ten grid with distinct undirected slopes has at most ten points. For cyclic triple packings by five-subsets of Z/31Z, a witness and an exhaustive search establish twelve base-block orbits as the maximum under reflection invariance; the unrestricted maximum remains between twelve and thirteen. The grid endpoint was already reported in TheoremDB on 19 August 2026, and Brouwer's tables already supply the cyclic lower bound of twelve. Peile and Taylor's earlier finite grid table remains unavailable for comparison. No first-endpoint claim is made. The contribution is an explicitly documented, independently replayed computational package: complete coverage arguments, witnesses, search implementations, logs, and a separate algorithmic audit.

AI contribution and verification scope. Byungwoong Yoo supplied the problem material and directed the project. ChatGPT Pro generated mathematical reductions, search code, computations, and an initial explanation. Codex inspected and replayed the supplied code, checked prior sources, performed additional computational audits, and prepared the note. AI assistance included substantive mathematics and computation, not only language editing. Separate implementations and assistant contexts are internal verification; no independent human peer review or formal proof-assistant certificate is claimed.

Rights. The CC BY 4.0 license applies to the original technical note. Original project verification and discovery software is separately offered under the MIT License as specified in PUBLICATION_LICENSES.md. No license is granted over third-party articles, externally sourced prose, database responses, or other third-party material; those retain their existing rights and attribution.