Arithmetic constraints and covering curves for rational distances in a square

This record contains the mathematical preprint and its partial Lean companion.

We study points in the strict interior of a square whose four vertex distances are rational. Primitive integral and Gaussian models yield exact denominator identities, prime-power bounds, and reciprocity restrictions on the squarefree classes assigned to opposite corners. We construct ordered elliptic sources and covering curves of genera five, nine and seventeen, retaining the four distances. A common squareclass descent gives infinite families of genus-nine curves without rational points, while a compensated source descent excludes an entire squarefree kernel. Quaternion evaluations yield a parity obstruction and an infinite family of complete genus-five failures of the Hasse principle. We describe nine elliptic factors of the genus-nine Jacobian and derive rank constraints. For the fixed opposite-distance difference 1/5, we determine all twenty-eight bitangents of a genus-three quotient, its odd-theta algebra and the class defining the original double cover. We also establish locally soluble lifts at every fixed finite Abel–Jacobi level through a rational boundary point.

Contents: the paper PDF, self-contained LaTeX sources, checked Lean source projects, pinned dependency manifests, reproduction instructions, a statement-level coverage table, and file checksums.

Lean coverage in this release: 34 of 69 numbered manuscript statements are fully formalized. The accompanying development is partial and will be extended in subsequent versions. The exact verified statements and permitted axiom dependencies are recorded in the code archive. The global square rational-distance problem remains open.

Licenses: paper and LaTeX sources, CC BY 4.0; original Lean code, Apache-2.0. Third-party sources retain their original copyright notices and licenses. Future development is intended for GitHub, with verified snapshots archived as Zenodo versions.