Minkowski Bounds for Pasten's Arithmetic Derivative Lattices in the Squarefree Subfamily

Abstract

This preprint studies the squarefree subfamily of the arithmetic-derivative lattices introduced by Hector Pasten. It proves an exact determinant identity and combines it with Vaaler's subspace lattice theorem to obtain explicit lattice-vector bounds. It further develops a structural classification of non-degenerate partition types, records unconditional upper bounds, identifies which sharpness claims depend on infinite prime-pattern families, and separates finite-verification candidate formulas from theorem-tier results. The accompanying Lean 4 development formalizes selected key inequalities with zero `sorry`, while deterministic Python scripts replay the finite evidence cited by the paper. The work also records a quality-rho structural boundary for squarefree triples. Important scope note: this paper does not prove the abc conjecture, does not prove Pasten's Small Derivatives Conjecture, and does not claim progress on the hard high-quality cases. Its contribution is a structural and reproducible analysis of Pasten's lattice geometry.