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

Abstract

For each coprime triple $(a,b,c)$ with $a+b=c$, Pasten (2021) constructs a universal derivative lattice $F(a,b)$ of rank $\omega(abc)-1$ from the linear constraint imposed by the weighted arithmetic derivative. We prove that for squarefree coprime triples, the lattice determinant satisfies $\det(F(a,b)) < \operatorname{rad}(abc)$. Combined with Vaaler's subspace lattice theorem, this yields a nonzero lattice vector with $\ell^\infty$-norm $\Vert{}\psi\Vert{}_\infty \le \operatorname{rad}(abc)^{1/(\omega-1)}$. We give a structural classification of partition types by whether the minimum non-degenerate norm ratio $\rho = \Vert{}\psi_{\mathrm{nd}}\Vert{}_\infty/R^{1/(\omega-1)}$ is universally bounded: nine types are bounded. We prove unconditional upper bounds, including $2^{-1/(\omega-1)}$ for balanced types at $\omega = 3, 4, 5$, $\rho < 1$ for two $\omega=5$ types with $a=1$, $\rho < 2^{-1/4}$ for type $(2,1,2)$, and $\rho < 6^{-1/4}$ for type $(2,2,1)$. For $\omega \ge 6$ we prove an explicit unboundedness criterion; applying it to every partition type requires the corresponding infinite prime-pattern family. The asymptotic constants are sharp only under the relevant prime-pattern conjectures. The inequalities have been checked over 44 000 squarefree triples and the key inequalities formalized in Lean 4 (zero sorry). We prove a clean quality boundary: for squarefree triples with $a \le b$, $\mathrm{quality} > 1/2$ if and only if $a = 1$. Combined with the classification, this shows that the triples hardest for abc (highest quality) are structurally opposite to the triples closest to the extremal $\rho$ bound in the surveyed families. The argument is self-contained and uses no abc-equivalent hypothesis.