Semiprime-Order Circulant Determinants: A Conductor-65 Counterexample, Reciprocal Unit Bases, and a Prime-Norm Classification

Abstract

This preprint addresses three questions raised by Paudel and Pinner concerning integer circulant determinants of semiprime order. It proves that ±5³ and ±13³ do not occur at order 65, establishes a reciprocal and skew-reciprocal unit-basis theorem for odd cyclotomic fields, and classifies the corresponding prime-norm problem. The deposit includes the complete source, verification materials, and reproducible exact computations.