Davis's Circular-Unit Signature Deficiency: Truncated Minus-Class Fitting Colengths and an Augmented-Ray Formula

Abstract

This preprint gives a uniform arithmetic interpretation of the polynomialthat computes Davis's circular-unit signature deficiency for primecyclotomic fields. An integral normalization of the odd-residue indexpolynomial is shown to encode truncated lengths of the 2-primary minus-classblocks through their zeroth Fitting ideals, for every odd prime and withoutcyclicity or plus-class assumptions. For primes p congruent to 3 modulo 4, the invariant simplifies to blocksupport and supplies an exact correction term for a circular-unit augmentedray residue quotient of conductor 4(1-zeta_p). When the plus class number isodd, the same correction applies to the full-unit quotient. The paper alsoidentifies the invariant's precise limitation: a single scalar cannot recoverthe elementary-divisor structure of a noncyclic block, illustrated atp=7687. The accompanying archive provides exact checks of the polynomialidentities, finite-field ranks, Bernoulli valuations, and Smith normal formsused as computational controls.