Integer Group Determinants for C2 x C14: An Exact Hadamard–Conductor Classification

Abstract

This preprint gives a complete effective classification of the integer group determinants for C2 x C14, equivalently C2^2 x C7. It derives the exact Walsh–Hadamard normalization image and conductor, reduces arbitrary-integer membership to a finite necessary-and-sufficient ideal-allocation test, and reconstructs an integral group-ring witness whenever the test accepts. It determines the complete signed pure-2 and pure-7 axes, the Lind–Lehmer minimum 3, the coprime spectrum, and the exact 49m face. For split primes q congruent to 1 modulo 7, it proves that +49q is always absent and gives an exact projected ray-packet criterion for -49q. The deposit includes matching source, a frozen finite certificate, an executable exact decision-and-witness interface, independent verification programs, PARI number-field checks, expected transcripts, and SHA-256 data.