The Integer Group Determinants for Q₂₀: Closing the Last Valuation-Five Face

Abstract

This preprint completes the classification of the integer group determinants of the dicyclic group Q₂₀ by resolving the last valuation-five face. For an integer m with gcd(m,10)=1, it proves that 32m occurs exactly when |m| is divisible by a prime p congruent to 13 or 17 modulo 20, by q² for a prime q congruent to 9 or 19 modulo 20, or by r⁴ for a prime r congruent to 3 or 7 modulo 20. Both signs occur. The proof combines an exact C₄ boundary calculation, a mod-5 bridge, a class-number-one definite Eichler order, simultaneous split-order lifting, and conductor descent. An independently reproducible standard-library verification package checks all displayed witnesses, the finite boundary and conductor arithmetic, and all 2,880 simultaneous residue targets.