Explicit 8-Rank Criteria in a Squarefree Composite Family: Rédei Kernels, Artin Symbols, and Continued-Fraction Centers

Abstract

We study the narrow class group of real quadratic fields with fundamental discriminant D = 4AB, where A is a prime congruent to 7 modulo 8 and B = p₁p₂ is a product of two distinct primes lying either both in 1 modulo 8 or both in 7 modulo 8. A single four-bit formula is obtained for the column-normalized Rédei matrix in these two residue layers. The formula simultaneously determines the 4-rank, decompositions of the second kind, and, when r₄ = 1, an explicit basis of the right kernel. We prove that right-kernel membership is exactly the ideal-class admissibility condition for the third arguments of Stevenhagen’s Artin–Rédei symbols. This yields an explicit criterion for r₈ = 1. We further connect the unique redundant right-kernel direction with the central complete quotient of the continued fraction of √(AB). When the central quotient Q_{L/2} divides 2AB, its ramified-prime support, together with the center-sign correction, generates the unique relation among ambiguous ideal classes. Consequently, the 8-rank criterion reduces to a single Artin–Rédei symbol evaluation. The deposit includes the article, LaTeX source, PARI/GP and Python implementations, complete finite-population data, calibration and class-field cross-check logs, environment information, and integrity checksums. The reconstructed main scan verifies the Artin criterion and twist invariance in 194 out of 194 cases. The unified Rédei matrix is verified on 19,927 inputs, and the explicit right-kernel formula on all 10,098 inputs with r₄ = 1. Complete class-number scans contain 4,440 and 5,658 cases with r₄ = 1 in the two residue layers. Earlier partial counts of 2,176 and 4,467 are superseded and are not used as complete populations. These finite computations support implementation correctness; they are not proofs of a limiting density.