U-Bit Collapse in Arnault Composites:Probing the Boundary of Strong Lucas Pseudoprimes

Abstract

We present a computational study of 200 composite integers of approximately 350 bits, engineered using the Arnault framework to pass all Miller-Rabin tests up to base 11. Generated at a rate of approximately 20 per hour from a high-throughput construction process producing ~7,700 Carmichael numbers per minute, all samples fail the strong Lucas probable prime test. We introduce the U-bit collapse metric delta = log_2(n) - log_2(U_d mod n) to quantify deviation from the expected uniform distribution of Lucas sequence terms. Analysis reveals minimal collapse values: mean delta = 1.61 bits, median delta = 1.0 bits, maximum delta = 8 bits, with 26% showing no measurable collapse. We analyze correlations with prime residue classes modulo 35, Arnault construction parameters (k,M), and composite bit-sizes. Our results demonstrate that composites engineered for Miller-Rabin resistance exhibit negligible Lucas sequence degeneracy, providing strong empirical evidence for the statistical independence of these two primality test components and supporting the continued robustness of Baillie-PSW-type tests.