Statistical Structure of Carmichael Numbers from a Complete Census: Spacings, Leading Digits, and Factor-Tuple Analysis

Carmichael numbers are the composite integers that satisfy the Fermat congruence for every base coprime to n. We study the complete census of the 1,401,644 Carmichael numbers below 1018, with external validation on the 306,878,295 numbers between 1018 and 1024 under an analysis plan deposited in advance. First, we ask whether their placement, once the varying density is divided out, departs from an inhomogeneous Poisson reference model. Monte Carlo tests that account for the estimated intensity and statistics of consecutive gap ratios that need no intensity estimate detect no departure at the 5% level in the primary or confirmatory analysis, and a power study states which departures could have been detected. An excess serial correlation seen below 1018 is not reproduced above it. Second, the leading-digit distribution is close to the generalized Benford law implied by the growth of the counting function, with deviations below 10−3 in every proportion, and at fixed scale, its exponent increases with the number of prime factors. Third, the factor tuples repeat often for three prime factors and are almost all distinct from five upward, and a sieve for a prescribed tuple reproduces the census on its validation domain. Code and outputs are public.