The localized Bost-Connes system attached to the prime family of Q47(n) = n^47 - (n-1)^47: fractional critical temperature, KMS uniqueness, and a congruence that the dynamics cannot see

Abstract

Let Q47(x) = x^47 - (x-1)^47, an irreducible polynomial of degree 46, and let S be the set of primes it represents. We analyse the localized Bost-Connes system A_{Q,S} in the framework of the author's paper "KMS state uniqueness and phase transitions in localized Bost-Connes systems" (DOI 10.5281/zenodo.22152101), whose partition function is the partial Euler product zeta_S(beta) = prod_{p in S} (1 - p^{-beta})^{-1} and whose symmetry group is the product of the groups of p-adic units over p in S. Three results are proved. First, the critical inverse temperature is the fractional value beta_c = 1/46, the reciprocal of the degree: the upper bound beta_c <= 1/46 is unconditional (Selberg sieve), the matching lower bound follows from the Bateman-Horn conjecture, and at the critical point the defining series diverges at the doubly logarithmic rate (A/46) log log X, so that zeta_S(1/46) is infinite and the critical point lies in the high-temperature phase, the opposite of the Brun-summable behaviour isolated in the foundational paper. Second, assuming Bateman-Horn with congruence conditions, the obstruction group vanishes for all beta <= 1/46, so the KMS_beta state is unique on the whole high-temperature half-line; the proof replaces Chebotarev equidistribution by Weil's bound for multiplicative character sums along Q47, together with a Hensel step for prime-power moduli. Third, we explain a mechanism we call congruence invisibility: every prime represented by Q47 satisfies the rigid congruence p = 1 mod 282, an extreme bias, yet no symmetry breaking results. The reason is structural: a modulus l can impose a deterministic congruence on the family only if Q47 is constant modulo l, which forces l not to be represented, hence removes the l-component from the symmetry group altogether. The hidden modulus 282 is the denominator of the Bernoulli number B_46, by von Staudt-Clausen. We close by observing that the quadruplet subsystem of the author's empirical study of Bateman-Horn for Q47 (DOI 10.5281/zenodo.20753750) has the same critical temperature 1/46 but the opposite behaviour at it, giving an explicit pair of nested systems whose critical points lie in opposite phases. Unrefereed preprint. The upper bound on the critical temperature and the algebraic results (Propositions 3.1, 3.2, 5.3 and Theorem 6.1) are unconditional; the equality beta_c = 1/46 and the uniqueness theorem are conditional on the Bateman-Horn conjecture, with congruence conditions for the latter, and are labelled as such in the text.