Explicit Twisted Hilbert Class Components Beyond Classical Irregularity

Abstract

Let $p \equiv 1 \pmod 6$ be prime and $K_p = \mathbf{Q}(\zeta_{3p})$. We study the reflected circular unit $\mu_p = (1+z\zeta_p)/(1+\bar z\zeta_p)$, $z = -\zeta_3^2$, and its character projections. A universal Stirling polynomial $P_m$ gives an exact identity between the anti-spectrum of $\mu_p$ and the primitive divided $\chi_{-3}$-twisted Stickelberger spectrum: $P_{p-j}(h) - P_{p-j}(1-h) = -(2h-1)(j-1)!\,b_j$, $h = z/(1+z)$. Thus the locally blind lines of the reflected unit are precisely the zeros of the corresponding divided twisted Bernoulli eigenvalues. For every $p<500$ we enumerate these zeros. Exactly twelve character lines occur. On each line an explicit integral idempotent product of $\mu_p$ is a local $p$-th power at the conductor primes but not a global $p$-th power. Small completely split primes provide finite Artin certificates. The generalized Bernoulli number has exact $p$-adic valuation one in every case; the character-wise Main Conjecture therefore proves that each radical generates the complete Hilbert-class-field component, which has order $p$. Seven of the twelve lines occur at classically regular primes, so twisted degeneracy below 500 is more often invisible to ordinary irregularity than aligned with it. The first case, $p = 67$, is worked out in full, and a deterministic integer-arithmetic program (included as an ancillary file) reproduces the enumeration and every certificate.