Character Fourier Spectra of Circular Units and Twisted Bernoulli Class Components

Abstract

Let $χ$ be a primitive odd Dirichlet character of conductor $f$. For the universal projector polynomials $P_m$ introduced in arXiv:2607.23177, defined by $\sum_m P_m(X)Y^m = -\log(1-X(1-e^{-Y}))$, we evaluate the character Fourier spectrum at $h_t = ζ_f^t/(ζ_f^t-1)$: for every odd $m$, $\sum_t \barχ(t)P_m(h_t) = τ(\barχ)B_{m,χ}/(m\,m!)$. After reduction at any prime above $p \nmid f$, the case $m = p-j$ identifies this spectrum, including its exact nonzero scalar, with the divided generalized Bernoulli value attached to $χω^{-j}$; the spectral-zero and Bernoulli-zero criteria therefore agree over every residue field, with no splitting hypothesis on the coefficient field. We connect the identity with the local Kummer spectrum of the circular unit $1-ζ_fζ_p$ and carry the programme through in the first non-real case: for the two primitive quartic characters modulo 5 and primes $p < 500$, $p \equiv 1 \pmod{20}$, exactly eleven zero lines occur. On each line an integral character projection of $1-ζ_5ζ_p$ is everywhere locally unramified, a finite split-prime Artin computation proves it is not a global $p$-th power, and the character-wise Main Conjecture shows the radical generates the complete order-$p$ Hilbert class component. Six of the eleven components occur at classically regular primes. At $p = 61$ the two conjugate characters contribute on different indices. A deterministic integer-arithmetic program (ancillary file) verifies the enumeration, the divided digits, and every certificate.