2026/07/29 by Peter Chocian
Mathematics · #math.GM #msc:11R18 #msc:11R23 #msc:11R29 #msc:11Y40
13 pages. Deterministic integer-arithmetic verification program included as ancillary file. Companion to arXiv:2607.23177
arxiv created 2026/07/29 · arxiv updated 2026/07/31
Let χ be a primitive odd Dirichlet character of conductor f. For the universal projector polynomials Pm introduced in arXiv:2607.23177, defined by ∑m Pm(X)Ym = -log(1-X(1-e-Y)), we evaluate the character Fourier spectrum at ht = ζft/(ζft-1): for every odd m, ∑t χ(t)Pm(ht) = τ(χ)Bm,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-ζfζp and carry the programme through in the first non-real case: for the two primitive quartic characters modulo 5 and primes p < 500, p ≡ 1 \pmod20, exactly eleven zero lines occur. On each line an integral character projection of 1-ζ5ζp 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.