AI 中文总结
该研究计算圆单位与扭伯努利类分量的特征傅里叶谱，建立谱零点与伯努利零点准则的等价性，在模5本原四次特征及相关素数情形中验证零点线数量，推进希尔伯特类分量的相关研究。
AI 中文摘要
设χ为导子f的本原奇狄利克雷特征。对于arXiv:2607.23177中引入的通用投影多项式Pₘ，其定义为∑ₘPₘ(X)Yᵐ = -log(1 - X(1 - e⁻<sup>Y</sup>))，我们在hₜ = ζ<sub>f</sub><sup>t</sup>/(ζ<sub>f</sub><sup>t</sup> - 1)处计算特征傅里叶谱：对每个奇数m，∑<sub>t</sub>χ̄(t)Pₘ(hₜ) = τ(χ̄)B<sub>m,χ</sub>/(m·m!)。在p ∤ f的任意素数处约化后，m = p - j的情形将该谱（包括其精确非零标量）与χω⁻ʲ对应的分划广义伯努利值等同；因此，在每个剩余域上，谱零点与伯努利零点准则一致，且无需对系数域做分裂假设。我们将该恒等式与圆单位1 - ζ<sub>f</sub>ζ<sub>p</sub>的局部库默尔谱关联，并在首个非实情形中推进该计划：对于模5的两个本原四次特征及素数p < 500、p ≡ 1 mod 20，恰好出现11条零点线。每条线上，1 - ζ₅ζ<sub>p</sub>的整特征投影在局部处处非分歧，有限分裂素阿廷计算证明其不是全局p次幂，而逐特征主猜想表明该根基生成完整的p阶希尔伯特类分量。11个分量中有6个出现在经典正则素数处；在p = 61时，两个共轭特征在不同指标上有贡献。确定性整数算术程序（辅助文件）验证了枚举、分划数字及所有证书。
英文摘要
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.
Comments13 pages. Deterministic integer-arithmetic verification program included as ancillary file. Companion to arXiv:2607.23177
