AI 中文总结
本文研究了超越经典不规则性的显式扭曲Hilbert类组件，通过特征投影和斯特林多项式分析，证明了特定线上的幂等乘积性质及广义伯努利数的p进估值，揭示了扭曲退化与经典正则性的关系。
AI 中文摘要
设p ≡ 1 mod 6为素数，K_p = Q(ζ_{3p})。我们研究反射圆单位μ_p = (1 + zζ_p)/(1 + \overline{z}ζ_p)，其中z = -ζ_3^2，及其特征投影。一个通用的斯特林多项式P_m给出了μ_p的反谱与原始除以χ_{-3}的扭曲Stickelberger谱之间的精确恒等式：P_{p-j}(h) - P_{p-j}(1-h) = -(2h-1)(j-1)!b_j，h = z/(1+z)。因此，反射单位的局部盲线恰好是相应除以扭曲伯努利特征值的零点。对于每个p < 500，我们列举这些零点。恰好有十二个特征线出现。在每条线上，显式的积分幂等乘积μ_p在导数素数处是局部p次幂，但不是全局p次幂。小完全分裂素数提供有限的Artin证书。广义伯努利数在每种情况下都有精确的p进估值为1；因此，特征-wise主猜想证明每个根生成完整的Hilbert类场组件，其阶数为p。十二条线中有七条出现在经典正则素数上，因此扭曲退化低于500时，比对普通不规则性更常不可见而非对齐。第一个案例，p = 67，被完整处理，且一个确定性的整数算术程序（作为附录文件包含）重现了列举和每个证书。
英文摘要
Let $p \equiv 1 \pmod 6$ be prime and $K_p = \mathbf{Q}(ζ_{3p})$. We study the reflected circular unit $μ_p = (1+zζ_p)/(1+\bar zζ_p)$, $z = -ζ_3^2$, and its character projections. A universal Stirling polynomial $P_m$ gives an exact identity between the anti-spectrum of $μ_p$ and the primitive divided $χ_{-3}$-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 $μ_p$ 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.
Comments16 pages. Deterministic integer-arithmetic verification program included as ancillary file
