The Fine-Structure Hierarchy of Prime Biases and the Universal Dominance of $-1(\mathrm{mod}N)$

Abstract

We investigate the deterministic hierarchy of prime distribution biases in arithmetic progressions modulo $N$ using a regularized spectral approach. Classical studies on Chebyshev's bias attribute prime races primarily to the accumulation of prime squares $p^2\equiv 1(\mathrm{mod}N)$, which creates a systematic deficit in quadratic residue classes. However, this classical mechanism fails to explain or distinguish any bias among residue classes sharing identical quadratic residue status (e.g., $3,5,7(\mathrm{mod}8)$). To overcome the long-standing analytical obstacles of jump discontinuities and non-convergent boundary fluctuations inherent in classical Perron-type step-function truncations, we introduce a smooth $C^{\infty }$ Gaussian mollifier into Weil's explicit formula for Dirichlet $L$-functions. By defining the spectrally normalized individual mollified sums ${\tilde{S}}_T(x,a)$ and adopting the virtual character ${\chi }_{1,a}(x):=1_{\{x\equiv 1(\mathrm{mod}N)\}}-1_{\{x\equiv a(\mathrm{mod}N)\}}$, the principal character component ${\chi }_0$ cancels identically since $1-\overline{{\chi }_0}(a)=0$. This automatic algebraic elimination erases both the universal logarithmic growth $\log x$ and the background noise $\log L(1,{\chi }^2)$. Under the Deep Riemann Hypothesis (DRH), we uncover a hitherto undetected \textbf{fine-structure bias} (or \emph{secondary bias}) strictly governed by the special values $\log L(1,\chi )$. We prove that ${\tilde{S}}_T(x,{\chi }_{1,a}):={\tilde{S}}_T(x,1)-{\tilde{S}}_T(x,a)=C_N\cdot \log L(1,{\chi }_{1,a})+\mathcal{O}((\log x)/\sqrt{x})$ as $x\to \infty$, where $C_N>0$ depends solely on $N$. Consequently, we establish a deterministic multi-way ranking (such as $7>3>5>1(\mathrm{mod}8)$) that completely transcends the classical quadratic residue framework.

10 pages, 4 tables. This is a revised and expanded version of the author's previous manuscript "A Hidden Hierarchy of Chebyshev's Bias and the Dominance of -1 (mod N)" (submitted April 26, 2026)