The Fine-Structure Hierarchy of Prime Biases and the Universal Dominance of $-1 \pmod N$

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 $p^2 \equiv 1 \pmod N$, 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., $3, 5, 7 \pmod 8$). 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 $\widetilde S_T(x, a)$ and adopting the virtual character $χ_{1,a}(x) := \mathbf{1}_{\{x \equiv 1 \pmod N\}} - \mathbf{1}_{\{x \equiv a \pmod N\}}$, the principal character component $χ_0$ cancels identically since $1 - \overline{χ_0}(a) = 0$. This automatic algebraic elimination erases both the universal logarithmic growth $\log x$ and the background noise $\log L(1, χ^2)$. 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 $\log L(1, χ)$. We prove that $\widetilde S_T(x, χ_{1,a}) := \widetilde S_T(x, 1) - \widetilde S_T(x, a) = C_N \cdot \log L(1, χ_{1,a}) + \mathcal{O}((\log x)/\sqrt x)$ as $x \to \infty$, where $C_N > 0$ depends solely on $N$. Consequently, we establish a deterministic multi-way ranking (such as $7 > 3 > 5 > 1 \pmod 8$) that completely transcends the classical quadratic residue framework.