Finite-Depth Hard Arithmetic Localization and Near-Cramér Chebotarev–Sato–Tate in Growing Arithmetic Progressions

Abstract

We develop a finite-depth hard arithmetic localization principle that converts a degree-controlled finite packet of twisted $L$-functions into normalized harmonic-mode estimates inside literal moving prime intervals with arithmetic selectors. The central mechanism separates the analytic packet, selector, hard-edge, and denominator layers. A two-scale plateau explicit formula is combined with full selector recombination before prime-power and hard-edge removal. This yields a general packet-to-selector interface in which the analytic cost remains linear in the spectral degree, while the hard-edge contribution is paid only after recombination. As a concrete external realization, let $E / \mathbf{Q}$ be a fixed non-CM elliptic curve and let $L / \mathbf{Q}$ be a fixed totally real $S_3$-Galois extension. For a conjugacy class $C \subset S_3$, we study primes satisfying simultaneously $$X < p \le X+h, \qquad p \equiv a \pmod q,$$ and $$\operatorname{Frob}_p(L/\mathbf{Q}) \in C.$$ We quantify the Sato–Tate distribution of the Frobenius angles of $E$ over this selected prime set. Let $$F = L^{A_3}$$ be the real quadratic resolvent. The nonabelian standard $S_3$-sector is realized through the induction identity $$\mathrm{std} \simeq \operatorname{Ind}_{G_F}^{G_{\mathbf{Q}}} \psi,$$ where $\psi$ is a nontrivial cubic character of $\operatorname{Gal}(L/F)$. The corresponding finite analytic packet is represented by symmetric-power automorphic $L$-functions over the totally real field $F$, twisted by finite-order Hecke characters. Under GRH for the finite packet up to the required truncation depth, the normalized Sato–Tate discrepancy satisfies $$D_{E,L,C;X,h;q,a} \ll_{E,L,C} \frac{1}{M} + M\,\mathcal{R}_{M,q}(X,h,\Delta).$$ At the balanced hard-edge scale $$\Delta = \varphi(q)\sqrt{X}\log X,$$ define the Cramér-normalized variable $$\mathfrak{G} = \frac{h}{\varphi(q)\sqrt{X}\log X}$$ and the phase parameter $$\eta = \frac{1+\log \mathfrak{G}}{\mathfrak{G}} + \frac{h}{X\log X}.$$ Choosing the truncation depth at the scale $$M \asymp \eta^{-1/2}$$ gives $$D_{E,L,C;X,h;q,a} \ll_{E,L,C} \eta^{1/2}.$$ In particular, for $$h = \sqrt{X}(\log X)^A, \qquad A>1,$$ and $$\varphi(q) = o\!\left((\log X)^{A-1}\right),$$ we obtain $$D_{E,L,C;X,h;q,a} \ll_{E,L,C,A} \left[ \frac{\varphi(q)}{(\log X)^{A-1}} \left( 1+ \log \frac{(\log X)^{A-1}}{\varphi(q)} \right) \right]^{1/2}.$$ The harmonic moment-to-discrepancy layer is compatible with existing compact-group Erdős–Turán theory. The emphasis here is instead on the finite-depth hard-mode production interface: $$\text{finite analytic packet} \longrightarrow \text{two-scale localization} \longrightarrow \text{selector recombination} \longrightarrow \text{single recombined correction} \longrightarrow \text{hard-edge recovery} \longrightarrow \text{denominator normalization}.$$ The totally real $S_3$-Chebotarev application provides a nonabelian realization beyond the arithmetic-progression-only settings from which the abstract mechanism was originally extracted. In the literature surveyed here, we have not found an exact precedent for this combined finite-depth hard-mode interface. Nevertheless, no absolute priority claim is made.