Finite-Depth Hard Arithmetic Localization and Near-Cramér Chebotarev–Sato–Tate in Growing Arithmetic Progressions
Year: 2026
Type: preprint
Abstract: This paper isolates a packet-to-selector mechanism that converts finite-depth analytic information on degree-controlled $L$-functions into normalized harmonic moments on literal moving prime intervals subject to arithmetic selectors. The analytic packet is required to have degree-linear prime-power coefficients, controlled conductor and archimedean complexity, no nonconstant main term, a controlled crossed-residue budget, and GRH at the finite spectral depth being used. The arithmetic selector is fully recombined before a single prime-power/source-replacement correction and before hard-edge removal. The resulting hard-mode estimate has the form $$\left\vert{} A^\sharp_{\lambda;\mathcal{S}} \right\vert{} \ll d_\lambda \mathcal{R}_{\mathcal{S},M},$$ where $$\mathcal{R}_{\mathcal{S},M} = \frac{\sqrt{X}}{\omega h} L_M \left( 1+\log\frac{h}{\Delta} \right) + \frac{\mathcal{E}_{\mathcal{S}}(X,\Delta)}{\omega h} + \frac{h}{X\log X}.$$ Here $\omega$ is the logarithmic density of the selector and $\mathcal{E}_{\mathcal{S}}(X,\Delta)$ is its selected first-prime edge mass. The harmonic discrepancy step is separated from the hard-localization theorem. If the harmonic approximation has error $O(M^{-1})$ and weighted spectral complexity $$\mathfrak{C}(M) \ll M^\kappa,$$ then $$D_{\mathcal{S}} \ll \frac{1}{M} + M^\kappa \mathcal{R}_{\mathcal{S},M}.$$ For $SU(2)^r$ box discrepancy this recovers the standard balance $$D^{(r)} \ll \frac{1}{M} + M^r \mathcal{R}_{\mathcal{S},M}.$$ The resulting exponent $1/(r+1)$ is a consequence of existing compact-group harmonic approximation and is not claimed as a new discrepancy phenomenon. As an 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. Fix a conjugacy class $$C \subset S_3,$$ and impose simultaneously $$X < p \le X+h, \qquad p \equiv a \pmod q,$$ and $$\operatorname{Frob}_p(L/\mathbf{Q}) \in C.$$ Writing $$F = L^{A_3},$$ the quadratic resolvent $F / \mathbf{Q}$ is real quadratic, and the two-dimensional standard representation satisfies $$\mathrm{std} \simeq \operatorname{Ind}_{G_F}^{G_{\mathbf{Q}}} \psi$$ for a nontrivial cubic character $\psi$ of $\operatorname{Gal}(L/F)$. Cyclic base change and the all-degree symmetric-power functoriality of Newton–Thorne over the totally real field $F$ realize the nonabelian standard-sector packet through $$L_F\!\left( s, \operatorname{Sym}^m \pi_F \otimes \psi\xi_F \right).$$ Under GRH for the finite packet at depth $M$, define $$\mathfrak{G} = \frac{h}{\varphi(q)\sqrt{X}\log X},$$ and $$\eta = \frac{1+\log\mathfrak{G}}{\mathfrak{G}} + \frac{h}{X\log X}.$$ The Sato–Tate discrepancy inside the combined Chebotarev/progression selector satisfies $$D_{E,L,C;X,h;q,a} \ll_{E,L,C} \eta^{1/2}$$ after optimization. Thus, for $$h = \sqrt{X}(\log X)^A, \qquad A>1,$$ and $$\varphi(q) = o\!\left((\log X)^{A-1}\right),$$ one obtains $$D_{E,L,C;X,h;q,a} \ll_{E,L,C,A} \left[ \frac{\varphi(q)}{(\log X)^{A-1}} \left( 1+ \log \frac{(\log X)^{A-1}}{\varphi(q)} \right) \right]^{1/2}.$$ Version v1.1 adds a critical Cramér-order Chebotarev–Sato–Tate occupancy theorem. For every fixed Sato–Tate interval $I \subset [0,\pi]$ of positive measure and every fixed conjugacy class $C \subset S_3$, there exist fixed constants $M_{C,I}$ and $G_{C,I}$ such that, under the corresponding fixed spectral-depth GRH hypotheses, every sufficiently large admissible window $$h = G\,\varphi(q)\sqrt{X}\log X, \qquad G \ge G_{C,I},$$ contains $$\gg_{E,L,C,I} \frac{h}{\varphi(q)\log X}$$ primes satisfying simultaneously $$p \equiv a \pmod q, \qquad \operatorname{Frob}_p(L/\mathbf{Q}) \in C, \qquad \theta_p \in I.$$ At the threshold Cramér order this gives $$\gg_{E,L,C,I} \sqrt{X}$$ target primes in every sufficiently large admissible moving window. The resulting two-phase interpretation is $$\text{fixed spectral depth} \longrightarrow \text{critical Cramér-order positive occupancy},$$ whereas the supercritical equidistribution theorem uses growing spectral depth. Fixed-depth Sato–Tate occupancy itself has earlier Hoheisel-scale precedents. The contribution here is the source-independent hard-localization interface and its transfer to selector-normalized Cramér order with a growing arithmetic progression and a fixed nonabelian Chebotarev condition. In the literature surveyed in the paper, no exact precedent is claimed absolutely for this combined interface. Version v1.1 strengthens the previous public version by adding the critical Cramér-order occupancy theorem, clarifying the finite-depth interpretation, updating the relation to short-interval Sato–Tate precedents, and explicitly positioning the joint Rankin–Selberg realization as a precursor to the abstract packet-to-selector framework.
Author: Lee, Byoungwoo
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204631335