Quantitative Sato--Tate in Near-Square-Root Prime Intervals under Symmetric-Power GRH

# Quantitative Sato--Tate in Near-Square-Root Prime Intervals under Symmetric-Power GRH **Version:** v3.0 **Specific Version DOI:** 10.5281/zenodo.22112048 **Author:** Byoungwoo Lee **Series relation:** Major-version successor to *Effective Entropy--Spectral Sato--Tate v2.2c: Quantitative Convergence under Symmetric--Power Analytic Inputs* (previous specific-version DOI: 10.5281/zenodo.17959752). ## Description Let E/Q be a fixed non-CM elliptic curve. This version rebuilds the earlier effective entropy--spectral Sato--Tate framework around a direct arithmetic localization theorem for the literal moving prime interval $(X,X+h]$. The central result is a finite-depth hard-window localization principle. For angular cutoff M, edge width Δ, and window length h, assuming RH for ζ(s) and GRH for L(s,Symᵐ E) only for 1≤ m≤ M, the paper proves I⊆[0,π]|μX,h^(I)-μST(I)|E1/M+M RM(X,h,Δ), where RM(X,h,Δ)=√ X/h\, LM,Δ(1+/Δ)+Δ/h+h/Xlog X. Here LM,Δ is logarithmic in the conductor, M, and X/Δ. In the near-square-root regime h=√ X(log X)A, A>1, choosing Δ=√ Xlog X, (log X)(A-1)/2√loglog X gives I⊆[0,π]|μX,h^(I)-μST(I)|E,A√loglog X(log X)(A-1)/2. The proof uses a two-scale plateau in the prime variable. Its Mellin transform has transition heights $X/h$ and X/Δ, leading to low-, middle-, and high-zero ranges in the symmetric-power explicit formula. The hard interval is recovered by Brun--Titchmarsh control of the edge layers. Secondary consequences include a truncated character-energy estimate and a heat-regularized relative-entropy estimate. The raw empirical measure is atomic, so no unsmoothed KL divergence is asserted. After the radial SU(2) heat semigroup is applied, Parseval gives an exact χ² identity and the heat kernel itself controls the high modes. ## Major changes from v2.2c Version v3.0 is a major mathematical rebuild rather than a minor revision. - Replaces the global empirical measure p≤ X by the literal moving hard prime interval $(X,X+h]$.- Replaces abstract low/high-mode and Rankin--Selberg packages by a direct finite-depth symmetric-power explicit formula.- Makes the analytic depth explicit: angular resolution through M uses symmetric-power GRH only through degree M.- Introduces independent window and edge parameters h and Δ, with Mellin transition scales $X/h$ and X/Δ.- Proves the near-square-root specialization h=√ X(log X)A for every fixed $A>1$.- Treats entropy only after explicit heat regularization; no artificial arithmetic evolution and no arithmetic high-frequency tail hypothesis are used.- Supplies a rigorous contour closure, absolute convergence of the zero sum, bad-prime coefficient control, and explicit trivial-zero bookkeeping. ## Relation to companion work The companion preprint *Mixed Frobenius Character Energies for Joint Chebotarev--Sato--Tate Data: Normally Monomial Realizations and Heat-Smoothed Entropy Decay*, Version v0.6r2, DOI 10.5281/zenodo.21743631, develops product-character energy and fixed-heat entropy for joint Chebotarev--Sato--Tate data. The present v3.0 paper is independent of that paper's automorphic-induction machinery; the overlap is the character/Parseval heat-regularization viewpoint. ## Claim boundary This work does **not** claim a new class of explicit-formula method or the first theorem on short-interval Sato--Tate. The claimed contribution is the finite-depth hard-window localization theorem, its near-square-root specialization under symmetric-power GRH, and the secondary character-energy and heat-entropy consequences. ## Suggested citation note For reproducibility, cite the specific v3.0 Version DOI **10.5281/zenodo.22112048**. The Zenodo Concept DOI may instead be used when citing the evolving series as a whole.