Mixed Frobenius Character Energies for Joint Chebotarev–Sato–Tate Data: Normally Monomial Realizations and Heat-Smoothed Entropy Decay

Overview This record contains Version v0.6r2 of the research article: Mixed Frobenius Character Energies for Joint Chebotarev--Sato--Tate Data: Normally Monomial Realizations and Heat-Smoothed Entropy Decay. The paper develops a harmonic and information-theoretic framework for the joint distribution of finite Galois Frobenius data and Sato--Tate angles associated with a fixed non-CM elliptic curve over Q. For a fixed finite Galois extension K/Q with Galois group G, the joint Frobenius data are represented by pairs (Cₚ,θₚ)∈ Conj(G)×[0,π], where Cₚ is the Frobenius conjugacy class and θₚ is determined by aₚ=2√pcosθₚ. The reference distribution is the Chebotarev--Sato--Tate product measure QG = qG ⊗ μST, where qG(C) = CG and dμST(θ) = 2/πsin²θ\, dθ. Using irreducible characters χ_ρ of G and the $SU(2)$ characters Uₖ(cosθ) = sin((k+1)θ)/sinθ, the paper introduces the mixed Fourier coefficients λ(ρ,k) = ∫ χ_ρ(C)̄ Uₖ(cosθ)\, dλ(C,θ). The resulting joint character energy admits the exact orthogonal decomposition Dⱼₒᵢₙₜ = DCheb + DST + Dₘᵢₓ. The mixed sector records correlations that cannot be detected from the Chebotarev and Sato--Tate marginal distributions separately. The paper also identifies a structural obstruction to using raw relative entropy. Since the empirical Sato--Tate distribution is atomic while μST is continuous, the unsmoothed Kullback--Leibler divergence is infinite. After applying the $SU(2)$ heat semigroup for a fixed time $t>0$, the full mixed character energy becomes exactly the chi-square divergence: χ²(λₜ QG) = ∑(ρ,k)≠(1,0) e⁻²ᵗᵏ⁽ᵏ⁺²⁾ λ(ρ,k) ². Consequently, DKL(λₜ QG) ≤ log( 1 + χ²(λₜ QG) ) ≤ χ²(λₜ QG), and the total variation distance is controlled by the same spectral energy. Arithmetic Realization and Main Results The arithmetic realization is proved unconditionally for fixed Galois extensions whose Galois groups are normally monomial. For every irreducible representation ρ of G, one chooses a normal subgroup H_ρ G and a linear character ψ_ρ such that ρ IndH_ρG ψ_ρ. The solvable quotient is decomposed into a tower of cyclic prime-degree extensions. Iterated automorphic induction then produces a cuspidal automorphic representation σ_ρ whose unramified local parameters agree with those of ρ. The proof explicitly establishes the chain: stagewise irreducibility Artin-orbit freeness automorphic-orbit freeness stagewise cuspidality. The coefficient λ(ρ,k) is governed by the contragredient Rankin--Selberg system LS(s, ρ̄ᵏ E) = LS(s, σρ̄×πₖ), where πₖ = SymᵏπE. At good primes, the first logarithmic-derivative coefficient is a_σρ̄×πₖ(p) = χ_ρ(Cₚ)̄ Uₖ(cosθₚ). Using conductor-uniform Rankin--Selberg prime number theorems, explicit exceptional-zero terms, analytic-conductor growth, and a growing symmetric-power window, the paper proves unconditional decay of the heat-smoothed information quantities. The logarithmically weighted empirical law is λₓˡᵒᵍ = 1/ϑS(x) ∑_p≤ x \\ p∉ S (log p)\, δ(Cₚ,θₚ), where ϑS(x) = ∑_p≤ x \\ p∉ S log p. For every fixed heat time $t>0$, there exist constants cₜ,Cₜ>0, depending at most on E, K, and t, such that χ²( (λₓˡᵒᵍ)ₜ QG ) ≤ Cₜ exp( -cₜ loglog x/logloglog x ). The same rate therefore holds for the Kullback--Leibler divergence: DKL( (λₓˡᵒᵍ)ₜ QG ) ≤ Cₜ exp( -cₜ loglog x/logloglog x ), and, after adjusting the constants, for total variation: (λₓˡᵒᵍ)ₜ - QG TV ≤ Cₜ exp( -cₜ loglog x/logloglog x ). The paper also proves weak convergence of the raw unweighted joint Frobenius distribution. For every conjugacy class C⊆ G and every interval I⊆[0,π] whose boundary has zero Sato--Tate measure, limx→∞ #\ p≤ x : p∉ S,\, Cₚ=C,\, θₚ∈ I \π(x) = CG μST(I). Thus, the raw joint Frobenius law converges weakly to qG⊗μST. Structure and Scope The article organizes the abelian, dihedral, cyclic-complement Frobenius, and normally monomial cases within one character-energy framework. It clearly separates the external automorphic and analytic inputs from the arguments proved in the paper, including: the exact Chebotarev, Sato--Tate, and mixed-sector decomposition; the raw relative-entropy obstruction; the heat-smoothed chi-square identity; the Artin-to-automorphic orbit-transfer argument; the multi-step cuspidal automorphic realization; the conductor-window and exceptional-zero bookkeeping; the growing-spectral-window summation; the information-theoretic decay estimates; and the raw joint product law. The results are established for fixed Galois extensions and fixed heat time $t>0$. Growing torsion levels, arbitrary nonnormal monomial induction, and vanishing heat scales t=t(x)→0 are not claimed. Version v0.6r2 incorporates the final proof-hardening and submission audit, including uniform conductor thresholds, partial Euler-product conventions, contragredient indexing, continuity-set treatment of interval counts, bad-prime control, external-input dependency tracking, and final notation and citation checks. Version: v0.6r2 Author: Lee Byoungwoo Date: August 2, 2026 Files included: Article PDF, LaTeX source, README file, and SHA-256 manifest.