Microlocal Prime Correspondences and Prime-Power Trace Identities in the TEBAC Hilbert–Pólya Framework (GS3 Companion)

This companion preprint isolates and closes the GS3 module in the TEBAC Hilbert--Pólya framework: a microlocal prime/prime-power correspondence package on the one-dimensional balanced $GL(1)$ channel and the associated prime-power heat-trace identities required in the GS5 exchange step.We work in the exponential confinement regime on R, modeled by the Schrödinger operator{equation*}LGL = -∂_u^2 + V(u),(u) = α e²ᵘ + β e⁻²ᵘ + R(u),α,β > 0,{equation*}where R is assumed trace-smoothing (for example, a smoothing operator with C^∞, compactly supported kernel).The centered heat trace is{equation*}Θ(t) := (e^{-tLGL}) - Pref(t),{equation*}with Pref(t) the fixed GS5 reference subtraction. On the analytic side (GS3-α), we define the prime-power correspondence operators Tpᵏ as normalized $GL(1)$ translations and prove the shifted-diagonal trace identity for the heat-regularized compositions e^-tLGL Tpᵏ in the fundamental-domain trace convention{equation*}(e^{-tLGL} Tp^k)= p-k/2 ∫F_p KGL(t; u, u + klog p),du,{equation*}where KGL(t;u,v) is the heat kernel of LGL and Fₚ is a fundamental domain of length log p.Using Gaussian domination bounds for KGL(t;u,v), we obtain uniform prime-power majorants on compact t-intervals, which yield absolute convergence and uniformity of the series {equation*} ∑ₚ ∑k ≥ 1 (e^{-tLGL} Tp^k), {equation*} and provide GS5-ready Tonelli/Fubini domination on the half-plane (s) > 1, including differentiation under the integral sign for the prime test family. On the structural side (GS3-β), we construct a microlocal branch decomposition on the $GL(1)$ channel and identify the prime-power contributions, matching the diagonal term to the GS5 subtraction Pref(t) and proving trace-negligibility of the microlocal remainder under explicit support assumptions.In particular, the structural identity takes the form{equation*}Θ(t) = -∑ₚ∑k≥ 1 (e^{-tLGL}Tp^k), t>0,{equation*}and the right-hand side is an absolutely convergent series for each $t>0$, uniformly on compact t-intervals. For completeness, an appendix titled “Sieve roadmap” records a legacy semigroupoid normal form and sieve-input interface (I1–I5) purely as roadmap; it is not used in any proof in this preprint.This upload is part of the broader TEBAC “unconditionality audit” and is intended to be cited as the GS3 companion module supporting the prime-power trace package used in GS5.