Centered Heat Traces and Zeta-Regularized Determinants in a Balanced Hilbert–Pólya Model: The GS5 Proof Package (TEBAC Companion)

We isolate and close the GS5 analytic module in the TEBAC Hilbert--Pólya framework, focusing on the one-dimensional balanced GL (1) channelequation* L₆₋ = -ᵤ² + V (u), V (u) = e^2u + e^-2u + R (u), , > 0, equation*and the associated centered heat traceequation* (t): = (e^-tL₆₋) - Pₑ₄₅ (t). equation*Working in the balanced exponential confinement regime, the paper assembles a referee-oriented, self-contained package for the centered heat trace, the centered spectral zeta function, and the zeta-regularized determinant, designed to be cited as the GS5 closure component in the broader TEBAC Hilbert--Pólya program. Under the explicit confining Schrödinger model on R and a trace-smoothing hypothesis on R, the paper proves the analytic backbone needed for the trace--determinant calculus and the prime-side exchanges, including: (I) Gaussian domination bounds for the heat kernel K₆₋ (t; u, v) via the Feynman--Kac formula. (II) Compact resolvent and trace-class properties of e^-tL₆₋ for all t > 0. (III) A small-time trace expansion with the characteristic logarithmic singularity equation* (e^-tL₆₋) = a-₁ \, t^-1/2 (1t) + a₀ \, t^-1/2 + a₁ + O (t^1/2), a-₁ = (4) ^-1/2, equation* together with a self-contained Duhamel remainder argument showing that trace-smoothing perturbations contribute only an O (t^1/2) correction after subtraction. (IV) Holomorphy at w=0 for the centered spectral zeta function equation* ₂₄₍ (w; s) = 1 (w) ₀^ t^w-1 e^- (s-1/2) ² t \, (t) \, dt, equation* and the resulting log-derivative identity for the centered determinant D₂₄₍ (s). (V) An exchange theorem on an explicit domain U that justifies Tonelli/Fubini and differentiation under the integral sign for the prime test family, yielding the Euler--von Mangoldt block equation* 2 (s-12) ₀^ e^- (s-1/2) ² t (t) \, dt = -₏ ₊ ₁ (p) \, p^-ks = -₍ ₁ (n) nˢ, (s) > 1. equation* The paper carefully separates what is proved internally (heat-kernel bounds, trace-class control, small-time structure, zeta/determinant calculus, and exchange on U) from what is imported as structural geometric input (the prime/Hecke correspondences Tₚ and the trace--prime decomposition of (t) ). This makes the GS5 component a checkable, standalone analytic module and reduces the remaining identification targets in the full TEBAC Hilbert--Pólya closure chain to explicit, auditable geometric/microlocal deliverables in the main preprint.