Balanced Normalization and Archimedean Matching in the TEBAC Hilbert–Pólya Framework (GS6 Companion)

A companion manuscript designed for referee-checkability and mechanical dependency auditing. This companion preprint isolates and discharges the balanced-normalization deliverable (GS6) in the TEBAC Hilbert–Pólya program of Karadzhov (Zenodo, 2026; DOI: 10.5281/zenodo.18202632). Working entirely within the balanced end package (Variant B), we prove that the confining end coefficients in the GL(1) normal form L₀(α,β) = −∂ᵤ² + α·e^(2u) + β·e^(−2u) admit a canonical scale fixing c₊c₋ = √π, equivalently αβ = π,where c₊ = √α and c₋ = √β. The argument is engineered to be firewall-safe: it does not invoke analytic continuation or functional-equation input for ζ or ξ. What this preprint delivers Haar/measure rigidity on the arithmetic end: a canonical identification L²((0,∞), dx/x) ≅ L²(ℝ, du) (up to a global scalar). Invariant scale parameter: admissible end rescalings x ↦ c·x preserve αβ = (c₊c₋)². Reference-package invariance: via unitary shift conjugation, the spectral invariants entering the zeta-regularized reference factor depend only on αβ. No hidden exp(b·s) ambiguity: compatibility with the diagonal heat-trace subtraction prevents residual s-dependent exponential renormalizations in the completed determinant. Imported convention (explicit) The completion convention uses the standard archimedean factor π^(−s/2)·Γ(s/2) (up to an s-independent constant). Under this convention, the balanced end scale is uniquely fixed by αβ = π. Scope and intended audience Written for readers in spectral theory, heat-kernel methods, and spectral / zeta-regularized determinants; the presentation is modular and referee-oriented. Companion to T. L. Karadzhov, A Hilbert–Pólya Program for the Riemann Hypothesis via TEBAC: Conditional Reduction, Trace Identities, and an Unconditionality Audit (Zenodo, 2026). DOI: 10.5281/zenodo.18202632.