What question did this study set out to answer?

To fix a canonical normalization for the frozen GL(1) reduced determinant and demonstrate the determinant–ξ identity.

February 19, 2026Open Access

TEBAC Hilbert–Pólya for GL(1): E3 determinant–ξ identity (canonical normalization and rigidity)

Key Points

To fix a canonical normalization for the frozen GL(1) reduced determinant and demonstrate the determinant–ξ identity.
Established a canonical normalization for the GL(1) determinant through a rigidity argument.
Utilized inputs from the frozen E2 determinant package for various determinant configurations.
Employed log-derivative matching on the region where Re(s) > 1.
Proved the identity D_{GL(1)}(s) = ξ(s) using a fixed basepoint for normalization.
Established clarity on the relationship between the GL(1) determinant and the ξ function as Re(s) approaches infinity.

Abstract

This module (E3) fixes a single canonical normalization for the frozen GL (1) reduced determinant D₆₋ (₁) () (primitive spectral shift, with 0) and proves the determinant identity in the form D₆₋ (₁) (s) = (s) (with D₆₋ (₁) (s): =D₆₋ (₁) (s (1-s) ) ) by a rigidity argument. All determinant inputs (HT2/HT3, kernel-centering, cutoffs, and subtraction Pₑ₄₅^final) are imported from the frozen E2 determinant package closure. The log-derivative match on (s) >1 is imported from the Trace Prime modules. The basepoint/normalization is fixed only via ₒ + Q (s) =1 for the quotient Q (s): =D₆₋ (₁) (s) / (s), eliminating any ambiguity. Appendices record the formal complex-analytic lemmas, E2 integral convergence facts, Trace Prime glue identities, a formal wedge continuation/rigidity package, and (optionally) standard -function facts.

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