What question did this study set out to answer?

The central aim is to discharge TP-specific assumptions from the TEBAC HP corpus and explore trace-prime conversion.

March 3, 2026Open Access

TEBAC HP III: The GL(1) Trace–Prime Package (TP), Fully Discharged

Key Points

The central aim is to discharge TP-specific assumptions from the TEBAC HP corpus and explore trace-prime conversion.
Discharged TP assumptions from TEBAC HP corpus, including T1, T3, and T5 assumptions.
Developed a justified trace-prime conversion for real values of s greater than 1.
Proved Tonelli/Fubini interchanges at the module level.
Established an upper bound for the Gaussian heat-kernel using truncation and Trotter domination.
Derived a representation via a Brownian-bridge model.
Produced an error term that holomorphically extends to the wedge defined by C9_c.
Validated interchanges at the module level, enhancing the foundational framework.
The analytic model supports the implications of the TEBAC Hilbert--PF3lya program through the explicit HP--II end operator.

Abstract

This module discharges the TP-specific assumptions from the TEBAC HP corpus (T1 arithmetic circle action/invariance, T3 end-translation model for prime correspondences, and T5 semigroupoid closure). It provides a justified trace--prime conversion on real s>1, producing an error term that extends holomorphically to the wedge \ Wc \;=\; \\, s C: ( (s-12) ²) > -c \, \ with all Tonelli/Fubini interchanges proved at module level. A key analytic ingredient is a self-contained Gaussian heat-kernel upper bound for the explicit HP--II end operator \ -ᵤ² + e^2u + e^-2u + R (u), \ proved via truncation and Trotter domination and re-derived via a Brownian-bridge (Feynman--Kac) representation in the appendix. This preprint is part of the TEBAC Hilbert--Pólya program.

TEBAC HP III: The GL(1) Trace–Prime Package (TP), Fully Discharged

Key Points

Abstract

Cite This Study