What question did this study set out to answer?

The aim is to refine the arithmetic generator in the TEBAC Hilbert–Pólya GL(1) framework to ensure it is a well-defined self-adjoint operator.

February 16, 2026Open Access

TEBAC Hilbert–Pólya for GL(1): E2-final operator choice and self-adjointness (Addendum)

Key Points

The aim is to refine the arithmetic generator in the TEBAC Hilbert–Pólya GL(1) framework to ensure it is a well-defined self-adjoint operator.
Adjusted the arithmetic generator to include a new confining operator term.
Applied quadratic forms to demonstrate lower semiboundedness.
Established unitarily equivalent representations with a one-dimensional Schrödinger operator.
Developed a consistent heat trace package across different sectors.
Confirmed the self-adjointness of the arithmetic generator via functional analysis.
Established a relation to a one-dimensional Schrödinger operator with a confining potential.
Prevented version drift in the GL(1) sector through consistent subtraction conventions.

Abstract

This addendum fixes a final confining version of the arithmetic generator \ (L₀ₑ₈ₓ₇ = L₀ₑ₈ₓ₇^ (0) + B₂₎₍₅\) used in the TEBAC Hilbert--Pólya \ (GL (1) \) construction (E2/GS5), and records a minimal functional-analytic package ensuring that \ (L₀ₑ₈ₓ₇\) is well-defined as a lower semibounded self-adjoint operator via quadratic forms (KLMN). On the zeta channel \ (H₂₄₍ = Ran (P₂₄₍) \) the reduced operator is unitarily equivalent to a one-dimensional Schrödinger operator \ (-ᵤ² + V₄₅₅ (u) \) with confining potential and compact resolvent. The note also fixes a final small-time subtraction convention \ (Pₑ₄₅^final (t) \) for the centered heat trace package used consistently across E2/E3/E4, thereby eliminating ``version drift'' in the \ (GL (1) \) sector of the programme.

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