From the TEBAC GL(1) Channel to the Arithmetic GL(1) Channel: A Bridge Toward a Hilbert–Pólya Route to the Riemann Hypothesis

This preprint isolates a bridge between an intrinsic GL(1) channel arising in the TEBAC Hilbert--Pólya framework and the arithmetic GL(1) channel for the Riemann zeta function in the adelic setting. On the intrinsic side we assume a scalar Hilbert--Pólya package (self-adjointness, a centered zeta determinant of order 1 with no linear Hadamard factor, and a Trace--Prime block), together with an intrinsic GL(1) translation channel after GL(1) compression. On the arithmetic side we use the canonical model\[\mathcal{H}_{\mathrm{arith}} := L^2(\mathbb{A}^\times / \mathbb{Q}^\times, d^\times x),\]with prime-power idele multiplication operators and a centered regularized trace. We formulate precise intertwining, Trace--Prime, and wedge analyticity/uniqueness hypotheses which, combined with canonical determinant normalization, yield a non-circular identification\[\det_{\zeta}(s - H) = \xi(s)\]of a zeta-regularized determinant with the completed zeta function; under these hypotheses this implies the Riemann Hypothesis.