The Archimedean Spectral Factor of the Riemann Zeta Function on the Bounded Manifold [1, 2]

This research demonstrates that the interval [1, 2], equipped with a singular Jacobi-type weight, provides a concrete geometric realization of the archimedean component of the completed Riemann zeta function. We prove that this weighted manifold is isometric to the flat interval [0, pi] via an explicit proper-distance map. By performing a von Neumann deficiency index analysis for the Berry-Keating Hamiltonian on this domain, we show that self-adjoint extensions exist as a one-parameter family with purely linear spectra. Key Results: Geometric Isometry: The Riemannian manifold ([1,2], w(x)dx^2) with weight w(x) = 1/((x-1)(2-x)) is isometric to [0, pi], establishing a finite proper length of exactly pi. Spectral Determinant: The spectral determinant of the associated Laplacian is shown to be sin(piE)/(piE), which relates to the Gamma factor of the zeta function via Euler's reflection formula. Boundary Scattering: The scattering phases at the singular endpoints reproduce the phase of the archimedean Gamma factor, accounting for the smooth part of the Riemann-von Mangoldt formula. Adelic Conjecture: We propose that while the interval [1, 2] encodes the archimedean component, the full spectral realization of the Riemann zeros requires an adelic manifold where each prime p contributes a p-adic geometric factor.