Discretizing the Archimedean place in the spectral realization of the Riemann zeta function

We revisit the operator-theoretic construction proposed in the earlier draft and present it in a more pedagogical form, with expanded definitions, intermediate logical steps, and detailed proofs of the genuinely rigorous parts. The core objects are the truncated arithmetic operatorsP = K₀ₑ₈ₓ^ (P) I + I L₎ₒ₂, on tensor products of arithmetic Hilbert spaces with the harmonic oscillator space. We prove carefully that the relevant truncated operators are self-adjoint, compute their spectra, and establish exact heat-trace factorisations in the range where the traces converge. We also explain in detail why the determinant identities suggested by the original draft are natural targets, but remain open analytic problems rather than established theorems. The goal of this expanded revision is not to strengthen the claims, but to make the logical status of each statement fully transparent and to remove unnecessary nontrivial jumps for readers.