The Riemann Hypothesis as a Unitary Constraint on the Analog-to-Digital Transition of Arithmetic: A Conditional Proof in Information Physics

This paper presents a conditional proof of the Riemann Hypothesis by recontextualizing the distribution of prime numbers within the framework of Information Physics and Spectral Signal Processing. We posit that the Number Line, operationally defined as a transmission channel for arithmetic information, is subject to physical constraints regarding information capacity and unitary evolution. By analyzing the Explicit Formula through the lens of Shannon-Nyquist Sampling Theory, we demonstrate that the non-trivial zeros of the Riemann Zeta function act as the discrete sampling frequencies of the arithmetic field. We introduce three physical postulates: (1) The Holographic Information Bound, which limits the spectral energy density of the error term; (2) Unitary Conservation, which forbids the generation of information ex nihilo; and (3) Operational Distinguishability, which requires a non-zero Signal-to-Noise Ratio (SNR) for the resolution of distinct integers. We establish that any zero off the critical line ("Re"(s)≠1/2) generates a "Hyper-Extensive" spectral noise that diverges asymptotically (X^2Θ). This divergence violates the Bekenstein bound for 1D manifolds and drives the SNR to zero, rendering the integer sequence operationally indistinguishable at the limit. Consequently, we conclude that the Riemann Hypothesis is a necessary condition for the number line to function as a physically realizable, unitary information system.