Since 1859, the Riemann Hypothesis has been primarily approached through analytic number theory. Following the spectral geometry insights of Alain Connes and the Hilbert-Pólya conjecture, this paper proposes a paradigm shift by redefining the Riemann Zeta function no longer as a static series, but as an operator of wave mechanics acting on the phase space of prime numbers. By applying the Central Limit Theorem to lacunary trigonometric sums, we establish a Stochastic Dispersion Lemma proportional to square root of N. The transition from discrete arithmetic to a continuous geometric topology is mathematically formalized via the Euler-Maclaurin wake integral. We demonstrate that to admit stable non-trivial zeros, the Zeta operator must satisfy a strict unitarity condition. Furthermore, by interpreting the Riemann Functional Equation as a "thermodynamic mirror", we prove that the critical line sigma = 1/2 is the unique axis of unitary invariance. This energetic conservation equilibrium formally connects the predictability of prime numbers to quantum chaos and wave dynamics, while redefining cryptographic complexity (RSA) as a geometric magnitude (artificial entropy). Files included: Full Manuscript (English) PDF v4.3 Full Manuscript (French) PDF v4.3 Python Source Code for Graphical Proofs (Vortex, Dispersion, Potential Well)
Jean-Philippe Annet (Fri,) studied this question.