The Riemann Zeta Function as a Conservative Dynamical System: A Heuristic Demonstration of the Riemann Hypothesis via Unitary Invariance

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)