On Certain Spectral Operators Associated with the Riemann Zeta Function - with consequences for the Riemann Hypothesis

We develop a spectral--operator framework for the Riemann zeta function in which the arithmetic explicit formula arises as a canonical renormalized trace identity associated to a self--adjoint scaling operator perturbed by prime dilation dynamics. The construction is unconditional and avoids any positivity, coercivity, or trace--class assumptions. At the distributional level, we obtain a complete spectral identification with the Weil distribution associated to $\zeta(s)$, thereby resolving the traditional ambiguity between operator--theoretic models and explicit formulae. We then isolate a single remaining analytic obstruction preventing operator--level spectral rigidity: the absence of \emph{polynomial Schatten control} for the renormalized prime kernel. The core contribution of this work is a structural reduction of this obstruction to a rigidity problem for the prime dilation action on an associated von Neumann algebra. Using Connes spectrum theory, weak Mourre estimates, and noncommutative prime mixing, we show that the dilation action admits a spectral gap, ruling out invariant Cartan subalgebras and eliminating the final source of exponential growth. As a consequence, polynomial Schatten control follows, excluding off--critical--line spectral contributions and yielding a reformulation of the Riemann Hypothesis as a rigidity statement for prime--generated operator algebras. The argument is entirely unconditional and highlights rigidity, rather than positivity, as the fundamental mechanism underlying the Riemann Hypothesis.