Spectral Boundaries of the Divisor Operator Framework: A Fredholm Determinant No-Go Theorem for the Riemann Hypothesis

This paper presents the decisive test of the algebraic-spectral framework established in four preceding works. By constructing a Fredholm determinant ∆(s) associated with the bounded inverse divisor operator T −1 , we subject the model to the rigorous requirements of analytic number theory: the Euler product and the functional equation. Through the calculation of the trace expansion, we rigorously demonstrate that the operator yields an irreducible non-prime residual structure due to the composite contributions encoded in the algebraic weight W(v(n)). Consequently, the model fails to reproduce the Euler product and thus cannot serve as an analytic representation of the Zeta function. However, we acknowledge that the spectral density accurately simulates the Riemann-von Mangoldt law, making this model a powerful statistical spectral tool rather than a proof of the Riemann Hypothesis. The paper concludes with a transparent assessment of the framework’s mathematical boundaries.