Rigid Modular Bounds and Exclusion Theorems in N = 2 Super-Schwarzian Holography: On the Non-Existence of Anormal Scales and the Critical Geometry of Riemann Zeros

We present a rigorous, non-linear theoretical framework connecting lower-dimensional quantum gravity with analytic number theory via the conformal bootstrap and N = 2 super-supersymmetric algebraic systems. By embedding an N = 2 Super-Schwarzian action into a rigid arithmetic background under the modular group SL(2, Z), we analyze the exact spectral distribution of scalar conformal primary operators. Rather than heuristically instantiating number-theoretic metrics, we deploy the exact RankinSelberg integral transform to demonstrate that the functional constraints of the torus partition function naturally yield the Riemann zeta function within the denominator of the spectral density. We explicitly derive the linear functionals Λ∗ optimizing the crossing equations and formalize their analytical boundary conditions in the Borel complex plane. This framework establishes a geometric exclusion theorem: the existence of any conformal primary operator possessing an anomalous scaling dimension within the region Re(∆) = 1/4 breaks the fundamental requirement of quantum unitariety, inducing negative-norm ghost states and destroying causality. By dualizing the U(1)R charge conservation and topological index constraints, we demonstrate how the critical line Re(s) = 1/2 is rigidly enforced by the non-local geometric structures of thequantum space-time.