An Operator-Geometric Framework for Dilation Dynamics and Spectral Structures Associated with the Riemann Zeta Function

This paper investigates a novel geometric framework that maps arithmetic operations into a non-Euclidean configuration space, termed a Task Space, to study the distribution of the non-trivial zeros of the Riemann zeta function. By combining prime-scaled dilations and differential shifts over a Hilbert state space, the framework establishes a continuous manifold where scalar curvature is governed by the local density of prime numbers. Using the Mellin transform to link operational path sequences directly to the zeta function, the research explores the connection between the Riemann Hypothesis and global geometric rigidity. The core argument examines how a hypothetical zero off the critical line would introduce a localized topological defect that conflicts with the uniform algebraic symmetry of the underlying group completion, suggesting a geometric constraint that requires all non-trivial zeros to lie strictly on the critical line. This paper does not set out to prove such a geometric constraint but to establish under what frameworks it can be established from.