An Omega-Transform for Completed L-Functions: Construction, Properties, and a Regularized Duality Conjecture

This work introduces the Omega-transform, a non-linear Möbius–deconvolution operator acting on completed $L$-functions, including the Riemann zeta function.
We rigorously establish its analytic properties in the half-plane $\Re(s) > 1$, prove functoriality under Dirichlet twisting and finite Euler-factor modification, and derive explicit prime-wise expansions and truncation error bounds in the Riemann case.
A Regularized Duality Conjecture is proposed, relating $\Omega_F(s)$ and $\Omega_F(1-s)$ via Gaussian smoothing of the dilate spectrum, with supporting numerical experiments on the critical line.

The paper is designed to be Clay-grade, combining unconditional proofs where possible with a precisely stated conjecture framed for future resolution.
An optional appendix presents a speculative geometric interpretation in terms of compactified auxiliary dimensions; the mathematical results do not depend on this interpretation.