Paper B-1: Towards the Statistical Weil Duality: Mellin Operator Construction, Residue Obstruction and Regularised Moment Convergence

Paper B established the Pythagorean-Golden Quantile Duality and conjectured that the prime gap ratio distribution is the push-forward of the GUE spacing ratio distribution under a Weil-type operator W. This paper undertakes the first systematic investigation of that conjecture. We construct W explicitly as a Mellin-space convolution operator and show that the golden and Pythagorean Mellin eigenbases are incommensurable, a consequence of Baker's theorem, forcing W to mix all frequencies. We identify a hierarchy of three nested obstructions -- incommensurability, residue normalisation, convergence -- each logically forcing the next. The tail-exponent gap between the two distributions (exactly 1) is shown to be the arithmetic signature of the simple pole of the Riemann zeta function at s=1, mediated by the Weil explicit formula. Regularisation by a Gaussian-damped test function yields moments that drift monotonically toward the prime gap ratio targets over 100 Riemann zeros, consistent with O(1/T) convergence. All verifiable algebraic content is machine-checked in Lean 4 (sorry-free). No new theorems are proved. The contribution is a precise identification of the obstructions and a concrete programme for resolving them.