Zero-Less Regularized Sum for the Riemann Zeta Function with n⁴ Cutoff: Computational Framework and Mellin Correction

We introduce a null-state regularization Phi_X(s) = sum(n^-s / (1 + n^4/X^4)) inspired by Ramanujan's denominator. We separate trivial zeros (s = -2k, divergence to All-state A) from non-trivial zeros (Null-state ∅). We prove the balance lemma where the conjugation of B(s) equals B(s) if and only if Re(s) = 1/2. We formulate Lemma 0 via the Mellin transform, showing Phi_X(s) approaches the Zeta function plus the leading pole X¹⁻ˢ. Numerically, we show |Phi_X| dips only at Re(s) = 1/2 for Im(s) < 50 with X = 500, and that the simple regulator without the Mellin term does not converge to Zeta (error growth slope +1.09). Crucially, we demonstrate that after subtracting the leading pole, the residual at the exact non-trivial zeros decays exactly as O(X^-4), confirming the validity of the full contour shift. We state a consistency conjecture equivalent to the Riemann Hypothesis as open. This is not a proof of RH.