V138 — The M = 1 Diagonal Barrier in Translated Mean-Square Estimates for the Logarithmic Derivative of the Zeta Function

Description: This paper examines translated vertical mean‑square estimates for the logarithmic derivative of the Riemann zeta function near the critical line. These estimates naturally appear when shifted‑contour remainders in smoothed Perron formulae are expressed as vertical convolutions of the logarithmic derivative.

The central observation is that each critical‑line zero contributes a local packet with diagonal mass proportional to the inverse of the distance from the critical line. Since there are about T log T zeros in a height interval of length T, the formal diagonal scale of the mean square is of order T log T divided by that distance. This corresponds to a loss exponent of M = 1.

Therefore, any translated mean‑square estimate with M less than 1 is stronger than what diagonal zero‑packet counting alone can supply. Such bounds must rely on additional structure: off‑diagonal cancellation, renormalization, zero‑spacing repulsion, or stronger zero‑distribution input. The paper formulates the relevant Gram matrices and shows that the translated vertical mean‑square barrier is the same zero‑packet Gram barrier encountered in high‑frequency packet formulations.

The contribution is diagnostic rather than constructive. It identifies the natural M = 1 diagonal barrier and clarifies why an M < 1 input is a genuinely stronger, zero‑sensitive hypothesis. No proof of the Riemann Hypothesis is claimed.