Amplitude and Fourth-Moment Obstructions for Prime-Corrected Sublevel Approximations to Riemann Zeta Zeros

The sublevel approximations τC k of an earlier IJNT manuscript use the Riemann–von Mangoldt main term and finitely many Euler factors. Their original truncation bound h(t) = log log max(t, ee) is too small for eventual containment between the imaginary parts of two consecutive neighboring zeros: an unconditional Ω°æ theorem for S(t) forces violations with arbitrarily large index displacement. We replace h by an explicit bound H(t) = 0.08382 log max(t, e) + 5.5, supplied by the author’s Mathematics of Computation submission, and prove that the modified approximations τCm k retain the IJNT fourth-moment and mean-square localization bounds. We then formalize a separate structural saturation obstruction: ordering, truncation, and the deterministic sublevel-transfer inequality alone do not exclude a rapidly oscillating construction whose values approach N0 °æ H with equal proportions. Such saturation has fourth moment at least R H4, which is incompatible with an O(T) bound uniform in both height and prime cutoff. For weighted prime phases with a pole compensator, a growing-prime cutoff X = To(1) necessarily has fourth moment diverging on the T scale; in contrast, the bound Oε(T) is proved uniformly for Tε ≤ X ≤ T1/4. A separate unconditional lemma controls the fourth moment of a smoothed sum over imaginary parts of the zeros for all logX ≥ ε log T, without an upper restriction on X. This last lemma does not control the full weighted Euler phase without a uniform explicit formula. We distinguish unconditional theorems, statements conditional on the IJNT input, and the remaining problems concerning unbounded X and off-critical-line zeros.