A Level-Four Sum Rule and the Resolution of the Small-Gap Deficit of the Riemann Zeros: A High-Precision Computational Report

Abstract

This computational report presents three novel findings regarding the local spacing and invariant structure of the nontrivial zeros of the Riemann zeta function, derived from high-precision evaluations at heights $\gamma$ ranging from $1.2 \times 10^4$ to $10^8$. Working within the mathematical framework of the jet tower $L^{(n)}=\mathbb{C}[b]/(b^{n})$, we report the following results: A Level-Four Sum Rule: We identify a sum rule linking the level-three one-sum ($S_1$) and the level-four zero-sum ($X$), demonstrating that $\mathbb{E}[S_1^2] = \mathbb{E}[X] - \pi^2/3$. The data reveals that finite-height arithmetic corrections enter both levels identically and cancel out exactly. This combination represents the first quantity in the tower that captures three-point correlations, moving beyond Montgomery's regime. Resolution of the Small-Gap Deficit: The previously reported scarcity of small gaps between zeros is proven to be strictly a finite-height effect. By comparing against the exact sine-kernel value, we show that the deficit decays according to $1 - A/A_{sine} = c/L^p$ with an exponent of $p \approx 2$ (specifically $c=17.49$ at $p=2$), statistically excluding the naive $1/L$ correction model with $\Delta\chi^2=10.0$. A Local Lehmer Criterion: We introduce a functional that certifies a Lehmer pair (anomalously close zeros) using solely two evaluations of the level-four jet, without requiring knowledge of any other surrounding zero. Using this criterion, a new tight pair is discovered at $u = 21137307.8265$ with $\Lambda=0.00214$. Additionally, the report documents four critical numerical traps encountered in high-height zeta computations, including the exact $3/2$ tail index (infinite variance) of the variable $X$, and details the mitigation of phase noise using a truncated Riemann-Siegel formula. All computations were performed and validated in MATLAB.