A Constrained Variational Problem for Majority Isolation of Zeta Zeros: An Explicit Optimizer, Quantitative Stability, and a Certified Majority Bound

Overview How far apart are the zeros of the Riemann zeta function, and how many of them can be guaranteed to be isolated from their neighbors? This paper investigates these questions through a constrained Fourier optimization problem. Building on recent unconditional results concerning the pair correlation and isolation of zeta zeros, we determine an explicit optimal Fourier kernel for a particular zero-counting criterion. The main contribution is an exact solution to the underlying variational problem, together with a quantitative stability theorem showing that kernels with nearly optimal energy must be close to the unique optimizer. Main Results 1. Explicit global optimization We derive a closed-form expression for the unique minimizing density under a prescribed Fourier constraint. The result establishes the optimality of the associated energy within the specified admissible class. 2. Improved quantitative isolation bound Applying this optimizer to an existing unconditional zero-counting inequality, we prove that more than half of the nontrivial zeros of the Riemann zeta function are isolated on both sides at the normalized gap parameter c = 0.17661804. More precisely, the asymptotic proportion exceeds 50.0000016%. A slightly smaller gap parameter, c = 0.17660, provides the more substantial certified proportion of over 50.004%. These bounds improve the guaranteed isolation distance while allowing the proportion of isolated zeros to approach one half. 3. Quantitative stability We establish an explicit stability inequality for the constrained optimizer. It shows that any admissible density with energy close to the minimum must also be close to the optimal density in the L² norm, with corresponding control of its Fourier transform. This strengthens the uniqueness result by quantifying how strongly the optimization problem constrains near-optimal kernels. Reproducibility The deposit includes two files: Research paper (PDF): The complete mathematical statements, proofs, quantitative estimates, and references. Certification programs (ZIP): Python source code, numerical verification routines, execution records, reproduction instructions, and SHA-256 checksums. The numerical certification uses rational interval arithmetic to establish rigorous bounds for the relevant constants. The accompanying programs make the computational component available for independent inspection and reproduction. Scope and Significance The results are unconditional: they do not assume the Riemann Hypothesis. The improvement concerns the guaranteed vertical isolation of zeta zeros within a specific Fourier-kernel method. It does not establish the Riemann Hypothesis, a zero-free region, or an absolute optimal isolation threshold. The broader contribution is a transparent connection between exact variational optimization, quantitative stability, and rigorous numerical certification in the study of zeta-zero statistics.