Small-Spacing Isolation of Zeta Zeros via Signed Multiset Bounds and a Certified Fourier Witness

Description We establish two explicit, unconditional computer-assisted lower bounds for the proportion of isolated simple zeros of the Riemann zeta function on the critical line, without assuming the Riemann hypothesis. A critical-line zero is counted as isolated if it is simple and its ordinate is separated from that of every other nontrivial zero, including zeros off the critical line, by more than $2\pi c / \log\gamma$, where $\gamma$ denotes its ordinate. Our principal results are $$\liminf_{T\to\infty} \frac{B_{\mathrm{loc}}(T, 0.00596804)}{N(T)} > 0.655200$$ and $$\liminf_{T\to\infty} \frac{B_{\mathrm{loc}}(T, 0.03799)}{N(T)} > 0.65026.$$ Here $N(T)$ counts nontrivial zeros with multiplicity, and $B_{\mathrm{loc}}(T, c)$ counts simple critical-line zeros satisfying the prescribed ordinate-isolation condition. The proof combines a signed Hilbert–Schmidt inequality for conjugation-invariant finite multisets, a nonreusing allocation of repeated-real and nonreal zero centers, and an explicit fifty-mode Fourier density. Rigorous interval enclosures and a smooth-square approximation provide the numerical bounds. The arithmetic argument uses unconditional fixed-test pair correlation and Conrey's integrated estimate for zeros off the critical line. The two results exhibit complementary quantitative bounds: a higher isolation proportion at a smaller spacing parameter, and a proportion exceeding $65\%$ at a substantially larger explicit spacing. This record contains the manuscript, its LaTeX source, exact Fourier coefficients, verification programs, interval certificates, and reproducibility documentation. The computer-assisted inequalities are supported by reproducible computations; the verification software is not formally verified. Version: v1.0 — Initial public preprint, 31 October 2026. Creator Lee Byoungwoo Independent Researcher, Daejeon, Republic of Korea Keywords Riemann zeta function; isolated zeros; critical line; zeta zeros; pair correlation; unconditioned bounds; computer-assisted proof; interval arithmetic; rigorous numerics; Hilbert-Schmidt inequality; Fourier density; zero spacing