An optimised inequality for finite conjugation-invariant multisets, with a conditional application to zeros of the Riemann zeta-function in short intervals

For a finite multiset Z ⊂ ℂ invariant under complex conjugation and a real even compactly supported weight η with ∫η² = 1, put K = (η²)^ and S(Z) = Σ_{z,s∈Z} K(z−s)². In the Hilbert-space formulation of Lamzouri, following Alpöge and Furman, we prove S(Z) ≥ x + (|Z|−x)²/N_m, where x is the number of simple real points of Z and N_m the number of distinct multiple real points plus the number of conjugate pairs; Lamzouri's printed inequality is the case τ = 2 of the scalar parameter we optimise, and the bound is exactly the optimum of the programme his own block estimates define. It yields U(Z) ≥ c*(A)|Z| whenever S(Z) ≤ A|Z|, where U counts the elements of Z that are simple or real and c* has a second branch 4/A − 1 on [2+√2, 4], which we have not found stated in prior work, extending the reach of the corollary from A < 5/2+√2 to A < 4. We also identify the quantities used in these proofs as the diagonal of a self-adjoint operator with trace |Z| and squared Hilbert–Schmidt norm S(Z), which names the only loss in the argument. This part is unconditional.\n\nAssume now the short-interval pair-correlation formula stated as Theorem 2.2 of B. Wang, arXiv:2609.07918v1 (2026) — a transposition, asserted there unconditionally, of a published theorem of Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh, but itself an unrefereed preprint; call it (W). Writing H = T^θ, C_θ for the Montgomery–Taylor functional of the interval and N_{s∪0} for the zeros of ζ in (T, T+H] that are simple or lie on the critical line, counted with multiplicity, we deduce liminf N_{s∪0}/N ≥ c*(C_θ), positive for θ > 0.2554312. We have found no source, refereed or self-published, that states a bound for this statistic in short intervals; the statistic is Lamzouri's, who bounds it globally by 0.8876200…, the value our bound tends to as θ → 1 and does not improve. The range in which we have found no prior every-T statement is (0.2554312, 7/22] (measured conservatively against the exponent 7/22 stated in the review of Rakhmonov's paper) — and there too a positive proportion is unconditional for almost all T and every θ, by a theorem of Karatsuba, so what it adds, as far as we can determine, is the every-T form with an explicit constant. Of that range, only the part below θ₂ = 0.2612864…, a window of width 0.0059, needs the new branch of c*; above θ₂ it follows from Lamzouri's printed inequality under (W). Two further corollaries under (W) — liminf N_d/N ≥ max{1/C_θ, (3−C_θ)/2} for distinct zeros, and liminf (N_s+N_0)/2N ≥ (3−C_θ)/2 — are weaker in novelty: the first can be qualitatively new only on 0 < θ ≤ 7/22, coincides with Wang's own bound for θ ≥ θ₀ = 0.5501939647…, where a self-published deposit resting on the same hypothesis certifies a strictly larger value, and the second adds an explicit constant in a range where positivity is unconditional.\n\nThe zeta-function statements are offered as consequences of an unrefereed preprint statement, and would become unconditional if (W) is confirmed externally; nothing here bears on the Riemann Hypothesis.