Simple critical zeros in short intervals: stability, pressure, parity, and localization

Abstract

Let C3 = 0.6567752140190419405677628751089899133... be the reproducible rank-three optimized Selberg constant. For every fixed 1/2 < theta < 1, localization of the Selberg detector to (T,T+Ttheta] gives a lower asymptotic proportion (theta-1/2)/(4eC3) of distinct odd-multiplicity critical zeros. Combining this estimate with a finite rank-trace inequality yields the simple-critical lower bound max{0,c(theta),(c(theta)+(theta-1/2)/(2eC3))/3}, where c(theta)=2-theta/2-cot(theta/sqrt(2))/sqrt(2). The combined numerator is strictly increasing. Exact interval arithmetic places its unique zero in (0.5459,0.546), below the positivity threshold 0.550193964744154... of the unaugmented cosine curve. At theta=0.546 the certified simple-critical lower proportion exceeds 0.0000976239413345; direct optimization of the displayed curve gives 0.0000994910410327. The localization uses the arbitrary-subinterval mean-value estimate in Pearce-Crump's optimized Selberg framework, together with a fixed mollifier length u < (theta-1/2)/2. The stability, pressure, and parity transfers are also developed with reproducible exact certificates. No effective starting height, global zero-density record, or proof of the Riemann hypothesis is claimed. Source code, data, and computational records are available at https://github.com/fsantibanezleal/CAOS_RESEARCH under problems/number-theory/riemann-hypothesis.