$\theta$ certificate, block 8

The kernel checker ThetaSieve.Seg.ok accepts every record of ThetaSieve.ThetaRun8.segs (block 8 of 102): for each segment, every window $[a,b)$ passes the two mesh checks $\theta(a)\ge b-b/(5\log^2 b)$ and $\theta(b-1)\le a+a/(5\log^2 a)$ in fixed point, and the bounds for $Q\,\theta$ advance from the stated start values to the stated end values.

The blocks chain from $3594641$ to beyond $3584912847$; with the soundness of the checker they give $|\theta(t)-t|\le t/(5\log^2t)$ on that range.

Retired (2026-10-17) Superseded: this mesh block is not used, because the full mesh of the Chebyshev certificate exceeds the platform's per-job memory and time limits. The full Chebyshev certificate and its proof Dusart.chebyshev_theta_k2_finite are in the release doi:10.5281/zenodo.23432990; the checker and base range on this platform remain ThetaCertCore, ThetaCertData_base and ThetaCert.base.