Ramanujan's prime counting inequality holds for all x ≥ 38,358,837,683: ranges from the zeros, a spectral barrier, and trade-offs
Ramanujan proved that $\pi(x)^2 7/8$ of a recent OpenAI preprint. Its main statement for $\zeta$ is formalized in Lean 4 and has been rebuilt independently; its written proof has not been peer reviewed. The deduction is our Theorem B: if $\zeta(s) \ne 0$ for $\mathrm{Re}s > 0.945092$, then $N_R = 38{,}358{,}837{,}683$. The proof of Theorem B uses only refereed inputs, together with Bhattacharjee's correction of a theorem of Büthe. Relying only on the peer-reviewed literature, and on preprints where stated, we also prove the following by rigorous computer-assisted interval arithmetic. (A) The…
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