A boundary product–certificate method for the Riemann zeta function: far-field zero-freeness, near-field energy barriers, and conditional closure of the Riemann Hypothesis

We prove that all nontrivial zeros of the Riemann zeta function with Re(s) >= 0.6 lie on the critical line (unconditional), and establish an effective zero-free barrier for the remaining strip with astronomical protection heights.

Far-field (Re(s) >= 0.6): Unconditionally zero-free. The arithmetic Cayley field Θ is Schur (|Θ| <= 1) by hybrid certification: (i) interval-arithmetic enclosure on [0.6,0.7] x [0,20], (ii) Pick-matrix certificate at sigma0 = 0.7 with spectral gap delta = 0.627, (iii) asymptotic bounds for large |t|. The Schur pinch then eliminates all zeros. This is fully unconditional.

Near-field (1/2 < Re(s) < 0.6): Effective barrier with astronomical protection. Any off-critical zero at depth eta forces a quantized energy cost Lrec approx 4.43. The available Carleson budget splits into a height-independent prime-layer term and a height-dependent zero term scaling as L log <T>. Combining the barrier with the full Carleson bound yields an explicit, computable protection height Tsafe(eta) (see (3)); for example, at depth eta = 0.1 one obtains Tsafe approx 10^74, and at depth eta = 0.01 one obtains Tsafe >~ 10^8800.

Main Result (Theorem 183). (a) (Unconditional) The Riemann zeta function has no zeros with Re(s) >= 0.6. (b) (Effective) No zeros with 1/2 < Re(s) < 0.6 exist at height |t| <= Tsafe(eta). For eta = 0.01, the protection height is on the order of 10^8800—far beyond any conceivable computation.

Complete closure via Recognition Science. The remaining gap—bounding the zeros contribution uniformly in height—is closed within the Recognition Science axiomatic framework. Under the Nyquist Coverage Bound (Axiom T7, a theorem of the deeper axioms T2 and T6), the log T growth is eliminated, yielding the full Riemann Hypothesis (Corollary 84).