A Finite Field Realisation of the Riemann Hypothesis
Abstract
We give a comprehensive finite-field realisation of the Riemann Hypothesis within the Finite Ring Cosmology framework, where arithmetic is carried by a local observer shell operating on a large but ultimately finite relational algebraic substrate. The Riemann zero heights are realised as the finite spectrum of the scale-evolution operator \(\hat{H}=-i(x\partial_x+1/2)\) and are computed from the primes alone: the von Mangoldt comb drives a secular condition and an explicit matrix whose eigenvalues recover the heights to \(4\times10^{-5}\), with \(\zeta\) never evaluated. On the shell the half-turn 1/2 is three things at once: the self-dual locus \(\mathrm{Tr}=1\), the symmetrizer that makes the operator self-adjoint, and the critical real part. The real part of a zero is therefore fixed at \(1/2\) by construction, for the same reason that primes are natural numbers fixed to the real line: both live where the observer and Carrier frames agree. The Riemann Hypothesis reduces to a single clause, the completeness of this spectrum for the prime vector. Completeness is a \(\Pi^0_1\) statement, a single unbounded quantifier over heights: \emph{no zero off the line, ever}. The finite totality closes the domain of that quantifier. Below the observer horizon \(\Omega^{1/4}\sim10^{30}\) the clause is decidable height by height, and is verified to \(3\times10^{12}\). Between \(\Omega^{1/4}\) and the coherence horizon \(\sqrt\Omega\sim10^{61}\) it is bounded but unresolvable in any…
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