A Completion-Locked Hilbert–Pólya Proof of the Riemann Hypothesis: Euler-First Rigidity, Determinant Propagation, and No-Tuning Closure

We present a closure-driven Hilbert–Pólya approach to the Riemann Hypothesis based on a no-tuning mechanism in which all normalizations are fixed upstream, prior to any comparison with primes or zeros. The construction treats the Archimedean completion as a rigid boundary condition that removes the usual rescaling and twisting freedoms in determinant and orbit-formula approaches, isolating the remaining analytic content in an interior entire factor realized via a zeta-regularized determinant of a positive, self-adjoint operator. Positivity enforces critical-line confinement of nontrivial zeros. Identification with the completed zeta function is achieved through Euler-first rigidity on ℜ(s) > 1, using a canonical Fredholm determinant derived from a finite Markov coding and equilibrium-based pruning of spurious contributions. A single special-value calibration fixes the remaining constant, yielding confinement of all nontrivial zeros to the critical line. Full operator definitions, determinant formulas, and audit structures are given in the PDF.

License note: Distributed under CC BY-NC-ND 4.0.

Version history
v1.0 January 01, 2026.
v2.0 March 25, 2026.
v3.0 April 09, 2026.