A Canonical Determinant Model for the Riemann Zeta Function and a Sharp Boundary Positivity Criterion Equivalent to the Riemann Hypothesis

This paper develops a canonical Fredholm–determinant model for the completed Riemann zeta function and analyzes its analytic and geometric consequences using fixed-aperture operator methods. The work proceeds in four stages. First, a determinant Delta(s) is constructed from an explicit operator–kernel model with certified symbol bounds, yielding uniform contraction on vertical strips and a well-defined logarithmic derivative. Second, fixed-height Poisson positivity is established in the interior of the critical strip, together with a proof that this mechanism cannot be pushed to the boundary within the same symbol class. Third, a separate boundary Fejér positivity package is developed on the critical line and combined with a DN Jensen–Green sweep on thin rectangles, producing explicit per-band exclusion results and a quantitative zero-density bound above the critical line. Fourth, a convolved half-plane identity and Hardy-space uniqueness are used to close the global identity Delta = C·xi. All analytic, geometric, and operator-theoretic components of the construction are unconditional. The final logical step is isolated as a single boundary coercivity axiom (Assumption A3), asserting Hardy-stable positivity of the boundary kernel associated with the determinant model. The main conceptual result is that Assumption A3 is sharp: within this framework, A3 holds if and only if the Riemann Hypothesis holds.Thus the Riemann Hypothesis is reduced to a precise and irreducible boundary positivity criterion, and the paper classifies exactly what additional structure would be required to force spectral reality. This work does not claim an unconditional proof of the Riemann Hypothesis. Instead, it provides a complete structural reduction and a rigorous equivalence between the hypothesis and a specific boundary positivity principle arising from the determinant model. Version note:An earlier version of this work framed the DN sweep and determinant closure as unconditional. The present version clarifies the logical structure by isolating the final boundary coercivity requirement (Assumption A3) and proving that it is equivalent to the Riemann Hypothesis. All analytic and geometric components remain unconditional; only the final spectral conclusion is conditional on A3.