Non-Circular Projection Undemonstrability of the Riemann Hypothesis

This paper develops a conditional no-go framework for a class of proof strategies aimed at the Riemann Hypothesis (RH). The central distinction is between finite projection/readout routes, which compress global zero geometry into finitely many data, and closure-complete routes, which prove a global principle strong enough to exclude all off-axis zeros. The main claim is not a classical model-theoretic independence theorem. It is a proof-architecture theorem: a finite, non-circular, projection/readout route cannot establish RH unless it is supplemented by a closure principle whose mathematical content is already RH-level. The framework is formulated through I/Z projection algebra, a finite-readout twin lemma, TERRA-Stieltjes diagnostics, shifted Hausdorff-Hankel positivity, Li/Keiper positivity, Nyman-Beurling closure, Hilbert-Polya spectral completeness, de Bruijn-Newman boundary control, and the Ouroboros problem represented by axiom systems of the form T = ZFC + RH. Keywords: Riemann Hypothesis; no-go theorem; finite readout; projection algebra; closure principle; Stieltjes functions; Hausdorff moments; Hankel positivity; Li criterion; Hilbert-Polya; Nyman-Beurling; de Bruijn-Newman; non-circular proof.