On the Nonexistence of Functorial Positivity-Selecting Operators in Boundary-Positivity Approaches to the Riemann Hypothesis

This paper establishes a limits-of-method result for boundary-positivity approaches to the Riemann Hypothesis.

Recent work has shown that the Riemann Hypothesis is equivalent, for a canonical admissible analytic realization, to a single sharp boundary positivity or coercivity condition on an associated sesquilinear form. All other analytic components of the reduction are unconditional, and the boundary condition is irreducible in the sense that no strictly weaker positivity notion suffices.

The present paper explains why this boundary positivity condition cannot be derived internally. A minimal categorical framework is introduced, encoding exactly the admissible boundary objects and transformations already present in the reduction. Within this framework, a positivity-selecting operator is defined as a functorial, order-monotone transformation acting on boundary forms.

The main result is a no-go theorem: there exists no nontrivial functorial, monotone positivity-selecting operator that forces boundary positivity without either trivializing the problem, breaking functoriality through non-canonical choices, or introducing genuinely new structural input. In particular, boundary positivity cannot be manufactured from admissible data alone by any internal selection mechanism.

This result does not assert that the Riemann Hypothesis is false or independent of standard axioms. Rather, it provides a structural classification of boundary-positivity methods, showing that any successful proof along these lines must introduce new positivity-certifying structure beyond the admissible closure framework.

This paper is intended as a companion to the author’s reduction work and isolates the precise obstruction underlying boundary-positivity strategies.