Hilbert-or-Denial: A Categorical Meta-Proof that Admissible Predictive Geometry Must Be Hilbertian

Title: Hilbert-or-Denial: A Categorical Meta-Proof that Admissible Predictive Geometry Must Be Hilbertian This paper proves a meta-closure result for predictive systems with independent composition. Under minimal structural constraints — Standing Conservation, the Impossibility Suite (No Generators, No Carriers, No Repairs), Zero-Parameter Admissibility, and symmetric monoidal composition — any non-degenerate predictive geometry is forced into a Hilbertian regime. Formally: Any predictive functor with compositional closure and admissible standing weight is either Hilbertian (projective complex Hilbert geometry with quadratic standing norm) or structurally denied: it induces a carrier under composition, requires a free tuning (parameter illusion), or violates standing invariance. The proof is categorical and classification-based. It does not assume: Born’s rule Measurement axioms Ontic wavefunctions Specific Hamiltonians Physical constants Instead, it shows that once independent composition and admissibility constraints are imposed, non-Hilbert predictive geometries fail closure. Hilbert geometry emerges as the unique non-degenerate admissible class. This result blocks the common objection that Hilbert space is “assumed” or “smuggled” in foundational derivations of quantum structure. Within an admissible compositional framework, Hilbert geometry is not a choice but a forced closure class. The theorem is meta-structural: it classifies admissible predictive geometries without committing to physical interpretation.