Some results on BZ structures from Hilbertian unsharp quantum physics

Gianpiero Cattaneo, Stanley Gudder

Gianpiero Cattaneo, Roberto Giuntini, Sylvia Pulmannová

PEKKA J. LAHTI

Stanley Gudder

Amos Maley

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.

Navjot Singh Gadda

There are infinite frames of reference at once while taking into account the motion of an object and all of them can be superposed to one another forming a dynamical net frame bundle. This results in a non-Hilbert space which represents a kind of state space of the object at hand. It will be shown that a particular type of this non-Hilbertian structure can be used to represent all elementary systems as the quantum counterpart of the net frame bundle. In order to achieve such a result, it is mandatory to adopt a nonstandard quantum field theoretical approach in which the mass distribution and the gauge conditions of a physical system are intrinsically connected.