Admissibility Forces Hilbert Geometry and Schrodinger-Form Dynamics

Title: Admissibility Forces Hilbert Geometry

This paper proves that Hilbert space geometry is a structural consequence of admissibility constraints applied to compositional predictive systems.

Starting from minimal assumptions — continuity, ray-invariance, independent composition, Standing Conservation, and the exclusion of hidden carriers and free tunings — we show that any admissible standing-weight functional must be quadratic. This forces the parallelogram identity and, by the Jordan–von Neumann theorem, uniquely induces a complex inner product structure.

Formally:

Any continuous, ray-invariant, multiplicative standing-weight functional on a compositional predictive state space is necessarily quadratic and reconstructs a complex Hilbert geometry.

No Born rule, measurement postulates, ontic wavefunctions, Hamiltonians, or physical constants are assumed. The result is purely structural.

The paper closes a common foundational gap: it demonstrates that Hilbert space is not an arbitrary modeling choice but the unique admissible geometry compatible with compositional closure and standing conservation. Any non-Hilbert predictive geometry either introduces contextual carriers, requires free tunable structure (parameter illusion), or violates standing invariance.

This work provides the geometric substrate upon which subsequent dynamical results — such as the necessity of unitary evolution and Schrödinger-form dynamics — are derived.