Ontic ψ and the Necessity of Geometry

An ontic wavefunctional ψ must define objective relational distinguishability in order to have physical consequences. An operative ontology requires more than presence: ψ must generate an invariant notion of how nearby relational configurations differ, so that dynamics, coarse-graining, and projection are well-defined operations rather than arbitrary choices. Without this, ψ is an inert label, a hidden object that leaves no invariant trace in dynamics, interference, or projection. Geometry is therefore not an added structure but the minimal condition for ψ to become physics. Without such an invariant notion, there is no well-defined way to compare neighboring configurations, no canonical notion of small perturbations, and therefore no way to define a generator of dynamics (e.g., a Laplace–Beltrami operator or transport law) that is independent of representation. On the amplitude sector, this requirement uniquely selects Fisher geometry within the class of monotone, coordinate-invariant coarse-grainings (by Čencov's theorem). It is the only Riemannian metric on configuration-space distributions that is monotone under all admissible coarse-grainings, requires no structure beyond ρ(X) itself — no background metric, no preferred coordinates, no additional fields — and directly generates the quantum potential through its Laplace–Beltrami operator. Fisher geometry is not chosen for convenience; it is the least-committed structure that renders ψ physically legible. Any richer geometry would smuggle in unjustified extra structure; any sparser one would fail to be simultaneously local and coordinate-invariant, and therefore cannot support a well-defined differential operator for dynamics. Fisher sits at precisely that threshold. In particular, once Fisher geometry is fixed, the differential structure of configuration space, the associated Laplace–Beltrami operator, and hence the quantum potential and coarse-graining flow are all determined, closing the amplitude sector as a self-contained dynamical geometry. On the phase sector, the corresponding structure governs Hamilton–Jacobi transport. The phase S defines a momentum 1-form dS on the relational configuration manifold. The amplitude defines the geometry of distinguishability, while the phase defines the momentum of its evolution. These roles are cleanly separated but dynamically coupled: the Fisher metric raises the index of the phase gradient, vᴵ = Gᴵᴶ ∂ⱼS, transforming the covector dS into a velocity field that generates Bohmian flow. In this way, the amplitude determines how distinctions are measured, while the phase determines how they propagate. One curves relational configuration space, the other moves through it. Together, amplitude and phase carry the full operative geometry of ψ. The philosophical and mathematical arguments are not independent: the requirement that ψ be operative forces it to survive coarse-graining in an invariant way, and that condition is precisely what Čencov's theorem acts on, making Fisher uniqueness the mathematical expression of ontological seriousness. This is not an additional assumption but a consistency condition: requiring ψ to remain physically meaningful under all admissible coarse-grainings is exactly the hypothesis on which Čencov’s theorem acts. Ontology forces geometry because without an invariant measure of how neighboring relational states differ, ψ has no mechanism by which to become dynamics.