A Coherence Connection on Fisher–Rao Probability Geometry

A mathematically explicit extension of quantum hydrodynamics is developed

in which a scalar coherence field ΦG modifies phase transport and is driven

by the information-geometric structure of the probability density together

with probability transport itself. The construction starts from the Madelung

representation Ψ = √ρ eiS/ℏ, augments the phase gradient by the covariant

shift ∇S ↦ ∇S − αℏ∇ΦG, and defines the dynamics through a variational (ρ, S)

sector together with a dissipative Onsager-type relaxation law for ΦG. This

yields a closed PDE system consisting of a continuity equation, a modified

Hamilton–Jacobi equation, and a coherence-field equation that reduces in the

overdamped limit to an elliptic closure. The source of ΦG is neither bare mass

density nor bare probability density: it is a local Fisher-information density

together with probability transport, so that ΦG measures coherence demand.

The field equations are written explicitly; the coupling sign is fixed by an

energy-stability argument, the transport-forcing sign by a dissipation

argument, and the induced coherence-pressure term is computed exactly. A

structural no-signalling condition is obtained by restricting the nonlinearity to

Fisher-marginal source functionals that are blind to the entanglement Fisher

excess under marginalisation. A coarse-grained Yukawa sector is shown to

admit a conditional Newtonian limit under explicitly stated screening and

additivity assumptions; this remains a conjectural macroscopic regime rather

than a completed derivation of gravity. The paper closes with a falsifiable

numerical programme—ablation studies, dispersion diagnostics,

Fisher-information monitoring, and an explicit pilot solver scaffold—together

with a claim-status ledger, a testability map, and a development roadmap that

separate what is defined, what is derived, what is heuristic, and what remains

unproven.

Keywords: quantum hydrodynamics; Fisher–Rao geometry; Madelung

dynamics; coherence field; no-signalling; information geometry.