The Converse Madelung Question

We define and address the converse Madelung question: not whether Fisher information can reproduce quantum mechanics, but whether it is necessary. We adopt minimal, physically motivated axioms on hydrodynamic variables: locality, probability conservation, Euclidean invariance with global U(1) phase symmetry, reversibility, and convex regularity. Within the ensuing explicitly restricted class of first-order local Hamiltonian field theories, the Poisson bracket is uniquely fixed to the canonical bracket on (ρ, S) under the Dubrovin-Novikov hypotheses for local first-order hydrodynamic brackets with probability conservation. Under a pointwise, gauge-covariant complexifier ψ = √ρ e^iS/ℏ, the only convex, rotationally invariant, first-derivative local functional of ρ whose Euler-Lagrange contribution yields a reversible completion that becomes exactly projectively linear is the Fisher functional.