Finite-Action Reconstructive Histories and the Fisher Structure of the Quantum Potential

This paper extends the causal reconstruction programme from static optimal states to finite-action dynamical histories. A reconstructive history is modelled as a curve of probability measures in the mass-scaled Wasserstein space (P(M), W2,m), and admissible histories are selected by a reconstructive action combining a Wasserstein kinetic term, a Fisher information term, and a causal constraint. The resulting Euler–Lagrange equations form a coupled system: a continuity equation for the evolving density and a generalized Hamilton–Jacobi equation for the velocity potential. The central result is the Fisher–Bohm identity. For any sufficiently regular positive density ρ, the functional derivative of Fisher information satisfies δI / δρ(x) = -4 Δ√ρ / √ρ, so that the Bohm–Madelung quantum potential is exactly Q(x) = (ℏ2 / 8m) · δI / δρ(x). Thus the quantum potential is the Fisher information gradient of the probability density, scaled by ℏ2/8m. When the Fisher coupling in the reconstructive action is set to λ = ℏ2/8m, the Euler–Lagrange equations coincide with the Madelung equations and are therefore equivalent to the Schrödinger equation, under the standard assumptions of positive density and irrotational flow. This establishes a structural correspondence between reconstructive dynamics and non-relativistic quantum mechanics. The result is not presented as a derivation of quantum mechanics: the coupling λ = ℏ2/8m is matched through the Madelung Hamiltonian rather than derived from the reconstruction hypotheses. The geometric origin of this coupling remains an open problem.