Superfluid Hydrodynamic Model (SHM) — Version 3: Corrected Madelung–Metric Closure and Canonical Density

This Version 3 presents a revised and mathematically corrected formulation of the Superfluid Hydrodynamic Model (SHM). The model postulates a compressible, inviscid quantum superfluid substrate whose local limiting propagation speed \(c(\mathbf{x},t)\) is a dynamical scalar field. Gravity is identified with the hydrostatic gradient \(\mathbf{g}=-\frac12\nabla(c^2)\). The local metrological invariant \(hc=h_0c_0\) is retained, implying that a local observer always measures the standard value of the speed of light; genuine spatial variations of \(c\) become accessible only through inter-regional comparisons of standards.Earlier derivations of a quadratic core profile from a constant-\(\hbar\), constant-\(m\) Bohm balance are withdrawn: the correct three-dimensional radial Laplacian does not support that monomial solution, and the sign of the quantum–metric matching condition is corrected. A canonical Noether density \(n=\rho/\xi\) (with \(\xi=c/c_0\)) is introduced, together with a metric connection \(\mathcal{D}_\xi=\nabla+\frac12\nabla\ln\xi\). Within this closure the generic smooth isotropic stasis branch is \(\xi\sim r^2\), yielding the regular local scalings \(n\sim r^6\) and \(\rho\sim r^8\).A screened radial effective theory built on this branch admits a thin transition layer whose numerical diagnostics produce three characteristic integrals\[(\mathcal{J}_Y,\mathfrak{M},\mathcal{N}_w)\simeq(1.57797\times10^{-5},\,2.9889\times10^{-2},\,1.5826\times10^{-5}).\] These quantities are internal consistency markers of the candidate mathematical closure; they are not presented as empirical confirmation of the model.Electromagnetic, subatomic, and cosmological extensions inherited from earlier drafts are retained only as hypotheses or research directions wherever they have not yet been derived from the canonical action. The present work therefore supplies a cleaner, more cautious, and explicitly testable mathematical core for the Superfluid Hydrodynamic Model.