Scretching–Planck Density and Phase and Bohmian Mechanics

The Scretching–Planck (SP) continuity and phase equations, obtained from ψ = √P e^(iS/ħ), are the same scalar equations that underlie de Broglie–Bohm mechanics [1–3]. The SP quantum term is Bohm’s quantum potential, and the SP flow velocity ∇S/m is the guidance velocity. Bohmian mechanics assigns positions to particles, while the SP programme uses (P, S) as working variables. The two-slit example illustrates equivariance with a Kolmogorov–Smirnov distance of 0.0178 for 4,000 trajectories, below the approximate 5% critical value of 0.0215. The free-state energy split conserves ⟨p²⟩/(2m) as the Fisher and flow contributions change. Further examples show no crossing, rigid coherent-state motion and the pointwise balance for real stationary states, where the guidance velocity vanishes and ⟨T⟩ = ⟨Q_SP⟩. A single-valued wave function imposes the circulation condition ∮∇S·dl = nh, which the local hydrodynamic equations alone do not supply [4]. Eight solved problems are included.