PASS nonlinear area-energy derivative agrees with independent finite difference
PASS nonlinear pressure Hessian agrees with independent finite difference
PASS variational equilibrium has the radius-derived constant Laplace pressure
PASS localized trial/test spaces are biorthogonal and retain boundary coenergy traces
PASS each region volume is an exact invariant direction of the projection
PASS full nonlinear coupled step preserves regional volumes with shared flux
PASS pressure plus explicit capillary shape work closes both regional energy budgets
PASS regional capillary-energy split independently matches the explicit shape-work formula
PASS localized reduced step preserves separate volume and interface-power ledgers
PASS unreduced two-block coordinate control matches the full physical solve
PASS reference-perturbation pressure agrees with extended-precision difference for weak states
PASS weak-drive fine-grid step and regional power budget converge without tolerance changes
12 passed, 0 failed