Variational proof 071 · fixed wire, living interior
Soap Film Forge
The gold wire is law. Everything inside it is negotiable. Rotate the object, catch a boundary handle, or deform the whole frame: the triangulated membrane can move only when the move lowers discrete area.
The twisted wire was released. Interior vertices may move; gold vertices may not.
gold boundary is exact · sparse mesh lines expose the triangulation
Keyboard wire rack
Fixed boundary handles
Spatial controls
Focus the film. `H` cycles handles; arrows move the selected handle in the viewing plane; Q/E move it in model depth; Shift makes larger moves. Space watches or pauses descent; S requests twenty monotone steps. Keys 1–3 load the three topologies. Dragging away from a handle rotates the object; the wheel zooms.
What the colour can and cannot prove
Each triangle contributes its exact Euclidean area. Free-vertex colour comes from the magnitude of the discrete area gradient divided by local lumped area—a mesh-dependent mean-curvature or tension residual. The descent uses simultaneous negative gradients, a displacement cap, orientation/nondegeneracy checks, and Armijo backtracking. Colour is illustrative sheen, not thin-film rainbow interference. A real film may self-intersect, remesh, change topology, or pinch into separate disks; this fixed triangulation cannot.
Ideal discrete area minimization on a fixed topology. No fluid flow, thickness, gravity, pressure, surfactant transport, Plateau-border physics, collision handling, remeshing, or calibrated time is modeled.