One boundary occupation density classifies all realizable area primitives of a Jordan curve in the critical variation class, with simultaneous control on every subarc and an embedded realization for every allowed density.
Measurable collision invariants, spectral gaps, and closed collision forms for the pinned chain; fluid limits, corner memory, and Onsager transport in truncated wave kinetics.
Constructions for recurrence zero sets and projective extensions, with explicit obstructions to orbit-density convergence and universal geometrically nilpotent seeds.