From structure to a testable question.

A graph is not a filled cell.

The complete graph K₄ has four vertices and six edges. The filled tetrahedron Δ³ adds four triangular faces and one 3-cell. Its graded chain space therefore has 15 real amplitude coordinates.

Topology fixes a language. A generator fixes a motion.

The boundary operator B takes an oriented simplex to its boundary. The identity B² = 0 means the boundary of a boundary vanishes. With an orthonormal simplex basis, UD studies the real skew generator A = Bᵀ − B.

Skew symmetry preserves the squared norm. Here s is a mathematical evolution parameter. Its relation to seconds, energy and laboratory readout requires a separate calibrated map.

A conserved total can still flow locally.

Adjacent chain coordinates exchange quadratic support through signed incidence currents. The total remains conserved in a closed fixed-carrier flow, while individual simplex stocks can rise or fall. Shared simplices are represented once; a shared face is an active reservoir.

“Support” here means squared chain amplitude. It is not automatically mass, thermodynamic entropy or energy.

An observation can hide a state.

Stocks discard signs and phases. Projective support receipts add signed or complex incidence information, plus the required component references, to reconstruct a normalized state. A valid reconstruction theorem depends on which observations were actually supplied.

Persistence needs its own test.

Agreement at one instant, retention over an interval, and preference for growing into a cluster are different observables. Prepared shells, memories and controllers can test conditional mechanisms. A source-selected shape law and a complete physical apparatus remain further questions.