Publication scope: Independent ordered-parent products, not fixed-vertex closure or a selected physical growth law. The view is a projection of supplied embedding coordinates. No particle identity or measured parameter is established.
Relational squaring: growth into product coordinates
R → R × R → (R × R) × (R × R) · ordered parents retained
Record colors identify the first primitive coordinate. Blue lines are relationships; gold trails show left-parent provenance during unfolding, not extra relationships. Reciprocal arrows are drawn as one undirected line.
Exact rule: two tuples are related precisely when they differ in every coordinate. The primitive relation contains all 12 off-diagonal arrows on A, B, C, D. Each squaring selects two independent parent records. A tuple (u, v) retains both; it is not a duplicate with the same identity.
Projection, not physical space: each tetrahedral coordinate occupies its own 3D block. The view projects 3, 6, then 12 embedding coordinates into three display coordinates. The associated Dowker homotopy models are S², S² × S², and (S²)⁴, of dimensions 2, 4, 8. Those are not dimensions inferred from the displayed line graph. Layout, expansion timing and projection are illustrative; no particular rotation angle or physical growth law is asserted.
This implements the prior relational-squaring experiment, not the fixed-vertex paw closure or line-graph promotion. Identification with an “identity-squared” physical operation remains to be made.
Constructor research · Product geometry · Two-paw closure · Three.js MIT