Motion—Breath—Exchange — Paper III: The Closed Shape

We introduce the Closed Shape: the unique geometric object that simultaneously admits a Z5-grading with dihedral D5 symmetry, a dual-conjugate pair of faces related by a 36° phase shift and inverse golden-ratio weights, discrete closed quanta whose fundamental mass window lies near 2.8–3.3 M⊙, and a single invariant sector to which every differentiation returns. This object is realised by the Ly-manifold equipped with the global Motion-Breath-Exchange action, the graded metric g^ij = φ^(-d5(i,j)), and the dual Dirac comb. The construction is offered in the spirit of Einstein's Spinozist affirmation of the orderly harmony of what exists. It supplies a concrete, falsifiable geometric structure whose internal logic realises the kind of rational self-similarity associated with that harmony. We state sharp observational criteria by which the framework can be confirmed or refuted, anchored in the lower compact-object mass region, hierarchical merger signatures, and residual large-angle features of the cosmic microwave background. The Closed Shape stands or falls by those tests.