lin-0219

18.2 Why a gluing loss is not the declaration

Earlier Prometheus and Odyssey realizations used a weighted gluing tension

\[ \mathcal L_{\mathrm{glue}} = \sum _{i,j} w_{ij} \left\lVert \rho ^i_{ij}(x_i)-\rho ^j_{ij}(x_j) \right\rVert ^2 \]

[ Mahadevan , 2026k , 2026j ] . It is a useful concrete architecture, but it presupposes subtraction, a norm, and a weight relative to prediction. It also vanishes on every compatible family, including families outside the admitted global class.

SID retains the cover, provenance, restrictions, and promotion semantics but makes the descent factorization primitive. A metric may choose among corrections; it does not define what gluing means.

Proposition 18.2 Restriction to a subcover

A global compatibility witness restricts naturally to every declared subcover, before and after tangent lift. Conversely, vanishing on selected subcovers implies global compatibility only if they jointly contain every declared simplex and their corrections agree on shared local objects.

Independently repairing each pair can assign incompatible updates to the same foundry. Localization must itself be a compatible family.