Proposition: Conditional affine gluing and composition

Suppose realized patches have matching common boundary realizations, canonical overlap maps, inverse reverse maps, consistent triple-overlap products, and the required marked peripheral data. Their union has the concatenated certificate.

A change of adapted charts $A_f\mapsto K_f{A}_f$ changes transitions by

and conjugates based closed words. It preserves reference-frame acceptance.

The proposition does not glue different deficits into one spacetime evolution merely because their separate endpoint packets are certified.

Proof. Matching boundary maps identify the same realized points. The inverse and triple conditions make these identifications consistent on overlaps. Every closed word in the union is computed by concatenating its ordered overlap components. The asserted chart law follows by substitution and telescoping. Different-deficit endpoints have not supplied the common realization and overlap hypotheses, proving the stated limit.