Weighted representatives and the boundary question

This is the supplied XII-B weighted-representative relation, now derived from the stated variation convention XIIB.

The XI-A statement that the linear Einstein–Hilbert coefficient is a divergence, together with its exclusion of metric boundary variation, does not supply a boundary action in this setup XIA. The narrow citation question is therefore settled:

The cited XI-A passage does not add a boundary action or authorize a metric boundary variation to the prepared cubic model.

This does not settle every possible boundary problem. XII-C fixes its cubic representative and uses compact interior variations XIIC. Fixing that representative excludes an unannounced change of model; it does not prove the weighted residual in (eq:boundaryrepresentative) vanishes for every other representative.

No boundary charge, boundary action, or additional variational principle is constructed here.