Simply weighting the source is not variational

Consequently the simultaneous demands “an action vertex varied before response” and “Euler source exactly ${\sigma }_{\epsilon }{J}_3$ on unrestricted off-shell tests” are not interchangeable specifications. Adding an unnoticed total divergence cannot evade Proposition prop:integrability.

A source-weighted forcing prescription may still be considered as a nonvariational model. A density-ramped action may also be considered, with its actual layer terms included. Neither possibility is ruled out. Choosing between them, and freezing the relevant identities, is outside this stopped construction.