Declarations

correction. This manuscript selects density weighting at cubic order only. It explicitly amends the full-density formula in the production brief because that formula is incompatible with the required unchanged lower-order handoff. No claim is made that the two actions are equivalent.

scope. All new products and variations are local classical expressions on smooth configurations. No new quantum product is defined.

scope. XII-B's weighted Euler distinction and pure-trace layer identity are copied as inputs and used as checks. The new general tensor source, changed identity, and coupled-jet verdict are proved here. XI-A's original jet theorem is not used as the new jet verdict.

scope. The cubic representative is fixed before weighting. All variational proofs use compact interior variations. No boundary action, boundary charge, or additional variational principle is constructed.

work. No metric retarded map, kernel admission, wavefront test, response ledger, interacting differential, antibracket, ghost loop, state, enlargement, or $\epsilon \to 0$ limit is included.

. No successful pdflatex execution, proof-assistant verification, independently executed symbolic computation, independent referee review, or journal acceptance is asserted.