References

XIIA

N. Cheung, Landmark XII-A: One graded polynomial carrier for the finite-ramp multiplet, with ordering-stable composites and a robust observation exclusion. Supplied manuscript XII-A-v2.tex. Carrier, ordering transport, and original-class exclusion input.

XIIB

N. Cheung, Landmark XII-B: Admission and one retarded response of the ramped cubic Einstein source, at fixed regulator width: handoff audit, variational consistency, and the unresolved release. Supplied manuscript XII-B.tex. Weighted Euler identity, pure-trace layer term, and off-shell integrability distinction.

XIA

N. Cheung, Landmark XI-A: A switched Einstein–scalar action for the ramp relational scalar: sign conventions, Noether identities, and an empty fixed-input Einstein jet. Supplied manuscript XI-A.tex. Original signed action, scalar switch, curvature convention, and original jet quantifier.

XIB

N. Cheung, Landmark XI-B: The free de Donder multiplet for the ramp Einstein–scalar sheet: causal propagators, residual gauge, and the layer dressing. Supplied manuscript XI-B.tex. Free normalization, plateau class, and support distinctions.

XIE

N. Cheung, Landmark XI-E: On-shell admission of the fixed-width tree, a localized first-order identity, and the held-fixed defect. Supplied manuscript XI-E.tex. Inherited scalar-completion signature only; no new completion is attributed to that manuscript.

XIG

N. Cheung, Landmark XI-G: The free-shell class of the cubic Einstein kernel: the onset covector is inherited by every representative. Supplied manuscript XI-G.tex. Closed original obstruction and non-reopening boundary.

SF

T. P. Sotiriou and V. Faraoni, “$f(R)$ theories of gravity,” Reviews of Modern Physics 82 (2010), 451–497, arXiv:0805.1726. Comparison for metric curvature variation and scalar-prefactor derivative terms; the present coefficient calculation is explicit.

BS

R. Bluhm and A. Sehic, “Noether identities in gravity theories with nondynamical backgrounds and explicit spacetime symmetry breaking,” Physical Review D 94 (2016), 104034, arXiv:1610.02892. Comparison for fixed-background Noether defects; the selected cubic-only identity is derived here.