Prerequisite sheet

These are the statements actually delivered by the earlier manuscripts, copied in their own conventions. None is a theorem about the regulated operator introduced below. Shared vocabulary is not a transfer.

-A, copied. Signature $(-,+,+,+)$. Minimally coupled scalar on the formal metric $\mathbf{g}=\eta +\kappa h+{\kappa }^{2}k$, compactly switched scalar action, unswitched harmonic coordinates with fixed Cauchy data. Admitted transformations are supported strictly inside a region where the interaction switch equals one and are the identity near the initial surface. On that restricted algebra: exact first-order cancellation and a second-order classical intertwiner, with all switch derivatives retained. The proof uses covariance of the switched density-valued Euler operator and uniqueness of the coefficientwise Cauchy problems. The first-order plateau identity is

for $\xi$ supported in the plateau. Outside that support,

and a higher-order coefficient cannot repair this first-order defect. The metric equations in the named Einstein–scalar jet are unswitched; the scalar equation is switched. A causal support bound and a classical comparison map between switches are proved. Equality for switches sharing only an observation plateau is not asserted. No renormalized product, quantum Ward identity, or counterterm separation.

, copied. Signature $(-,+,+,+)$. Second-order classical descent for a specified relational Einstein–scalar jet, and a separate exact first-order compact-switch defect for the identification $(h,{\varphi }_0)\sim (h+K\xi ,{\varphi }_0)$:

with $[P,g]u=({\square }_{\eta }g)u+2({\partial }^{\mu }g){\partial }_{\mu }u$. No correction beginning at second order removes it while the identification and the switch remain as stated. No renormalized response and no quantum Ward identity.

, copied; a different equation space. Signature $(-,+,+,+)$. An unchanged free-product quotient does not receive the Poisson-dressed symbol. Moving the scalar does not preserve the massive free shell for general compact gauges. On a specified coupled first-order jet with forced scalar correction the functional descends, with a unique retarded harmonic section. A compact coupled Ward pair is nonzero. No quantum representation.