Theorem: X-E-ST local Grade S release

For the fixed supported host and the fixed-width sheet declared before the calculation:

1. the Palatini variation, switched metric and scalar Euler equations, and exact Noether defect are constructed on the completion;

2. the plateau local action is invariant and its Euler densities are covariant; the inherited classical scalar identity holds through ${\kappa }^2$, including on the invariant two-jet surface when that specified-input fibre is nonempty;

3. the Bianchi condition (eq:bianchi) is necessary; for a nonconstant switch, an open dense set of free scalar inputs has an empty coupled-jet fibre;

4. the Einstein–Hilbert quadratic form and Feynman-weight de Donder operator have the stated sign and retarded normalization; the compact residual algebra is trivial;

5. the isolated matter tree is the admitted scalar Wick cubic $\tau$, with support in $C^3$, wavefront in $Z_3$, fixed-support continuity, and normalization $-2$;

6. the one-tree background identity holds through order $\kappa$, with the precise order-${\kappa }^2$ defect (eq:defect); that class is nonzero for the supported instance of Theorem thm:witness;

7. the installed finite dictionary has a complete destructive-clock instrument, scheduled locality, a regular reference-preserving route, and a declared resource/error ledger.

This is local Grade S, not spacetime emergence.

Proof. Lemmas lem:palatini and Theorem thm:noether, with Proposition prop:euler, prove (i). Proposition prop:covariance and Lemma lem:scalaridentity prove (ii). Theorem thm:bianchi proves (iii). Proposition prop:quadratic, Theorem thm:free, and Proposition prop:residual prove (iv). Proposition prop:treesupport, Theorems thm:treeadmission, thm:normalization, and thm:continuity prove (v). Theorems thm:defect and thm:witness prove (vi). Sections sec:Enter–sec:transport prove (vii), completing Definition def:grade.