Theorem: X-I-ST mathematical release

Under the retained supported-host contract:

1. ${{\mathcal{S}}_{\lambda }}^{[3]}$ is a specified classical relative BV Hamiltonian modulo ${\kappa }^4$, and ${{Q_{\lambda }}^{[3]}}^2=0$ in that ring;

2. its new scalar Euler term is $\lambda {\kappa }^{3}a{\varphi }^3/6$, with compact support;

3. coefficients through order two agree with X-H-ST; the field free limit is exactly X-G-ST's $s_0$, and the full free BV limit is exact in the centered sector;

4. the off-shell background tadpole remains visible outside that centered sector;

5. the chosen polynomial lift has the exact new residual $\lambda {\kappa }^4{\mathcal{B}}_a$, with the nonzero real evaluation (eq:positiveB);

6. the natural interacting field rule still fails descent to the unchanged free equation quotient;

7. the earlier three-tree coefficient packet is not changed through its retained order, and the specified positive reciprocity diagnostic remains positive;

8. the independent convolution obstruction and the free fish/null-wavefront conditionals retain their separate scopes;

9. every released finite supplied residual report has the complete instrument and finite-composition theorems proved above;

10. scheduled locality and regular reference-preserving transport hold under the stated finite permissions.

These results form local Grade S.

They are not interacting quantum BRST, not a characteristic coefficient, not loop-anomaly cancellation, not an ultraviolet completion, and not a derived metric or spacetime emergence.

Proof. Theorem thm:deformation proves (i)–(ii). Corollary cor:recoverH, Proposition prop:freecausal, and Proposition prop:tadpole prove (iii)–(iv). Theorem thm:residual and Proposition prop:positiveB prove (v). Theorem thm:nondescent proves (vi). Propositions prop:trees and prop:reciprocity prove (vii). Section sec:diagnostics proves precisely the non-interference statements in (viii). Theorems thm:instrument and thm:composition, with Proposition prop:noacquisition, prove (ix). Theorems thm:locality and thm:route prove (x).