Both internal placements of K 5

No ${\hat{s}}_0$, $s_{\mathrm{ext}}$, $s_{\Psi }$, or $s_0$ is applied to the absent coefficient. No Ward defect is formed.

and exact release

For the unchanged frozen pair and the route fixed in I1:

1. the fixed internal retarded multiplier has the exact pole decomposition (eq:pole);

2. the finite amplitude module has the uniform estimates, continuous reconstruction, and exact lower-degree remainder proved in Section sec:amplitudespace;

3. both $K_5$ placements have the amplitude maps (eq:A5), reconstructing their complete untested kernels with the stated wavefront bounds;

4. the retained six-branch tree kernel has the defined precomposition presentation (eq:presum);

5. no common characteristic operator map for the grouped target on tests meeting $N$ is constructed;

6. no ${R_1}^{(1)}$ is admitted in ${\mathcal{E}}_H$.

The named internal amplitude ending is Constructed at I2. The grouped characteristic coefficient remains Unresolved at I3, within G2/F2/E1/D2/J2. XV-B remains closed.

No universal obstruction to the class fixed in I1, full cancellation theorem, or admitted quantum output is asserted.

Proof. Lemma lem:multiplier and Proposition prop:pole prove (i). Theorems thm:dyadic and thm:reconstruction, with Corollary cor:remainder, prove (ii). Propositions prop:both and prop:WF prove (iii), summarized in Theorem thm:I2.

Proposition prop:precomposition proves (iv). Theorem thm:I3 identifies (v). Sections sec:I4–sec:I6 record the undischarged subsequent gates and establish (vi). Corollary cor:XVB gives the exit consequence.

What has advanced