Theorem: Exact binary release

For the frozen displayed lifts and fixed regulator width:

1. The vertex ${\mathcal{V}}_g$ is admitted. The tree coefficient (eq:tree-definition) exists and defines one selected causal datum $Q_H{R}_1^{(0)}\in {\mathcal{E}}_H$.

2. Its inherited inverse signs, properness, compact microcausal kernel, and first-completion branch overlap are proved.

3. Causal factorization has the values (eq:ordered-values). Replacing them by $R_1={\mathcal{V}}_g{\star }_H\mathcal{O}$ is Proved obstructed under the printed contraction normalization.

4. The corrected coefficient through $\lambda \hslash$ is Unresolved until the hypotheses of Theorem thm:conditional are established.

5. Classical restriction is applied after admission. The inherited cancellation of ${\Delta }_{\xi }\mathcal{O}$ remains an evaluation identity and does not imply a quantum restriction quotient or a zero interaction response on all of ${\mathcal{C}}_W$.

Proof. Proposition prop:vertex, Theorems thm:literal, thm:tree, and thm:conditional, Propositions prop:overlap, prop:properness, and prop:zeros, together with the stated inherited restriction input.