Theorem: Regularized map and reconstruction descent

For each $\delta >0$, (eq:F) is a continuous linear map from the quotient space ${\mathcal{K}}_{\mathrm{int}}$ to strong distributions. Its lift satisfies

It is defined on every compact joint test, including tests meeting $N$. Its value on $K_{\mathrm{ret}}$ uses the complete branch and vertex inventory.

Proof. The array $C_{\delta }$ is smooth. The test formula (eq:vertex-test) uses smooth multiplication, finite component contractions, and output differentiation. These operations, for fixed smooth coefficients, are continuous on strong distributions.

Composing with Lemma lem:reconstruction gives a continuous map on presentations. It depends only on their actual reconstructed distribution and therefore descends continuously for the quotient topology. A null reconstruction pairs to zero with every smooth test in (eq:vertex-test). This proves all null identities simultaneously.

Linearity in $K$, together with (eq:Kret), retains every term. Appendix app:inventory specifies the finite vertex expansion without a leading-term truncation.