Remark: Analytic boundary and dependency on (S-bv)

An identity of rational functions away from the null cone is not automatically a distributional identity on $\Omega$. Lemma lem:bv is about the model family $T_m$ and does not by itself identify $S$ with $c{T}_1$. The identification of the numerical kernels ${q^{\circ }}_{\lambda }$ with the corresponding products of $T_m$'s—and hence the numerical conclusions of M.2–M.4 that use the graded presentation—rests on Input (S-bv), which is certified by Theorem thm:wpA-Sbv; these conclusions therefore hold unconditionally on $\Omega$. Proposition prop:wave remains the part that needs no kernel identification. The linear Helmholtz lemma (Lemma lem:helm) is pure variational calculus under Gate V and never depended on (S-bv).