B5: the epsilon count

The first coordinate source already contains $-{\sigma }_{\epsilon }^{\prime }{a_1}^{0\mu }[h]$. For $r\ge 1$,

A direct differentiated-source estimate gives

The seminorms are restrictions of ambient smooth derivatives.

To see the derivative count, use the massless Kirchhoff formula for the smooth, flat-at-zero coordinate source $s$:

Spatial derivatives commute with this formula. Repeated time differentiation produces no lower-limit data terms because $s$ is flat at $t=0$. On the compact causal region, derivatives of order $N+1$ of ${\mathsf{X}}_1$ are bounded by source derivatives of order $N+1$. The source has one additional derivative of ${\sigma }_{\epsilon }{a}_1[h]$, giving at most $N+2$ ramp derivatives. Differentiating $c_1=\partial (f{\mathsf{X}}_1)$ therefore yields (eq:eps-estimate).

Thus this direct upper bound can cost ${\epsilon }^{-N-2}$, not merely ${\epsilon }^{-1}$. It is not a lower bound or a necessary divergence rate. For backgrounds vanishing near the edge, the corresponding terms disappear; cancellations can also improve estimates. Higher coefficients require their own estimates.

No uniform sharp-limit continuity, convergence theorem, or admission topology is inferred from these bounds. They do not establish VII-C's limiting hypotheses. The limit $\epsilon \to 0$ is unopened.