Manuscript IX-B-ST@1 · IX-B-ST.tex · corollary · f6af1523

Corollary: Smooth spectator rule

Observation derivatives and smooth spectators · lines 1096–1118 · proof included or linked

IX-B-ST@1#cor:spectators

Suppose that, after fixing smooth configurations $Ψ = (h, ϕ)$ , a selected numerical coefficient is exactly

a [Ψ] D T_{ε}^{ρ}

with $a [Ψ] \in C^{\infty} (X)$ . If $Ψ \mapsto a [Ψ]$ is continuous in the compact smooth seminorms through order $64 + d$ , then its numerical finite part is

a [Ψ] D u_{μ}^{ρ} .

Convergence is uniform on configuration sets for which those multiplier seminorms are bounded.

Proof. Insert $a = a [Ψ]$ in Theorem thm:descendants and use its uniform coefficient bound.

Displayed formulas in this passage

unlabeled · line 1100
$a [Ψ] D T_{ε}^{ρ}$
unlabeled · line 1107
$a [Ψ] D u_{μ}^{ρ} .$

Nearby in this section

section: Observation derivatives and smooth spectators
lemma: Lemma: Explicit test-order bound
theorem: Theorem: Derivative–spectator finite part
section: Observation derivatives and smooth spectators
corollary: Corollary: Descendant conversion and scale dependence
section: Observation derivatives and smooth spectators
section: Observation derivatives and smooth spectators