Manuscript IX-B@2 · IX-B-v2.tex · section · 8ce915d6

Three derivative operations and three different claims

Observation derivatives and smooth spectators › Three derivative operations and three different claims · lines 920–962

IX-B@2#eq:testderivative

For product tests,

⟨ (- \partial_{x^{0}})^{n} T_{ε}^{ρ}, f \otimes g \otimes f^{'} ⟩ = ⟨ T_{ε}^{ρ}, (\partial_{t}^{n} f) \otimes g \otimes f^{'} ⟩ .

This is the observation-test descendant calibration for $B_{\partial_{t}^{n} f}$ .

Operation	Established claim and boundary
Observation differentiation	Differentiation of the joint numerical distribution, equivalently transfer to its test. Theorem thm:descendants proves the finite-part estimate.
Configuration differentiation	Fixed smooth directional differentiation of an actual polynomial. IX-A proves closure and continuity on $P_{Σ}$ (`thm:closure`, `prop:top-basic`).
Functional derivative slots	An unsmeared functional derivative exposes or deletes field slots in subsequent contractions. The resulting decorated graph need not be an observation derivative of $T_{ε}^{ρ}$ ; no general extension theorem for these graphs is proved here.

Directional derivatives are evaluations of functional derivatives on directions, but the three extension claims above do not follow from one another. In particular, IX-A's directional closure does not place the observation residues $r_{n}$ in a finite-dimensional distribution space. Conversely, the descendant finite part does not establish directional differentiation of an unconstructed sharp triple output.

Displayed formulas in this passage

IX-B@2#eq:testderivative · line 923
$⟨ (- \partial_{x^{0}})^{n} T_{ε}^{ρ}, f \otimes g \otimes f^{'} ⟩ = ⟨ T_{ε}^{ρ}, (\partial_{t}^{n} f) \otimes g \otimes f^{'} ⟩ .$