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

Three derivative operations

Observation derivatives and smooth spectators › Three derivative operations · lines 1128–1171

Operation	Established statement	Not inferred
Observation derivative	$D$ acts on the joint numerical distribution. For example, $⟨ (- \partial_{t})^{n} T_{ε}^{ρ}, f \otimes g \otimes f^{'} ⟩ = ⟨ T_{ε}^{ρ}, (\partial_{t}^{n} f) \otimes g \otimes f^{'} ⟩ .$	A quantum product on carrier elements.
Configuration derivative	$D_{η} F$ is differentiation of an actual polynomial along a fixed smooth field direction; its carrier closure is inherited.	Differentiability of an unconstructed sharp triple output.
Functional-slot derivative	Unsmeared functional derivatives expose field slots for later contractions.	Existence of the resulting decorated distributional graphs.

An observation derivative such as $\partial_{t}$ uses the time coordinate of the developing chart. Thus its coordinate vector is related to $\partial_{T}$ by the fixed calibration. Nevertheless, differentiating a kernel argument is not evolving the host metric, implementing a clock operation, or constructing quantum evolution without external scheduling. This is the precise meaning of the time-axis separation.

Displayed formulas in this passage

unlabeled · line 1139
$⟨ (- \partial_{t})^{n} T_{ε}^{ρ}, f \otimes g \otimes f^{'} ⟩ = ⟨ T_{ε}^{ρ}, (\partial_{t}^{n} f) \otimes g \otimes f^{'} ⟩ .$