Manuscript XVI-L@1 · XVI-L.tex · theorem · 35081099

Theorem: Source derivative budget

L.1 The complete source as a jet polynomial › The derivative budget, established · lines 453–472 · proof included or linked

XVI-L@1#thm:budget

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The complete source $J_{D}$ of XII-C.tex, eq:actualsource, has the following budget, where “slot order” is $∣ a ∣ + ∣ b ∣$ in (eq:jet):

Summand	Preparation factor	Slot order	Occurring $(∣ a ∣, ∣ b ∣)$	Status
$C_{2}$	$\partial\partial σ_{ε}$	exactly $0$	$(0, 0)$	Grading Proved locally; attainment Computationally verified
$M_{2}$	$\partial σ_{ε}$	exactly $1$	$(1, 0)$ , $(0, 1)$	Grading Proved locally; attainment Computationally verified
$J_{3}$	$σ_{ε}$	exactly $2$	$(2, 0)$ , $(1, 1)$ , $(0, 2)$	Grading Proved locally; attainment Computationally verified

In particular, the maximum source-slot derivative order of the complete source is $2$ , attained only in the $σ_{ε} J_{3}$ summand; and the anticipated table of the XVI-L plan ( $C_{2}$ : $0$ ; $M_{2}$ : $\leq 1$ ; $J_{3}$ : $\leq 2$ ) holds, in the sharper form that each summand is homogeneous in the slot order.

Proof. Grading and preparation factors: Proposition prop:jet(4). Attainment of every listed $(∣ a ∣, ∣ b ∣)$ : Lemma lem:L1comp(C4).

Nearby in this section

remark: Remark: What the budget does and does not yet give