Manuscript XX-F@1 · XX-F.tex · proposition · de0bd24f

Proposition: Bracket of the retained classical operations

The perturbative-order correction and the surviving quotient test › Closure of the classical gauge jets · lines 839–860 · proof included or linked

XX-F@1#prop:bracket

With commutators of operators on functionals,

[V_{ξ}, V_{η}] = - U_{[ξ, η]}, [V_{ξ}, U_{θ}] = 0.

These are the retained pullback, or anti-action, sign conventions.

Proof. On $z$ , the first commutator is zero. On $ζ$ , it is

2 A_{η} K ξ - 2 A_{ξ} K η = - 2 K [ξ, η] .

The operations are ordinary polynomial derivations, so their values on generators determine them. The second commutator is zero on both generators.

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