Manuscript XIII-A@1 · XIII-A.tex · theorem · f56e91c7

Theorem: Cancellation on the fixed subclass

A1: the classical evaluation identity · lines 663–689 · proof included or linked

XIII-A@1#thm:classical, XIII-A@1#eq:classical-cancellation

Under {(UCP $_{W}$ )},

Δ_{ξ} O (u) = 0 for every u \in C_{W} and every ξ \in g_{W} .

This is an identity of evaluations of the prescribed off-shell lift.

Proof. Every term of $D s_{ξ} [γ]$ is supported in $supp ξ$ . The inherited support statement gives

supp (K ρ_{ξ}) \subset J^{+} (supp ξ) \cap M_{T} \subset J^{+} (W) \cap M_{T} .

Thus {(UCP $_{W}$ )} implies (CP) for every smooth $γ$ . The companion vanishes for the free scalar in $u$ .

The auxiliary term vanishes by Proposition prop:inclusion. Finally, the additional scalar-sourced term in (eq:full-object) is unchanged by a metric-only shift. Equation (eq:copied-remainder) proves the result.

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XIII-A@1#eq:classical-cancellation · line 666
$Δ_{ξ} O (u) = 0 for every u \in C_{W} and every ξ \in g_{W} .$
unlabeled · line 678
$supp (K ρ_{ξ}) \subset J^{+} (supp ξ) \cap M_{T} \subset J^{+} (W) \cap M_{T} .$

Theorem: Cancellation on the fixed subclass

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