Manuscript X-F-ST@1 · X-F-ST.tex · proposition · e71f03ef

Proposition: The response remainder is in the ideal

F4: the graviton-slot ideal and direct cancellation › An explicit compact preimage of the remaining term · lines 1757–1791 · proof included or linked

X-F-ST@1#prop:remainderideal, X-F-ST@1#eq:preimage

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For $ξ \in g_{p}$ ,

R_{ξ} := A_{1} [Δ_{gr}^{R} L_{ξ}^{d} P_{gr} r] \in J .

A compact mixed preimage is

v_{ξ} (x, y) = - L_{ξ, y}^{c} N_{0} (x, y), supp v_{ξ} \subset C \times supp ξ, WF (v_{ξ}) \subset Z_{2},

with

R_{ξ} = (ϕ \otimes r) ((1 \otimes P_{gr}_{y}) v_{ξ}) .

Proof. The expression before final field pairing is

\int N_{0} (x, y) (L_{ξ}^{d} P_{gr} r) (y) d^{4} y .

Apply (eq:Lieadjoint) in $y$ , then formal self-adjointness of $P_{gr}$ . The coefficients of $L_{ξ}^{c}$ vanish outside $supp ξ$ . Thus $v_{ξ}$ is compact even though $N_{0}$ alone need not be compact in $y$ . Its wavefront is still in $Z_{2}$ , so Lemma lem:gslot applies.

Displayed formulas in this passage

X-F-ST@1#eq:preimage · line 1765
$v_{ξ} (x, y) = - L_{ξ, y}^{c} N_{0} (x, y), supp v_{ξ} \subset C \times supp ξ, WF (v_{ξ}) \subset Z_{2},$
unlabeled · line 1760
$R_{ξ} := A_{1} [Δ_{gr}^{R} L_{ξ}^{d} P_{gr} r] \in J .$
unlabeled · line 1772
$R_{ξ} = (ϕ \otimes r) ((1 \otimes P_{gr}_{y}) v_{ξ}) .$
unlabeled · line 1781
$\int N_{0} (x, y) (L_{ξ}^{d} P_{gr} r) (y) d^{4} y .$