Manuscript X-G-ST@1 · X-G-ST.tex · lemma · 3c97567f

Lemma: Necessary reciprocity on free inputs

Stage 3: variational reciprocity and the positive witness › The excluded upgrade class · lines 1369–1394 · proof included or linked

X-G-ST@1#lem:reciprocity, X-G-ST@1#eq:reciprocity

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For an upgrade in Definition def:upgrade,

D_{w} (u, v) := \int (u_{μν} (N_{w} v)^{μν} - v_{μν} (N_{w} u)^{μν}) d^{4} x = 0

for every pair of smooth free metric solutions and every compact background $w$ .

Proof. Apply $P_{gr}$ to (eq:exactresponse). The right-inverse identity yields $L_{w} u = N_{w} u$ for each free $u$ . Symmetry of the bosonic second variation gives

⟨ u, L_{w} v ⟩ = ⟨ v, L_{w} u ⟩ .

Both pairings exist by (U3). Substitute the equalities on the free inputs to obtain (eq:reciprocity).

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X-G-ST@1#eq:reciprocity · line 1372
$D_{w} (u, v) := \int (u_{μν} (N_{w} v)^{μν} - v_{μν} (N_{w} u)^{μν}) d^{4} x = 0$
unlabeled · line 1388
$⟨ u, L_{w} v ⟩ = ⟨ v, L_{w} u ⟩ .$

Lemma: Necessary reciprocity on free inputs

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