Manuscript XX-E@1 · XX-E.tex · proposition · cf7a8f40

Proposition: Evaluated classical Ward image of the family

Explicit Newton-contact images and comparison Ward images › A declared comparison counterterm family · lines 1044–1076 · proof included or linked

XX-E@1#prop:counterimage, XX-E@1#eq:counterimage

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Its image under the coupled classical operation, through the retained order, is

V_{ξ} C_{m, T; f} = s^{2} \int f (m (z, K ξ) - R_{T} (K ξ)^{a} \partial_{a} ℓ z) d t d^{3} x (mod s^{3}) .

When $T$ is presented by contact kernels, Theorem thm:contactimage supplies its complete Newton image before this pairing.

Proof. Differentiate the symmetric bilinear term and the two rod factors in (eq:Ccandidate). This gives

s^{2} \int f [m (z, K ξ) - R_{T} (K ξ) \cdot \nabla ℓ z - R_{T} z \cdot \nabla ℓ (K ξ)] .

The last term vanishes by Lemma lem:rodgauge. The source map is fixed and linear, so (eq:RT) commutes with this variation on its declared smooth domain.

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XX-E@1#eq:counterimage · line 1048
$V_{ξ} C_{m, T; f} = s^{2} \int f (m (z, K ξ) - R_{T} (K ξ)^{a} \partial_{a} ℓ z) d t d^{3} x (mod s^{3}) .$
unlabeled · line 1066
$s^{2} \int f [m (z, K ξ) - R_{T} (K ξ) \cdot \nabla ℓ z - R_{T} z \cdot \nabla ℓ (K ξ)] .$

Proposition: Evaluated classical Ward image of the family

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