Manuscript VII-A-ST@1 · VII-A-ST.tex · theorem · c39a67a3

Theorem: Exact first-order plateau cancellation

First-order repair and the larger-support residual · lines 789–815 · proof included or linked

VII-A-ST@1#thm:first, VII-A-ST@1#eq:first-three

For every smooth $h$ , every free $ϕ_{0}$ , and $ξ \in g_{U}$ ,

g K ξ = K ξ, u_{1, g} [h + K ξ] - u_{1, g} [h] = X_{ξ} ϕ_{0}, w [h + K ξ] - w [h] = ξ .

Consequently $A_{1}^{(g)}$ , and hence $C_{1} [g]$ , is invariant. No metric field equation is required.

Proof. All derivatives of $ξ$ have support in the plateau. Linearity and (eq:QK-shell) give

δ_{ξ} u_{1, g} = - E_{m} Q_{K ξ}^{(m)} ϕ_{0} = E_{m} P_{m} (X_{ξ} ϕ_{0}) = X_{ξ} ϕ_{0} .

The last equality uses the vanishing near $Σ_{0}$ , not an unrestricted inverse identity. The coordinate identity follows from Lemma lem:linear. Substitution in (eq:A12) cancels the two variations.

Displayed formulas in this passage

VII-A-ST@1#eq:first-three · line 793
$g K ξ = K ξ, u_{1, g} [h + K ξ] - u_{1, g} [h] = X_{ξ} ϕ_{0}, w [h + K ξ] - w [h] = ξ .$
unlabeled · line 806
$δ_{ξ} u_{1, g} = - E_{m} Q_{K ξ}^{(m)} ϕ_{0} = E_{m} P_{m} (X_{ξ} ϕ_{0}) = X_{ξ} ϕ_{0} .$

Theorem: Exact first-order plateau cancellation

Displayed formulas in this passage

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