Manuscript XII-E@1 · XII-E.tex · lemma · 9471ea56

Lemma: Held-fixed defect of the scalar completion

E3: the ledger, printed before addition · lines 708–724 · proof included or linked

XII-E@1#lem:heldfixed, XII-E@1#eq:heldfixed

For every smooth $k$ ,

A_{2}^{sc} [f, g] (γ + K ξ, k; ϕ) - A_{2}^{sc} [f, g] (γ, k; ϕ) = - A_{1} [f, g] (s_{ξ} [γ]; ϕ) .

Proof. Input statement input:XA with $H = γ$ gives $A_{2}^{sc} (γ + K ξ, k + s_{ξ} [γ]; ϕ) = A_{2}^{sc} (γ, k; ϕ)$ , since $L_{ξ} γ + \frac{1}{2} L_{ξ} K ξ = s_{ξ} [γ]$ . Lemma lem:affine with $r = s_{ξ}$ gives $A_{2}^{sc} (γ + K ξ, k; ϕ) = A_{2}^{sc} (γ + K ξ, k + s_{ξ}; ϕ) - A_{1} (s_{ξ}; ϕ)$ .

Displayed formulas in this passage

XII-E@1#eq:heldfixed · line 711
$A_{2}^{sc} [f, g] (γ + K ξ, k; ϕ) - A_{2}^{sc} [f, g] (γ, k; ϕ) = - A_{1} [f, g] (s_{ξ} [γ]; ϕ) .$

Nearby in this section

section: E3: the ledger, printed before addition
section: E3: the ledger, printed before addition
lemma: Lemma: Variation of the gravitational slot
section: E3: the ledger, printed before addition