Manuscript X-A-conv@1 · X-A-conv.tex · proposition · c76fffa3

Proposition: Pure-gauge relational defect through second order

The complete second-order pure-gauge test › The relational coefficients on that jet · lines 1973–2026 · proof included or linked

X-A-conv@1#prop:pureobservable, X-A-conv@1#eq:pureOone, X-A-conv@1#eq:pureOtwo

Substitution of (eq:puregauge) into the regulated relational functional gives

Φ_{λ}^{[1]} Φ_{λ}^{[2]} = λ X_{ξ - a} ϕ_{0}, = λ^{2} [\frac{1}{2} X_{ξ}^{2} - X_{a} X_{ξ} + \frac{1}{2} X_{a}^{2} - X_{E_{0} C d_{ξ}}] ϕ_{0} = λ^{2} [\frac{1}{2} X_{ξ - a}^{2} + \frac{1}{2} X_{[ξ, a]} - X_{E_{0} C d_{ξ}}] ϕ_{0} .

For $ξ = ξ_{*}$ , $ϕ_{0} = x^{1}$ , and the frozen $f$ , the first smearing is $λ c_{ε}$ .

Proof. The coefficient substitution in (eq:evalone)–(eq:evaltwo), with (eq:purescalarcoeff) and (eq:purecoordcoeff), gives the first line and

\frac{1}{2} X_{ξ}^{2} - X_{a} X_{ξ} + J (a) - X_{\frac{1}{2} X_{a} a + E_{0} C d_{ξ}} .

The identity

J (a) = \frac{1}{2} X_{a}^{2} + \frac{1}{2} X_{X_{a} a}

reduces this to the first expression in (eq:pureOtwo). Finally,

X_{ξ}^{2} - 2 X_{a} X_{ξ} + X_{a}^{2} = X_{ξ - a}^{2} + [X_{ξ}, X_{a}] = X_{ξ - a}^{2} + X_{[ξ, a]} .

The smearing statement is Lemma lem:positivegap.

Displayed formulas in this passage

X-A-conv@1#eq:pureOone · line 1977
$Φ_{λ}^{[1]} Φ_{λ}^{[2]} = λ X_{ξ - a} ϕ_{0}, = λ^{2} [\frac{1}{2} X_{ξ}^{2} - X_{a} X_{ξ} + \frac{1}{2} X_{a}^{2} - X_{E_{0} C d_{ξ}}] ϕ_{0} = λ^{2} [\frac{1}{2} X_{ξ - a}^{2} + \frac{1}{2} X_{[ξ, a]} - X_{E_{0} C d_{ξ}}] ϕ_{0} .$
X-A-conv@1#eq:pureOtwo · line 1977
$Φ_{λ}^{[1]} Φ_{λ}^{[2]} = λ X_{ξ - a} ϕ_{0}, = λ^{2} [\frac{1}{2} X_{ξ}^{2} - X_{a} X_{ξ} + \frac{1}{2} X_{a}^{2} - X_{E_{0} C d_{ξ}}] ϕ_{0} = λ^{2} [\frac{1}{2} X_{ξ - a}^{2} + \frac{1}{2} X_{[ξ, a]} - X_{E_{0} C d_{ξ}}] ϕ_{0} .$
unlabeled · line 2007
$\frac{1}{2} X_{ξ}^{2} - X_{a} X_{ξ} + J (a) - X_{\frac{1}{2} X_{a} a + E_{0} C d_{ξ}} .$
unlabeled · line 2013
$J (a) = \frac{1}{2} X_{a}^{2} + \frac{1}{2} X_{X_{a} a}$
unlabeled · line 2018
$X_{ξ}^{2} - 2 X_{a} X_{ξ} + X_{a}^{2} = X_{ξ - a}^{2} + [X_{ξ}, X_{a}] = X_{ξ - a}^{2} + X_{[ξ, a]} .$