Manuscript X-A-ST@1 · X-A-ST.tex · proposition · d9ccbde1

Proposition: Pure-gauge two-jet

Mandatory diagnostics · lines 1448–1487 · proof included or linked

X-A-ST@1#prop:puregauge, X-A-ST@1#eq:puregauge

Take $ξ \in g_{p}$ , $ζ = 0$ , and

h = K ξ, k = \frac{1}{2} L_{ξ}^{2} η, ϕ = F_{κ}^{*} ϕ_{0} .

Then

w = ξ, v^{μ} = \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ}, Φ^{(1)} = Φ^{(2)} = 0.

The coordinate coefficients are independent of any ramp width for which the same gauge support remains strictly post-ramp.

Proof. The flat coordinate solution is $x$ , so Theorem thm:coordinate gives $Co_{F^{*} η} = F^{*} x$ . This gives $w, v$ . In (eq:relationalcoeff),

ϕ_{1} = X_{ξ} ϕ_{0}, ϕ_{2} = \frac{1}{2} X_{ξ}^{2} ϕ_{0} .

The first cancellation is immediate. At second order use

X_{ξ}^{2} ϕ_{0} = (ξ^{ν} \partial_{ν} ξ^{μ}) \partial_{μ} ϕ_{0} + ξ^{μ} ξ^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .

The terms cancel exactly. The result depends only on the post-ramp support condition, not on the width inside the initial edge zone.

Displayed formulas in this passage

unlabeled · line 1451
$h = K ξ, k = \frac{1}{2} L_{ξ}^{2} η, ϕ = F_{κ}^{*} ϕ_{0} .$
X-A-ST@1#eq:puregauge · line 1457
$w = ξ, v^{μ} = \frac{1}{2} ξ^{ν} \partial_{ν} ξ^{μ}, Φ^{(1)} = Φ^{(2)} = 0.$
unlabeled · line 1473
$ϕ_{1} = X_{ξ} ϕ_{0}, ϕ_{2} = \frac{1}{2} X_{ξ}^{2} ϕ_{0} .$
unlabeled · line 1479
$X_{ξ}^{2} ϕ_{0} = (ξ^{ν} \partial_{ν} ξ^{μ}) \partial_{μ} ϕ_{0} + ξ^{μ} ξ^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .$