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

Proposition: Exact preflight on post-ramp gauges

Preflight and the retained edge obstruction · lines 875–897 · proof included or linked

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

If $ξ \in C_{c}^{\infty} (M_{T_{*}}^{\circ}; T M)$ and $supp ξ \subset {t \geq ε}$ , then

Co_{1} [K ξ] = w [K ξ] = ξ .

In particular, this holds for every $ξ \in g_{p}$ .

Proof. All derivatives of $ξ$ , and hence $a_{1} [K ξ]$ , are supported where $σ = 1$ . Therefore

□ w^{μ} [K ξ] = - \partial_{α} (σ a_{1}^{α μ} [K ξ]) = C^{μ} [K ξ] = □ ξ^{μ} .

Both sides have zero Cauchy data. Uniqueness proves the assertion.

Displayed formulas in this passage

X-A-ST@1#eq:preflight · line 879
$Co_{1} [K ξ] = w [K ξ] = ξ .$
unlabeled · line 890
$□ w^{μ} [K ξ] = - \partial_{α} (σ a_{1}^{α μ} [K ξ]) = C^{μ} [K ξ] = □ ξ^{μ} .$