Manuscript VII-A@1 · VII-A.tex · proposition · 11da3fe6

Proposition: Equality under causal agreement of switches

Support and comparison of switches › A valid equality criterion · lines 1624–1645 · proof included or linked

VII-A@1#prop:cutoffequal, VII-A@1#eq:cutoffequal

Fix $h, k, ϕ_{0}$ and the same harmonic Cauchy prescription. If two switches $g_{1}, g_{2}$ agree on an open neighbourhood of $D_{f}$ , then

O_{f} [g_{1}] = O_{f} [g_{2}] (mod κ^{3}) .

Proof. Their first- and second-order scalar operators agree on that neighbourhood, including all derivatives of the switches. Retarded support and the coefficient formulas then give equality of the scalar coefficients on $supp f$ . The harmonic coordinates do not depend on the switch. Equation (eq:cutoffequal) follows.

Displayed formulas in this passage

VII-A@1#eq:cutoffequal · line 1629
$O_{f} [g_{1}] = O_{f} [g_{2}] (mod κ^{3}) .$

Nearby in this section

section: A valid equality criterion