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

Proposition: Scalar intertwiner with unchanged free input

Relational expansion and covariance: Route I › Naturality and plateau covariance · lines 1141–1165 · proof included or linked

X-A-ST@1#prop:scalar, X-A-ST@1#eq:scalarintertwiner, X-A-ST@1#eq:rholaws

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For $F \in G_{p}$ ,

ρ_{g} [F^{*} q; ϕ_{0}] = F^{*} ρ_{g} [q; ϕ_{0}] (mod κ^{3}) .

Thus

ρ_{1, g}^{'} ρ_{2, g}^{'} = ρ_{1, g} + X_{ξ} ϕ_{0}, = ρ_{2, g} + X_{ξ} ρ_{1, g} + X_{ζ} ϕ_{0} + \frac{1}{2} X_{ξ}^{2} ϕ_{0} .

Proof. Lemma lem:plateaueuler makes the pulled-back solution solve the transformed equation. Its Cauchy data are exactly those of the same $ϕ_{0}$ , because the transformation is initial-trivial. Coefficientwise Cauchy uniqueness proves (eq:scalarintertwiner); expansion gives (eq:rholaws).

Displayed formulas in this passage

X-A-ST@1#eq:scalarintertwiner · line 1144
$ρ_{g} [F^{*} q; ϕ_{0}] = F^{*} ρ_{g} [q; ϕ_{0}] (mod κ^{3}) .$
X-A-ST@1#eq:rholaws · line 1149
$ρ_{1, g}^{'} ρ_{2, g}^{'} = ρ_{1, g} + X_{ξ} ϕ_{0}, = ρ_{2, g} + X_{ξ} ρ_{1, g} + X_{ζ} ϕ_{0} + \frac{1}{2} X_{ξ}^{2} ϕ_{0} .$

Proposition: Scalar intertwiner with unchanged free input

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