Manuscript VII-I@1 · VII-I.tex · proposition · 907e96ab

Proposition: Plateau covariance at fixed width

Route B1b: a divergence-form density preparation › Classical covariance of the prepared model · lines 933–964 · proof included or linked

VII-I@1#prop:rampcov

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Assume

U \subset int {g = 1} \cap {t > ε} .

For the same restricted transformations as before,

X_{F^{*} q, ϵ} = X_{q, ϵ} \circ F (mod κ^{3}),

and

O_{f, ϵ}^{D} [F^{*} q, F^{*} ϕ] = O_{f, ϵ}^{D} [q, ϕ] (mod κ^{3}) .

Proof. Write

A_{ϵ} [q] = (1 - σ_{ε}) A [η] + σ_{ε} A [q] .

On the support of the transformation, $σ_{ε} = 1$ , and outside that support the transformation and its derivative are the identity. Hence this density transforms naturally under the restricted pullback. The transformation is the identity near the initial surface, so Cauchy uniqueness proves the coordinate identity. Formal inversion and scalar pullback prove the observable identity.

Displayed formulas in this passage

unlabeled · line 936
$U \subset int {g = 1} \cap {t > ε} .$
unlabeled · line 940
$X_{F^{*} q, ϵ} = X_{q, ϵ} \circ F (mod κ^{3}),$
unlabeled · line 945
$O_{f, ϵ}^{D} [F^{*} q, F^{*} ϕ] = O_{f, ϵ}^{D} [q, ϕ] (mod κ^{3}) .$
unlabeled · line 954
$A_{ϵ} [q] = (1 - σ_{ε}) A [η] + σ_{ε} A [q] .$

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section: Classical covariance of the prepared model