Manuscript VI-B@1 · VI-B-v1.tex · corollary · 33798723

Corollary: A useful exact scaling family

An explicit nonempty sourced family · lines 1266–1289 · proof included or linked

VI-B@1#cor:scaling

For the preceding tuple, let

ϕ_{0, λ} = λ ϕ_{0}, j_{λ} = λ^{2} j, ϕ_{2, λ} = λ^{3} ϕ_{2}, h_{λ} = ϕ_{1, λ} = 0.

For every real $λ$ this is a jet, and

O_{f_{s}}^{[0]} (λ) = λ a_{0}, O_{f_{s}}^{[1]} (λ) = 0, O_{f_{s}}^{[2]} (λ) = λ^{3} a_{2},

where $a_{0}, a_{2} > 0$ .

Proof. The stress is quadratic, the metric response and its initial preparation are linear in that stress, and $L_{j} ϕ_{0}$ is linear in each argument. The zero-data scalar response therefore scales cubically. Here $w = v = 0$ .

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unlabeled · line 1269
$ϕ_{0, λ} = λ ϕ_{0}, j_{λ} = λ^{2} j, ϕ_{2, λ} = λ^{3} ϕ_{2}, h_{λ} = ϕ_{1, λ} = 0.$
unlabeled · line 1276
$O_{f_{s}}^{[0]} (λ) = λ a_{0}, O_{f_{s}}^{[1]} (λ) = 0, O_{f_{s}}^{[2]} (λ) = λ^{3} a_{2},$

Corollary: A useful exact scaling family

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