Manuscript X-I-ST@1 · X-I-ST.tex · proposition · 9745fd40

Proposition: Incremental scalar response

Section A: the compact classical interaction › A compact-source response comparison · lines 1152–1187 · proof included or linked

X-I-ST@1#prop:response, X-I-ST@1#eq:response

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In the centered sector, suppose a smooth formal reference Euler solution and a deformed Euler solution have the same free input and zero difference Cauchy data. Write their scalar difference as

ϕ_{λ} - ϕ_{ref} = κ^{3} δ ϕ_{3} (mod κ^{4}) .

Then its new scalar contribution is uniquely

δ ϕ_{3} = \frac{λ}{6} E_{m}^{R} (a ϕ_{0}^{3}) .

The added contribution to the metric–auxiliary response at this order is zero. The new scalar dependence is contained in $J_{η}^{+} (supp a)$ .

Proof. Coefficients below order three agree by Lemma lem:orderprotection. At order three, subtraction of the two scalar Euler equations leaves

- P_{m} δ ϕ_{3} + \frac{λ}{6} a ϕ_{0}^{3} = 0.

The source is smooth and compact. The completion retarded inverse proves (eq:response), uniqueness, and causal support. The vertex has no metric or auxiliary derivative.

Displayed formulas in this passage

X-I-ST@1#eq:response · line 1163
$δ ϕ_{3} = \frac{λ}{6} E_{m}^{R} (a ϕ_{0}^{3}) .$
unlabeled · line 1158
$ϕ_{λ} - ϕ_{ref} = κ^{3} δ ϕ_{3} (mod κ^{4}) .$
unlabeled · line 1180
$- P_{m} δ ϕ_{3} + \frac{λ}{6} a ϕ_{0}^{3} = 0.$

Proposition: Incremental scalar response

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