Manuscript X-Enter@1 · X-Enter.tex · theorem · 484b71bc

Theorem: Second-order test

Selective obstruction and retained-order test · lines 653–690 · proof included or linked

X-Enter@1#thm:secondorder, X-Enter@1#eq:secondorder

Keep the encoding and program fixed. Suppose

a (κ, λ) = κλ c_{ε} + κ^{2} b (λ) + O (κ^{3}), b (0) = 0.

Rec_{P} (τ_{1} (a (κ, λ))) - Rec_{P} (τ_{1} (0)) = O (κ^{3})

for every $λ$ , then $B_{y} = C_{y} = 0$ for every $y$ . The remainder assertion is pointwise in $λ$ , in trace norm on the finite-dimensional record space.

Proof. Each selective block has expansion

Φ_{y}^{P} (τ_{1} (a)) - Φ_{y}^{P} (τ_{1} (0)) = \frac{κλ c _{ε}}{σ} B_{y} + κ^{2} (\frac{b ( λ )}{σ} B_{y} - \frac{λ ^{2} c _{ε} ^{2}}{2 σ ^{2}} C_{y}) + O (κ^{3}) .

The first coefficient forces $B_{y} = 0$ ; the second forces $C_{y} = 0$ , independently of $b$ .

Displayed formulas in this passage

X-Enter@1#eq:secondorder · line 677
$Φ_{y}^{P} (τ_{1} (a)) - Φ_{y}^{P} (τ_{1} (0)) = \frac{κλ c _{ε}}{σ} B_{y} + κ^{2} (\frac{b ( λ )}{σ} B_{y} - \frac{λ ^{2} c _{ε} ^{2}}{2 σ ^{2}} C_{y}) + O (κ^{3}) .$
unlabeled · line 656
$a (κ, λ) = κλ c_{ε} + κ^{2} b (λ) + O (κ^{3}), b (0) = 0.$
unlabeled · line 663
$Rec_{P} (τ_{1} (a (κ, λ))) - Rec_{P} (τ_{1} (0)) = O (κ^{3})$

Theorem: Second-order test

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