Manuscript X-Enter-ST@1 · X-Enter-ST.tex · theorem · f76acc1f

Theorem: Second-order encoding test

Stage C: the retained finite amplitude trial › The retained-order diagnostic · lines 778–813 · proof included or linked

X-Enter-ST@1#thm:secondorder

Keep $σ$ and the instrument fixed. Suppose, for each fixed $λ$ ,

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

Rec_{Φ} (τ (a (κ, λ))) - Rec_{Φ} (τ (0)) = O (κ^{3})

in trace norm for every fixed $λ$ , then $B_{y} = C_{y} = 0$ for all $y$ .

Proof. The expansions

t (a) = a / σ + O (a^{3}), s (a) = 1 - a^{2} / (2 σ^{2}) + O (a^{4})

give, in every selective block,

\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 gives $B_{y} = 0$ ; the second then gives $C_{y} = 0$ .

Displayed formulas in this passage

unlabeled · line 782
$a (κ, λ) = κλ c_{ε} + κ^{2} b (λ) + O (κ^{3}), b (0) = 0.$
unlabeled · line 789
$Rec_{Φ} (τ (a (κ, λ))) - Rec_{Φ} (τ (0)) = O (κ^{3})$
unlabeled · line 799
$t (a) = a / σ + O (a^{3}), s (a) = 1 - a^{2} / (2 σ^{2}) + O (a^{4})$
unlabeled · line 804
$\frac{κλ c _{ε}}{σ} B_{y} + κ^{2} (\frac{b ( λ )}{σ} B_{y} - \frac{λ ^{2} c _{ε} ^{2}}{2 σ ^{2}} C_{y}) + O (κ^{3}) .$