Manuscript X-A-conv-ST@1 · X-A-conv-ST.tex · proposition · 8ca47088

Proposition: Second-order encoding diagnostic

Stage C: readout closure of the retained coefficient › The retained amplitude family · lines 1564–1587 · proof included or linked

X-A-conv-ST@1#prop:encodingsecond

For a fixed instrument and encoding, let

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

If its complete output agrees with that of $τ (0)$ through order two for every fixed $λ$ , then $B_{y} = C_{y} = 0$ .

Proof. Every selective difference has expansion

\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}) .

Successive coefficient comparison gives the result.

Displayed formulas in this passage

unlabeled · line 1567
$a (κ, λ) = κλ c_{ε} + κ^{2} b (λ) + O (κ^{3}), b (0) = 0.$
unlabeled · line 1579
$\frac{κλ c _{ε}}{σ} B_{y} + κ^{2} (\frac{b ( λ )}{σ} B_{y} - \frac{λ ^{2} c _{ε} ^{2}}{2 σ ^{2}} C_{y}) + O (κ^{3}) .$