Manuscript XXII-F@1 · XXII-F.tex · section · 67821705

Standing approximation failure tests

Readout, limiting tests, and dynamics separation › Standing approximation failure tests · lines 1601–1638

• Endpoint limits versus continuum evolution. A terminal comparison does not supply intermediate channels, a composition law, or a unique interpolation. For example,

Φ_{n}^{c} = (1 - c / n) id + (c / n) R_{0}, n \geq 2,

has

(Φ_{n}^{c})^{⌊ n t ⌋} ⟶ R_{0} + e^{- c t} (id - R_{0})

uniformly on compact time intervals. Two different $c$ 's can be assigned the same coarse $n = 1$ channel and have different limits. They choose different missing refinements; no exact subdivision compatibility with that coarse member is asserted. This is the scope of XIX-C's refinement warning CheungXIXC (prop:two-limits, prop:interpolation).

An exact edge can also disappear at a process limit. Mix a nonsignalling $W_{0}$ with a definite $A \to B$ identity process $W_{\to}$ :

W_{δ} = (1 - δ) W_{0} + δ W_{\to} .

The contribution of $W_{0}$ to each signalling difference is zero, so $Δ_{W_{δ}} (A, B) = δ$ . The edge is present for every $δ > 0$ and absent at the limit. A convergent sequence of probability tables therefore need not supply compatible component-quotient maps CheungXXIIC (prop:strength-stability).

Displayed formulas in this passage

unlabeled · line 1607
$Φ_{n}^{c} = (1 - c / n) id + (c / n) R_{0}, n \geq 2,$
unlabeled · line 1611
$(Φ_{n}^{c})^{⌊ n t ⌋} ⟶ R_{0} + e^{- c t} (id - R_{0})$
unlabeled · line 1629
$W_{δ} = (1 - δ) W_{0} + δ W_{\to} .$

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section: Standing approximation failure tests