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

Theorem: Exact tolerance–visibility trade-off

Stage F: bounded segments and excluded prescription changes › A separately imposed comparison restriction · lines 1306–1352 · proof included or linked

X-Enter-ST@1#thm:tradeoff, X-Enter-ST@1#eq:segment, X-Enter-ST@1#eq:scales, X-Enter-ST@1#eq:ratio

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For direct readout of (eq:encoding),

∣ λ ∣ \leq Λ sup d_{TV} (Law_{read} (λ c_{ε}), Law_{read} (0)) = \frac{1}{2} tanh (d / σ) \leq \frac{d}{2 σ} .

Let $0 < δ < 1/2$ be an allowed variation and $0 < v < 1/2$ a desired visibility for a separately specified benchmark magnitude $A_{sig} > 0$ . Both

\frac{1}{2} tanh (d / σ) \leq δ, \frac{1}{2} tanh (A_{sig} / σ) \geq v

hold precisely when

\frac{d}{artanh ( 2 δ )} \leq σ \leq \frac{A _{sig}}{artanh ( 2 v )} .

A permitted scale exists if and only if

\frac{A _{sig}}{d} \geq \frac{artanh ( 2 v )}{artanh ( 2 δ )} .

For $d > 0$ , zero tolerance is impossible at finite $σ$ .

Proof. The binary-law distance is the absolute difference of its success probabilities. Monotonicity and oddness of $tanh$ give (eq:segment); $tanh u \leq u$ for $u \geq 0$ gives its upper bound. Applying the increasing inverse $artanh$ to the two requirements gives (eq:scales). The interval is nonempty exactly under (eq:ratio).

Displayed formulas in this passage

X-Enter-ST@1#eq:segment · line 1309
$∣ λ ∣ \leq Λ sup d_{TV} (Law_{read} (λ c_{ε}), Law_{read} (0)) = \frac{1}{2} tanh (d / σ) \leq \frac{d}{2 σ} .$
X-Enter-ST@1#eq:scales · line 1326
$\frac{d}{artanh ( 2 δ )} \leq σ \leq \frac{A _{sig}}{artanh ( 2 v )} .$
X-Enter-ST@1#eq:ratio · line 1333
$\frac{A _{sig}}{d} \geq \frac{artanh ( 2 v )}{artanh ( 2 δ )} .$
unlabeled · line 1321
$\frac{1}{2} tanh (d / σ) \leq δ, \frac{1}{2} tanh (A_{sig} / σ) \geq v$

Theorem: Exact tolerance–visibility trade-off

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