Manuscript Hawk01-A@1 · Hawk01-A.tex · corollary · 11979653

Corollary: Thermal-limit weight and conditional threshold

An explicit Hawking-mode completion and its nonidentifiability › Exact coherent, number-record, and referenced-recovery optima · lines 1775–1811 · proof included or linked

Hawk01-A@1#cor:threshold, Hawk01-A@1#eq:thermalweightlimit, Hawk01-A@1#eq:finiteRACthreshold, Hawk01-A@1#eq:thermalRACthreshold

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As $L \to \infty$ with fixed $0 < q < 1$ ,

p_{L} (q) ⟶ p_{\infty} (q) = \frac{2 q}{1 + q} = \frac{2 n}{1 + 2 n} .

For a finite, energy-admissible $L$ , coherent access exceeds the trusted measure-before-query bound if and only if

p_{L} (q) > \frac{1}{2} .

In the mathematical untruncated limit this condition becomes

q > \frac{1}{2 2 - 1}, equivalently n > \frac{2 + 1}{2} .

Proof. In (eq:pLclosed), let $Z_{L} \to 1$ and the remaining geometric tail tend to zero. Then

1 - p_{\infty} = \frac{( 1 - q ) ^{2}}{1 - q ^{2}} = \frac{1 - q}{1 + q} .

The threshold follows by solving $1/2 + p / (22) > 3/4$ . Substitute $q = n / (1 + n)$ for the final form.

Displayed formulas in this passage

Hawk01-A@1#eq:thermalweightlimit · line 1778
$p_{L} (q) ⟶ p_{\infty} (q) = \frac{2 q}{1 + q} = \frac{2 n}{1 + 2 n} .$
Hawk01-A@1#eq:finiteRACthreshold · line 1786
$p_{L} (q) > \frac{1}{2} .$
Hawk01-A@1#eq:thermalRACthreshold · line 1791
$q > \frac{1}{2 2 - 1}, equivalently n > \frac{2 + 1}{2} .$
unlabeled · line 1802
$1 - p_{\infty} = \frac{( 1 - q ) ^{2}}{1 - q ^{2}} = \frac{1 - q}{1 + q} .$

Corollary: Thermal-limit weight and conditional threshold

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