Manuscript Hawk02-D@1 · Hawk02-D.tex · corollary · e6e4b72f

Corollary: Explicit nonempty recovery regime

Actual complete-residual recovery · lines 891–934 · proof included or linked

Hawk02-D@1#cor:window, Hawk02-D@1#eq:time-choice, Hawk02-D@1#eq:packet-choice

Fix $0 < ε < 1$ . Choose any

t \geq \frac{2 K}{c ε}, x = 2 c t,

and an integer $m = M + 1$ satisfying

m \geq 2 e x, m \geq \frac{x + lo g ( 2 K / ε )}{lo g 2} .

Then

ε_{rec} (t, M) \leq ε, δ_{full} (t, M) \leq ε .

Proof. The time choice gives $q (t) \leq ε^{2} / (4 K)$ . The factorial bound $m! \geq (m / e)^{m}$ gives

e^{x} \frac{x ^{m}}{m !} \leq e^{x} (\frac{e x}{m})^{m} \leq e^{x} 2^{- m} \leq \frac{ε}{2 K} .

Thus

ℓ_{M} (t) \leq \frac{ε ^{2}}{4 K}, s_{M} (t) \leq \frac{ε ^{2}}{2 K} .

Apply (eq:simple-beta).

Displayed formulas in this passage

Hawk02-D@1#eq:time-choice · line 894
$t \geq \frac{2 K}{c ε}, x = 2 c t,$
Hawk02-D@1#eq:packet-choice · line 900
$m \geq 2 e x, m \geq \frac{x + lo g ( 2 K / ε )}{lo g 2} .$
unlabeled · line 908
$ε_{rec} (t, M) \leq ε, δ_{full} (t, M) \leq ε .$
unlabeled · line 918
$e^{x} \frac{x ^{m}}{m !} \leq e^{x} (\frac{e x}{m})^{m} \leq e^{x} 2^{- m} \leq \frac{ε}{2 K} .$
unlabeled · line 928
$ℓ_{M} (t) \leq \frac{ε ^{2}}{4 K}, s_{M} (t) \leq \frac{ε ^{2}}{2 K} .$