Manuscript Hawk02-C@2 · Hawk02-C.tex · lemma · bb332d66

Lemma: The geometric-tail majorant used by the script

Finite-interface and predictive bridges › The coupled autonomous chain · lines 1616–1633 · proof included or linked

Hawk02-C@2#lem:geometric, Hawk02-C@2#eq:geometric

If $z = 2 c T$ and $z / (N + 2) < 1$ , then

S_{N} (T) \leq G_{N} (T) := \frac{2 z ^{N + 1}}{( N + 1 )! [ 1 - z / ( N + 2 )]} .

Thus $min {1, G_{N} (T)}$ is a channel allowance. The uncapped $G_{N} (T)$ is an isometry-norm allowance.

Proof. Starting at $m = N + 1$ , every successive series-term ratio is at most $z / (N + 2)$ . Sum the geometric majorant.

Displayed formulas in this passage

Hawk02-C@2#eq:geometric · line 1619
$S_{N} (T) \leq G_{N} (T) := \frac{2 z ^{N + 1}}{( N + 1 )! [ 1 - z / ( N + 2 )]} .$