Manuscript VII-G@1 · VII-G.tex · section · 54f561e8

A finite certified geometric family

An explicit resource-budgeted feedback witness › A finite certified geometric family · lines 1747–1782

VII-G@1#eq:example-labels, VII-G@1#eq:slice-height, VII-G@1#eq:example-successors

Fix an integer budget $B \geq 1$ , a tick duration $τ > 0$ , and $0 < δ < τ$ . Set

G_{B} = {(k, q) : 0 \leq k \leq B, q \in Z, ∣ q ∣ \leq k, q \equiv k (mod 2)} .

The resource is $r (k, q) = B - k$ . The controller tick is $k τ$ . The selected slice height is

T (k, q) = k τ + δ q .

These two quantities are distinct.

Use the fixed ultrastatic spinless cone

M = R_{T} \times C_{π /2}, d s^{2} = - d T^{2} + d r^{2} + r^{2} d θ^{2}, 0 \leq θ \leq 3 π /2,

with its wedge boundary rays identified. Each label denotes the horizontal slice $T = T (k, q)$ , with the same fixed sector cellulation, stationary axis, and any fixed annular collar.

For $k < B$ , define

Γ_{a} (k, q) = (k + 1, q + a), a \in {- 1, + 1} .

At $k = B$ , the label is absorbing.

Displayed formulas in this passage

VII-G@1#eq:example-labels · line 1751
$G_{B} = {(k, q) : 0 \leq k \leq B, q \in Z, ∣ q ∣ \leq k, q \equiv k (mod 2)} .$
VII-G@1#eq:slice-height · line 1759
$T (k, q) = k τ + δ q .$
VII-G@1#eq:example-successors · line 1777
$Γ_{a} (k, q) = (k + 1, q + a), a \in {- 1, + 1} .$
unlabeled · line 1766
$M = R_{T} \times C_{π /2}, d s^{2} = - d T^{2} + d r^{2} + r^{2} d θ^{2}, 0 \leq θ \leq 3 π /2,$

A finite certified geometric family

Displayed formulas in this passage

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