Manuscript En06-B@1 · En06-B.tex · section · ee886ae8

Local junctions and the global partition schedule

Local junctions and the global partition schedule · lines 652–685

En06-B@1#eq:framebudget, En06-B@1#eq:vertexclock

Fix the finite causal graph after the zero-duration contraction. Let $p_{v}$ be the number of original payload maps at macrovertex $v$ , and let $d_{v}^{-}$ , $d_{v}^{+}$ be its incoming and outgoing wire counts. Choose

B \geq v max (d_{v}^{-} + p_{v} + d_{v}^{+}) + 1.

All constants $L$ and $B$ depend only on this fixed graph.

Let

t_{*} = v min t_{v}, T_{Γ} = v max t_{v} - t_{*}, η_{n} = n^{- 1/2} .

Retard the geometry slightly by

f_{n} (t) = t_{*} + (1 + η_{n}) (t - t_{*}) .

The hub frame at vertex $v$ begins at

k_{v} τ_{n}, k_{v} = ⌊ \frac{f _{n} ( t _{v} )}{τ _{n}} ⌋ .

Its duration is $B τ_{n}$ .

Displayed formulas in this passage

En06-B@1#eq:framebudget · line 658
$B \geq v max (d_{v}^{-} + p_{v} + d_{v}^{+}) + 1.$
unlabeled · line 666
$t_{*} = v min t_{v}, T_{Γ} = v max t_{v} - t_{*}, η_{n} = n^{- 1/2} .$
unlabeled · line 674
$f_{n} (t) = t_{*} + (1 + η_{n}) (t - t_{*}) .$
En06-B@1#eq:vertexclock · line 678
$k_{v} τ_{n}, k_{v} = ⌊ \frac{f _{n} ( t _{v} )}{τ _{n}} ⌋ .$