Manuscript VII-J@1 · VII-J.tex · theorem · d8c01b04

Theorem: Finite-time cone gate certificate

The fixed cone and VII-F's slicing event › Physical ports, not abstract cell labels · lines 1463–1489 · proof included or linked

VII-J@1#thm:conegates, VII-J@1#eq:conegate

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For an input mark $x$ at $s$ and output mark $y$ at $s^{'} \geq s$ , it is sufficient that

s^{'} - s \geq (1 + λ_{0}) d_{C} (x, y)

to certify $p_{x} (s) ⪯ p_{y} (s^{'})$ . Consequently every complete network satisfying this inequality for all dependency edges, including its encoding, decoding, controls, and transition, obeys (eq:main-containment) across $s = 0$ .

Proof. By Lemma lem:slice-estimates,

F_{s^{'}} (y) - F_{s} (x) \geq (s^{'} - s) - λ_{0} d_{C} (x, y) .

Condition (eq:conegate) makes the right side at least $d_{C} (x, y)$ . Apply Lemma lem:conecausal and then Theorem thm:physical. The estimates hold on both sides and at the event.

Displayed formulas in this passage

VII-J@1#eq:conegate · line 1467
$s^{'} - s \geq (1 + λ_{0}) d_{C} (x, y)$
unlabeled · line 1481
$F_{s^{'}} (y) - F_{s} (x) \geq (s^{'} - s) - λ_{0} d_{C} (x, y) .$

Theorem: Finite-time cone gate certificate

Displayed formulas in this passage

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