Manuscript En08-B@1 · En08-B.tex · proposition · 23cac52a

Proposition: Characterization of finite-scale null attainment

Failure tests and finite-scale attainment › Exact discrete null transport is exceptional, not forbidden · lines 1388–1428 · proof included or linked

En08-B@1#prop:exact-null, En08-B@1#eq:exact-null

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Fix $j \geq 1$ , $n \geq 1$ , and $b \in Λ_{j}$ with

∥ b ∥_{2} = n a_{j} .

There exists an admissible $n$ -tick protocol transmitting payload influence from $0$ to $b$ if and only if

b = n h_{j} s for some s \in Z^{d} satisfying ∥ s ∥_{2} = j .

Whenever (eq:exact-null) holds, exact referenced identity transport is attainable.

Proof. Influence requires an ancestor walk from $0$ to $b$ . Write its $n$ displacements, including possible zero steps, as $v_{1}, \dots, v_{n}$ , where

∥ v_{k} ∥_{2} \leq a_{j}, k \sum v_{k} = b .

The null equality gives

n a_{j} = ∥ b ∥_{2} \leq k \sum ∥ v_{k} ∥_{2} \leq n a_{j} .

Hence every step has length $a_{j}$ , and equality holds in the Euclidean triangle inequality. All steps have the same direction and, because their lengths agree, the same vector. Each is a lattice displacement $h_{j} s$ with $∥ s ∥_{2} = j$ . This proves necessity.

Conversely, initialize the $n + 1$ sites $0, h_{j} s, \dots, n h_{j} s$ , and apply a SWAP along each successive edge in one tick. Lemma lem:identity proves complete identity transport.

Displayed formulas in this passage

En08-B@1#eq:exact-null · line 1396
$b = n h_{j} s for some s \in Z^{d} satisfying ∥ s ∥_{2} = j .$
unlabeled · line 1391
$∥ b ∥_{2} = n a_{j} .$
unlabeled · line 1410
$∥ v_{k} ∥_{2} \leq a_{j}, k \sum v_{k} = b .$
unlabeled · line 1414
$n a_{j} = ∥ b ∥_{2} \leq k \sum ∥ v_{k} ∥_{2} \leq n a_{j} .$

Proposition: Characterization of finite-scale null attainment

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