Manuscript Oai265-A@2 · Oai265-A.tex · theorem · c0a34a45

Theorem: Quantitative terminal area-law certificate

Stage 2: a quantitative terminal certificate › The terminal constant, conditional on actual information charges · lines 666–711 · proof included or linked

Oai265-A@2#thm:terminal

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Let $ψ$ be pure on a finite induced square-lattice domain. For every positive-boundary cut $A$ , suppose one supplies a partition satisfying Lemma lem:cancellation, together with constants

t \in N, g_{1}, K \geq 0, a \geq 0, δ > 0

independent of the domain and the cut, such that:

$D \subseteq Z_{A}^{[t]}$ ;
the tiles have integer levels $j \geq 0$ ;
the number of level- $j$ tiles is at most $g_{1} ∣ \partial_{Λ} A ∣ 2^{aj}$ ;
each level- $j$ information charge obeys $ε_{i} \leq K 2^{- (a + δ) j}$ .

Then

S_{ψ} (A) \leq [2 v_{t} lo g q + \frac{g _{1} K}{2 ( 1 - 2 ^{- δ} )}] ∣ \partial_{Λ} A ∣.

Proof. Each endpoint of a crossing edge contributes at most $v_{t}$ sites to $Z_{A}^{[t]}$ . Since $∣ Z_{A} ∣ \leq 2∣ \partial_{Λ} A ∣$ ,

∣ D ∣ \leq 2 v_{t} ∣ \partial_{Λ} A ∣.

Thus $S (D) \leq 2 v_{t} ∣ \partial_{Λ} A ∣ lo g q$ . Summing the information charges gives

i \sum ε_{i} \leq g_{1} K ∣ \partial_{Λ} A ∣ j \geq 0 \sum 2^{- δ j} = \frac{g _{1} K}{1 - 2 ^{- δ}} ∣ \partial_{Λ} A ∣.

Apply Lemma lem:cancellation.

Displayed formulas in this passage

unlabeled · line 672
$t \in N, g_{1}, K \geq 0, a \geq 0, δ > 0$
unlabeled · line 685
$S_{ψ} (A) \leq [2 v_{t} lo g q + \frac{g _{1} K}{2 ( 1 - 2 ^{- δ} )}] ∣ \partial_{Λ} A ∣.$
unlabeled · line 697
$∣ D ∣ \leq 2 v_{t} ∣ \partial_{Λ} A ∣.$
unlabeled · line 702
$i \sum ε_{i} \leq g_{1} K ∣ \partial_{Λ} A ∣ j \geq 0 \sum 2^{- δ j} = \frac{g _{1} K}{1 - 2 ^{- δ}} ∣ \partial_{Λ} A ∣.$

Theorem: Quantitative terminal area-law certificate

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