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

Proposition: Scaling, monotonicity, and benchmark values

Stage 2: a quantitative terminal certificate › Interaction counting and dimensionless parameters · lines 624–652 · proof included or linked

Oai265-A@2#prop:Cstar

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For $a > 0$ ,

C_{*} (q, R, a J, a Δ) = C_{*} (q, R, J, Δ) .

The quantity is nondecreasing in $J$ and $R$ , and nonincreasing in $Δ$ . For $R = 0$ , it is zero. For $R \geq 1$ and $0 < Δ \leq J$ ,

C_{*} (q, R, J, Δ) \geq lo g q .

Proof. Scaling every Hamiltonian by $a$ is a bijection between the two classes and preserves its ground state. The monotonicity statements follow from class inclusion.

When $R = 0$ , all terms are on-site. A unique ground state is a product, so all cut entropies vanish. For the lower bound, take a two-site induced edge and

H = J (I - ∣ Ω_{q} ⟩ ⟨ Ω_{q} ∣), ∣ Ω_{q} ⟩ = q^{- 1/2} j = 0 \sum^{q - 1} ∣ j ⟩ ∣ j ⟩ .

Its gap is $J$ , and its one-edge cut entropy is $lo g q$ .

Displayed formulas in this passage

unlabeled · line 627
$C_{*} (q, R, a J, a Δ) = C_{*} (q, R, J, Δ) .$
unlabeled · line 634
$C_{*} (q, R, J, Δ) \geq lo g q .$
unlabeled · line 647
$H = J (I - ∣ Ω_{q} ⟩ ⟨ Ω_{q} ∣), ∣ Ω_{q} ⟩ = q^{- 1/2} j = 0 \sum^{q - 1} ∣ j ⟩ ∣ j ⟩ .$

Proposition: Scaling, monotonicity, and benchmark values

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