One gap for the whole class

Theorem 7.8 · thm:G · page 21

What is happening

For each cutoff g, spec H(g) ⊆ {0} ∪ [γ, ∞): the ground state is simple and the rest of the spectrum starts at γ. The content of Theorem 7.8 is the word uniform — one γ that works for every g in the class, not a γ(g) that may shrink as the cutoff grows.

Everything downstream needs exactly that. The inverse-energy filter of Lemma 9.1 is built from a function that vanishes on [−γ/2, γ/2], so it inverts H only on the complement of the kernel, and its time kernel decays at a rate set by γ. If γ were allowed to shrink with the cutoff, the kernel would spread, the two-regime bound would weaken, and the rate would degrade to nothing in the limit.

Switch on the counterfactual to see it: a gap closing like 1/n is still positive for every single cutoff, and still useless, because no fixed γ serves the whole net.

In §11 the abstract parameter γ is instantiated as the GJS mass m₁.