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Proposition 18 ([Uniform local counts and conditional fields]). In either geometry, every normalized δ -state in a finite sector satisfies P≤C . Fix y∈ Y . Suppose the initial finite sequence of cuts has total out support covered by at …

Lemma 17 ([Local second moments]). For every normalized δ -state in any finite sector and every y∈ Y , ny2≤C((ay-6 ,1)+δay). In particular the number P in is at most a universal constant.

Lemma 15 ([Conditional positive fields]). Fix C0≥1 . Suppose the initial sequence is absent, or its total out support is covered by at most C0 balls Bz and the sum of its cuts' squared gradients is bounded by ∑z Dz2az-2 \1Bz , …

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The count assertion is Lemma 17 ([Local second moments]). Comparable scales have comparable m and e=m^2/a , uniformly in δ , so e_*≤C(C_0)e_y in Lemma 15 ([Conditional positive fields]). Substitute P≤C into that lemma to obtain the field assertion.

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