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Lemma 13 ([Two peelings]). Take the constants in, and choose the reference accuracy in with ε<ζ/4 . For every law σ satisfying, the following decompositions are available. After discarding the exception in Lemma 12 and at most 2^ -ζN …

Lemma 13 ([Two peelings]). Take the constants in, and choose the reference accuracy in with ε<ζ/4 . For every law σ satisfying, the following decompositions are available. After discarding the exception in Lemma 12 and at most 2^-ζN further mass, the law is partitioned into leaves. Each leaf has fixed pin spaces with total dimension at most K₁ , and its normalized law σ_λ satisfies σ_λa specified exact-image constraint of rank t modulo the pins ≤2^-(1-ζ)tN (t≥1). The normalized leaf laws satisfy a common bound 2^O(N) μ² , with the implied constant determined by the early parameters. Suppose a restriction of the mixed law has relative mass 2^-o(N) . One may discard an exponentially small fraction of that restricted law so that every remaining normalized leaf restriction satisfies σ_λa specified exact-image constraint of rank t modulo the pins ≤2^-(1-2ζ)tN (t≥1), and still has density 2^O(N) relative to μ² . For several successive restrictions, the same statement applies to their combined relative cost. In addition, suppose at most u₀ further coefficient directions per unit must be pinned, their coefficient choices and all required scalar metadata having at most 2^O(n) possibilities per old leaf. After splitting by these choices and their exact images, discarding exponentially small mass, and peeling again, the total pin dimension is at most K . The fresh leaves satisfy; after further restrictions as in (b), they satisfy. They retain a uniform 2^O(N) μ² density bound. In all parts the mixed law uses each leaf's actual mass, normalized only after the indicated discards. No equal reweighting of leaves is used. After a restriction, σ_λ denotes the normalized surviving law on that leaf.