Proposition

Lemma 19 ([Integrated cross comparison]). Fix two simple star cell paths from a to b , their label tuples matching the outer Y -cells and having a common first label, two skeletons, and endpoint data Y,S^C,x,z as in. With B , the near …

Lemma 19 ([Integrated cross comparison]). Fix two simple star cell paths from a to b , their label tuples matching the outer Y -cells and having a common first label, two skeletons, and endpoint data Y,SC,x,z as in. With B , the near occurrences, G , and HF(B) defined above, one has II,I' ≤, eC| h|+σρJ 1occ ∏near occurrences; in cell Q [(1-ε)nQ(Y)]-1 ; , 8mm·∫W(SB,EB;dω)\,1G HF(B). Here C<∞ is fixed and σρ≥0 tends to zero as ρ0 after the cutoffs are fixed, uniformly in all permitted volumes and temperatures. A cell on both paths is counted twice in the product. The right side is zero when the occupancy condition fails.

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Lemma 18 ([Path lists and their encounter bound]). For every original witness…Lemma 17 ([Stability under every cut replacement]). Fix feasible endpoint and…Proposition
The preceding inner-integral comparison proves the displayed bound on every feasible fiber. On an infeasible fiber the original contribution is zero. We explain the remaining bookkeeping to specify precisely how the full integral is recovered. Keep L,B_ l ,B_ j j_n fixed and integrate the extensions in j_n, d as in the displayed ledger. Whenever the original domain is nonempty, all the nearest-kernel factorizations apply. Restore the baseline kernels from , then integrate the fixed paths. Finally drop the remaining cut and nearest-lift restrictions from the baseline integral by positivity, obtaining. Its predicates are measurable functions of the baseline paths, specified by finite cell algorithms. This use of the conditional comparison is valid pointwise at the fixed data: a positive original inner integral has a witness, and otherwise the required upper bound is automatic.
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