Prove
Lemma 4 ([Rank-layer limit]). There is an integer D_0 such that, for every…Proposition 3. The complex has the following properties. Every pair in E lies…Proposition
Conditional on all types, sample an activation law independently at each incidence when expanding the product in Equation. Conditional on these samples, the matrix integral factors across coordinates. At coordinate κ apply Lemma 4 ([Rank-layer limit]) with the active set of incidences for which κ∈S , and with their sampled signs. Use Equation at each coordinate. There are only finitely many activation and sign configurations. The model expectations are bounded, and their approximating matrix expectations are consequently bounded for large q . A finite product and then the finite average therefore preserve the o(1) error. Expand these finite parity products, and let α_e be the set of coordinates in which the four-sign term for e is selected. A coordinate selected for e requires activation at all four corners in U_e and contributes the product of their signs. At (i,j) the two subsets on its incident pairs give exactly φ_ α_ e(i,j) φ_ α_ f(i,j) ; averaging that incidence's activation and signs gives its factor A . Thus the limiting contribution at the fixed type assignment is ∑_ α_e⊆ K (e∈E) ∏_ (i,j)∈ C A_ a(i)b(j) (α_ e(i,j) ,α_ f(i,j) ). Repeated vertex types leave the matrix draws at distinct source vertices independent. Repeated type pairs use independent samples from the same activation law at their distinct source incidences. The all-empty labeling contributes one. In any other nonzero term, no face can have both an empty and a nonempty pair label: the two meet at a corner, where Equation makes the factor zero. Hence any face with one nonempty label has all three labels nonempty. The shared-pair labels and connectedness of the face-neighbor graph propagate this conclusion to every face. Thus the remaining labels are exactly those summed in Equation. This proves Equation. For the transpose, the same argument uses the last assertion of Lemma 4 ([Rank-layer limit]).