Two Latent Types on Arbitrary Covers

Abstract The lab for latent actors [1] proved its two-type results mostly on the complete pair cover of three or four variables. This companion asks the same questions for an arbitrary graph G of observed pairs on m binary variables. When do the pair tables admit a two-type explanation? Which freedoms remain? What does the set of explanations look like? Part I answers these at the pair layer. Existence (theorem 2). A factorisation cij= vivj exists iff three conditions hold. No zero moment may join two variables that carry non-zero moments. On each component of the non-zero graph G∗, the sign cocycle must be a Z/2 coboundary, and log |c|must lie in the image of the unsigned incidence matrix. The last two are the lab’s H1 + = H1(Γ; Z/2) ⊕coker B, extended to any graph. Freedom. rank BG = m−b(G), with b(G) the number of bipartite components (isolated vertices included). Odd cycles pin the magnitudes. Each bipartite component keeps a scale λ±1. Each component keeps its own sign, so the labels can be swapped on one component alone, invisibly to the pairs. Hiding polytopes (theorem 5). The two-type explanations fall into sign sheets. In the coordinates y = log t (the weight) and xk = log λk, each sheet is a convex polytope whose facets all have slope ±1, two per variable: the positivity walls a1i,a2i ∈{0,1}. For a connected bipartite graph the sheet is a rectangle in (log λt,log λ/t). The generic dimension is b(G) + 1. The label swap identifies sheets in pairs. Positivity can empty a sheet. Strata. A coincidence vi = 0 deletes vertex i from G∗. On the triangle it leaves a single edge, so the lab’s hiding segment becomes a two-dimensional region. Nine checks (CV1–CV9) verify every statement on a zoo of eleven graphs, by sampling each polytope and rebuilding the law. Higher windows (Phase III), the identifiability notions (Phase IV), walls and strata in general (Phase V), and the three-type proof of concept (k= 3, m= 5) follow in Part II.