Sharp exponential edge bounds and flag constructions for multipartite intersecting hypergraphs

Let H be a finite, nonempty, simple r-partite r-uniform hypergraph in which any two distinct edges meet in at least t vertices, with t <= r <= 3t. We prove |E(H)| >= 2^tau(H)-1. For tau(H) >= 2, equality forces r = 3t, exact intersection size t, and positive degrees 2^(tau(H)-1), ..., 1 in every part. Complete flags in binary vector spaces attain equality with arbitrarily large cover number at ratio r/t = 3; the intersection parameter grows with the construction. An incidence-matrix inequality relates minimum positive degree and cover number to the maximum, over edges, of the sum of their intersection excesses above t. At r = 3t, it implies tau(H) <= t+1 in the strictly t-intersecting case and yields necessary counterexample conditions for varying intersections. Version 1.0.0 is the first public preprint release, prepared after two internal AI-assisted review rounds and two manuscript revisions. These are not external human peer review. Analytic proofs are primary; two separate finite verification implementations, their outputs, source, and review/revision history are provided in the supplement. The general mixed-intersection boundary question, equality classification, and comprehensive priority assessment are not settled by this release. Documents and result data: CC BY 4.0. Python verification programs: Apache License 2.0, as specified in LICENSES.txt.