Finite Bipartite Graphs as Induced Subgraphs of Subfactor Principal Graphs

Every finite bipartite simple graph occurs as a vertex-induced unrooted subgraph of the principal graph of an irreducible finite-depth inclusion of hyperfinite type II1 factors. For part sizes m and n, an explicit neighborhood-multiplicity parameter q gives index q3^n, with q at most m+1, and depth at most four. The selected vertices have depths two and three. The complete depth ranks and adjacency spectrum are computed. A separate amplification realizes every finite bipartite multigraph as an ordinary subgraph, with edge deletion allowed. Scope: complete theorem for finite unrooted simple subgraphs across variable indices, not closure of the full AIM polymer programme, fixed-index rooted extension, infinite subgraphs or induced multigraph universality. The finite-group subfactor construction and representation-theoretic ingredients are classical. Historical novelty confidence is low, possibly folklore; no absolute priority is claimed. AI-assisted, self-audited, unrefereed preprint. No independent peer review or formal verification is claimed. Author: Alper Ferudun, Mercury Software GmbH. Corpus identifier: AIM-OTHER-0060, UnsolvedMath v1.6.0. Paper page: https://eulersolve.org/papers/aim-other-0060/