# Verification and scope report

The complete all-input proof realizes every finite bipartite simple graph as a vertex-induced unrooted subgraph of an irreducible finite-depth hyperfinite subfactor principal graph. The index is q3^n with the stated neighborhood-sensitive q, the ambient depth is at most four, and all depth ranks and adjacency eigenvalues are computed. Ordinary bipartite multigraph realization is a separate theorem allowing edge deletion.

The originating researcher self-audited the fixed-point orientation, outer action, representation irreducibility, exact zero entries, distinct tags, isolated vertices, empty parts, root membership, depth, rank counts, Gram matrix spectrum and Q8 amplification. The written proof covers all finite inputs.

The final direct-product checker passed 441,362 exact assertions under Python 3.13.5 and 3.9.6, with identical reports apart from the runtime. It covers 689 exhaustive small inputs, 128 repeated-neighborhood controls, 817 full principal graphs with 47,192 vertices, and 5,460 exact Walsh eigenvector-coordinate checks. Two preserved earlier constructions passed 199,403 and 282,170 assertions per runtime. These finite diagnostics are not a formal or independent proof.

Classical finite-group subfactor theory, tensor representations, the S3 character table and Frobenius amplification are credited. Historical novelty confidence is low; no exhaustive novelty clearance or absolute priority is claimed.

The full AIM polymer programme, fixed-index or prescribed-root extension, infinite subgraphs and induced multigraph universality remain unresolved here. This is an AI-assisted, self-audited, unrefereed preprint, not an independent-review certificate.