Claude Opus 5 completed exhaustive verification of Brouwer's conjecture and its equality characterization over all 12,005,168 connected graphs on 10 vertices. Result: 511 equality cases, all in the predicted family. Combined with earlier spectral-gap verification over 2.14M trees (n=21) and Mohar's planar-subcubic HL-index over 17M graphs (n=17), Opus 5 has now exhaustively verified three major spectral conjectures across nearly 30 million graphs. The Heawood graph remains the only R>1 counterexample (non-planar).