The Graded Classification Conjecture holds for graphs with disjoint cycles

The paper proves GCC for countable composition S-NE graphs via explicit canonical forms and bespoke graph operations, establishing the equivalence (1)⇔(2)⇔(3) and realizing any POMD-isomorphism by a concrete graded *-isomorphism. The candidate solution outlines a conceptually similar pipeline—functorial identification M^Γ_E ≅ V^Γ(L_K(E)), canonicalization by invariant-preserving moves, and faithfulness on canonical objects—culminating in the same equivalences and realization. While the model’s move set (expansions/contractions, ‘fan’ normalizations) differs from the paper’s specific operations (1-S-NE moves, out-splits, blow-ups, exit moves, tail-cuts), the logical structure aligns. The model’s outline omits some technical combinatorics (e.g., connecting matrices and tail accounting) that the paper supplies, but it does not contradict the paper.