Perfect play has different arithmetical complexity in semifinite and type III algebras

For finite rational nonlocal games, existence of perfect normal play in a type II₁ algebra, a type II∞ algebra, or any semifinite algebra is Σ⁰₂-complete. For each fixed λ∈[0,1], existence in a type IIIλ factor is equivalent to commuting perfection and is Π⁰₁-complete. Existence of commuting perfect play together with absence of every normal semifinite perfect strategy is Π⁰₂-complete. Its hardness holds on a computable family whose every member is commuting perfect and has finite-dimensional quantum value one. The lower bound combines a critical binary-tree selector, an effective group embedding, and a Boolean compiler on arbitrary faithful normal supports. The selector finds a finite terminal depth even in diffuse semifinite algebras. A finite-corner overlap criterion gives the upper bound. The perfect-play languages for type I and type II₁ are incomparable, and both are proper subsets of the type II∞ language. The type-I complexity used here is established in the companion paper. Two directional differences are D₂(Σ⁰₂)-complete: perfect play in type I but not in type II₁ (I-ONLY), and perfect spatial play attainable only with infinite entropy (INF-ENTROPY-ONLY). For semifinite-but-not-type-I play (II-ONLY) and its finite-entropy variant, completeness remains open; both are Σ⁰₂-hard and Π⁰₂-hard.