ASYMPTOTIC RATIO SET OF ITPFI FACTORS

The paper proves the same two-part classification of type-III ITPFI factors as the candidate solution: (A) for limsup k_n = ∞, type-III_1 iff 0 is a cluster point of normalized eigenvalue ratios or the multiplicative group generated by nonzero cluster points is (0,∞); type-III_λ iff λ is a cluster point and all other cluster points lie in a finite subset of {λ^n}; type-III_0 iff the only cluster point is 1 (Theorem 2.14) . (B) For ITPFI2, parameterizing weights via λ_n and a vanishing sequence ε_n, type-III_1 occurs iff inf Λ > 0 and either some tail ∑_{n∈N(λ)} ε_n diverges or else all such sums converge and ⟨Λ⟩ = (0,∞); type-III_λ occurs iff λ ∈ Λ, Λ\{1} is a finite subset of {λ^n}, and all tail sums converge; type-III_0 iff Λ = {0,1} (Theorem 2.20) . The model reaches the same conclusions using Araki–Woods η-sequences and the structure of closed subgroups of R_{>0}, while the paper uses Bernoulli schemes, the Maharam extension, and product-type arguments (Proposition 2.1, Theorems 2.16, 2.19) . Minor imprecision in the model’s aside about the two-point case at 0 (the paper shows 0 ∈ Λ forces type-III_0: Proposition 2.17) does not affect its main conclusions .