Dynamical pair assignments

The paper establishes F(PX) is Borel and claims R(PX) is Borel iff the P-rank is bounded, via a coanalytic-rank argument. However, its Γ-operator is defined as Γ(E)=E+ (or E+ ∪ ΔX earlier), which need not send closed sets to closed sets; yet Γ is treated as a map K(X×X)→K(X×X) and as a Borel expansion, creating a definitional gap that affects Corollary 5.10 and Theorem 5.11 (see the Γ(E)=E+ line and Proposition 5.9’s proof sketch) . Moreover, the proof of Theorem 3.3 cites the boundedness theorem to conclude boundedness “on X,” but the argument given shows boundedness only on R(PX), not on all of C(X,X) . The candidate solution fixes the closure/invariance issue by building a closed, T×T-invariant equivalence via a transfinite derivative and gets the right descriptive conclusions, but it leaves key measurability steps (e.g., Borelness of composition at the hyperspace level) only sketched and asserts, without a solid argument, that boundedness of the rank on R(PX) implies boundedness on all of C(X,X). Therefore, both the paper and the model contain correct core ideas but have nontrivial gaps that need repair.