Logically Contractive Mappings: Fixed Points and Event-Indexed Rates

The paper’s Theorem 2.2 proves the same claim under the same hypotheses, using the same core idea: pick the first event n1 so T^{n1} is a strict contraction with a unique fixed point z, use the commuting identity T^{n1}(Tz) = T(T^{n1}z) to deduce Tz = z, obtain the eventwise estimate d(T^{nk}x, z) ≤ λ^k d(x, z), and extend convergence to all n via nonexpansiveness. The model solution mirrors this argument step-by-step, differing only in that it spells out the Picard iteration proof for the contraction step; the rest (use of nonexpansiveness and the commuting powers of T) aligns with the paper’s proof and assumptions, including that completeness is used only for the contraction argument (Remark 2.3). Hence both are correct and essentially the same proof.