HOW LONG IS THE CHAOS GAME?

The candidate solution reproduces the paper’s approach almost step-for-step: (i) reduce the δ-denseness problem to covering all δ-cylinders via an OSC⇒SOSC geometric lemma (Proposition 2.2 uses Schief’s SOSC under OSC), and sandwich E(W_{δ,v}) between covering times for a finite Markov chain on P_δ up to fixed rescaling of δ (Proposition 2.1) ; (ii) build the chain whose transitions prepend a symbol and “reduce” back to P_δ, with stationary weights π(w)=p_w and irreducibility (Proposition 2.3) ; (iii) identify the slow states via t=max_i log p_i/log r_i, using Lemma 3.2 (min stationary mass ≈ δ^t) and Kac’s formula (Proposition 3.5) to get the base scale δ^{-t} ; (iv) apply Matthews-type covering-time bounds (Propositions 3.1, 3.3, 3.7) to harvest logarithmic factors, distinguishing the non-unique case (matching δ^{-t} log(1/δ) bounds) from the unique case (lower δ^{-t} log log(1/δ) vs upper δ^{-t}(log log(1/δ))^2) . These are precisely Theorem 1.1’s statements (equations (2)–(3)) . Any minor wording differences (e.g., the model’s description of the “hard states” as long runs of the maximizing digit versus the paper’s B/B′ partition) are cosmetic; the proof skeleton and conclusions coincide.