Variations of Topological Theory and Ergodic Theory via Gap Function in Non-uniform Specification

The candidate’s three-step argument proves exactly Theorem A(1)–(3) of the paper. Step 1 matches the paper’s critical case (Theorem 3.2), giving h_B^top(f, I_φ) ≥ (1/(1+τ)) sup_{(μ,ν)∈D_f(φ,τ)} min{h_μ,h_ν} , and the paper’s detailed proof uses a gluing/separation lemma to pass product counts to Bowen entropy . For the general case τ ≤ σ < 1, the paper proceeds by mixing measures with a high-entropy measure (λ) and then applying the critical case to (μ̃,ν̃) ∈ D_f(φ,τ) to obtain the convex-combination bound in Theorem A(1) , whereas the model inserts “free” top-entropy pieces; these are methodologically different but yield the same estimate. The passage from functions to IR(f) uses a standard separation lemma for observables (Lemma 3.3/Cor. 3.4) and the limit σ→1 to get Theorem A(2) , and the paper then supplies the needed entropy supply (Proposition 3.5) to conclude Theorem A(3) , matching the model’s Step 3. In particular, the “τ=0” corollary is identical in both . Overall, the paper is correct and complete on these claims; the candidate’s proof is also correct and essentially equivalent in spirit but not identical in technique.