UNIFORM ESTIMATES FOR RANDOM MATRIX PRODUCTS AND APPLICATIONS

The paper proves the uniform spectral gap inf_{µ∈K}(λ1(µ)−λ2(µ))>0 on compact K⊂P^δ_{s-log}(SL_d(K)) with strongly irreducible and contracting (S.I.C.) dynamics (Theorem 1.18) by combining continuity of λ1 on the S.I.C. locus in W^1_log (Proposition 1.12) with upper semi-continuity of λ1+λ2 (Theorem 1.19), which itself is obtained via continuity of the pushforward µ↦∧^2_*µ from W^δ_{s-log} to W^1_log (Theorem 2.11). This yields lower semi-continuity of the gap and hence a positive minimum on compact K, exactly as stated in the paper . By contrast, the candidate solution asserts full continuity of both λ1 and λ1+λ2 in W^δ_{s-log} and uses this to conclude continuity of the gap. The paper does not claim continuity of λ1+λ2—only upper semi-continuity—and carefully avoids assumptions needed to guarantee uniqueness of stationary measures on Gr_2 or proximality in the ∧^2-representation. In general, uniqueness on the Grassmannian and continuity of λ1+λ2 can fail without additional spectral gap assumptions (e.g., λ2>λ3). Thus, while the candidate’s conclusion matches the paper’s theorem, their argument overreaches by (i) claiming continuity of λ1+λ2 without ensuring the needed dynamical hypotheses and (ii) implicitly using uniqueness of stationary measures on Gr_2. The paper’s argument is correct and complete for the stated generality; the model’s is not .