The paper establishes constructive equivalence between Brouwer's fixed-point theorem and weak König's lemma, but it does so within RCA0 as a base theory. A genuinely open question is whether this equivalence holds at lower base theories, particularly those that do not assume full induction or comprehension for recursive sets. The difficulty lies in the fact that the combinatorial encoding of continuous functions on higher-dimensional domains requires more complex approximation arguments, and it is unclear whether those arguments can be carried out without the resources of RCA0. Dimension by dimension, the proof complexities diverge, and no systematic classification exists for the constructive strength of Brouwer's theorem across all dimensions under weak base assumptions. The problem sits at the intersection of reverse mathematics, constructive analysis, and proof theory, where even small changes in the base theory can dramatically alter which theorems are provably equivalent.