Basic Algebraic Properties of Brattka's Inverse Limit Operator: Coproducts, Products, and Two Structural Obstructions

We study how Brattka's inverse limit (infinite loop) operator $(\cdot)^{\infty}$ interacts with the standard coproduct and product operations on Weihrauch problems. First, we prove a universal one-sided inequality $f^{\infty}\sqcup g^{\infty}\le_{sW}(f\sqcup g)^{\infty}$. We then isolate a natural tag-stability hypothesis and show that, under tag-stability, $(f\sqcup g)^{\infty}$ decomposes (up to $\equiv_{sW}$) as a coproduct of inverse limits of tagged components, thereby avoiding the usual detagging pitfall caused by the universal evaluator. Second, we record a pointed upper bound $f\sqcup g\le w f\times g$ and lift it through $(\cdot)^{\infty}$ using Brattka's monotonicity theorem. Finally, we formalize a projection-based lower bound $f^{\infty}\times g^{\infty}\le_{sW}(f\times g)^{\infty}$ under explicit stability conditions, and we give a self-contained failure example showing that such conditions are genuinely additional.