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 () ^ interacts with the standard coproduct and product operations on Weihrauch problems. First, we prove a universal one-sided inequality f^ g^ₒₖ (f g) ^. We then isolate a natural tag-stability hypothesis and show that, under tag-stability, (f g) ^ decomposes (up to ₒₖ) 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 g w f g and lift it through () ^ using Brattka's monotonicity theorem. Finally, we formalize a projection-based lower bound f^ g^ₒₖ (f g) ^ under explicit stability conditions, and we give a self-contained failure example showing that such conditions are genuinely additional.