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Reasons from function-space topology, convergence modes, and constant-dependent inequalities (Hölder, Sobolev, Gronwall) through compactness theorems (Rellich-Kondrachov, Banach-Alaoglu), Calderon-Zygmund and Schauder estimates, and Lax-Milgram while treating limit-integral swaps without dominated convergence, boundary-degenerating constants, weak-versus-classical regularity gaps, and concentration-compactness loss as first-class failure modes.