Remark: Regulators are not silently removed

The estimates below take finite observation time and refine a frequency discretization and a particle cutoff at fixed $\mu ,\Lambda$. They do not prove the limits $\mu \downarrow 0$ or $\Lambda \uparrow \infty$. Indeed the explicit constants in (eq:f-norm)–(eq:f-weighted) diverge in those limits. Failure of these estimates to be uniform is not a theorem that every renormalized or alternative infrared construction is impossible.