What is flat ΛCDM, and may we choose it?

The Universe is neither homogeneous nor isotropic, but it is close enough that we can reasonably approximate it as such on suitably large scales. The inflationary-Λ-Cold Dark Matter (ΛCDM) concordance cosmology builds on these assumptions to describe the origin and evolution of fluctuations. With standard assumptions about stress-energy sources, this system is specified by just seven phenomenological parameters, whose precise relations to underlying fundamental theories are complicated and may depend on details of those fields. Nevertheless, it is common practice to set the parameter that characterizes the spatial curvature, Ω_K, exactly to zero. This parameter-fixed ΛCDM is awarded distinguished status as separate model, "flat ΛCDM." Ipso facto this places the onus on proponents of "curved ΛCDM" to present sufficient evidence that Ω_K ≠ 0, and is needed as a parameter. While certain inflationary model Lagrangians, with certain values of their parameters, and certain initial conditions, will lead to a present-day universe well-described as containing zero curvature, this does not justify distinguishing that subset of Lagrangians, parameters and initial conditions into a separate model. Absent any theoretical arguments, we cannot use observations that suggest small Ω_K to enforce Ω_K = 0. Our track record in picking inflationary models and their parameters a priori makes such a choice dubious, and concerns about tensions in cosmological parameters and large-angle cosmic-microwave-background anomalies strengthens arguments against this choice. We argue that Ω_K must not be set to zero, and that ΛCDM remains a phenomenological model with at least 7 parameters.