I'm learning about Hardy-Weinberg equilibrium (HWE) and how we can use statistical tests (e.g. chi-squared) to compare observed allele frequencies with those expected under the null hypothesis of HWE. It's my understanding that, since the assumptions used to derive HWE essentially translate to "no evolutionary forces are acting" (among other things), we are testing for a significant presence of evolutionary forces acting. This is considered a proxy for "is the population evolving?", where "evolving" means "changing allele frequencies over time".

However, I'm wondering if deviation from HWE necessarily means evolutionary change, and conversely, whether being at HWE necessarily means no evolutionary change, and if neither, then is this test strictly valid (even ignoring the possibilities of Type I/II errors as in any statistical test)?

First example - imagine a population where the allele frequencies are instantaneously at HWE (at one moment in time), but the actual allele frequencies have some non-zero rate…