Why do we need to model lower-order effects in models with interactions? I recently saw a paper with a four-way interaction. That already is difficult to interpret (maybe if you have 1 or more categorical variables but definitely near impossible to interpret if all continuous) and you’re almost certainly underpowered for it.
But beyond the four-way, the researcher modeled only one main effect, two 2-way interactions, and two 3-way interactions. So the authors dropped a bunch of lower-order effects in their (already difficult to interpret) four-way interaction.
I know this is a problem. In terms of consequences, it will inflate the likelihood of your interaction effect appearing as a false positive when you omit the lower-order effects.
But I still struggle a bit as to why mathematically this is a problem? Why is it a bad idea or mathematically a problem to not simultaneously model the lower-order effects of an interaction? Can someone break down or explain to me why it is essential to model your lower-order effects in an interaction? Why is it necessary? Any other details or input would be appreciated.
Thank you!

You could consider a model that doesn't contain the linear effects but does contain the interaction (quadratic) effects. There is mathematically no problem with it.
But usually there is some linear dependency and then the model would have to crank the linear dependencies into the quadratic effect. This is similar to fitting a linear model without an offset. E.g., consider linear data like $y = 2\cdot x + 100$. Fitting this without an offset in you model would result in very bad results.
