Permutation Symmetry

Swap two neurons in a hidden layer, and swap the corresponding weights that read from them, and the network computes exactly the same function. So every trained model is one of an enormous number of identical twins, differing only in the order things are written down — and two models that look completely different in weight space may be the same model in disguise.

This is why averaging two independently trained networks gives garbage. They are not in conflict about what to compute; they have written equivalent answers in different orders, and averaging across orderings destroys both. Align one to the other first — find the permutation that best matches their units — and the average becomes reasonable. That alignment step is what separates "merging does not work across runs" from "merging needs a correspondence".

A layer of width n has n! equivalent orderings, so the number of weight-space points computing one function grows factorially with width and multiplicatively across layers. That is why the loss landscape looks like it has an astronomical number of distinct minima and why so many of them turn out to be the same one: quotient out the symmetry and much of the apparent multiplicity disappears, which is the observation Mode Connectivity depends on.