Resolution-Indexed Strata

The formalism for the stratified view.

What the view asks for

A mechanism’s stratum is settled by measurement, not by assumption. Three quantities fix it: effective dimensionality, read off the participation ratio; localizability in specific components; and whether either holds up when the measurement resolution changes. Two mechanisms are the same when they sit in the same stratum and are locally equivalent within it.

That is the whole operational content. A mechanism that looks one-dimensional at a coarse probe and five-dimensional at a finer one has not changed — the resolution has, and the view’s claim is that a mechanism description is incomplete until the resolution is stated.

Whitney stratification, the antecedent

The mathematical idea the name comes from. A stratified space is decomposed into smooth pieces — strata — of different dimensions. The pieces are not disjoint: a sequence of 2-dimensional subspaces can converge to a 1-dimensional one as a direction collapses, so the lower-dimensional strata sit on the boundary of the higher-dimensional ones. Whitney’s two regularity conditions, A and B, constrain how the strata are allowed to meet, so that tangent planes behave continuously in the limit and the decomposition survives perturbation.

The strata used here are constructed rather than assumed, and no claim in the stratified view depends on the regularity conditions holding. Whitney supplies the picture — pieces of different dimension meeting in a controlled way — and the measurements above supply the content. Stratification carries formal weight once the stratified object is a finite constructed complex; extending that treatment to activation space directly remains open.

See Whitney conditions and Stratification on Wikipedia.