Representations of Computation and Structural Cores

Two systems can behave identically forever and still be built completely differently inside — this paper works out exactly how much internal structure "same behavior" actually pins down, and under what extra assumptions you can pin down more.

Start with the classical fact: if two deterministic systems produce the same output after every possible input sequence (not just the one history you happened to observe), you can merge their states by future-indistinguishability and get a canonical "predictive core" — essentially automaton minimization. This is a clean, provable result, but it's also a warning: this core only keeps the state distinctions needed to predict outputs. It can silently throw away components, internal wiring, and the very operations either system actually uses to compute. So "same behavior" does not imply "same mechanism," and treating a minimized model as evidence of shared internal organization is a category error unless you can show more.

The paper's main technical content is about when you can say more. For linear systems, there's a clean classical theorem: any two minimal linear realizations of the same behavior are related by an invertible change of coordinates — nothing hidden, nothing extra, just a basis change. Some nonlinear system classes have analogous (weaker) uniqueness results. The paper's own contribution is a "component rigidity" proposition: if you're willing to test not just ordinary inputs but every single-component overwrite (i.e., you can reach in and clamp any one part of the state, in both systems, under a proposed correspondence between their parts), and if that richer experiment set distinguishes all states, then agreement under this stronger regime forces the two systems' component-wise dynamics to match under that correspondence. That's a real result, but notice the price: you need product-structured state spaces, a proposed matching of components already in hand, and access to every atomic intervention. It doesn't discover the components — it verifies a guessed alignment.

Alongside this, the paper surveys a softer, complementary approach: looking at geometry and topology of reduced neural state spaces (rings, tori, attractors) as partial invariants. These can rule things out or constrain models without ever building a full core, and they connect to real neuroscience findings like toroidal grid-cell structure. But their meaning depends entirely on which transformations you allow, and linking them to actual subjective experience needs an extra, separate argument the math alone doesn't supply.

The upshot: this is a careful map of what's proven, what's assumed, and what's still missing — especially a general, computable method for finding shared structural cores with bounded cost and error. It's directly motivated by (but doesn't resolve) questions like whether a behaviorally identical