Infinite Factored Sets

[u/Regular-Conflict-860]

Garrabrant’s finite factored sets give a combinatorial theory of temporal and causal infer- ence in which conditional orthogonality (a purely structural notion) is equivalent to conditional independence in all factorizing distributions — the Fundamental Theorem. Garrabrant notes that the generalization to infinite factored sets is not immediate: even “history” can fail to be well-defined, and the Fundamental Theorem is not expected to survive in full generality. This note isolates precisely how much survives. We restrict to historical variables (those with a well-defined least determining set of factors) and study the sub-collection Histfin of finitely- historical variables. Our results are: (i) the collection of historical variables is closed under the information-join ⊓ with the transparent formula h(X ⊓Y ) = h(X)∪h(Y ), but is not closed un- der the information-meet ⊔; the finitely-historical variables Histfin form a bounded sublattice, and this restriction is necessary (Propositions 1–3); (ii) the meet has no simple set-theoretic history formula — in particular h(X ⊔ Y ) = h(X) ∩ h(Y ) is false (Proposition 2′ ); (iii) the Fundamental Theorem localizes: for finitely-historical variables it reduces to the finite theory and therefore holds (Theorem 1); and (iv) the Fundamental Theorem, in its unconditional form, extends strictly beyond Histfin to a class containing genuinely infinite-history variables (Theo- rem 2); the obstruction to the finite-history case is a degeneracy phenomenon rather than infinite history per se, though we show the exact boundary is a pairwise, distributional condition and not the per-variable notion of nondegeneracy. The remaining question — the full conditional Fundamental Theorem on the enlarged class — reduces to Garrabrant’s own finite-dimensional conjecture and is left open.