Paper 8: NTLA-O VII: Complete Separation, Canonical Invariants, Locally Finite Reconstruction, and the Continuous Separation Problem

Complete Separation, Canonical Invariants, Locally Finite Reconstruction, and the Continuous Separation Problem

NTLA-O series Paper 8. Directly poses the Complete Separation Problem: if NTLA claims two structures are different, can some observation actually be constructed that necessarily separates them? The first part restricts to finite relational structures, defining a canonical form as the lexicographically smallest encoding over all admissible relabelings, and proves two structures' canonical forms are equal if and only if they are isomorphic — i.e., a complete separator genuinely exists for finite structure domains (the document explicitly notes: the existence of a complete invariant does not imply an efficient canonization algorithm — these are different problems). The second part handles connected, rooted, locally finite, countable structures, extending the result with finite observation balls plus a König-type compactness argument. The document explicitly does not presume the Continuous Separation Problem (the version for continuous topological/geometric object classes) has a simple solution in general, and leaves it as an open problem.