Connectedness without geometry

This extension asks whether a finite observation space splits into disconnected pieces. The connectedness discussion was present in the earlier manuscript and is not a standalone claim of the v0.9 core. We retain it as labelled enrichment using the specialization convention already established. Use the original X,Y assignments: A is one-zero, B one-one, C zero-one and D zero-zero. A selected signature gives the truth answers to the chosen formula slots; its support lists the true slots. One point precedes another when all its selected positives are retained. Different original points may have the same signature. One means true and zero means false. A space is disconnected if it is the union of two disjoint nonempty open subsets. Each side is then also closed because its complement is the other open side. A subset both open and closed is called clopen. Use each original point as a vertex. Join two distinct points whenever either precedes the other in the specialization preorder. Make the edge undirected for this connectedness test, even though the underlying information comparison may be one-way. Distinct points with equal signatures precede each other and must also be joined. Omitting these edges would falsely split an indiscrete block into isolated graph vertices. We have not assumed that the original space is T zero. The positive X,Y cover diagram links D to A and C, and those to B. The full comparability graph also joins D directly to B, but adding that edge does not change the connected components. Following undirected edges connects every point. Direct comparability between A and C is unnecessary: a path through another point is enough. The graph has one component, and the topology is connected. A lack of an open separating A in one direction from B does not mean the whole space is disconnected. Connectedness and pairwise distinguishability answer different questions. Keep the four valuations and make both signs of X available. A and B form one equal-signature block, and C and D another. Each pair is joined internally; there is no comparison edge crossing between the two blocks. The two blocks are clopen and are the graph components. Each block is connected internally despite containing two indistinguishable points. Confusing connected components with singleton distinguishability would lose exactly this example. If a point lies in one graph component, every comparable point is in that component. The component is therefore both an upper and a lower set. In the finite upper-set topology this makes it clopen. If two clopen sides had comparable points across them, upward closure of one side or of its complement would force a point across the separation. No comparison can cross. A graph component therefore cannot be divided by a clopen separation. For a new check, use three tests: X, Y, and the negation of the whole conjunction X and Y. Can B connect by a comparison edge to A, C or D under this menu? Use the displayed signatures and predict the connected components before the answer. B has signature one, one, zero. The others have the third coordinate true, so neither inclusion direction connects B to them. D precedes A and C, connecting those three. The components are B alone and A, C and D together. The undirected comparison graph detects connected components of these finite spaces, including equivalent distinct points. It does not measure metric distance, and its components must not be identified with the separate test-compatibility complex introduced next.