Better observations: refining the topology

X means dot and Y means stripe on our original cards: A has only a dot, B both, C only a stripe, and D neither. A selected signature lists answers to the chosen formula slots. Keep its headers distinct from the fixed X,Y valuation annotations. Begin with the one-test menu X and all four valuations. Refining the observations means adding available tests, not declaring their outcomes true. The carrier remains fixed, and the old coordinate keeps its identity in the enlarged indexed list. Add Y as a second available test. The old three-open topology is contained in the new six-open topology. Every old certificate construction remains available, and new combinations may become expressible. Finer means more opens, not smaller state cards. The old generators are among the new generators, so any old union of finite intersections is still open. This proves topology refinement. An added formula need not create a strictly finer topology, because its region might already have been open. If two longer signatures agree, deleting the new coordinates leaves equal old signatures. Classes may split when tests are added, but cannot merge. Similarly a comparison preserving every new positive coordinate must preserve every old one. The support is the set of selected slots answered true. Our information arrow goes from a support to one containing all of it. The refined information preorder is a subrelation of the old preorder. Some one-way comparisons may disappear. The arrow direction is the same as before: from a smaller positive support to a support containing it. Symmetric distance adds weights at all differing slots. Directed loss adds them only for true-to-false changes from the source. The labelled coordinate weights are part of the chosen presentation. Distances can only increase under indexed extension when the old coordinates keep their old weights. New coordinates add nonnegative contributions. If we also lower old weights, that argument no longer proves a nondecreasing distance. These are weak inequalities: a particular pair may agree on every added coordinate and keep the same distance. A general refinement theorem does not promise that every pair becomes more distinguishable or numerically farther apart. For a new comparison, start from X, Y with unit weight on each coordinate. Add their conjunction as a third indexed observation, also with unit weight. Its truth region is B alone, already an open. The topology and its qualitative order therefore stay unchanged. Before the addition, B to A loses only Y, costing one. After the addition, B also loses the newly indexed conjunction. That is an additional coordinate contribution even though its region added no new open. With unit weight on the new coordinate, the directed loss becomes two. Topological redundancy is not numerical redundancy. The same distinction applies to literal repeated coordinates with their own weights. The topology remembers which events are open, but generally does not reconstruct the chosen list or its weights. Two presentations can have the same opens and different distances. This is a feature to label, not a contradiction to repair by changing the topology. Throughout these refinements, no state has been removed and no observed result asserted. Learning that Y is true would be a different operation, restricting possible valuations. We will reset the two operations to the same starting point next. Does adding the conjunction strictly refine the topology? Can it nevertheless change directed loss? Explain with B as source and A as destination under unit weights. The conjunction region was already open, so there is no strict topology refinement. Its indexed coordinate supplies an extra loss from B to A, increasing that distance from one to two. Qualitative sameness does not force numerical sameness.