The specialization preorder: preserving positive information

An open neighborhood is an open region containing the point. A selected signature gives the truth answers to our chosen formula slots; its support lists the indices answered true. Our original X,Y cards are A with a dot only, B with both features, C with a stripe only, and D with neither. Their first bit answers X and the second Y. The paper chooses this direction: v precedes w when every open neighborhood of v contains w. Read that sentence before reading the arrow. Other sources may reverse the specialization convention, so memorizing an arrow without its meaning is unsafe. We will prove that w preserves every selected positive formula true at v. It may satisfy additional selected formulas. This compares complete semantic profiles; it does not say that a sequence of certificates was delivered or that a causal transition occurred. Suppose every open containing v contains w. Each selected truth region is an open. Whenever its formula is true at v, that region contains v and therefore w. Every positive coordinate of v is consequently positive at w. Suppose instead that v's positive support is included in w's. Every basic conjunction true at v remains true at w. Any union containing v has one such basic witness, which also contains w. Thus every open containing v contains w. Under the menu X, Y, D has empty support, A has the first index, C the second, and B both. Put D at the bottom, A and C in the middle and B at the top. Upward means preserving positive information. A precedes B because its first positive coordinate survives. B does not precede A because the second coordinate would be lost. A and C are incomparable, since each lacks one positive coordinate supplied by the other. In a finite space, intersecting all open neighborhoods of a point still gives an open neighborhood. There are only finitely many distinct opens. This smallest neighborhood consists exactly of points above the chosen point in our information comparison. A requires X, so its smallest neighborhood contains A and B. B requires both tests, so its smallest neighborhood is B alone. A point can have a larger smallest neighborhood even though the carrier is unchanged. C requires Y, giving B and C. D requires no positive test. The empty conjunction imposes no restriction, so D's smallest neighborhood is the whole carrier. D is still a point, not the empty set. Support inclusion is reflexive and transitive, so this relation on valuations is a preorder. It can compare two distinct points in both directions when their signatures coincide. Antisymmetry would require that two-way comparison happens only for the same point. That requirement fails for distinct valuations with equal signatures. Identify points only when both directions hold. The classes then carry a partial order, which is also antisymmetric. A single comparison in one direction is not a reason to merge its endpoints. An open is upward closed in this order. Conversely an upward-closed subset is a union of the smallest neighborhoods of its points, hence open. This supplies the finite upper-set representation of the topology. Under only X, do A and B precede each other? Does D precede A? Does A precede D? Explain which pair is merged in the quotient. A and B have equal supports, so both directions hold and they merge. D's empty support is included in A's, so D precedes A. The reverse fails because X is true at A and false at D. The order, smallest neighborhoods and opens encode the same finite qualitative structure. Directed distance will recover this particular direction through zero loss, while its positive numerical values will require additional presentation data.