Boundary is ambiguity, not XOR

Interior asks whether the whole smallest open neighborhood fits inside an event. Closure asks whether that neighborhood meets the event at all. We are using the original four-state carrier and only positive X and Y. The fixed X,Y values are A: one-zero, B: one-one, C: zero-one and D: zero-zero. Our first event is classical X implies Y. One means true and zero means false. The topological boundary is the closure of an event with its interior removed. It can be understood as observational ambiguity: neighborhoods cannot settle membership wholly inside or wholly outside. It is not defined by the edge of a shaded shape on the screen. Retain the implication calculation from Lecture 18. Its interior is B and C and its closure is the whole carrier. Removing the interior from the closure leaves A and D as the boundary. A is outside the implication truth region. Its smallest neighborhood contains A outside and B inside. It cannot settle the event in either direction, so a boundary point need not satisfy the event. D satisfies implication, yet its neighborhood contains both satisfying and falsifying valuations. It also lies on the boundary. Truth at the centre and ambiguity of its available neighborhood are different properties. For the next event, XOR means exactly one of X and Y is true. The carrier and the positive observation menu stay fixed. The XOR truth region contains A and C. Under positive X and Y, A's and C's smallest neighborhoods both include B, which is outside this event. No nonempty open set fits inside XOR. The interior is empty. A, C and D have neighborhoods meeting XOR, while B's singleton neighborhood does not. The closure, and hence the boundary in this case, is A, C and D. Make both signs of X and Y available on the same carrier. Each complete minterm now certifies a singleton, so the topology is discrete. The truth table of XOR has not changed; A and C remain its true points. In a discrete topology every subset is both open and closed. XOR's interior and closure both equal its own truth region, so its boundary is empty. The Boolean event stayed fixed while its observational boundary changed. A point is on the boundary exactly when its smallest neighborhood meets both the event and its complement. The two witnesses lie in that same neighborhood: one inside the event and one outside. Neither witness has to be the centre. That neighborhood cannot certify either alternative completely. Symmetric difference compares two truth sets and removes their overlap. Boundary compares an event with the topology of available neighborhoods. Even a nonempty XOR region has empty boundary when the topology is discrete, so the two operations cannot be identified. When we replace positive X and Y by both signs, which changes: the XOR truth region, its interior, its closure, or its boundary? State the old and new boundary. The XOR truth region stays A and C. Its interior becomes A and C, its closure becomes A and C, and its boundary becomes empty. The earlier boundary was A, C and D. The interface changed, not classical evaluation. A boundary statement must identify its event, carrier and topology. Without those labels, a viewer may mistake Boolean disagreement for observational uncertainty or infer a physical edge that has no mathematical role in this finite example.