One observation table, the whole framework

Our original X,Y values are A: one-zero, B: one-one, C: zero-one and D: zero-zero. A selected signature lists answers under the chosen formula headers. Its support lists the true slots. A point’s smallest neighborhood contains exactly the worlds preserving those selected positives. Keep the current tuple, weights and possible worlds as separate inputs. One means true and zero means false. Begin with the full four-state carrier and the menu X alone. Keep presentation and weights as separate labels. We will reconstruct several structures and then change one ingredient at a time so that their dependencies remain visible. The selected X column gives two realized signatures and two equal-signature classes. The smallest neighborhood of A or B is A and B. The smallest neighborhood of C or D is the whole carrier, because no positive requirement is imposed there. Our order preserves selected positives. Symmetric distance counts all disagreements with their weights. Directed loss counts only true-to-false changes, keeping the source first. The false-signature class precedes the true-signature class. With unit weight the symmetric distance across classes is one, while directed loss is zero from false to true and one in the reverse direction. The direction identifies loss, not a temporal learning process. Positive X generates three opens on the four valuations. Symmetric signature balls give the four-open partition topology. The two constructions use the same signatures but answer different questions about one-sided versus two-sided distinctions. Add Y as an available test. All four signatures now differ. The six positive opens and the D–A/C–B information diagram follow from those signatures. The carrier remains fixed because no premise has been learned. Interior asks whether a whole smallest neighborhood fits in the event. Closure asks whether it meets the event. Boundary is closure with interior removed. Here the event is classical X implies Y, false only at true X and false Y. Implication's interior is B and C, while its closure is the whole carrier and its boundary A and D. D satisfies the formula without a sufficient positive certificate under this interface. Truth evaluation and certifiability remain distinct. Now accept two premises: X implies Y, and X. Their common truth region is B. Restrict the ambient topology to that surviving carrier rather than declaring a new unrelated topology. Every surviving world satisfies Y, so the conclusion follows by inclusion. The menu determines signatures and the qualitative topology and order. Weights additionally determine numerical distances. Premises determine a possibility subregion. Keeping these arrows separate prevents treating a topology as a complete record of every modelling choice. For this two-slot exercise, use unit weight on each selected coordinate. Read the new formula headers when building the signatures; keep the original X,Y valuation annotations fixed. Return to the full carrier and select X as observation one and X or Y as observation two. Find the signatures, equal-signature classes, smallest neighborhoods and the directed losses between C and A. Pause and work from the table. A and B have both selected formulas true. C has only the second true, and D has neither. The classes are A and B together, C alone and D alone. Their quotient supports form a three-level inclusion chain. Above the class A and B there are no additional signatures. Above C lies that class too; above D lies everything. These upward neighborhoods reconstruct the topology with four opens: empty, A and B, A and B and C, and the carrier. C loses no selected positive when moving to A. In the reverse direction A loses the first coordinate but keeps the second, costing one. Distinct complete valuations A and B still have identical signatures under this new tuple. Making both signs available for this tuple opens its three original classes, yielding eight opens on the four valuations. It does not split A from B. This is another reason both signs need not mean discreteness on the original carrier. A successful reconstruction names each input, computes the resulting objects and explains their dependencies. If a remembered answer disagrees with the displayed formula table, return to the table. The framework is a chain of constructions, not a single diagram to recite.