What the paper establishes, and what remains open

The paper assembles established finite Boolean, topological, order and distance constructions in one propositional model. Its contribution is organizational and explanatory. It does not claim a new class of topological spaces merely because formulas provide a convenient finite presentation. A restricted family of positive tests determines which truth events are certifiable, which valuations are indistinguishable and which one-way comparisons preserve positive information. Selecting all definable subsets would instead give a discrete topology and remove that restriction. A signature is the selected truth-answer vector; its support lists the true positions. A signature fiber is a class of worlds with the same selected signature. Our information order preserves those positives. Opens describe what positive tests can certify. Numerical distances also need the indexed test list and its weights. Adding tests changes the interface, while accepted premises restrict the surviving worlds. Signatures recover topological indistinguishability and the specialization preorder. In this finite setting the preorder determines the opens as upper sets. These are different descriptions of the same qualitative structure under the chosen convention. Numerical distances require the indexed formula family and weights. Adding a redundant coordinate or changing a weight can change numbers without changing the topology. A topology by itself does not determine a unique formula list or distance table. Positive observation generates opens. Adding both signs opens original signature fibers. Adding tests refines the interface on a fixed carrier. Learning premises restricts possible valuations and uses the induced subspace topology. Those operations must not be merged into one picture of information gain. Classical consequence remains inclusion of truth regions. An inclusion map keeps a subspace point as the same ambient point. Its continuity says that ambient-open events restrict to open events in that subspace. The topology and its continuous inclusions organize this reasoning but do not create a new validity rule. In the optional open-set lessons, negation uses the interior of a complement, and implication uses the interior of its classical candidate. Each rule is applied at its own connective in the expression. Heyting open-set operations were explicitly a different recursive interpretation, not a retroactive change to classical evaluation. A relation nerve records which indexed tests can be simultaneously satisfied. In this diagram, vertices stand for indexed tests. An edge means two tests share one satisfying world; a filled triangle requires one world satisfying all three. Any such compatible group is called a simplex. Homology is being used here to discuss holes, such as an unfilled loop. The diagram’s vertices are test slots, rather than worlds. It can distinguish pairwise from joint compatibility. But it is not determined by the observation topology, and its holes need not describe the homology of the valuation space. A tautology is a formula true at every valuation. Adding it supplies a constant true coordinate, leaving generated opens and both distances unchanged. In the test-compatibility complex it adds a new vertex joined to every old simplex, forming a cone. Any homological claim must therefore identify the presentation and state the intended invariance separately. The extension lessons discuss connectedness, a compatibility complex, local consistency and observation design. Some are historical enrichment removed or narrowed in v0.9. They are labelled accordingly; taking them does not turn suggested directions into established new results of the manuscript. The notes distinguish complete Boolean truth sets, two-sided indiscernibility partitions and positive observation opens. For this optional comparison, a conjunctive extent is the set of worlds satisfying a chosen group of tests together. The formal concept lattice organizes these extents by inclusion. The generated opens also allow unions of the conjunction regions. A union of conjunctive extents need not itself be an extent, so the open-set lattice is not generally the formal concept lattice. These are comparisons, not new duality theorems. Does the observation topology alone determine numerical weights, the homology of an indexed compatibility complex, or a new classical inference rule? Explain one concrete reason for each answer before checking the notes. Weights are extra quantitative choices. A tautology can change the nerve while preserving the topology. Classical validity still asks whether every premise survivor satisfies the conclusion. None of those stronger claims follows merely from naming the same finite topology. The useful outcome is knowing which distinctions a chosen interface can support and which conclusions require additional data. New algorithmic results, new invariants or validated applications would require further work. This course has taught the finite constructions and their boundaries, not claimed those further achievements.