Local contexts and global consistency

This extension illustrates extra structure needed for local-to-global questions. It was a deferred direction in the earlier manuscript, not an established new application of the v0.9 theory. We will specify contexts, allowed assignments and overlap rules rather than treating a topology alone as that data. A context lists variables considered together. Use the first context X,Y and the second context Y,Z. Their overlap is Y. Restricting an assignment from X,Y to that overlap means forgetting X and keeping the Y value. Choose X one and Y zero on the first context. Choose Y zero and Z one on the second. The shared Y value agrees, so these assignments combine into X one, Y zero, Z one. No variable receives two conflicting values. When local assignments agree on every overlap, their values define a unique assignment on the union of contexts. This elementary gluing fact holds for unrestricted assignments. An obstruction requires additional restrictions on which local assignments are allowed. Require equal bits in the X,Y context and equal bits in the Y,Z context. Each context separately has allowed assignments. But together these requirements imply all three bits are equal in any global assignment. Now require unequal bits in the X,Z context. It is separately satisfiable, yet conflicts with the equality forced by the first two. There is no global assignment meeting all three context constraints simultaneously. Each individual variable can be zero or one in the allowed assignments of every context containing it. The allowed value sets on overlaps agree. Yet we cannot choose one allowed assignment in each context so that all chosen assignments agree on their overlaps. Agreement of available local values is weaker than a global solution. This is a deterministic allowed-assignment obstruction. It is useful to label that level explicitly. The mere existence of one support-compatible global assignment would settle this particular existence question, but would not settle every probabilistic consistency question. A finite probability distribution assigns nonnegative weights to assignments, totalling one. A marginal distribution on fewer variables sums weights over the omitted variables. It contains more information than the list of assignments that merely have nonzero weight. Given local distributions, ask for one global distribution whose marginals equal all of them. Overlap agreement is necessary but is not sufficient in general. Finding one allowed global assignment does not automatically reproduce the requested collection of local probabilities. Can these three context constraints hold in one global assignment? If we remove the third constraint, does a global assignment exist? Explain the difference without appealing to a picture of a hole. The first two equalities force X equal to Z, contradicting the third. Removing the inequality allows all-zero and all-one assignments. This direct check identifies a support obstruction; it makes no claim about a physical system or a general probability model. Contexts, restrictions, allowed assignments and possibly distributions are additional modelling choices. The observation topology alone does not provide them or establish a contextuality result. This extension teaches the distinction and a small consistency calculation, leaving general theory beyond scope.