WEBVTT

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This extension illustrates extra structure needed for local-to-global questions.

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It was a deferred direction in the earlier manuscript, not an established

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new application of the v0.

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9 theory.

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We will specify contexts, allowed assignments and overlap rules rather than treating

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a topology alone as that data.

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A context lists variables considered together.

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Use the first context X,Y and the second context Y,Z.

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Their overlap is Y.

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Restricting an assignment from X,Y to that overlap means forgetting X

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and keeping the Y value.

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Choose X one and Y zero on the first context.

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Choose Y zero and Z one on the second.

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The shared Y value agrees, so these assignments combine into X one,

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Y zero, Z one.

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No variable receives two conflicting values.

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When local assignments agree on every overlap, their values define a unique

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assignment on the union of contexts.

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This elementary gluing fact holds for unrestricted assignments.

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An obstruction requires additional restrictions on which local assignments are allowed.

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Require equal bits in the X,Y context and equal bits in

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the Y,Z context.

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Each context separately has allowed assignments.

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But together these requirements imply all three bits are equal in any

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global assignment.

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Now require unequal bits in the X,Z context.

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It is separately satisfiable, yet conflicts with the equality forced by the

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first two.

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There is no global assignment meeting all three context constraints simultaneously.

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Each individual variable can be zero or one in the allowed assignments

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of every context containing it.

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The allowed value sets on overlaps agree.

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Yet we cannot choose one allowed assignment in each context so that

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all chosen assignments agree on their overlaps.

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Agreement of available local values is weaker than a global solution.

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This is a deterministic allowed-assignment obstruction.

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It is useful to label that level explicitly.

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The mere existence of one support-compatible global assignment would settle this

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particular existence question, but would not settle every probabilistic consistency question.

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A finite probability distribution assigns nonnegative weights to assignments, totalling one.

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A marginal distribution on fewer variables sums weights over the omitted variables.

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It contains more information than the list of assignments that merely have

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nonzero weight.

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Given local distributions, ask for one global distribution whose marginals equal all

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of them.

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Overlap agreement is necessary but is not sufficient in general.

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Finding one allowed global assignment does not automatically reproduce the requested collection

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of local probabilities.

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Can these three context constraints hold in one global assignment?

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If we remove the third constraint, does a global assignment exist?

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Explain the difference without appealing to a picture of a hole.

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The first two equalities force X equal to Z, contradicting the third.

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Removing the inequality allows all-zero and all-one assignments.

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This direct check identifies a support obstruction; it makes no claim about

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a physical system or a general probability model.

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Contexts, restrictions, allowed assignments and possibly distributions are additional modelling choices.

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The observation topology alone does not provide them or establish a contextuality

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result.

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This extension teaches the distinction and a small consistency calculation, leaving general

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theory beyond scope.