The claim
Two local descriptions can assign different values to what look like the same observables. That alone shows no incompatibility. There are three cases, and they should not be conflated:
- Coordinate difference: compatible transition maps reconcile the descriptions. Inertial frames that disagree about temporal order are the example (CASE-039).
- Structural transformation: the descriptions stay comparable but are no longer equivalent.
- Global obstruction: they cannot be jointly extended under the stated compatibility conditions, as in sheaf contextuality.
Which case holds can be decided only relative to a specified class of admissible correspondences between the descriptions’ contexts and outcomes, each coherent with restriction. The order of tests is the claim: specify the class, then ask whether what remains is a transformation or an obstruction. A single correspondence is one such class; a verdict can also hold over a whole class without any one member being fixed (note of 2026-10-19).
The proviso. A correspondence admitted only because it makes the disagreement disappear decides nothing. The class must be justified or validated independently of the disagreement it is used to assess.
Where the record applies it
CLAIM-143 applies the order of tests to source and target judgements in translation, where the correspondence is the context part of a transport. It rests on CLAIM-011, CLAIM-004 and CLAIM-035 for that application, and now on this claim for the order of tests.
What it does not say
- It does not say every domain has transition maps. Relativity has them; that another domain does is a separate question.
- It does not say any map that removes a disagreement shows a mere change of description. The map has to be anchored first.
Note of 2026-10-19: why the verdict is relative to a class
Promotion audit, batch 040. Write V(D1, D2, C) for the verdict on two local descriptions D1 and D2 under a class C of correspondences between their contexts and outcomes, each coherent with restriction. D1 and D2 are compatible relative to C if some member of C carries D1 onto a description that agrees with D2 on every shared context. They obstruct relative to C if no member does.
- Disagreement does not entail incompatibility. The inertial frames of CASE-039 assign different simultaneity relations, and the Lorentz maps carry one onto the other. One case settles a negated entailment.
- The verdict varies with the class. Take one shared context with two outcomes. D1 assigns (1/2, 1/2) and D2 assigns (1/3, 2/3). Let C be the bijections of outcomes. Both bijections carry D1 to (1/2, 1/2), which is not (1/3, 2/3), so D1 and D2 obstruct relative to C. Let C’ be the stochastic maps of outcomes (Markov kernels). The kernel that sends each outcome to (1/3, 2/3) carries D1 to (1/3, 2/3), which is D2, so D1 and D2 are compatible relative to C’. The same pair gets opposite verdicts, so there is no verdict until a class is specified.
- No single correspondence need be fixed. The obstruction in step 2 holds for every member of C at once, and no one bijection was chosen. So “only once the correspondence is fixed”, in the singular, is stronger than what holds. A fixed correspondence is the special case of a class with one member.
- The proviso. Relative to the stochastic maps of outcomes, every pair of descriptions of a single shared context is compatible, by the constant kernel onto the second description. That a reconciling correspondence exists therefore says nothing about a given pair unless the class was specified first, independently of the disagreement. Inside a class so specified, finding a reconciling member is the compatibility verdict for that class. So the proviso holds for correspondences admitted only because they remove the disagreement, and no further.
Steps 1 to 4 establish the title as restated. They leave the three-way classification and its application to translation (CLAIM-143) as they were.