MATH.9:4.5 - Repair an obstructed choice

Choose the change that answers the receiving problem.

A supplied distinguished attribute can reduce the stabilizer. Include that attribute and its transformation in the new input, then repeat the fixed-output test. The marked-cycle case below constructs such a rule.

A different output may retain the alternatives the input leaves open. Returning the whole set A(d) respects its induced transformation from :4.1. When A(d) is finite and nonempty, a uniform probability distribution on it does too, if a distribution is the requested result. Sampling a single value is a further calculation with its own randomness and comparison requirement.

An average is usable only when it belongs to the permitted output set and has the required transformation law. The mean of two opposite unit directions is zero, which fails a unit-direction requirement.

The receiving task may instead allow coordinate-dependent choice or a narrower input range. State that changed requirement and construct the corresponding rule. Adding distinctions or weakening a requirement is useful only when the resulting answer serves that task; C.11.DUA helps decide whether obtaining further input is worth its cost when that question is unresolved.