MATH.8:4.5 - Use an orbit representative without losing the requested result

A quantity constant on each orbit can be calculated from any representative. To define additional operations on orbit classes, use MATH.2’s representative-independence condition; an orbit partition by itself only supplies classes.

When a problem is solved using representative data d0 and solution x0, retain a transformation g with g*d0=d for the requested data d. Return g*x0. If two transformations satisfy g1*d0=g2*d0=d, an answer independent of that choice requires g1*x0=g2*x0. A disagreement identifies information that the representative alone does not supply.

MATH.13 supplies the fixed-point restriction on a unique solution and the test for an impossible equivariant choice. Those questions require the output action and, for the unique-solution deduction, a justified uniqueness premise. Generating a solution set here does not select one member from it.

Stop with the requested related solution, complete orbit, orbit classification or identified failure of preservation. When data, constraints or the output change, reopen the affected action and preservation argument before reusing the family. A more costly enumeration or symmetry computation is useful only if it can improve the receiving result.