MATH.8:4.2 - Establish the solution-preserving relation

Show that S(d,x) implies S(g*d,g*x) for every transformation and candidate in the claimed range. Applying the same implication to the inverse gives the converse. The transformation therefore pairs the two solution sets.

For an equation, substitute the transformed variables and data. For several constraints, preserve each one used by the solution. An optimization result needs both transformed feasibility and the corresponding criterion: if feasible candidates are paired bijectively and J_(g*d)(g*x)=J_d(x), a better transformed candidate would return a better original candidate. Thus a minimizer transfers.

Preservation for generating transformations and their inverses extends to every finite composition by applying the implications successively. Use MATH.4’s finite-construction argument when that extension needs explanation. A check on a few candidate values establishes only those cases unless a general argument is also available.

If a transformation fails, retain the first changed condition and decide whether the changed problem is useful. MATH.6 can exhibit the failure; FPF C.29 relates the mathematical correspondence to another subject when one is involved.