MATH.22:4.6 - Use the revised theory

Return the changed assumptions together with the construction, theorem or obstruction needed by the receiving work. Carry the conditions under which it can be used. An unresolved consistency or consequence question can be handed to a collaborator without presenting it as settled.

Choose between revised theories by the work they enable and the costs they impose. One can admit more objects, another can make a needed construction available, and another can retain an effective procedure. FPF’s ordinary characterization and choice methods apply when these alternatives must be compared; the present method supplies their mathematical consequences.

Use the changed result to reformulate a model, modify a method, or pose the next mathematical problem. For a new conjecture, state which further relation might hold under the retained assumptions. A later failure reopens the assumption or inference on which the failed use depends.