MATH.6:10 - Architectural Rationale

The method joins logical failure conditions with mathematical construction. Negating the conclusion tells the worker what to build; satisfying the assumptions makes the constructed case relevant; carrying the result back into the argument makes the criticism usable.

Separating these operations exposes two different errors: an inadmissible example and a failure condition that does not defeat the actual quantified claim. The finite and infinite inverse cases show how the same question can move between refutation and proof as the domain changes.

B.5.RA recovers an available argument, and B.5.RR revises reasoning when its premises or question change. This pattern supplies the mathematical countermodel they can consume. MATH.2’s failed-identification witness is a direct use, while the relation and inverse cases give this construction independent mathematical entries.

A known case or a short direct construction is preferable when it already settles the implication. Constraint solving helps when the assumptions interact enough to make that construction difficult. Neither route requires finding a smallest example unless its size matters to the receiving question.