MATH.19:10 - Architectural Rationale

Backward obligations expose what would suffice; forward constructions expose what can be obtained. A useful intermediate claim joins the two. This organization supports informal reasoning and formal derivations while preserving the mathematical question as the governing object.

Induction, countermodels, interpretation and witness extraction contribute different operations. The stalled inference determines which contribution the argument needs: the list example requires a varying parameter, while the composition example requires a separately proved commutation relation.

The proof plan and its completion have different uses. A plan can locate a lemma for a collaborator; a completed argument supports the mathematical conclusion. Making that distinction visible permits useful incomplete work without treating the remaining obligation as already solved.