MATH.4:10 - Architectural Rationale

The input’s formation rules determine the cases. Pairing each output rule with its property argument links object construction, reasoning and computation without requiring a proof-assistant language.

The stronger specification is part of the construction method. It lets a smaller-input result provide what a larger input needs, as the root-color parameter shows. Returning the witness separately from optional proof packaging keeps the result usable in different mathematical and computational descriptions.

Structural recursion was selected because finite constituent structure supplies a local termination argument. Finite enumeration is a serious alternative for one small instance, and an existing operation can be the cheapest way to obtain its answer. Inductive construction earns its extra work when a family, reusable operation or premise-sensitive argument is needed.

The division and tree cases use different objects while sharing the base-and-constructor move.