MATH.5:10 - Architectural Rationale

The method connects three mathematical contributions: recursive construction, preservation of operations, and compatibility with imposed equations. Their combination answers a different question from constructing the source objects alone: how a chosen interpretation extends across that whole source.

The free-expression step makes evaluation and uniqueness accessible. The quotient step then exposes the obligations introduced by an identification. Keeping them separate lets a failed equation leave the valid free-expression map available.

For one small expression, direct evaluation can be sufficient. The general construction earns its cost when many expressions, substitutions or a reusable representation are needed. A ready homomorphism can supply the same result without reconstructing its proof.

The affine-pair case gives the same composition a second mathematical description. Its equation connects the descriptions.