MATH.21:10 - Architectural Rationale

Approximation is a construction through a relation between stages and a result. The relation determines what information accumulates and what a finite stage can supply. This explains the common method across rational intervals, prefixes and functions without making their convergence definitions interchangeable.

Existence, identification of equivalent constructions, preservation of operations and finite return are connected tasks. Quotient construction handles identification; proof construction supplies missing implications; convergence and continuity supply the limiting passage. Keeping their contributions explicit lets a changed operation reopen the relevant mathematical step.

A stronger notion of convergence is useful when it supports a needed consequence. A weaker one is sufficient when it already supplies the receiver’s observation. This permits economical use of mathematical results while preserving their conditions.