MATH.9:10 - Architectural Rationale

The stabilizer condition is necessary because transformations invisible at the input must leave a deterministic answer unchanged. It is sufficient for extension over one orbit because those same transformations account for every ambiguity in how that orbit is reached. The construction supplies both the impossibility result and its positive counterpart.

MATH.8 generates solution families from an action. This method selects a compatible member and extends that selection across related inputs. A unique solution supplies the fixed-member condition automatically, but several permitted solutions can still contain a suitable fixed member.

A canonical input and a normalizing transformation are different outputs. Choosing the same input representative throughout an orbit is an invariant operation. Recovering an oriented answer additionally uses a transformation and its output action; :5.3 shows why that transformation may not be unique.

Orbit-by-orbit construction is preferable when its fixed-point test settles existence or produces a useful rule. A direct formula can avoid representative search. Set or distribution outputs answer different questions, and geometric continuity can require a construction beyond the pointwise rule.