The writable-region idea packages the remaining invisible freedom after you pin down a chosen observable family. It is the natural place to talk about low-loss transport directions.
The key warning is that refinement shrinks writable freedom. So a candidate transport direction that disappears under better observables was not a trustworthy symmetry direction to begin with.
Lean anchors.
boundaryWritableRegion_antitone,
fake_symmetry_shrinks_writable_region,
writableRegion_monotone_under_scale
Math statement.
In English. Refining the observable family can only remove invisible moves. So the only transport directions worth trusting are the ones that survive better measurement.
Physical intuition. As you measure more, the set of invisible internal moves can only shrink. Real quotient directions are the ones that survive this squeeze.
The outer rings are freedoms that looked invisible only because the observables were too weak. The inner core is what remains after better measurement.
Large parameter motion can correspond to tiny behavioral motion if you move along a true quotient fiber.
Different internal allocations can remain nearly equivalent because the objective only cares about coarse summaries.
If refinement kills it, it was never a trustworthy route between minima.
This is the cleanest way to connect theorem language to physics and engineering: quotient directions are interesting because they can turn huge coordinate drift into small functional drift.
This is a practical criterion for sorting real symmetries from probe artifacts while keeping the engineering goal in view.