A candidate symmetry direction becomes physically interesting only if it survives three filters: better observables, realistic slack budget, and stepwise transport through the dynamics.
This is the page that turns the symmetry survey into an optimization picture. The goal is not “any invisible move,” but “a move that stays nearly invisible under the resolutions and budgets that matter.”
Lean anchors.
boundaryWritableRegion_antitone,
ApproximateTransportCriterion,
approximateTransportCriterion_of_exact_depthGauge,
refined_budgeted_step_transport_criterion,
approximate_interlingua_step_transport
Math statement.
In English. A transport move counts as a serious candidate when it survives a finer observable test, fits inside a chosen scalar rewrite budget, and still sits inside a controlled one-step latent transport bound.
Physical intuition. Real low-loss fibers are the ones that survive refinement, remain affordable under realistic rewrite budget, and do not blow up under the actual stepwise dynamics. The new theorem package makes that filter explicit without relying on the broken legacy budget file.
This is the optimization reading of the survey. Geometry is not just a quotient-space slogan. It is a filter for which rewrites stay behaviorally cheap under the constraints that matter.
If a candidate fiber disappears under a richer observable set, it was probe artifact.
If a rewrite needs more slack than the system can supply, it is not an operational symmetry direction, even if it looks invisible on paper.
If stepwise mismatch compounds too fast, the direction is not a robust path between nearby functions.
Feynman reading: a “possible motion” only becomes a real physical degree of freedom when the apparatus cannot see it, the system can afford it, and the dynamics does not immediately tear it apart.
Slide the three tests and see when a candidate direction deserves to be called a real low-loss transport fiber rather than a pretty but useless quotient direction.