a method that recovers the correct quantity but is blind to its place — the value is right, the location or global form unexhibitable

The recurring signature failure of the hardest cases: a method gets the quantity right and is blind to its place. Curvature fixes the local metric datum but not the global form — SL₂ and PGL₂ share the hyperbolic plane, and which cover you inhabit lives in the lattice, not the metric. The spectrum fixes the eigenvalues but not the operator's placement — isospectrality, until the minor spectra are added back. The anonymous obstruction has local data at every place and no global witness; the residue of a group action gives a value you can produce by performing the group element but never exhibit as a certificate. In each, the magnitude is correct and the localization is lost.

Calibration question: when an invariant or a method "determines" a thing, ask separately whether it fixes the value and whether it fixes the place. A method can be right about the quantity and silent about where it sits — and that silence, not a wrong number, is where the hardest problems live. Match a new "X is determined by Y" claim against this shape: is Y blind to X's global form the way curvature is blind to the isogeny class?