In fragment-2-rdm-sheaf the active space moves over a grassmannian and the natural map between fibres is parallel transport of the tautological connection. In fragment-2-rdm-sheaf the map between fragments is orbital compression
The tension
Isometry versus information loss
Grassmann parallel transport is an isometry. It rotates a fibre onto a neighbouring one without distortion. Compression is deliberately not an isometry. Its adjoint carries the overlap
, and its non-invertibility is exactly what lets the sheaf residual measure something, since a local system would have zero separation gap. The transport connection and the compression restriction are therefore different maps in general. On which submanifold of , if any, do they agree, and how large is their discrepancy elsewhere?
Why it matters
- If they agree on the relevant region, the B2 Grassmannian picture and the A sheaf picture share one operator, and the connection-laplacian and sheaf Laplacian become two readings of one object, a genuine unification.
- If they disagree, the transport picture must fall back to the sheaf Laplacian
wherever compression loses information, and B2 is only a smooth-geometry heuristic, not a substitute for consistency.
A concrete test transports a fixed two-particle Reduced Density Matrix around a small loop of active spaces and compares the holonomy against the composition of compressions along the same loop. The gap is the obstruction to identifying the two.