Meaning
The relative linking obstruction tests whether selected local source–destination spatializers can fit into one coherent graph comparison.
At finite total-nerve truncation, it minimizes the linking compatibility and residual energies over the allowed local spatializers. In the cofinal system, vanishing requires a family with energy tending to zero and uniform linking, heat, and graph bounds sufficient for the prescribed compactness modulo zero-cost normalizers.
Example (illustrative)
Local spatializers may each solve their corner equations while disagreeing on overlaps. A nonzero compatibility residual then obstructs their assembly. Driving the residual to zero only helps if the necessary uniform bounds prevent the spatializers from escaping.
Scope and limits
This is relative to the chosen source and destination sector. It does not assert that the original correspondence or vacuum gerbe is globally trivial.
Source
GAT I — Grassmannian Gluing, Spectral Trichotomy, and Conditional Sectorial Type II. GAT I, Definition 5.54.