Meaning
The filling obstruction tests whether a prescribed mismatch can be filled by an allowed correction.
For a closed mismatch Δζ and a comparison map Q, the obstruction is o_fill=[QΔζ] in the declared degree-two cohomology. It vanishes precisely when QΔζ=dτ in that complex. Interpreting τ as a physically allowed filling also requires the coefficient model to represent all and only the permitted corrections appropriately.
Example (illustrative)
If the allowed differential d maps every degree-one cochain to zero, a nonzero degree-two mismatch QΔζ cannot be filled. If QΔζ=dτ for an allowed τ, the cohomological filling test passes.
Scope and limits
Existence of a filling does not imply that the relative class [Δζ,τ] vanishes. It also does not certify operator domains, detector rank, or a positive excitation bound.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 9.4.