Meaning
The total comparison form measures the combined defect of a proposed comparison in the declared comparison space.
For a comparison variable V and homogenizing parameter λ, GAT I defines q_tot(V,λ)=‖D̂(V,λ)‖² on its stated domain. The operator D̂ packages the comparison equations into one squared norm. A closure is used only when it exists.
Example (illustrative)
For a toy defect map D(x,y)=(x−y,2y), the form is (x−y)²+4y² on real pairs. It is nonnegative and vanishes exactly when both recorded equations hold. The actual comparison form uses the source’s full defect map and domain.
Scope and limits
A small defect is not an exact solution, and a formal squared norm does not prove closability. Zero modes must still be interpreted using the certified transport and destination structure.
Source
GAT I — Grassmannian Gluing, Spectral Trichotomy, and Conditional Sectorial Type II. GAT I, Definition 4.1.