Meaning
A coherent certificate is a family of certificates whose witnesses agree across the retained faces and compositions.
Restriction to an overlap must match the certificate already assigned there, up to the specified twist. Successive refinement comparisons must match the corresponding composite. Hamiltonian-bearing faces must also preserve their typed mixed-square cells and residual budgets.
Example (illustrative)
A projection certified on a large region and a projection certified on a smaller region may each be valid separately. Coherence additionally checks that the declared restriction relates them as required, with the same source and branch data.
Scope and limits
A collection of individually correct certificates is not automatically a coherent family. Their overlap and composition comparisons carry mathematical content.
Source
GAT I — Grassmannian Gluing, Spectral Trichotomy, and Conditional Sectorial Type II. GAT I, Definition 13.2.