Meaning
A Hamiltonian descent certificate accounts for how the full Hamiltonian changes under gauge comparisons and regional restrictions.
It retains electric, magnetic, and seam/boundary descent, mixed-square comparison cells, and ultraviolet, infrared, and mixed residual budgets. The residual on a regional restriction must be located on its declared cut, collar, or seam carrier. Higher comparisons must compose coherently.
Example (illustrative)
Restricting a region can cut a plaquette or create a boundary term. The certificate records that term and bounds its contribution instead of asserting that the restricted Hamiltonian is unchanged.
Scope and limits
A comparison of carriers or algebras alone is not a Hamiltonian descent certificate. Residual budgets cannot be discarded when transferring a coercive estimate.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 14.3.