Meaning
The coercive cone is the set of two-block lower-bound data that certify a chosen floor κ after the coupling between blocks is included.
For nonnegative u,v,m, membership in C(κ) means u≥κ, v≥κ, and m²≤(u−κ)(v−κ). Equivalently, the comparison matrix [[u,−m],[−m,v]] is at least κI. The numbers u and v bound the diagonal blocks from below; m bounds the coupling between them.
Example (illustrative)
With u=v=3 and m=1, the comparison matrix has eigenvalues 2 and 4, so it certifies κ=2. With the same diagonal values and m=3, its smallest eigenvalue is zero: positive diagonal bounds alone no longer give a positive floor.
Scope and limits
The inequalities apply to the exact excitation splitting and verified block estimates. A finite floor does not by itself establish an invariant mass gap.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 14.11.