Buffered comparison and stopping-band resolution in critical Ising

We prove pointwise comparison and finite stopping-band approximation theorems for the critical nearest-neighbor Ising model on the square lattice. Arbitrary common pinned spins are allowed in the comparison: the likelihood ratio is controlled by a zero-field FK connection probability on the graph with those vertices deleted. For an interface exploration across a band of fixed positive width, finitely many regular signed barriers approximate every good conditional target law in total variation, uniformly over bounded observables and sufficiently fine meshes.