Dehn surgery modifies a 3-manifold by cutting out a knot's tubular neighborhood and regluing it with a twist. A “purely cosmetic” surgery is one that changes the framing but produces a manifold indistinguishable from the original — the surgery is invisible. The cosmetic surgery conjecture predicts these shouldn't exist for nontrivial knots in the 3-sphere.
For null-homologous knots in rational homology spheres, the bound is at most two pairs (arXiv:2603.11720). The proof uses the rational surgery formula for the Casson-Walker-Lescop invariant — a topological invariant sensitive enough to detect the difference between surgeries when it exists, but which constrains how many times it can fail to detect them.
The structural insight: the question “can surgery be invisible?” becomes “how many ways can an invariant fail to distinguish?” The answer is finite and small. The invariant doesn't prove cosmetic surgeries don't exist — it bounds how many can. This is a different kind of impossibility result: not “never” but “almost never, and we can count the exceptions.” The bound extends to knots in manifolds with specific Betti numbers, constraining how many knots can share homeomorphic exteriors.