The purely cosmetic surgery conjecture for knots in the three-sphere

Theorem 1.1: for every nontrivial smooth knot $K\subset S^3$ and all $r,s\in\mathbb{Q}\cup\{\infty\}$, an orientation-preserving homeomorphism $S^3_r(K)\cong S^3_s(K)$ forces $r=s$; the homeomorphism need not preserve the surgery core or the knot exterior. After the cited reduction to $g(K)=2$ and slopes $\pm 2$, the paper derives a contradiction by comparing a nonzero integer instanton count $\Omega$ on a parameterized cobordism with a one-dimensional count of solutions with spinors whose boundary would force $2^\eta\Omega=0$. It does not address chirally cosmetic (orientation-reversing) surgeries or knots in other three-manifolds.

Theorem 1.1: for every nontrivial smooth knot $K\subset S^3$ and all $r,s\in\mathbb{Q}\cup\{\infty\}$, an orientation-preserving homeomorphism $S^3_r(K)\cong S^3_s(K)$ forces $r=s$; the homeomorphism need not preserve the surgery core or the knot exterior. After the cited reduction to $g(K)=2$ and slopes $\pm 2$, the paper derives a contradiction by comparing a nonzero integer instanton count $\Omega$ on a parameterized cobordism with a one-dimensional count of solutions with spinors whose boundary would force $2^\eta\Omega=0$. It does not address chirally cosmetic (orientation-reversing) surgeries or knots in other three-manifolds.