Fix a finite time horizon $T>0$ and Hamilton-Ivey constants $A_T,B_T,C_T>0$. Let $g^-$ be the pre-surgery metric at a surgery time $t_0\le T$ on a three-dimensional flow with surgery. Write $\nu\le\lambda\le\mu$ for the eigenvalues of the curvature operator on $\Lambda^2TM$ and set

\begin{align*} \rho:=\nu+\lambda+\mu. \end{align*}

With the scalar-curvature convention used above, the scalar curvature is $R_{g^-}=2\rho$. Assume $g^-$ satisfies the Hamilton-Ivey alternative expressed in this half-scalar trace: at every point either $\nu\ge0$ or

\begin{align*} \rho\ge(-\nu)\left[-A_T+B_T\log(C_T(-\nu))\right]. \end{align*}

Fix a tolerance $\eta>0$. Suppose the surgery replaces pairwise disjoint strong $\delta$-necks of radius $h$ by standard caps, leaves the old metric unchanged outside the cutoff regions, and, after scaling every cap and transition region by $h^{-2}$, the inserted and transition metrics satisfy the same Hamilton-Ivey alternative with constants $A_T+\eta$, $B_T$, and $C_T-\eta$. Then there is $\delta_T>0$, depending only on $T$, the model cap, and $\eta$, such that for every $0<\delta\le\delta_T$ the post-surgery metric $g^+$ satisfies the tolerated Hamilton-Ivey alternative: writing $\rho_{g^+}:=\nu_{g^+}+\lambda_{g^+}+\mu_{g^+}$, so that $R_{g^+}=2\rho_{g^+}$, at every point either $\nu_{g^+}\ge0$ or

\begin{align*} \rho_{g^+}\ge(-\nu_{g^+})\left[-(A_T+\eta)+B_T\log((C_T-\eta)(-\nu_{g^+}))\right]. \end{align*}