[proofplan] We first fix the finite-time Hamilton-Ivey pinching datum and the tolerated constants, so that the quantitative target is explicit. The post-surgery manifold decomposes into the untouched region, the inserted standard caps, and the transition annuli where the cap is glued to the remaining neck. On the untouched region the metric is identical to the pre-surgery metric, and on the caps the normalized admissibility of the standard cap gives a stronger lower curvature bound. In the transition annuli, the rescaled metric is $C^2$-close, with closeness parameter $\delta>0$, to a compact family of normalized model transition metrics that satisfy the tolerated Hamilton-Ivey inequality with a uniform scale-adjusted positive margin; curvature continuity, eigenvalue continuity, and the explicit scaling law for curvature transfer this estimate back to the physical surgery scale once $\delta$ is chosen uniformly small on $[0,T]$. [/proofplan] [step:Decompose the post-surgery manifold into unchanged, cap, and transition regions] Fix constants $A_T,B_T,C_T>0$ defining the finite-time Hamilton-Ivey pinching estimate on $[0,T]$, and fix the allowed tolerance $\varepsilon_T>0$ with $C_T-\varepsilon_T>0$. For a three-dimensional metric $g$ on a manifold $M$ and a point $x\in M$, let \begin{align*} \operatorname{Rm}_g(x): \Lambda^2T_xM \to \Lambda^2T_xM \end{align*} denote the curvature operator, let $\nu_g(x)\leq \lambda_g(x)\leq \mu_g(x)$ denote its ordered eigenvalues, and define the scalar curvature quantity $R_g:M\to\mathbb R$ by \begin{align*} R_g(x):=\nu_g(x)+\lambda_g(x)+\mu_g(x). \end{align*} This fixes the curvature convention used throughout the proof: the scalar curvature quantity in the Hamilton-Ivey estimate is the trace of the curvature operator with the normalization above. Define the physical Hamilton-Ivey defect functional $\mathcal{P}_{T,\varepsilon_T}:\{(a,b,c)\in\mathbb{R}^3:a\leq b\leq c,\ a<0\}\to\mathbb{R}$ by \begin{align*} \mathcal{P}_{T,\varepsilon_T}(a,b,c):=(a+b+c)-(-a)\left[-(A_T+\varepsilon_T)+B_T\log\left((C_T-\varepsilon_T)(-a)\right)\right]. \end{align*} The finite-time Hamilton-Ivey estimate with tolerated constants means that, at every point $x$, \begin{align*} \nu_g(x)\geq 0 \end{align*} or else \begin{align*} \mathcal{P}_{T,\varepsilon_T}(\nu_g(x),\lambda_g(x),\mu_g(x))\geq 0. \end{align*} For a surgery component with surgery scale $h>0$, define the scale-adjusted normalized defect functional $\mathcal{P}_{T,\varepsilon_T,h}^{\mathrm{norm}}:\{(a,b,c)\in\mathbb{R}^3:a\leq b\leq c,\ a<0\}\to\mathbb{R}$ by \begin{align*} \mathcal{P}_{T,\varepsilon_T,h}^{\mathrm{norm}}(a,b,c):=(a+b+c)-(-a)\left[-(A_T+\varepsilon_T)+B_T\log\left((C_T-\varepsilon_T)h^{-2}(-a)\right)\right]. \end{align*} This is the normalized expression equivalent to the physical estimate after the constant rescaling $\tilde g=h^{-2}g$, because the smallest curvature-operator eigenvalue rescales as \begin{align*} a=\nu_{\tilde g}=h^2\nu_g. \end{align*} Let $M^+$ denote the post-surgery manifold at the surgery time $t_0$, equipped with the post-surgery metric $g^+ := g(t_0^+)$. The standard Ricci-flow-with-surgery construction gives finitely many surgery components on $[0,T]$; for each component indexed by $j$, let $h_j>0$ denote its surgery scale, let $U_{\mathrm{cap},j}$ denote its inserted cap region, and let $U_{\mathrm{tr},j}$ denote its transition annulus. Set \begin{align*} U_{\mathrm{cap}}:=\bigcup_j U_{\mathrm{cap},j},\qquad U_{\mathrm{tr}}:=\bigcup_j U_{\mathrm{tr},j}. \end{align*} The surgery construction gives a disjoint decomposition, up to shared boundary hypersurfaces, into three types of regions: \begin{align*} M^+ = U_{\mathrm{old}} \cup U_{\mathrm{cap}} \cup U_{\mathrm{tr}}, \end{align*} where $U_{\mathrm{old}}$ is the part inherited from the pre-surgery manifold, $U_{\mathrm{cap}}$ is the union of inserted standard caps, and $U_{\mathrm{tr}}$ is the union of transition annuli. The phrase scale-adjusted admissible surgery model means the following two quantitative assertions are part of the surgery hypotheses on $[0,T]$. First, every normalized cap metric, after inserting the actual surgery scale $h_j$ into $\mathcal{P}_{T,\varepsilon_T,h_j}^{\mathrm{norm}}$, satisfies the tolerated Hamilton-Ivey alternative at each cap point. Second, every normalized transition model satisfies the tolerated Hamilton-Ivey alternative with a margin $m_T>0$, and there is a scale-adjusted robustness modulus for the actual allowed surgery scales on $[0,T]$; explicitly, perturbing the ordered curvature triple by at most $r$ in normalized units changes the scale-adjusted defect by at most a quantity $\omega_T(r)$ with $\omega_T(r)\to0$ as $r\downarrow0$, uniformly over all transition models and all allowed surgery scales. This modulus is uniform in the component and remains valid even when the physical surgery scales $h_j$ approach $0$. On $U_{\mathrm{old}}$, the metric is unchanged: \begin{align*} g^+\big|_{U_{\mathrm{old}}} = g(t_0^-)\big|_{U_{\mathrm{old}}}. \end{align*} Hence, for every $x \in U_{\mathrm{old}}$, the curvature operators agree: \begin{align*} \operatorname{Rm}_{g^+}(x) = \operatorname{Rm}_{g(t_0^-)}(x), \end{align*} and therefore their ordered eigenvalues agree: \begin{align*} \nu_{g^+}(x)=\nu_{g(t_0^-)}(x), \qquad \lambda_{g^+}(x)=\lambda_{g(t_0^-)}(x), \qquad \mu_{g^+}(x)=\mu_{g(t_0^-)}(x). \end{align*} Since $g(t_0^-)$ satisfies the [Hamilton-Ivey pinching estimate](/theorems/6022), the same estimate holds at every point of $U_{\mathrm{old}}$. [/step] [step:Use scale-adjusted admissibility of the standard caps to obtain the tolerated pinching estimate] Let $x \in U_{\mathrm{cap},j}$ lie in a cap inserted at surgery scale $h_j$. Let $C_j$ denote the normalized standard cap domain used for this surgery component. Let \begin{align*} \Psi_j: C_j \to U_{\mathrm{cap},j} \end{align*} be the cap parametrization from the normalized standard cap domain $C_j$ onto the physical cap region, and define the normalized pulled-back cap metric \begin{align*} \hat g_j:=h_j^{-2}\Psi_j^*g^+. \end{align*} The scale-adjusted $(T,\varepsilon_T)$-admissibility of the standard cap, together with the cap robustness modulus for sufficiently small normalized $C^2$ perturbations, says that the actual normalized cap metric $\hat g_j$ satisfies the tolerated cap alternative: for every $y\in C_j$, either \begin{align*} \nu_{\hat g_j}(y)\geq0, \end{align*} or else \begin{align*} \mathcal{P}_{T,\varepsilon_T,h_j}^{\mathrm{norm}}(\nu_{\hat g_j}(y),\lambda_{\hat g_j}(y),\mu_{\hat g_j}(y))\geq0. \end{align*} Set $y:=\Psi_j^{-1}(x)$. Curvature eigenvalues and scalar curvature scale by the same factor under constant rescaling: \begin{align*} \nu_{\hat g_j}(y)=h_j^2\nu_{g^+}(x),\qquad \lambda_{\hat g_j}(y)=h_j^2\lambda_{g^+}(x),\qquad \mu_{\hat g_j}(y)=h_j^2\mu_{g^+}(x),\qquad R_{\hat g_j}(y)=h_j^2R_{g^+}(x). \end{align*} If $\nu_{\hat g_j}(y)\geq0$, then $\nu_{g^+}(x)\geq0$. If $\nu_{\hat g_j}(y)<0$, then dividing the normalized inequality by $h_j^2$ and using the definition of $\mathcal{P}_{T,\varepsilon_T,h_j}^{\mathrm{norm}}$ gives \begin{align*} R_{g^+}(x)\geq (-\nu_{g^+}(x))\left[-(A_T+\varepsilon_T)+B_T\log\left((C_T-\varepsilon_T)(-\nu_{g^+}(x))\right)\right]. \end{align*} Therefore the post-surgery metric satisfies the tolerated Hamilton-Ivey estimate throughout $U_{\mathrm{cap}}$. [/step] [step:Control the curvature eigenvalues in the transition region by $C^2$ closeness] Let $\mathcal{V}_T$ denote the compact finite family of normalized model transition domains used by the surgery algorithm on the time interval $[0,T]$; each $V\in\mathcal{V}_T$ is a compact smooth three-manifold with boundary carrying a fixed normalized transition model metric $g_{\mathrm{mod},V}$. For each transition component $U_{\mathrm{tr},j}$ at surgery scale $h_j$, choose the surgery parametrization \begin{align*} \Phi_j: V_j \to U_{\mathrm{tr},j}, \end{align*} where $V_j\in\mathcal{V}_T$. Define the rescaled pulled-back transition metric \begin{align*} \hat g_j := h_j^{-2}\Phi_j^* g^+. \end{align*} By the surgery hypothesis, \begin{align*} \|\hat g_j - g_{\mathrm{mod},V_j}\|_{C^2(V_j,g_{\mathrm{mod},V_j})} \leq \delta. \end{align*} The normalized transition models are assumed scale-adjusted $(T,\varepsilon_T)$-admissible with uniform robustness: for every $V\in\mathcal{V}_T$, every surgery scale $h$ actually allowed on $[0,T]$, and every $y\in V$, the curvature triple of $g_{\mathrm{mod},V}$ satisfies the physical-scale Hamilton-Ivey alternative with a uniform positive margin $m_T>0$. Concretely, either $\nu_{g_{\mathrm{mod},V}}(y)\geq m_T$, or \begin{align*} \mathcal{P}_{T,\varepsilon_T,h}^{\mathrm{norm}}(\nu_{g_{\mathrm{mod},V}}(y),\lambda_{g_{\mathrm{mod},V}}(y),\mu_{g_{\mathrm{mod},V}}(y)) \geq m_T. \end{align*} Moreover, the robustness is uniform for the actual scale-adjusted surgery data: the constants below do not require the physical scales $h$ to range in a compact subset of $(0,\infty)$. Because the curvature tensor depends continuously on the metric and its first two derivatives, and because the [Courant-Fischer min-max principle](/theorems/553) gives continuous dependence of ordered eigenvalues for self-adjoint endomorphisms, compactness of the finite family $\mathcal{V}_T$ gives a single constant $C_{\mathrm{eig}}=C_{\mathrm{eig}}(T,\mathcal{V}_T)>0$ such that, for every $V\in\mathcal{V}_T$ and every $y \in V$, \begin{align*} \left|\nu_{\hat g}(y)-\nu_{g_{\mathrm{mod},V}}(y)\right| + \left|\lambda_{\hat g}(y)-\lambda_{g_{\mathrm{mod},V}}(y)\right| + \left|\mu_{\hat g}(y)-\mu_{g_{\mathrm{mod},V}}(y)\right| \leq C_{\mathrm{eig}}\delta. \end{align*} On the negative branch, the function $z\mapsto z\log((C_T-\varepsilon_T)h^{-2}z)$ extends continuously to $z=0$ for each fixed $h>0$ by assigning value $0$, because $z\log z\to0$ as $z\downarrow0$. Since the factor $\log(h^{-2})$ need not be uniformly bounded when the physical surgery scales become small, the required uniformity is exactly the scale-adjusted robustness assumption in the surgery model, not a consequence of compactness of the physical scale range. Thus there is a nondecreasing robustness modulus $\omega_T:[0,\infty)\to[0,\infty)$, depending only on $T,\mathcal{V}_T,A_T,B_T,C_T,\varepsilon_T$ and the chosen scale-adjusted surgery model, with $\omega_T(r)\to0$ as $r\downarrow0$, such that changing each of the three curvature eigenvalues by at most $r$ changes the scale-adjusted Hamilton-Ivey defect by at most $\omega_T(r)$ whenever the perturbed triple remains on the negative branch. [guided] The transition region is the only part where the metric is neither exactly old nor exactly the standard cap. The proof has three separate tasks here: compare the transition metric with the normalized model, use the model's positive Hamilton-Ivey margin, and then scale the normalized conclusion back to the physical metric. Let $\mathcal{V}_T$ be the compact finite family of model transition domains used on $[0,T]$. For one transition component, let \begin{align*} \Phi: V \to U_{\mathrm{tr}} \end{align*} be the surgery coordinate map, where $V\in\mathcal{V}_T$. Let $g_{\mathrm{mod},V}$ be the corresponding normalized model metric on $V$. Since the surgery takes place at scale $h$, we compare metrics after rescaling by $h^{-2}$ and define \begin{align*} \hat g := h^{-2}\Phi^* g^+. \end{align*} The surgery construction assumes \begin{align*} \|\hat g - g_{\mathrm{mod},V}\|_{C^2(V,g_{\mathrm{mod},V})} \leq \delta. \end{align*} The decisive input is not mere closeness to a model; it is closeness to a model with spare room in the desired inequality. The normalized transition models are $(T,\varepsilon_T)$-admissible with scale-adjusted robustness, meaning that for every $y\in V$ either \begin{align*} \nu_{g_{\mathrm{mod},V}}(y)\geq m_T, \end{align*} or \begin{align*} R_{g_{\mathrm{mod},V}}(y)-(-\nu_{g_{\mathrm{mod},V}}(y))\left[-(A_T+\varepsilon_T)+B_T\log\left((C_T-\varepsilon_T)h^{-2}(-\nu_{g_{\mathrm{mod},V}}(y))\right)\right] \geq m_T. \end{align*} This is exactly where the hypothesis that the surgery parameters are chosen from a scale-adjusted admissible finite-time surgery model with uniform positive margin is used. The uniformity is an assumption on the normalized surgery model for the actual allowed surgery scales; it is not obtained by assuming that the physical scales $h$ stay bounded away from $0$. Now curvature varies continuously with the metric in the $C^2$ topology. In coordinates on the compact model transition domain $V$, the coefficients of the curvature tensor are universal smooth functions of the metric coefficients, inverse metric coefficients, first derivatives, and second derivatives. Since the family $\mathcal{V}_T$ is compact and finite, this continuity is uniform over all transition components. The curvature operator is a self-adjoint [linear map](/page/Linear%20Map) on the three-dimensional [vector space](/page/Vector%20Space) $\Lambda^2 T_yV$. For two [self-adjoint operators](/page/Self-Adjoint%20Operators) $A$ and $B$ on a finite-dimensional [inner product space](/page/Inner%20Product%20Space), the Courant-Fischer min-max principle gives \begin{align*} |\nu_A-\nu_B| \leq \|A-B\|_{\mathrm{op}}, \end{align*} and the same estimate holds for the other ordered eigenvalues. Applying this to \begin{align*} A=\operatorname{Rm}_{\hat g}(y), \qquad B=\operatorname{Rm}_{g_{\mathrm{mod},V}}(y), \end{align*} gives a uniform constant $C_{\mathrm{eig}}>0$ such that \begin{align*} \left|\nu_{\hat g}(y)-\nu_{g_{\mathrm{mod},V}}(y)\right| + \left|\lambda_{\hat g}(y)-\lambda_{g_{\mathrm{mod},V}}(y)\right| + \left|\mu_{\hat g}(y)-\mu_{g_{\mathrm{mod},V}}(y)\right| \leq C_{\mathrm{eig}}\delta. \end{align*} The Hamilton-Ivey expression contains a logarithm, so we must be precise at the branch point where the smallest eigenvalue approaches $0$. On the negative branch, write $z=-\nu>0$. For each fixed $h>0$, the term that appears is $z\log((C_T-\varepsilon_T)h^{-2}z)$, and this extends continuously to $z=0$ by assigning value $0$, since $z\log z\to0$ as $z\downarrow0$. However, because $\log(h^{-2})$ may be unbounded along small surgery scales, the uniform perturbation estimate is supplied by the scale-adjusted robustness hypothesis in the surgery model. Let $\omega_T$ be the uniform scale-adjusted robustness modulus for the Hamilton-Ivey defect on the compact finite model family and actual allowed surgery scales. Choose $\delta>0$ small enough that \begin{align*} C_{\mathrm{eig}}\delta\leq \frac{m_T}{2},\qquad \omega_T(C_{\mathrm{eig}}\delta)\leq \frac{m_T}{2}. \end{align*} If the model satisfies $\nu_{g_{\mathrm{mod},V}}(y)\geq m_T$, then $\nu_{\hat g}(y)\geq m_T/2>0$, so the nonnegative branch survives. If the model satisfies the negative-branch margin estimate, then there are two cases. If $\nu_{\hat g}(y)\geq0$, the nonnegative branch already proves the tolerated alternative. If $\nu_{\hat g}(y)<0$, then the normalized defect is defined at the perturbed curvature triple, and the modulus estimate leaves margin at least $m_T/2$; hence $\hat g$ satisfies the scale-adjusted tolerated Hamilton-Ivey inequality at $y$. It remains to return from $\hat g$ to $g^+$. Under the constant rescaling $\hat g=h^{-2}\Phi^*g^+$, all curvature eigenvalues and scalar curvature multiply by $h^2$: \begin{align*} \nu_{\hat g}=h^2\nu_{\Phi^*g^+},\qquad \lambda_{\hat g}=h^2\lambda_{\Phi^*g^+},\qquad \mu_{\hat g}=h^2\mu_{\Phi^*g^+},\qquad R_{\hat g}=h^2R_{\Phi^*g^+}. \end{align*} If $\nu_{\hat g}\geq0$, then $\nu_{\Phi^*g^+}\geq0$. Otherwise the scale-adjusted normalized inequality is \begin{align*} R_{\hat g}\geq (-\nu_{\hat g})\left[-(A_T+\varepsilon_T)+B_T\log\left((C_T-\varepsilon_T)h^{-2}(-\nu_{\hat g})\right)\right]. \end{align*} Dividing by $h^2$ and using $h^{-2}(-\nu_{\hat g})=-\nu_{\Phi^*g^+}$ gives exactly \begin{align*} R_{\Phi^*g^+}\geq (-\nu_{\Phi^*g^+})\left[-(A_T+\varepsilon_T)+B_T\log\left((C_T-\varepsilon_T)(-\nu_{\Phi^*g^+})\right)\right]. \end{align*} Thus the normalized scale-adjusted estimate is equivalent to the physical-scale estimate for $g^+$ on the transition component. Choose $\delta_T>0$ so small that, for every $0<\delta\leq\delta_T$, \begin{align*} C_{\mathrm{eig}}\delta\leq \frac{m_T}{2},\qquad \omega_T(C_{\mathrm{eig}}\delta)\leq \frac{m_T}{2}. \end{align*} Then for every $0<\delta\leq\delta_T$, if the model point satisfies $\nu_{g_{\mathrm{mod},V_j}}(y)\geq m_T$, the eigenvalue estimate gives \begin{align*} \nu_{\hat g_j}(y)\geq m_T-C_{\mathrm{eig}}\delta\geq \frac{m_T}{2}>0. \end{align*} If the model point lies on the negative Hamilton-Ivey branch, then either $\nu_{\hat g_j}(y)\geq0$, in which case the nonnegative branch of the tolerated alternative holds, or $\nu_{\hat g_j}(y)<0$. In the second case the normalized defect functional is defined at the perturbed curvature triple, and the modulus estimate gives that the perturbation of the scale-adjusted Hamilton-Ivey margin is at most $\omega_T(C_{\mathrm{eig}}\delta)\leq m_T/2$, so \begin{align*} \mathcal{P}_{T,\varepsilon_T,h_j}^{\mathrm{norm}}(\nu_{\hat g_j}(y),\lambda_{\hat g_j}(y),\mu_{\hat g_j}(y))\geq \frac{m_T}{2}>0. \end{align*} Hence the rescaled metric $\hat g_j$ satisfies the scale-adjusted tolerated Hamilton-Ivey estimate on every model transition domain $V_j\in\mathcal{V}_T$. Finally, the constant rescaling relation $\hat g_j=h_j^{-2}\Phi_j^*g^+$ gives \begin{align*} \nu_{\hat g_j}=h_j^2\nu_{\Phi_j^*g^+},\qquad \lambda_{\hat g_j}=h_j^2\lambda_{\Phi_j^*g^+},\qquad \mu_{\hat g_j}=h_j^2\mu_{\Phi_j^*g^+},\qquad R_{\hat g_j}=h_j^2R_{\Phi_j^*g^+}. \end{align*} If the nonnegative branch holds for $\hat g_j$, it holds for $\Phi_j^*g^+$. If the negative branch holds, dividing the scale-adjusted normalized inequality by $h_j^2$ gives exactly the physical Hamilton-Ivey inequality for $\Phi_j^*g^+$, because \begin{align*} h_j^{-2}(-\nu_{\hat g_j})=-\nu_{\Phi_j^*g^+}. \end{align*} Pullback by the diffeomorphism $\Phi_j$ preserves curvature eigenvalues pointwise, so the Hamilton-Ivey pinching estimate, with constants adjusted by no more than $\varepsilon_T$, holds throughout $U_{\mathrm{tr}}$. [/guided] [/step] [step:Choose the surgery parameter uniformly on the finite time interval] The constants and functions entering the preceding estimates are $A_T,B_T,C_T,\varepsilon_T$, the compact finite model family $\mathcal{V}_T$, the admissibility margin $m_T$, the eigenvalue-continuity constant $C_{\mathrm{eig}}$, and the scale-adjusted Hamilton-Ivey robustness modulus $\omega_T$. They do not depend on the particular surgery time $t_0 \in [0,T]$ or on the particular surgery component, because the surgery algorithm uses the same normalized cap and transition models, the same finite-time pinching datum, and the same scale-adjusted admissibility data throughout $[0,T]$. Define $\delta_T>0$ to be the minimum of the smallness thresholds required for the cap and transition estimates above, including the requirements that, for every $0<\delta\leq\delta_T$, \begin{align*} C_{\mathrm{eig}}\delta\leq \frac{m_T}{2},\qquad \omega_T(C_{\mathrm{eig}}\delta)\leq \frac{m_T}{2}. \end{align*} Then, whenever $0<\delta\leq\delta_T$, the post-surgery metric satisfies the tolerated Hamilton-Ivey pinching estimate on $U_{\mathrm{old}}$, on $U_{\mathrm{cap}}$, and on $U_{\mathrm{tr}}$. Since these regions cover $M^+$, the estimate holds everywhere on the post-surgery manifold with constants changed by no more than the prescribed tolerance $\varepsilon_T$. This proves the stated quantitative preservation of Hamilton-Ivey pinching through surgery. [/step]