% \providecommand{\Lbox}{\Lambda^{\circ}} \providecommand{\Hset}{\mathbb{X}} \providecommand{\Mset}{\mathbb{M}} \providecommand{\Uf}{\mathbb{U}} \providecommand{\Fset}{\mathfrak{F}} \providecommand{\Kc}{\mathfrak{K}} \providecommand{\Sbar}{\bar S} \providecommand{\Gp}{G'} \providecommand{\Ereg}{\mathcal{E}} \providecommand{\dist}{\operatorname{dist}} \providecommand{\supp}{\operatorname{supp}} \providecommand{\good}{\operatorname{good}} \providecommand{\Qc}{\mathcal{Q}} \providecommand{\Uc}{\mathcal{U}} \section{The regularization: construction and main properties}\label{sec:3} In this section we define a deterministic map $D\mapsto\Gp(D)$ from sets $D$ of frozen plaquettes to slit domains and state its main properties, Theorems~\ref{thm:3.20} (terminal state) and~\ref{thm:3.21} (history), which we prove in Section~\ref{sec:4}; the details of the proof of Lemma~\ref{lem:3.9}, including the lattice of minimizers, are in Section~\ref{S4-S:S4.1}. Throughout Sections~\ref{sec:3} to~\ref{sec:5} we fix $K>K_*$, $\beta\ge\beta_0(K)$, $\tau=K\beta$ and the constants fixed before $K$, for which we refer to Section~\ref{sec:8} and, for their values, to Section~\ref{S:constants}. The loop $C=\partial\Xi$ bounds the rectangle $\Xi:=[r_1,r_1+R]\times[t_1,t_1+T]\times\{z_0\}$ ($r_1,t_1,z_0\in\mathbb Z$, fine units), with $\min(R,T)\ge R_1$. We treat other planes by permuting the axes. All statements hold for arbitrary $D\subseteq\Lbox_2\setminus\partial_2\Lbox$. Units are coarse (one site is one coarse cell of the grid) and distances are coarse $\ell^\infty$ distances unless ``fine'' or ``Euclidean'' is written. We write $Q_\infty(u,r):=\{x:|x-u|_\infty\le r\}$. Before the formal definitions we outline the design of the construction. The Phase-1 hole set is the smallest minimizer of a cut functional, given in Definition~\ref{def:3.forced}, and in Section~\ref{ssec:4.9} we pay its cost by comparison with competitors that replace a region, at a bounded cost per mark. Smallest minimizers are canonical, so the definition of Phase~1 involves no tie-breaking rule. Fills add whole blocks and are triggered by a count of marks. Penalizing defective interfaces or weak links instead fails by cascades, since freezing the bonds of one defect can create defects of the same kind in other blocks, cells being defined through windows, whereas filling blocks changes no bond inside other blocks. We enforce $(\mathrm{NF}_*)$ by a rule of Phase~2, the cut rule, and not by a term of the functional, since the deletion competitor used in that comparison can create a fragmented split. The local seal of Definition~\ref{def:3.seal} fills the thin pieces of blocks that minimality does not exclude. \subsection{Scales, the box, the grid}\label{ssec:3.1} \begin{definition}[Scales and units]\label{def:3.1} Let $\tilde m\in[m_D/2,2m_D]$ be the screening mass of Section~\ref{sec:stage1} (fine units). The scales are: the fine side $r_0\ge1$ of a coarse cell, which is at most $\bar r_0:=2r_0-1$; the block scale $L_c$, the largest even integer $\le c_L\tilde m^{-1}/r_0$ (coarse), and $L:=r_0L_c$ (fine); $\rho_\lambda:=L_c/12$ and the hull penalty $\lambda:=1/\rho_\lambda=12/L_c$; $V_*:=(1-\eta_w)^3\rho_\lambda^3/2$ with $\eta_w:=1/10$; the window bound $k_P:=304L_c$; the removal scale $\ell:=\ell_D/r_0$ with $\ell_D:=L\lceil\tilde m^{-1}/L\rceil$; and $R_1:=(2r_\square+5)L$ (fine) with $r_\square:=(k_P+3)/L_c\le305$. \end{definition} The lower bounds on $\tau$ used here are $L_c\ge4$ and $\tilde mL\ge c_L/2$ (Section~\ref{S:B1.scales}). Since $c_L\le1/308$, we have \begin{equation}\label{eq:3.Hell} \ell\ge k_P+4L_c=308\,L_c,\qquad \tilde mL\le c_L\le\tfrac12 . \end{equation} \begin{definition}[The box $\Lbox$]\label{def:3.2} Put $m:=\max(R,T)$, $M_\Lambda:=\max\bigl(3m,\ r_0(\ell+3)\bigr)$ (fine), $M'_\Lambda$ the even integer in $\{M_\Lambda,M_\Lambda+1\}$, and $o=(o_1,o_2,o_3):=(r_1+R/2,\ t_1+T/2,\ z_0)$, the centre of $\Xi$. Then $\Lbox:=\prod_{\mu=1}^3[a_\mu,b_\mu]$ with $a_\mu:=\lfloor o_\mu\rfloor-M'_\Lambda/2$ and $b_\mu:=a_\mu+M'_\Lambda$, a cube whose faces are lattice planes; $\Lbox_2$ is its set of plaquettes and $\partial_2\Lbox$ the set of those contained in $\partial\Lbox$. \end{definition} \begin{lemma}\label{lem:3.3} (a) $\Lbox$ and its grid $\mathfrak G(C)$ depend only on $C$ and the constants. (b) Consecutive special planes are at fine distance $\ge L$. (c) Every coarse side of $\Lbox$ is $\ge\ell\ge k_P+4L_c$, and $\Lbox$ is a union of blocks and of removal cells. (d) The coarse sides $n_\mu$ of $\Lbox$ satisfy $\max_\mu n_\mu\le2\min_\mu n_\mu$. (e) $\Lbox$ contains the fine $\ell^\infty$-neighbourhood of radius $m$ of $\Xi$, so $\dist(C,\partial\Lbox)\ge\max(R,T)$, $C\subset\Lbox\setminus\partial\Lbox$, and the fine side of $\Lbox$ is $\ge3\max(R,T)$. \end{lemma} \begin{proof} Parts (a) to (d) are proved in Section~\ref{S:B2.lem33}, and for (e) we use $M'_\Lambda\ge3m$ and $o_\mu-\tfrac12\le\lfloor o_\mu\rfloor\le o_\mu$. These inequalities give $r_1-a_1\ge(M'_\Lambda-R)/2\ge m$, and the integer $b_1-(r_1+R)$ is at least $o_1-\tfrac12+M'_\Lambda/2-r_1-R=(M'_\Lambda-R-1)/2\ge m-\tfrac12$, hence at least $m$. Direction~2 is the same with $T$, and $z_0-a_3=b_3-z_0=M'_\Lambda/2\ge m$. \qed \end{proof} \begin{definition}[The loop-adapted grid $\mathfrak G(C)$]\label{def:3.4} The grid $\mathfrak G(C)$, constructed in Section~\ref{S:B2.grid}, is a function of $C$ and the constants. Its \emph{special planes} are the planes of $C$ and the faces of $\Lbox$. Its \emph{sites} are boxes of fine sides in $[r_0,\bar r_0]$ forming, combinatorially, a box of $\mathbb Z^3$, with nearest-neighbour bonds $E$. The coarse face dual to a bond is the common face of its two sites. \emph{Blocks} are the boxes between consecutive \emph{block planes} of the three directions. A \emph{block face} $F'$ is the common face of two adjacent blocks, and $|F'|$ is the number of bonds across it. The \emph{block distance} of two blocks is the $\ell^\infty$ distance of their indices. For a \emph{removal cell} $\Delta$, $|\partial\Delta|$ is the number of bonds of $\mathbb Z^3$ with exactly one end in $\Delta$, bonds leaving $\Lbox$ included. The \emph{window} $\hat Q(Q)$ of a block $Q$ is the union of the blocks at block distance at most $75$ from $Q$, and $\Qc$ is the set of windows. For a block face $F'$ between blocks $Q$, $Q'$, in the plane $P'$, the \emph{tube zone} $Z(F')$ is the set of sites of $Q\cup Q'$ in the $L_c/2$ layers nearest to $P'$ on each side, and the \emph{tube} $\tau_b$ of a bond $b$ across $F'$ is the segment of its line formed by its $L_c$ sites in $Z(F')$. The grid has the following properties. (G1) Consecutive parallel block planes are at coarse distance in $[L_c,2L_c-1]$. Blocks have coarse sides in $[L_c,2L_c-1]$, fine sides in $[L,2L-1]$, volume in $[L_c^3,8L_c^3)$, and $|F'|\in[L_c^2,(2L_c-1)^2]$. (G2) Every block is a box with six faces, each a block face or contained in $\partial\Lbox$. A line in direction $\mu$ crosses only block planes perpendicular to $e_\mu$. Every special plane is a block plane, and $\Lbox$ and every removal cell are unions of blocks. (G3) Removal cells tile $\Lbox$, have coarse sides in $[\ell,2\ell+2L_c-3]\subseteq[\ell,3\ell)$ and fine sides at most $3\ell_D$ (a block of coarse side $a\ge L_c\ge4$ has fine side at most $r_0a+r_0-1$, and $\ell\ge308L_c$; Section~\ref{S:B2.G}), and $|\partial\Delta|\in[6\ell^2,54\ell^2)$. (G4) Every window $\hat Q(Q)$ is a box of coarse sides less than $k_P$ that contains $Q_\infty(u,75L_c)\cap\Lbox$ for every site $u\in Q$, and $Q\cup Q'\subseteq\hat Q(Q)$ for every block $Q'$ adjacent to $Q$. Every site lies in at most $151^3$ windows. (G5) $Z(F')\subseteq Q\cup Q'$, tube zones of distinct parallel block faces are disjoint, and a tube crosses no block plane other than $P'$. \end{definition} We prove (G1) to (G5) in Section~\ref{S:B2.G} and list the sizes in Table~\ref{tab:4.1}. \begin{lemma}[Alignment]\label{lem:3.5} By (G2) the planes of $C$ are block planes; so, with $P:=\{x_3=z_0\}$: (A1) every plaquette of $\Xi$ lies in the block plane $P$, and no bond with both ends in one block crosses $\Xi$; (A2) every block face lies in $\Xi$ or contains no plaquette of $\Xi$; (A3) every block lies on one side of $P$, and $C$ lies on lines of block planes. Every plaquette of $\Xi$ is interior to $\Lbox$. \end{lemma} \begin{proof} By the construction of $\mathfrak G(C)$ (Section~\ref{S:B2.grid}, Steps 1--3) the special planes are block planes, and in each direction the block intervals subdivide each special interval. (A1) A bond crossing $\Xi$ joins two fine cubes on opposite sides of the block plane $P$, which lie in different blocks. (A2) A block face in a plane $x_\mu=\mathrm{const}$ with $\mu\ne3$, or in a plane $x_3\ne z_0$, contains no plaquette of $\Xi$. A block face in $P$ is $I^1\times I^2\times\{z_0\}$ with block intervals $I^1$, $I^2$; since $r_1$, $r_1+R$, $t_1$, $t_1+T$ are block planes, either $I^1\times I^2\subseteq[r_1,r_1+R]\times[t_1,t_1+T]$, and the face lies in $\Xi$, or the interiors are disjoint. (A3) A block lies between consecutive block planes and $P$ is one; each side of $C$ lies on the intersection of $P$ with one of the block planes $x_1\in\{r_1,r_1+R\}$, $x_2\in\{t_1,t_1+T\}$. The last claim is Lemma~\ref{lem:3.3}(e). \qed \end{proof} \begin{definition}[Marks]\label{def:3.6} For a set $Y$ of sites, $F(Y)$ is the set of fine plaquettes that are faces of fine cubes contained in sites of $Y$. A site is \emph{marked} iff one of its fine cubes has a face in $D$. The set of marked sites is $\Mset=\Mset(D)$, and $N(P):=|P|$ for $P\subseteq\Mset$. Then $D\subseteq F(\Mset)$, and a plaquette of $D$ marks at most two sites, since it is a face of at most two fine cubes. \end{definition} \subsection{States, functionals and steps}\label{ssec:3.2}\label{ssec:3.3} For a set $Y$ of sites, we write $E(Y)$ for the set of bonds with both ends in $Y$, $\partial Y$ for the set of bonds of $E=E(\Lbox)$ with exactly one end in $Y$, and $V(A)$ for the set of endpoints of a bond set $A$. A bond \emph{touches} $Y$ if it has an end in $Y$. Bonds leaving $\Lbox$ are not in $E$, that is, the faces of the box are free. \begin{definition}[States]\label{def:3.7} A \emph{state} is a triple $\sigma=(\Hset,\Uf,S)$: the \emph{holes} $\Hset\subseteq\Lbox$, the \emph{filled blocks} $\Uf\subseteq\Hset$, a union of blocks, and the \emph{frozen bonds} $S\subseteq E(\Lbox\setminus\Hset)$, each crossing a block face and each of one \emph{kind}: $S=S_1\sqcup S_{\rm cut}$, seal bonds and cut bonds (Definitions~\ref{def:3.seal} and~\ref{def:3.17}). A block is \emph{surviving} if it is not contained in $\Uf$. A frozen bond that $\Hset$ grows to touch leaves $S$. Such a bond is \emph{converted}, and from then on it is an ordinary hole face. The \emph{slit graph} is $\Gp(\sigma):=\bigl(\Lbox\setminus\Hset,\ E(\Lbox\setminus\Hset)\setminus S\bigr)$, and $\mathcal R(\sigma):=S\cup\{b\in E:b\text{ touches }\Hset\}$ is the set of bonds missing from it. Along a history, that is, a finite sequence of states produced by the steps below (Definition~\ref{def:4.1}), the \emph{base} is $B_t:=\Mset\cup\Uf_t$, and the \emph{ever-frozen set} $\Sbar_t$ is the set of bonds frozen at some time $\le t$ that do not touch $B_t$, converted bonds included, each keeping its kind. \end{definition} \begin{definition}[Hull functional]\label{def:3.8} Let $S\subseteq E$ carry weights $w_b\in[0,2]$. For $X'\subseteq\Lbox$, \begin{equation}\label{eq:3.F} F^S_\lambda(X'):=|\partial X'|+\sum_{b\in S\cap E(\Lbox\setminus X')}w_b+\lambda|X'| . \end{equation} Every frozen bond has the \emph{hull weight} $w_b:=2-\eta_w=\tfrac{19}{10}$. \end{definition} \begin{lemma}\label{lem:3.9} Let $S\subseteq E$ carry weights $w_b\in[0,2]$. (a) $F^S_\lambda$ is the sum of the cut function of a graph on $\Lbox$ with nonnegative capacities and of a modular function; in particular it is submodular. (b) For every $Y\subseteq\Lbox$, the minimizers of $F^S_\lambda$ over $\{X':Y\subseteq X'\subseteq\Lbox\}$ are closed under union and intersection; the smallest, $H^S_\lambda(Y)$, is the source side of a minimum $s$\textendash$t$ cut in a network with nonnegative capacities. Bonds of $S$ that touch $Y$ may be added to $S$ or removed from it, with any weights, without changing $H^S_\lambda(Y)$. (c) $H^S_\lambda$ is extensive and idempotent: $Y\subseteq H^S_\lambda(Y)=H^S_\lambda(H^S_\lambda(Y))$. (d) Let $S$, $S'$ carry weights that agree on $S\cap S'$. If $Y\subseteq Y'$ and $\mathcal R(Y,S)\subseteq\mathcal R(Y',S')$, where $\mathcal R(Y,S):=S\cup\{b:b\text{ touches }Y\}$, then $H^S_\lambda(Y)\subseteq H^{S'}_\lambda(Y')$. In particular $H^S_\lambda$ is a closure operator for fixed $S$ and monotone in $S$. \end{lemma} \begin{proof} For (a), we write a frozen bond of weight $w\le2$ as a modular term plus a cut term of capacity $\frac{2-w}2\ge0$. Indeed, with $c_b(X'):=\mathbf 1[\text{exactly one end of }b=\{u,v\}\text{ in }X']$, the three cases give $w_b\mathbf 1[u,v\notin X']+c_b(X')=w_b-\tfrac{w_b}2(\mathbf 1[u\in X']+\mathbf 1[v\in X'])+\tfrac{2-w_b}2c_b(X')$ for $b\in S$, hence \begin{equation}\label{eq:4.Fcut} F^S_\lambda(X')=\sum_{b\in E}\kappa_b\,c_b(X')+\sum_{x\in X'}\mu(x)+\sum_{b\in S}w_b, \end{equation} with $\kappa_b:=1$ for $b\notin S$, $\kappa_b:=(2-w_b)/2$ for $b\in S$, and $\mu(x):=\lambda-\frac12\sum_{b\in S,\,b\ni x}w_b$. This proves (a) and explains why frozen bonds weigh at most $2$: weights above $2$ would give negative capacities. Parts (b) and (c) are the lattice and closure properties of the minimizers of a submodular function on $\{X'\supseteq Y\}$. The network has capacities $\kappa_b$ on the bonds, $\mu(x)^\pm$ to the sink or from the source, and $\infty$ from the source to $Y$. For the last assertion of (b) we observe that, on $\{X'\supseteq Y\}$, a bond of $S$ touching $Y$ contributes $1$ or $0$ whatever its weight. (d) By (b) we may add the bonds of $S\setminus S'$, which touch $Y'$, to $S'$ with their $S$-weights, so that $S\subseteq S'$. Then $g:=F^{S'}_\lambda-F^S_\lambda=\sum_{b\in S'\setminus S}w_b\mathbf 1[\text{no end of $b$ in }X']$ is non-increasing. With $M:=H^S_\lambda(Y)$ and $M':=H^{S'}_\lambda(Y')$, submodularity gives $F^S_\lambda(M\cap M')+F^{S'}_\lambda(M\cup M')\le F^S_\lambda(M)+F^S_\lambda(M')+g(M\cup M')\le F^S_\lambda(M)+F^{S'}_\lambda(M')$. Each term on the left is at least the corresponding one on the right. Hence $M\cap M'\supseteq Y$ is a minimizer, and $M\subseteq M'$. \qed \end{proof} \begin{definition}[Hull step]\label{def:3.10} In a history, the \emph{hull step} at time $t$ replaces the hole set by $H^{\Sbar_t}_\lambda(B_t)$, the hull from the base computed with the ever-frozen set and the hull weight, and converts the frozen bonds that touch it. A time $t$ (or its state) is \emph{hull-closed} if $\Hset_t=H^{\Sbar_t}_\lambda(B_t)$. \end{definition} \begin{definition}[Forced blocks and the regularization functional]\label{def:3.forced} For a block $Q$ let $3Q$ be the union of the at most $27$ blocks at block distance at most $1$ from $Q$, and $N(3Q):=|\Mset\cap3Q|$. The block $Q$ is \emph{forced} if $N(3Q)\ge\vartheta L_c^2$, where $\vartheta:=3\cdot10^{-5}$ is the \emph{forcing threshold}, and $\mathfrak B_F=\mathfrak B_F(D)$ is the set of forced blocks. Put $\lambda_U:=2c_s/\ell$ and $\mu_{\rm B}:=25$. For a set $\Sbar$ of bonds that cross block faces, each with the hull weight, and for pairs $(\Hset,\Uf)$ with $\Mset\subseteq\Hset\subseteq\Lbox$ and $\Uf\subseteq\Hset$ a union of blocks, \begin{equation}\label{eq:3.E} \Ereg^{\Sbar}(\Hset,\Uf):=F^{\Sbar}_\lambda(\Hset)-(\lambda-\lambda_U)|\Uf|+\mu_{\rm B}L_c^2\,\#\{Q\in\mathfrak B_F:\ Q\not\subseteq\Uf\}. \end{equation} \end{definition} Filling a block $Q$ costs at most $|\partial Q|+\lambda_U|Q|<24.6L_c^2<\mu_{\rm B}L_c^2$, filling the forced blocks costs at most $27\cdot24.6/\vartheta$ per mark, and filling a removal cell $\Delta$ costs at most $w_*$ (we give the arithmetic behind these bounds in Section~\ref{S:B1.forced}). In Lemma~\ref{lem:4.forced} we deduce from the first bound that forced blocks are filled. The last bound implies that a minimizer fills each cell whose insulation exceeds $w_*$, as Theorem~\ref{thm:4.30} shows. \begin{lemma}\label{lem:3.minE} Let $\Sbar$ be as in Definition~\ref{def:3.forced}. (a) $\Ereg^{\Sbar}$ is, up to a constant, the cut function of an $s$\textendash$t$ network with nonnegative capacities; its minimizers are closed under componentwise union and intersection, and the smallest one, $(\Hset^*,\Uf^*)$, is the source side of a minimum cut, a deterministic function of $\Mset$ and $\Sbar$. (b) $\Hset^*=H^{\Sbar}_\lambda(\Mset\cup\Uf^*)$. (c) The smallest minimizer is componentwise nondecreasing in $\Sbar$. \end{lemma} \begin{proof} (a) We add to the network of Lemma~\ref{lem:3.9}(b) for $F^{\Sbar}_\lambda$ (with $\infty$ from the source to the marks) a node $u_Q$ per block $Q$, with capacity $\infty$ from $u_Q$ to each site of $Q$ and $(\lambda-\lambda_U)|Q|+\mu_{\rm B}L_c^2\mathbf 1[Q\in\mathfrak B_F]$ from the source to $u_Q$. Putting $u_Q$ on the source side means $Q\subseteq\Uf$. A finite cut has $\Mset\cup\Uf\subseteq\Hset$, and since the blocks tile $\Lbox$ by (G2), its capacity is $\Ereg^{\Sbar}(\Hset,\Uf)+(\lambda-\lambda_U)|\Lbox|$ up to the constant of \eqref{eq:4.Fcut}. The capacities are nonnegative, as $\lambda_U<\lambda$. Hence the minimum cuts form a lattice, and the smallest source side is the set reachable from the source in the residual network of a maximum flow. (b) For fixed $\Uf^*$, $\Ereg^{\Sbar}(\cdot,\Uf^*)-F^{\Sbar}_\lambda$ is constant on $\{\Hset\supseteq\Mset\cup\Uf^*\}$, and a smaller minimizer there would give a smaller joint minimizer. (c) Let $\Sbar\subseteq\Sbar'$, and $m$, $m'$ the smallest minimizers of $\Ereg:=\Ereg^{\Sbar}$ and $\Ereg':=\Ereg^{\Sbar'}=\Ereg+g$, $g(\Hset,\Uf):=\sum_{b\in\Sbar'\setminus\Sbar}w_b\mathbf 1[\text{no end of $b$ in }\Hset]$, which does not increase when the pair grows. Minimality of $m'$ for $\Ereg'$ and $g(m\cup m')\le g(m')$ give $\Ereg(m')\le\Ereg(m\cup m')$. Combined with submodularity, this yields $\Ereg(m\cap m')\le\Ereg(m)$. Thus $m\cap m'$ is a minimizer of $\Ereg$, and $m\subseteq m'$. \qed \end{proof} For a state $(\Hset,\Uf,S)$ and a bond set $A$, we define the \emph{two-sided insulation} by $\mathcal A_2(A):=\#\{b\in A:\ b\text{ has exactly one end in }\Hset\}+2\,\#(S\cap A)$, in which a frozen bond counts $2$ whatever its weight. \begin{definition}[Removal]\label{def:3.11} Let $\sigma$ be a hull-closed state with base $B$. For a removal cell $\Delta\not\subseteq B$ put $w(\Delta):=\mathcal A_2(E(\Delta))$, with $E(\Delta)$ the set of bonds with both ends in $\Delta$ and $S$ the current frozen set. The \emph{removal step} removes, all at once, every removal cell $\Delta\not\subseteq B$ with $w(\Delta)>w_*:=(1+c_s)|\partial\Delta|$, where $c_s\in[3.3\cdot10^{-7},10]$ is the removal slack: $\Hset:=\Hset\cup\Delta$, $\Uf:=\Uf\cup\Delta$, and the frozen bonds touching $\Delta$ are converted. A hull step follows every removal step. \end{definition} \begin{definition}[Cells, interfaces]\label{def:3.12} Let $\sigma$ be a state, $\Gp=\Gp(\sigma)$, and for $\hat Q\in\Qc$ let $\Gp\cap\hat Q$ be the subgraph of $\Gp$ induced on $\hat Q\setminus\Hset$, with components $\hat\Kc$. For a block $Q\not\subseteq\Hset$, the \emph{cells} of $Q$ are the sets $\Omega_{Q,\hat\Kc}:=Q\cap\hat\Kc$ for the components $\hat\Kc$ of $\Gp\cap\hat Q(Q)$ that meet $Q$. For a block face $F'$ between blocks $Q,Q'\not\subseteq\Hset$, an \emph{interface} $I$ is a nonempty set consisting of all $\Gp$-bonds across $F'$ that join a given cell of $Q$ to a given cell of $Q'$, and $F(I):=F'$. Frozen bonds and hole faces belong to no interface. A bond $b\in I$ is \emph{good} if its tube $\tau_b$ is a $\Gp$-path (its sites lie in $\Lbox\setminus\Hset$ and its bonds are $\Gp$-bonds), and $\good(I)$ is the number of good bonds of $I$. \end{definition} \begin{definition}[Weak links]\label{def:3.13} Let $\hat Q\in\Qc$, $\hat\Kc$ a component of $\Gp\cap\hat Q$, and $f_{\Gp}(A,A')$ the number of $\Gp$-bonds between disjoint site sets $A$, $A'$. For $\chi'>0$, $A\subseteq\hat\Kc$ is a \emph{$\chi'$-weak link} of $\hat\Kc$ if $0<|A|\le|\hat\Kc|/2$ and $f_{\Gp}(A,\hat\Kc\setminus A)<\chi'|A|/k_P$. The \emph{weak-link constants} are $\chi:=2.0\cdot10^{-8}$ and $\chi_{\rm N3}:=\chi/3$. \end{definition} Weak links are excluded by Theorem~\ref{thm:4.rest}(c). \begin{definition}[Auxiliary material and the seal]\label{def:3.seal} Let $\sigma=(\Hset,\Uf,S)$ be a hull-closed state with ever-frozen set $\Sbar$, let $j$ be a surviving block, and put $V_j:=j\setminus\Hset$. The \emph{auxiliary functional} $F^{\rm aux}_j$ is $F^{\Sbar\cup\partial j}_\lambda$, every bond of $\partial j$ counted as frozen with the hull weight, on the sets $Y$ with $\Hset\subseteq Y\subseteq\Hset\cup V_j$. Its minimizers are closed under union and intersection (Lemma~\ref{lem:3.9}(a)). The smallest one is $M_j$, and $A_j:=M_j\setminus\Hset$ is the \emph{auxiliary material} of $j$. The \emph{seal step} freezes every $\Gp$-bond that crosses a face between two surviving blocks and has an end in $\bigcup_jA_j$. These bonds form $S_1$ and carry the hull weight. A hull step follows every seal step. \end{definition} The set $M_j$ is obtained from one maximum flow inside $j$. By Lemma~\ref{lem:4.thin}, a block with a thin piece, attached to the rest of the block by few bonds however large its contacts across the faces of $j$, has auxiliary material, and Lemma~\ref{lem:4.aux} shows that after the seal the next hull step fills the auxiliary material. \subsection{Phase 1 and Phase 2}\label{ssec:3.4}\label{ssec:3.5} \begin{definition}[Phase~1 and the terminal properties]\label{def:3.14}\label{def:3.15} \emph{Phase~1} starts from the state $(\Mset,\emptyset,\emptyset)$ with $\Sbar:=\emptyset$ and repeats the following \emph{outer step} until an outer step changes neither $\Uf$ nor $\Sbar$, that is, until both loops are at rest: (1) $(\Hset,\Uf):=$ the smallest minimizer of $\Ereg^{\Sbar}$ (Lemma~\ref{lem:3.minE}); (2) with $\Uf$ fixed, as long as some surviving block has auxiliary material, perform a seal step, add its bonds to $\Sbar$ (in Phase~1, $\Sbar$ is not pruned: it keeps the seal bonds that touch the base, unlike $\Sbar_t$ of Definition~\ref{def:3.7}; the hulls are the same, as the two sets differ only by bonds touching $B_t$, but step (1) and Theorem~\ref{thm:4.30} use this unpruned set), and perform the hull step $\Hset:=H^{\Sbar}_\lambda(\Mset\cup\Uf)$. Its final state $(\Hset^1,\Uf^1,S^1)$, with $S^1:=\Sbar\cap E(\Lbox\setminus\Hset^1)$, base $B^1:=\Mset\cup\Uf^1$ and ever-frozen set $\Sbar^1:=\Sbar$, is the \emph{Phase-1 output}. The \emph{terminal properties} of a hull-closed state with base $B$, frozen set $S$ and ever-frozen set $\Sbar$ are: (S-hull) $\Hset=H^{\Sbar}_\lambda(B)=H^S_\lambda(\Hset)$; (S-rem) no removal cell $\Delta\not\subseteq B$ has $w(\Delta)>w_*$; (S-face) every interface $I$ has $\good(I)\ge c_g|F(I)|$; (S-plug) no component of any $\Gp\cap\hat Q$, $\hat Q\in\Qc$, has a $\chi_{\rm N3}$-weak link. \end{definition} We show in Theorem~\ref{thm:4.30} that Phase~1 terminates and that its output is the smallest minimizer of $\Ereg^{\Sbar^1}$, has no auxiliary material and no frozen bond, and has the four terminal properties. As $\Sbar$ does not change in the last outer step, the output is the pair returned by step (1) of that outer step, and step (2) finds no auxiliary material. \begin{definition}[Scale cost, fragmented splits, the cut rule, Phase~2]\label{def:3.17} Let $\sigma$ be a state with slit graph $\Gp$ and cells as in Definition~\ref{def:3.12}. Every site of $\Lbox\setminus\Hset$ lies in exactly one cell, and every cell in one component of $\Gp$. A \emph{cell set} is a union $U$ of cells. Its cut $\partial_{\Gp}U$ is the set of $\Gp$-bonds with exactly one end in $U$, and $|\partial_{\Gp}U|$ is its number of (coarse) bonds. $U$ is \emph{c-connected} if it is connected in the graph on $U$ whose edges are the $\Gp$-bonds inside $U$ and the pairs of sites in a common cell. The distance of two interfaces is the least Euclidean distance (fine units) between the coarse faces dual to their bonds. For a finite set $\mathcal B$ of interfaces and $s>0$, $\varphi_s(\mathcal B)$ is the number of classes of the equivalence relation generated by ``distance $\le s$''. The \emph{scale cost} is $J(U):=\sum_{j\ge0:\ 2^jL\ge R_{\rm NF}}(\varphi_{2^jL}(\partial_{\Gp}U)-1)$, where $R_{\rm NF}:=L\,e^{c_3^{\rm NF}F_{\rm unit}}$ (fine), $F_{\rm unit}:=\sigma_FL^2$ with $\sigma_F$ the wall tension \eqref{eq:7.sigmaF}, and $\theta_{\rm NF},c_3^{\rm NF}>0$ are the cut-rule constants. A pair $(\Kc,U)$ is \emph{fragmented} if $\Kc$ is a component of $\Gp$, $U\subsetneq\Kc$ is a nonempty c-connected cell set, $\Kc\setminus U$ is c-connected, and $J(U)>\theta_{\rm NF}F_{\rm unit}|\partial_{\Gp}U|/L_c^2$. \emph{The cut rule} acts at a hull-closed state at which no removal cell is over threshold and no surviving block has auxiliary material: a firing freezes $\partial_{\Gp}U$ for the first fragmented pair $(\Kc,U)$ in a fixed total order of the sets $U\subseteq\Lbox$ that depends only on $C$ and the constants (for instance: order the coarse sites lexicographically by their coordinates, and let $U\prec U'$ iff the least site of $U\triangle U'$ lies in $U$; any such order works, since the proofs below use only that a fragmented pair is cut, never which one; see Section~\ref{S4-S:S4.1}); these bonds form $S_{\rm cut}$, have the hull weight and count $2$ in $\mathcal A_2$ and in $w(\Delta)$; nothing is filled. \emph{Phase~2} starts from the Phase-1 output, with base $B^1$ and ever-frozen set $\Sbar^1$ minus the bonds touching $B^1$, the ever-frozen set of Definition~\ref{def:3.7} (the unpruned $\Sbar^1$ is used only in Theorem~\ref{thm:4.30} and Section~\ref{ssec:4.9}), and repeats the following pass until a full pass changes nothing: (1) a hull step; (2) a removal step, then a hull step; (3) a seal step, then a hull step; (4) if (1) to (3) changed nothing, one firing of the cut rule, then a hull step. Its terminal state $(\Hset,\Uf,S)$ is the output of the construction. \end{definition} The scale cost is well defined, since $\partial_{\Gp}U$ is a union of whole interfaces (Lemma~\ref{lem:3.18}(a)). The trigger of the cut rule uses the coarse cut size, and the non-fragmentation condition that the rule enforces implies the same condition with the fine cut size. Since a coarse face carries at least $r_0^2$ fine plaquettes, we have $|\partial_{\Gp}U|_{\rm fine}/L^2\ge|\partial_{\Gp}U|/L_c^2$, and hence a split fragmented with the fine cut size is fragmented. \begin{proposition}\label{prop:3.19} Phase~2 terminates. Its terminal state has (S-hull), (S-rem), (S-face), (S-plug) on the global slit graph and (S-NF): for every component $\Kc$ of the final $\Gp$ and every nonempty c-connected cell set $U\subsetneq\Kc$ with $\Kc\setminus U$ c-connected, \begin{equation}\label{eq:3.NFstar} J(U)\le\theta_{\rm NF}\,F_{\rm unit}\,|\partial_{\Gp}U|/L_c^2 .\tag{$\mathrm{NF}_*$} \end{equation} If the Euclidean diameter of $\Lbox$ (fine units) is at most $R_{\rm NF}$, then Phase~2 makes no firing and the output is the Phase-1 output. \end{proposition} \subsection{The output and the regularization theorem}\label{ssec:3.6} The output $(\Hset,\Uf,S)$, with base $B=\Mset\cup\Uf$ and $S=S_1\sqcup S_{\rm cut}\subseteq E(\Lbox\setminus\Hset)$, defines the slit domain $\Gp=\Gp(D):=\Gp(\Hset,S)$, the free set $\Fset$ of \eqref{eq:2.FF} and the min-cut area $A_{\Gp}(C)$ of Definition~\ref{def:2.11}. The fine slit graph is that of Section~\ref{ssec:2.5}, and cells and interfaces are those of Definition~\ref{def:3.12}. The \emph{cell graph} of a component $\Kc$ of $\Gp$ has the cells in $\Kc$ as vertices and the interfaces as edges. The weight of a cell $\Omega$ is $|\Omega|_{\rm fine}$, its number of fine cubes. The $\Uc$-cells are the parts of a partition of each cell graph given by Lemma~\ref{lem:4.19} with $V=V_{\Uc}:=\tilde m^{-3}$, one fixed choice being made for each $\Gp$. A component of weight $