% Copyright (c) 2026 Huang Man. Author-owned work: CC BY 4.0. % https://creativecommons.org/licenses/by/4.0/ % Options for packages loaded elsewhere \PassOptionsToPackage{unicode}{hyperref} \PassOptionsToPackage{hyphens}{url} \PassOptionsToPackage{dvipsnames,svgnames,x11names}{xcolor} \documentclass[ 11pt, a4paper, ]{article} \usepackage{xcolor} \usepackage[margin=2.2cm]{geometry} \usepackage{amsmath,amssymb} \setcounter{secnumdepth}{-\maxdimen} % remove section numbering \usepackage{iftex} \ifPDFTeX \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{textcomp} % provide euro and other symbols \else % if luatex or xetex \usepackage{unicode-math} % this also loads fontspec \defaultfontfeatures{Scale=MatchLowercase} \defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1} \fi \usepackage{lmodern} \ifPDFTeX\else % xetex/luatex font selection \setmainfont{lmroman10-regular.otf}[BoldFont=lmroman10-bold.otf,ItalicFont=lmroman10-italic.otf,BoldItalicFont=lmroman10-bolditalic.otf] \setmonofont{lmmono10-regular.otf} \setmathfont{latinmodern-math.otf} \fi % Use upquote if available, for straight quotes in verbatim environments \IfFileExists{upquote.sty}{\usepackage{upquote}}{} \IfFileExists{microtype.sty}{% use microtype if available \usepackage[]{microtype} \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts }{} \makeatletter \@ifundefined{KOMAClassName}{% if non-KOMA class \IfFileExists{parskip.sty}{% \usepackage{parskip} }{% else \setlength{\parindent}{0pt} \setlength{\parskip}{6pt plus 2pt minus 1pt}} }{% if KOMA class \KOMAoptions{parskip=half}} \makeatother \setlength{\emergencystretch}{3em} % prevent overfull lines \providecommand{\tightlist}{% \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} \usepackage{amsmath,amssymb} \usepackage{microtype} \allowdisplaybreaks \setlength{\emergencystretch}{2em} \AtBeginDocument{\hypersetup{pdftitle={An exact moment criterion for survival under renewed spatial catastrophes},pdfauthor={HUANG Man}}} \usepackage{bookmark} \IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available \urlstyle{same} \hypersetup{ pdftitle={An exact moment criterion for survival under renewed spatial catastrophes}, colorlinks=true, linkcolor={blue}, filecolor={Maroon}, citecolor={Blue}, urlcolor={blue}, pdfcreator={LaTeX via pandoc}} \title{An exact moment criterion for survival under renewed spatial catastrophes} \author{} \date{} \begin{document} \maketitle \textbf{HUANG Man}\\ Independent Researcher, Shanghai, China\\ Corresponding author: huangman@technocentrics.com\\ ORCID: https://orcid.org/0009-0003-3809-5105 \textbf{Preprint \textbar{} Version 1 \textbar{} 3 November 2026} \textbf{Repository preparation (10 November 2026).} Mathematical text and references follow Zenodo v1, \url{https://doi.org/10.5281/zenodo.23911961}. This package uses portable TeX fonts and adds provenance disclosures. The original source and finite diagnostics remain available at the same DOI. \subsection{Abstract}\label{abstract} We prove an exact moment criterion for survival in a binary growth process subject to independently renewed spatial catastrophes. Occupied sites spread through a finite set of allowed increments on the integer lattice, and Poisson catastrophes with unbounded cube sizes empty all occupied sites in each affected cube. Under finite one-site hitting intensity, a positive finite-seed survival phase exists if and only if the cube-size measure has a finite moment of order one more than the spatial dimension. The divergent endpoint yields extinction at every positive intensity. The proof centres on a one-dimensional positive-phase construction for connected interval errors under a finite second moment. Extremal-path resampling gives a conditional correlation inequality, which controls exact interval insertion counts and a finite-history potential. Coordinate restriction and envelope covering then establish the criterion in every dimension. We also prove the bulk-to-seed reversal identity and extend the criterion to uniformly fat shapes. For the one-sided one-dimensional rule, the stationary hole density has a quadratic dilute expansion under the same second-moment condition. Its excess over independently eroded Poisson events is strictly positive at every positive intensity, with an explicit quadratic coefficient in the dilute limit. \textbf{Keywords:} interacting particle systems; spatial catastrophes; survival criterion; dependent percolation; probabilistic cellular automata \textbf{MSC2020:} 60K35; 60G55; 82B43. \subsection{1. Introduction}\label{introduction} Spatially extended catastrophes couple the loss of sites that could otherwise sustain one another. We study how their size distribution controls survival in a binary growth process. At each integer time, occupied sites spread by a finite additive recovery rule. An independent Poisson field of cube-shaped catastrophes then empties the affected sites. This update combines finite-speed recovery with the spatial dependence created by a shared catastrophe. Destructive geometry affects survival in contact processes with cube and hyperplane catastrophes {[}8{]}. Poisson hail connects spatial extent to stability through jobs that occupy regions and retain service requirements {[}1{]}. Foss, Konstantopoulos and Mountford establish small-arrival-rate stability under moments of order dimension plus one plus a positive margin, together with lower-moment instability examples {[}3{]}. Mountford and Wang further analyse power-law stability regions {[}10{]}. These results supply the space-time moment perspective for the binary survival problem studied here. For the binary survival problem, we obtain the exact moment order \(d+1\): finiteness gives a low-intensity survival phase, and divergence gives extinction at every positive intensity. We prove a necessary-and-sufficient moment condition that includes the divergent endpoint. For cubes in dimension \(d\), the standing local-finiteness assumption is a finite \(d\)th size moment, while a nonempty low-intensity survival phase exists exactly when the \((d+1)\)st size moment is finite. A covering catastrophe can kill every possible descendant of a finite seed, giving necessity. Sufficiency reduces to the one-dimensional process through a coordinate line. The central estimate establishes a positive phase for unbounded interval errors under a finite second moment. A bounded law and a borderline heavy-tailed law can have the same one-site hitting probabilities while lying on opposite sides of this existence criterion. Stability under local noise supplies the methodological starting point. Toom constructs stable local automata under small independent update noise {[}12{]}. Gács and Törmä treat errors satisfying an exponential bound on every prescribed finite set of error sites {[}4{]}. Hartarsky reproves the one-dimensional eroder criterion under sufficiently small independent deaths (Theorem 1.9) {[}5{]}. Hartarsky and Toninelli give exponential tail bounds for space-time zero-state cluster diameters in the upper invariant trajectory under small independent deaths (Corollary 9.5) {[}6{]}, and Swart, Szabó and Toninelli develop Toom-contour estimates for local noisy dynamics {[}11{]}. Polynomial interval tails can violate this exponential error bound. For example, in one dimension take size masses \(\nu_r=Cr^{-4}\) with \(C>0\). The second size moment is finite, yet at any positive intensity \(\rho\) a specified interval of integer length \(L\ge2\) is emptied with probability at least \(1-e^{-\rho CL^{-4}}\) (a single anchor gives this bound; summing over all anchors gives a covering intensity of order \(L^{-2}\)), which eventually exceeds \(\epsilon^L\) for every fixed \(0<\epsilon<1\). The construction below supplies a positive-phase estimate uniform in the interval-length cutoff. Hilário, Sá, Sanchis and Teixeira establish directed survival in a layered environment with geometric defects and a common random edge-opening propensity within each forward layer, for sufficiently weak defects and sufficiently high base opening probability (Theorem 8.2) {[}7{]}. In our process, spatial dependence is generated by connected Poisson deletion events of unbounded sizes. To handle those events, we adapt the alternating leftmost/rightmost-path resampling argument of van den Berg, Häggström and Kahn {[}2{]}. Connectedness permits a left/right classification relative to a surviving path. The resulting conditional correlation bound controls an exact insertion count, and a finite-history potential gives the positive phase before a stationary limit is used. Section 2 states the model, the moment criterion and the reversal identity. Section 3 develops the one-dimensional positive-phase proof; Section 4 combines it with covering and path counting to prove the cube criterion. Sections 5 and 6 give the stationary density expansion and the shape extension. \subsection{2. Model and main results}\label{model-and-main-results} \subsubsection{2.1. Space, size measure and catastrophe field}\label{space-size-measure-and-catastrophe-field} Let \(d\) be a positive integer. Sites are the points of the lattice \(\mathbb Z^d\), and \(0=(0,\ldots,0)\) denotes its origin. Write \(e_1=(1,0,\ldots,0)\) for the first coordinate unit vector. Fix a finite set \(G\subset\mathbb Z^d\) containing \(0\) and \(e_1\). The elements of \(G\) are the allowed displacement vectors in one growth step. In particular, the displacement \(0\) allows occupancy to persist at its present site before the next catastrophe. Let \(\mathcal R=\{2,3,\ldots\}\), equipped with the power-set sigma-algebra, be the space of catastrophe sizes. Fix a nonnegative measure \(\nu\) on \(\mathcal R\) and write \(\nu_r=\nu(\{r\})\). The measure is not required to be a probability measure. Define its moments, allowing the value \(+\infty\), by \[ S_j=\sum_{r\in\mathcal R}r^j\nu_r,\qquad j\in\mathbb Z_{\ge0}. \tag{1} \] Our standing model assumption is \(00:s_\infty^G(\rho)>0\}. \tag{3} \] The limit exists because extinction is absorbing, so the survival events decrease with \(t\). We set the supremum of the empty set equal to zero. Poisson thinning couples all intensities so that increasing \(\rho\) adds catastrophes and decreases occupancy; consequently \(s_\infty^G(\rho)\) is nonincreasing. Translation invariance gives the same single-seed survival probability from any site. \subsubsection{2.3. The upper stationary field}\label{the-upper-stationary-field} Use the two-sided time environment already defined. At a fixed observation time \(t\), the sets \(A_{m,t}^{\mathbb Z^d,G}\) decrease as \(m\) tends to \(-\infty\). Indeed, the process started at an earlier time has a subset of \(\mathbb Z^d\) when it reaches the later initial time, and the update map preserves set inclusion. Define \[ A_t^{*,G}=\bigcap_{m0\quad\Longleftrightarrow\quad S_{d+1}<\infty. \tag{4} \] If \(S_{d+1}<\infty\), set \(A_*^{(d)}=(S_d+S_{d+1})/2\). Then \[ \frac1{32A_*^{(d)}}\le\rho_c^G\le\frac{\log|G|}{S_d}. \tag{5} \] If \(S_{d+1}=\infty\), then for every \(\rho>0\) and every deterministic finite initial set \(B\), the process \(A_t^{B,G}\) reaches the empty set in finite time almost surely. \textbf{Lemma 2.2 (reversal).} For every \(\rho\ge0\) and integer \(T\ge1\), \[ \Pr(0\in A_T^{\mathbb Z^d,G})=e^{-\rho S_d}s_{T-1}^G(\rho),\qquad \theta_d^G(\rho)=e^{-\rho S_d}s_\infty^G(\rho). \tag{6} \] \textbf{Proof.} The terminal site \(0\) must avoid \(D_T\). This event has probability \(e^{-\rho S_d}\) and is independent of all earlier layers. After removing this terminal test, reverse and reflect a path \((x_0,\ldots,x_T)\) ending at \(x_T=0\) by putting \(y_k=-x_{T-k}\) for \(0\le k\le T-1\). Its initial site is \(y_0=0\), and its increments remain in \(G\), because \(y_k-y_{k-1}=x_{T-k+1}-x_{T-k}\). Its tested layers are the reflected layers \(-D_{T-1},\ldots,-D_1\), in that order. Reflection sends \(B(l,r)\) to \(B(-l-(r-1)\mathbf1,r)\), where \(\mathbf1=(1,\ldots,1)\). This is a bijection of anchors at each size. Reflection and reversal of the independent layers therefore preserve their joint law. Conversely, a single-seed path with \(T-1\) tests determines \(x_1,\ldots,x_T\); taking \(x_0=x_1\) supplies the remaining initial step because \(0\in G\) and the initial state is fully occupied. Thus the two path-existence events agree under the transformation, proving the finite-time identity. The limits in (3) and Section 2.3 give the density identity. For completeness, survival at every finite time is equivalent to an infinite open path from the seed. The tree of finite open paths is finitely branching because \(G\) is finite. If it has a vertex at every depth, König's lemma gives an infinite branch. \(\square\) \textbf{Corollary 2.3 (power-law boundary and equal local exposure).} If \(\nu_r=Cr^{-p}\) with \(C>0\), the standing assumption holds exactly when \(p>d+1\). Within that domain, \[ \rho_c^G=0\quad(d+10\quad(p>d+2). \tag{7} \] There are two size measures with the same value of \(S_d\), and hence the same one-site no-hit probability at every intensity, for which the answers to whether \(\rho_c^G>0\) differ. \textbf{Proof.} The series defining \(S_j\) converges exactly when \(p>j+1\). At \(p=d+2\), the series for \(S_{d+1}\) is harmonic. To obtain equal exposure, rescale a nonzero bounded-support measure and the measure with masses \(r^{-(d+2)}\) to give them the same \(S_d\). Theorem 2.1 yields a positive critical intensity for the former and zero for the latter. \(\square\) \subsection{3. Positive phase for interval catastrophes}\label{positive-phase-for-interval-catastrophes} \subsubsection{3.1. Hole process and finite histories}\label{hole-process-and-finite-histories} For this section, \(d=1\) and \(G=\{0,1\}\). All spatial intervals have integer endpoints: \([a,b]\) means \(\{a,a+1,\ldots,b\}\) when \(a\le b\), and the empty set otherwise. Starting from full occupancy, define the hole process by \(H_t=\mathbb Z\setminus A_t^{\mathbb Z,\{0,1\}}\) for integers \(t\ge0\). A site is called closed if it is a hole and open if it is occupied. For a set \(H\), put \(H+1=\{x+1:x\in H\}\). Taking complements in (2) gives \(H_0=\varnothing\) and \[H_{t+1}=E(H_t)\cup D_{t+1},\qquad E(H)=H\cap(H+1).\tag{8}\] Replacing the interval field by independent single-site errors gives the classical Stavskaya rule {[}13,14{]}. Here the independent error coordinates are interval types. Let \(\beta_t=\Pr(0\in H_t)\), \(u_t=1-\beta_t\), and \(h_{j,t}=\Pr([0,j-1]\subset H_t)\). The one-time laws of \(H_t\) increase with \(t\): they can be represented by the nested fields \(\mathbb Z\setminus A_{-t,0}^{\mathbb Z,\{0,1\}}\) of Section 2.3. This is a coupling of different initial times at one observation time; successive fields along a forward sample path need not be nested. Their limit is \(H^*\); write \(\beta=\Pr(0\in H^*)=\lim_t\beta_t\) and \(u=1-\beta\). Define \[V(r)=\frac{r(r+1)}2,\qquad A_*=\sum_r\nu_rV(r)=\frac{S_2+S_1}2,\qquad c=\rho A_*.\] \textbf{Proposition 3.1 (uniform finite-history estimate).} If \(S_2<\infty\) and \(0\pi(t),\quad x\in I_e\quad\Longrightarrow\quad e\in\mathcal C_\pi\ \hbox{or}\ e\in\mathcal R_\pi,\] with the corresponding left-hand implication. An interval in \(\mathcal C_\pi\) is absent whenever \(\pi\) is open. A right update resamples all variables in \(\mathcal R_L\) from their original independent laws. It preserves \(L\). Indeed, if an updated open path \(p\) went strictly left of \(L\) in any row, then \(\min(p,L)\) would be an open path everywhere at or left of \(L\). Its sites are unaffected by the resampling, so it would have been open before the update, contradicting leftmostness. Also, \(\{L=\pi\}\) depends only on the variables in \(\mathcal L_\pi\cup\mathcal C_\pi\). Any path extending left of \(\pi\) can be replaced by its minimum with \(\pi\), so its existence can be tested using just these variables. Conditional on these variables and \(L=\pi\), the variables in \(\mathcal R_\pi\) therefore retain their product law. Resampling them preserves \(\mu\). A left update resamples \(\mathcal L_R\); the pointwise maximum argument preserves \(R\) and gives the same invariance. Alternating right and left updates gives a finite Markov chain with invariant law \(\mu\). Each state can reach the all-1 configuration in one complete right-then-left step with positive probability. For this event, set every resampled absence variable to 1. Every allowed path to 0 lies at or left of the vertical path at 0. After the right step, an interval hitting that vertical path either intersects the old \(L\) and was already absent, or is strictly right of \(L\) and has just been removed. The vertical path is consequently open and is the rightmost path. The subsequent left step removes every type strictly left of it; types intersecting it are absent and types to its right were resampled in the first step. All types are now absent. Since \(\mu\) is invariant and positive on every state of \(\Gamma\), all states are recurrent. Their common access to the all-1 state makes the chain irreducible. Each state has a positive self-loop probability, so the chain is aperiodic and converges to \(\mu\). \textbf{Step 2: comparison, common updates and coordinate monotonicity.} Compare configurations in \(\Gamma\) by writing \(\widehat\omega\preceq\omega\) when \[\widehat L\le L,\quad\widehat R\le R\quad\hbox{rowwise},\] \[\widehat Y_e\ge Y_e\quad(e\in\mathcal L_{\widehat L}),\qquad \widehat Y_e\le Y_e\quad(e\in\mathcal R_R).\] The remaining type variables are unrestricted. This is a comparison relation between two configurations; no order on the unrestricted coordinates is imposed. To define all random updates on a fixed product space, draw independent Bernoulli variables \(\xi_{n,e}\) for every update number \(n\) and every type \(e\in\mathcal E\), with \(\Pr(\xi_{n,e}=1)=p_e\). At a right update, use \(\xi_{n,e}\) only for \(e\in\mathcal R_L\); at a left update, use it only for \(e\in\mathcal L_R\). Ignore the other entries of that row of the array. Thus the set of entries used can depend on the current state, but the underlying independent array is fixed in advance. In two chains driven by a common array, any type resampled in both chains receives the same value. For a common right update, put \(\alpha=\widehat L\le L=\gamma\). The two leftmost paths remain fixed. If \(p\) is open in updated \(\widehat\omega\), then \(\max(p,\gamma)\) is open in updated \(\omega\). At a row where \(p(t)>\gamma(t)\), an interval covering that site either intersects \(\gamma\) and is absent, or lies strictly right of \(\gamma\). In the latter case it lies right of both \(\alpha\) and \(\gamma\) and has the same resampled variable in both configurations. At every other row the maximum follows the open path \(\gamma\). Hence \(\widehat R_{\rm new}\le R_{\rm new}\). Variables strictly left of \(\alpha\) are untouched. Every type strictly right of \(R_{\rm new}\) is right of \(\gamma\), so its variables were identically resampled. These observations preserve all three comparison conditions. For a common left update, put \(\alpha=\widehat R\le R=\gamma\). The two rightmost paths remain fixed. If \(p\) is open in updated \(\omega\), then \(\min(p,\alpha)\) is open in updated \(\widehat\omega\). At a row where \(p(t)<\alpha(t)\), an interval covering that site either intersects \(\alpha\) and is absent, or is strictly left of both \(\alpha\) and \(\gamma\) and has a common resampled variable. At other rows the minimum follows \(\alpha\). Thus \(\widehat L_{\rm new}\le L_{\rm new}\). Every type strictly left of \(\widehat L_{\rm new}\) is left of \(\alpha\) and was identically resampled. Types strictly right of \(\gamma\) are untouched. The comparison again persists. Changing a single fresh right-update variable from 1 to 0, while keeping the previous state and all other fresh variables fixed, produces a configuration further left in this comparison. The leftmost path is preserved, the rightmost path can only move left, and all changed absence variables have decreased. Changing a single fresh left-update variable from 0 to 1 likewise moves the configuration left: the rightmost path is preserved, the leftmost path can only move left, and the changed absence variable has increased. Common subsequent updates preserve these comparisons. More explicitly, change one entry \(\xi_{n,e}\) in a fixed finite array of updates. The two histories agree through update \(n-1\), so their resampling sets at update \(n\) are identical. If \(e\) is outside that set, the change has no effect. If it is inside, the single-update comparison just proved applies. Inducting over subsequent common updates proves the stated monotonicity of the terminal configuration, even though their later resampling sets can differ. The comparison orders terminal reachability on each side of 0. If a path to \(x<0\) is open in \(\omega\), its minimum with \(\widehat L\) ends at \(x\) and is open in \(\widehat\omega\). At a site strictly left of \(\widehat L\), a covering interval either intersects \(\widehat L\) and is absent, or belongs to \(\mathcal L_{\widehat L}\), where the absence variables satisfy the required inequality. Thus left reachability increases as a configuration moves left. For \(x>0\), the maximum of a path in \(\widehat\omega\) with \(R\) gives an open path in \(\omega\), using \(\mathcal R_R\) in the same way. Right reachability therefore decreases as a configuration moves left. \textbf{Step 3: association and limits.} Start the chain at the deterministic all-1 configuration. After any finite number of updates, a left increasing hole function decreases with each left-update absence variable and increases with each right-update absence variable. A right increasing hole function has the opposite monotonicities. Replace every right-update coordinate \(\xi_{n,e}\) by \(1-\xi_{n,e}\), including the entries that a given trajectory ignores. In the resulting independent ordered coordinates, the left function is decreasing and the right function is increasing. The product association inequality therefore gives nonpositive covariance. Convergence of the finite chain to \(\mu\) proves the conditional inequality for the finite model. For the infinite model, fix the time horizon \(T\) and the finitely many terminal coordinates on which the functions depend. Their backward cones contain finitely many sites. Retain types of length at most \(K\) that intersect these cones; only finitely many types remain. The total mean of omitted types hitting the cones is bounded by \[C\rho T\sum_{r>K}\nu_r(r+C)\longrightarrow0,\] for a finite constant \(C\) depending on the cones, by \(S_1<\infty\). One may take \(C\) to be the width of the union of the cones, at most \(w+T\) for observation sites and the conditioning origin in a window of width \(w\), since a type of length \(r\) meets the window for at most \(r+C\) anchors in each of the \(T\) rows. Under the common Poisson coupling, the truncated and full fields therefore agree on the observations with probability tending to one. The conditioning event has probability at least that of the open vertical path, \(e^{-\rho TS_1}>0\), uniformly in the cutoff. Expectations of the bounded side functions and their product, and the conditioning probabilities, converge. Dividing by the conditioning probabilities passes the conditional inequality to the limit. Deterministic absence variables follow by parameter limits whenever the conditioning event has positive probability. Every step permits type-dependent parameters, including the final interpolation row. \(\square\) Connected interval supports make the extremal-path resampling strategy of {[}2{]} applicable here. For a disconnected error closing \(\{-1,1\}\) with probability \(1/2\) while leaving 0 open, the two side-hole indicators have joint probability \(1/2\) and product of marginals \(1/4\). The conclusion is therefore false for arbitrary spatially correlated row errors. \subsubsection{3.3. Full interval insertion and phase selection}\label{full-interval-insertion-and-phase-selection} \textbf{Lemma 3.3 (full insertion count).} Let \(F\) be a translation-invariant random subset of \(\mathbb Z\), interpreted as the holes. Inserting an interval means replacing \(F\) by \(F\cup[0,r-1]\). Define the run lengths \[ L_z=\sup\{k\in\mathbb Z_{\ge0}:[z-k,z-1]\subseteq F\},\qquad R_a=\sup\{k\in\mathbb Z_{\ge0}:[a,a+k-1]\subseteq F\}, \] with values in \(\mathbb Z_{\ge0}\cup\{\infty\}\). The empty-interval convention makes \(k=0\) admissible. In particular, \(L_z\) counts holes immediately left of \(z\), and \(R_r\) counts holes starting at \(r\). Here \(z\notin F\) means that \(z\) is open. Each summand below is defined to be zero when \(z\in F\), including configurations with infinite closed runs. The number of newly closed, nonempty finite intervals is \[\Delta_r(F)=\sum_{z=0}^{r-1}\mathbf{1}_{\{z\notin F\}}(L_z+1)(r-z+R_r). \tag{10}\] A newly closed interval is a nonempty finite interval contained in \(F\cup[0,r-1]\) but not in \(F\). It has a unique leftmost site \(z\) that was previously open; this site belongs to \([0,r-1]\). Its left endpoint has \(L_z+1\) choices and its right endpoint has \(r-z+R_r\) choices. This decomposition counts each newly closed interval exactly once. Write \(\Delta_{r,j}(F)\) for the number of newly closed intervals of length \(j\), and \(\Delta_{r,\le M}(F)=\sum_{j=1}^M\Delta_{r,j}(F)\) for the count up to length \(M\). Set \(u_F=\Pr(0\notin F)>0\), \(\beta_F=1-u_F\), \(h_0(F)=1\) and \(h_k(F)=\Pr([0,k-1]\subseteq F)\) for \(k\ge1\). Let \(\zeta=\lim_{k\to\infty}h_k(F)\). Assume that the conditional inequality of Lemma 3.2 holds for \(F\). For \(k\ge1\), translation invariance gives \[ \Pr(z\notin F,L_z\ge k)=h_k(F)-h_{k+1}(F). \] Summing this identity, with truncation before passage to the limit, gives \[\mathbb E[\mathbf{1}_{\{z\notin F\}}(L_z+1)]=u_F+\beta_F-\zeta\le1.\] For \(m=r-z\), let \(w\) be the last open site in \([z,r-1]\). For each \(b\ge1\), the event \(\{z\notin F,[r,r+b-1]\subset F\}\) implies that \(w\) is open and \([w+1,r+b-1]\) is closed. Dropping the other restrictions gives \[\Pr(z\notin F,R_r\ge b)\le\sum_{w=z}^{r-1} \bigl(h_{r+b-w-1}(F)-h_{r+b-w}(F)\bigr).\] Sum over \(b\ge1\). For each \(w\), the telescoping sum is \(h_{r-w}(F)-\zeta\le\beta_F-\zeta\). There are \(m\) possible sites \(w\), so \[\mathbb E[\mathbf{1}_{\{z\notin F\}}R_r]\le m\beta_F,\qquad \mathbb E[\mathbf{1}_{\{z\notin F\}}(m+R_r)]\le m.\] Conditional on \(z\notin F\), the variables \(L_z+1\) and \(m+R_r\) are increasing functions of holes strictly to the left and right of \(z\). Apply Lemma 3.2 to \(\min\{L_z,n\}+1\) and \(m+\min\{R_r,n\}\), which are bounded functions of finitely many sites, and then let \(n\to\infty\) by monotone convergence: \[\mathbb E[\mathbf1_{\{z\notin F\}}(L_z+1)(m+R_r)] \le\frac{\mathbb E[\mathbf1_{\{z\notin F\}}(L_z+1)] \mathbb E[\mathbf1_{\{z\notin F\}}(m+R_r)]}{u_F} \le\frac m{u_F}.\] Summing over the first old open point \(z\) gives \[\mathbb E\Delta_r(F)\le V(r)/u_F. \tag{11}\] \textbf{Proof of Proposition 3.1.} For each anchor and size, take a Poisson process in the parameter \(\theta\ge0\) with rate \(\nu_r\), independently over types and independently of the first \(t\) catastrophe layers. Let \(D^{(\theta)}\) be the union of intervals with at least one mark by parameter time \(\theta\). Thus these fresh catastrophe sets increase with \(\theta\). The last-layer interpolation field is \[F_{t,\theta}=E(H_t)\cup D^{(\theta)},\qquad 0\le\theta\le\rho,\] where the first \(t\) rows have intensities \(\rho\nu_r\) and the last row has intensities \(\theta\nu_r\). It is exactly a finite-history process of Lemma 3.2 with non-identical type parameters; its open density is at least \(u_{t+1}\ge e^{-\rho(t+1)S_1}>0\), using the disaster-free vertical path. Work first on the infinite integer line with only lengths \(2\le r\le K\) retained. Write \(H_t^{(K)}\) for that truncated process and \(F_{t,\theta}^{(K)}=E(H_t^{(K)})\cup D^{(\theta,K)}\) for its interpolation, where \(D^{(\theta,K)}\) retains the same sizes. The law remains translation invariant on the full integer line. The density \(u_{t+1}\) in the bound below refers to the untruncated process. Each fixed finite observation and its derivative depend on finitely many types in a backward cone. Write \(h_{j,t}=\Pr([0,j-1]\subset H_t)\). Define \(\phi_M(F,x)=\sum_{j=1}^M\mathbf{1}_{\{[x,x+j-1]\subset F\}}\). For a finite family of Poisson variables with means \(\theta\nu_r\), differentiating its product law gives the insertion formula: the derivative of a bounded functional is the sum, over types, of its expected change when one extra mark of that type is added, weighted by \(\nu_r\). Here only finitely many types affect \(\phi_M\) when lengths are bounded. To identify the sum over anchors, fix \(r\) and let \(C_{l,x,j}(F)\) be the indicator that inserting \([l,l+r-1]\) newly closes \([x,x+j-1]\). Translation invariance gives \(\mathbb EC_{l,0,j}=\mathbb EC_{0,-l,j}\). Summing over \(l\) converts the insertion effect at the fixed observation window to the number of newly closed windows caused by an insertion at anchor \(0\). Hence Poisson differentiation and translation invariance give \[\frac{d}{d\theta}\mathbb E\phi_M(F_{t,\theta}^{(K)},0) =\sum_{r=2}^K\nu_r\,\mathbb E\Delta_{r,\le M}(F_{t,\theta}^{(K)}) \le A_*/u_{t+1}. \tag{12}\] Here \(\Delta_{r,\le M}\) counts new intervals of length at most \(M\). The upper bound is independent of \(K\): the truncated sum of \(\nu_rV(r)\) is at most \(A_*\), and the truncated process has open density at least the full-process value \(u_{t+1}\). Integrate over \(\theta\) and pass to \(K\to\infty\) for each fixed \(M\). The omitted-type estimate from Lemma 3.2 gives local convergence, uniformly in \(0\le\theta\le\rho\); the integrated bound is uniform. Erosion sends the closure probability of a \(j\)-interval to \(h_{j+1,t}\), so the resulting finite-\(M\) inequality is \[ \sum_{j=1}^M h_{j,t+1}\le\sum_{j=1}^M h_{j+1,t}+c/u_{t+1}. \] Define \(\Phi_t=\sum_{j\ge1}h_{j,t}\). Since \(\Phi_0=0\), induction and monotone convergence in the two nonnegative sums show that \(\Phi_t<\infty\) at each finite \(t\). Only after obtaining this finiteness do we subtract the two potentials. It follows that \[\Phi_{t+1}-\Phi_t\le c/u_{t+1}-\beta_t. \tag{13}\] Adding past rows makes the hole marginals larger, so \(\beta_t\) is nondecreasing. A single new row gives \(\beta_{t+1}\le\beta_t+q\) with \(q=1-e^{-\rho S_1}\le c\). Independent time-block concatenation gives \(u_{a+b}\ge u_a u_b\): choose, by a fixed rule using only the last \(a\) rows, an open path to 0 and its initial site \(z\) when one exists. The first \(b\) rows reach that site with probability \(u_b\), independently. The shared boundary row is checked by the first block only. Thus \(\beta_{a+b}\le\beta_a+\beta_b\). Sum \(\beta_T\le\beta_t+\beta_{T-t}\) over \(1\le t\le T-1\). Next sum (13) over \(t=0,\ldots,T-1\), use \(\Phi_0=0\), \(\Phi_T\ge0\) and \(u_{t+1}\ge u_T\). These two sums give \[\frac{T-1}{2}\beta_T\le\sum_{t1/8\ge4c\), contradicting (14). Hence the crossing never occurs. Taking \(T\to\infty\) in the sharper version of (14) gives \(\beta(1-\beta)\le2c\) on the small branch, proving (9). This first-crossing argument gives the sufficient constant \(1/32\), without optimization. \(\square\) Lemma 2.2 gives positive survival from one prescribed root whenever \(\rho A_*\le1/32\). The construction establishes the positive phase through finite-history estimates. \subsection{4. Proof of the cube criterion}\label{proof-of-the-cube-criterion} \subsubsection{4.1. Extinction from envelope-covering catastrophes}\label{extinction-from-envelope-covering-catastrophes} Let \(B\) be a nonempty finite seed and let \(w_0=1+\max_i(\max_{x\in B}x_i-\min_{x\in B}x_i)\) be the largest number of lattice sites in a coordinate projection of its smallest axis-aligned enclosing box. Put \(K=\max_{g\in G}\|g\|_\infty\ge1\), where \(\|x\|_\infty=\max_i|x_i|\). At time \(t\), every possible descendant lies in the deterministic box obtained by extending each coordinate interval by \(Kt\) sites on both sides. Each side therefore contains at most \(w_0+2Kt\) lattice sites. Writing \((v)_+=\max\{v,0\}\), the number of anchors of side-\(r\) cubes that contain this box is at least \[n_{t,r}=(r-w_0-2Kt+1)_+^d.\] The corresponding Poisson mean in row \(t\) is \[\lambda_t\ge\rho\sum_r\nu_r n_{t,r}.\tag{15}\] For all sufficiently large \(r\), there are a positive constant times \(r\) integer times \(t\ge1\) with \(w_0+2Kt\le r/2\). At each such time, \(n_{t,r}\ge(r/2)^d\). Hence \[\sum_{t\ge1}\lambda_t\ge C_{d,K,w_0}\rho\sum_{r\ge r_0}r^{d+1}\nu_r.\] If \(S_{d+1}=\infty\), this sum diverges. The covering events use independent catastrophe rows, so the probability that no covering mark ever occurs is \(\exp(-\sum_t\lambda_t)=0\). The first such mark removes every possible descendant. This proves almost-sure extinction from each finite seed at every positive intensity. \subsubsection{4.2. Sufficiency by a coordinate restriction}\label{sufficiency-by-a-coordinate-restriction} Choose a seed site \(x\) and retain only paths on the coordinate line \(L_x=x+\mathbb Ze_1\), with increments \(0,e_1\). For fixed longitudinal anchor and side \(r\), precisely \(r^{d-1}\) transverse anchors yield cubes intersecting this line: for each coordinate \(i=2,\ldots,d\), the anchor lies in \(\{x_i-r+1,\ldots,x_i\}\). Their projections give the same length-\(r\) interval. Summing these independent Poisson types gives an interval model with \[\widetilde\nu_r=r^{d-1}\nu_r,\qquad \widetilde S_1=S_d,\quad \widetilde S_2=S_{d+1},\quad \widetilde A_*=(S_d+S_{d+1})/2.\tag{16}\] For distinct projected time-anchor-size types, the source type sets are disjoint; the projected types are therefore independent. Proposition 3.1 and Lemma 2.2 show survival of this restricted process when \(\rho\widetilde A_*\le1/32\). The unrestricted process contains it under the same catastrophe field. This proves sufficiency and the lower bound in (5). \subsubsection{4.3. Upper bound and finite seeds}\label{upper-bound-and-finite-seeds} A fixed increment path through \(t\) tested rows avoids the catastrophe at its tested site with probability \(e^{-\rho S_dt}\). There are at most \(|G|^t\) such paths from one root, so \[s_t^G(\rho)\le\bigl(|G|e^{-\rho S_d}\bigr)^t.\tag{17}\] For \(\rho>\log|G|/S_d\), the right side tends to zero. Since the survival events decrease, their limiting probability is zero, proving the upper bound; no conclusion at equality is needed for this bound on the supremum. The process is additive in its initial set: at each finite time the descendants of a finite union are the union of the descendants of its individual sites. If every single-site process dies almost surely, the extinction times of finitely many such processes have an almost surely finite maximum. Conversely, survival from one site implies survival from any finite seed containing that site. This completes Theorem 2.1. By Lemma 2.2, the bulk and finite-seed positivity sets and critical intensities coincide. The criterion applies to bidirectional and other finite recovery rules containing \(0,e_1\). Their numerical upper bounds depend on \(|G|\); their existence boundary is the same moment condition. \subsection{5. Stationary density in one dimension}\label{stationary-density-in-one-dimension} Return to \(d=1\), \(G=\{0,1\}\), and assume \(S_2<\infty\). Use the observation-time field \(H^*=H_0^*\) of Section 2.3 and \(\beta,u,A_*,c\) as in Section 3. Set \(h_j=\Pr([0,j-1]\subset H^*)\), \(h_0=1\). The local flux identity below yields a quadratic refinement of the stationary density. \textbf{Proposition 5.1 (quadratic density response).} For a fixed nonzero measure \(\nu\) with \(S_2<\infty\), \[\beta=c+\tfrac12c^2+o(c^2)\qquad(\rho\downarrow0),\qquad c=\rho A_*.\tag{18}\] The expansion holds for each fixed measure under the stated second-moment condition. \subsubsection{5.1. A local flux identity}\label{a-local-flux-identity} \textbf{Lemma 5.2.} At every intensity for which \(u>0\), let \(D^{(\theta)}\) be a fresh catastrophe row independent of \(H^*\), coupled increasingly for \(0\le\theta\le\rho\), and put \(F_\theta=E(H^*)\cup D^{(\theta)}\). Then \[h_j-h_{j+1}=\int_0^\rho\sum_r\nu_r\mathbb E\Delta_{r,j}(F_\theta)\,d\theta,\qquad j\ge1,\tag{19}\] Here \(\Delta_{r,j}\) is the count defined in Lemma 3.3 for insertion at anchor \(0\). For a general hole field \(F\), we write \(h_j(F)=\Pr([0,j-1]\subseteq F)\) and \(u_F=\Pr(0\notin F)\). Moreover, \[\beta=\int_0^\rho\sum_r\nu_r\mathbb E\Delta_r(F_\theta)\,d\theta, \qquad c\le\beta\le\frac c{u}.\tag{20}\] \textbf{Proof.} Group the Poisson environment by spatial anchor: \(\mathcal X_l=(N_{t,l,r})_{t\le0,r\ge2}\) for \(l\in\mathbb Z\). These columns are independent and identically distributed. The construction of \(H^*\) is a measurable function of these columns and commutes with spatial translation. The shift of an i.i.d. sequence is ergodic, and a translation-equivariant function preserves ergodicity; hence \(H^*\) is spatially ergodic. When \(u>0\), the spatial ergodic theorem gives open-site density \(u\) in each direction almost surely. Thus open sites occur arbitrarily far to the left and right. In particular all closed runs are finite almost surely, and \(h_j\to0\) by continuity of probability for decreasing events. Since \(F_\theta\subset F_\rho\) and \(F_\rho\) has the law of \(H^*\), the same properties hold for the interpolation fields; their open densities satisfy \(u_\theta\ge u\). For fixed \(j\), an inserted length-\(r\) interval affects at most \(r+j-1\) windows. This is integrable against \(\nu\) under \(S_1<\infty\). Poisson differentiation in finite histories and finite length cutoffs therefore passes to the stationary local insertion difference by dominated convergence. The integrable window bound controls the insertion differences uniformly. At zero last-row intensity the closure probability is \(h_{j+1}\); at intensity \(\rho\) it is \(h_j\). This proves (19). For bounded side functions, conditional correlation passes from finite histories and size cutoffs to \(F_\theta\) by local convergence; the conditioning probabilities are at least \(u>0\). For the run functions used below, first make this passage with their finite-coordinate truncations, and only then remove those truncations by monotone convergence. For any such interpolation field \(F\), inclusion in \(F_\rho\) implies \(h_k(F)\to0\). The boundary-tail calculation of Lemma 3.3, now with \(\zeta=0\), gives \(\mathbb E[\mathbf1_{\{z\notin F\}}(L_z+1)]=1\). Lemma 3.3 consequently yields \[\mathbb E\Delta_r(F)=V(r)+\sum_{z=0}^{r-1}\mathbb E[\mathbf1_{\{z\notin F\}}(L_z+1)R_r]\ge V(r).\] Sum (19) over \(j\) by Tonelli. Its left side telescopes to \(\beta\), and the nonnegative insertion counts sum to \(\Delta_r\). This gives the identity and lower bound in (20). Lemma 3.3 gives \(\mathbb E\Delta_r(F_\theta)\le V(r)/u_\theta\le V(r)/u\), proving the upper bound. The insertion decomposition gives the lower bound directly. \(\square\) \subsubsection{5.2. The quadratic coefficient}\label{the-quadratic-coefficient} \textbf{Proof of Proposition 5.1.} Proposition 3.1 gives \(\beta=O(\rho)\) and \(u\to1\). Lemma 5.2 then implies \(\beta/\rho\to A_*\). Put \[b_i=\sum_r\nu_r(r-i)_+,\qquad a_j=\sum_{i\ge j-1}b_i\quad(j\ge1),\qquad a_1=\sum_{i\ge0}b_i=A_*.\] For \(0\le s\le1\), write \(F_s=E(H^*)\cup D^{(s\rho)}\). Its finite-window hole probabilities vanish uniformly in \(s\) because \(F_s\subset F_1\) and \(\beta=O(\rho)\). For fixed \(r,j\), an insertion at anchor 0 can affect only length-\(j\) windows whose left endpoints are in \([-j+1,r-1]\), so \(\Delta_{r,j}\) depends on finitely many sites and is at most \(r+j-1\). Consequently \(\mathbb E\Delta_{r,j}(F_s)\to(r-j+1)_+\) uniformly in \(s\). The bound \(r+j-1\) in (19) gives \(q_j/\rho\to b_{j-1}\), where \(q_j=h_j-h_{j+1}\). Also \[\sum_{j\ge1}\frac{q_j}{\rho}=\frac\beta\rho\to A_*=\sum_{j\ge1}b_{j-1}.\] For clarity, write \(x_j^{(\rho)}=q_j/\rho\) and \(x_j=b_{j-1}\). For each finite \(J\), the first \(J\) entries converge. Also \(\sum_jx_j^{(\rho)}\to\sum_jx_j=A_*\). Given \(\varepsilon>0\), choose \(J\) with \(\sum_{j>J}x_j<\varepsilon\); the convergence of the total mass and of the first \(J\) entries then bounds \(\sum_{j>J}x_j^{(\rho)}\) by \(2\varepsilon\) for small \(\rho\). This proves \(\sum_j|x_j^{(\rho)}-x_j|\to0\). Since \(h_j=\sum_{k\ge j}q_k\), the same conclusion gives the required tail limits. Summing the tails and using (19) up to intensity \(s\rho\) gives \[\frac{h_j}{\rho}\to a_j,\qquad \frac{h_j(F_s)}\rho\to a_{j+1}+s b_{j-1},\tag{21}\] for each fixed \(j\), with the second convergence uniform in \(s\). Fix an insertion length \(r\) and \(z\in[0,r-1]\). For a field \(F=F_s\), use the run length \(R_a\) from Lemma 3.3, and put \(T_{z,r}=\mathbb E[\mathbf1_{\{z\notin F\}}R_r]\) and \(m=r-z\). Partition by the last old open site \(w\in[z,r-1]\), using \[J_{w,r}=\mathbf1_{\{w\notin F\}}(R_{w+1}-(r-w-1))_+.\] Put \(m_w=r-w\ge1\). For \(b\ge1\), \[ \Pr(w\notin F,R_{w+1}\ge m_w-1+b) =h_{m_w-1+b}(F)-h_{m_w+b}(F). \] The nonnegative tail sum telescopes to \(\mathbb EJ_{w,r}=h_{m_w}(F)\) because \(h_j(F)\to0\). The exact decomposition is \(\mathbf1_{\{z\notin F\}}R_r=\sum_{w=z}^{r-1}\mathbf1_{\{z\notin F\}}J_{w,r}\). For \(w>z\), condition on \(w\notin F\) and use Lemma 3.2 to bound the omitted term by \[\mathbb E[\mathbf1_{\{z\in F\}}J_{w,r}] \le\frac{\beta_F h_{r-w}(F)}{u_F}\le\frac{\beta_F^2}{u_F}.\] Hence \(0\le\sum_{j=1}^m h_j(F)-T_{z,r}\le(m-1)\beta_F^2/u_F\). Conditional correlation at \(z\), together with \(\mathbb E[\mathbf1_{\{z\notin F\}}L_z]=\beta_F\), also gives \(\mathbb E[\mathbf1_{\{z\notin F\}}L_zR_r]\le m\beta_F^2/u_F\). Substitute these estimates into (10): for fixed \(r\), \[\mathbb E\Delta_r(F_s)-V(r)=\sum_{j=1}^r(r-j+1)h_j(F_s)+O_r(\beta^2/u).\] Integration over \(r\) uses the separate upper and lower insertion bounds, which give \[0\le\frac{\mathbb E\Delta_r(F_s)-V(r)}\rho \le V(r)\frac\beta{\rho u}\le C_\nu V(r)\tag{22}\] for all sufficiently small \(\rho\), uniformly in \(s\). The final bound is integrable against \(\nu\) by \(S_2<\infty\). Dominated convergence in (20), using (21), now gives \[\lim_{\rho\downarrow0}\frac{\beta-\rho A_*}{\rho^2} =\sum_{i\ge0}b_i\left(\sum_{k>i}b_k+\frac12b_i\right) =\frac12 A_*^2. \] To see the first equality, use \(\sum_r\nu_r\sum_{j=1}^r(r-j+1)f_j=\sum_{i\ge0}b_if_{i+1}\) for nonnegative \(f_j\), and insert the limits (21), integrated over \(s\in[0,1]\). For the second equality, all series are nonnegative and \(\sum_i b_i=A_*<\infty\), so \[ \sum_{ij\), exactly \(r-j\) anchors yield a residual interval containing \(0\). The total intensity of residual shadows hitting a site is therefore \[\rho\sum_r\nu_r\sum_{j=0}^{r-1}(r-j)=c.\] The Poisson void calculation gives \(\Pr(0\notin K^*)=e^{-c}\) {[}9{]}. Since \(E\) of a union contains the union of the individual erosions, \(K^*\subset H^*\) under the common construction. \textbf{Corollary 5.3 (density excess).} Under \(00\quad(\rho>0).\tag{23}\] For the fixed measure, \(\Gamma(\rho)=c^2+o(c^2)\) as \(\rho\downarrow0\). \textbf{Proof.} If \(u>0\), Lemma 5.2 gives \(\beta\ge c\); if \(u=0\), then \(\beta=1\). Thus \(\beta\ge\min\{c,1\}\). Strict positivity follows from \(e^{-c}>1-c\) for \(00\) for \(c\ge1\). Finally subtract \(1-e^{-c}=c-c^2/2+o(c^2)\) from (18). \(\square\) The same argument shows that \(c\ge1\) forces \(u=0\), so Lemma 2.2 gives the additional one-dimensional bound \(\rho_c^{\{0,1\}}\le1/A_*\). Since \(S_2\ge2S_1\) for sizes \(r\ge2\), one has \(A_*\ge\tfrac32S_1\), so \(1/A_*\le\tfrac{2}{3S_1}<\log2/S_1\): in dimension one this bound is always sharper than the path-count bound in (5) for \(|G|=2\). The bound and density expansion hold for the one-sided rule. \subsection{6. Uniformly fat catastrophe shapes}\label{uniformly-fat-catastrophe-shapes} The cube result extends by comparison. Let each supported shape \(\mathcal B_r\subset\mathbb Z^d\) be finite and nonempty. Assume that, for every sufficiently large \(r\), it contains a translated cube of side at least \(ar\) and is contained in a translated cube of side at most \(\kappa r\), where \(a,\kappa>0\). The finitely many shapes \(\mathcal B_r\) at sizes below the threshold have finite diameters, so the outer bound extends to all sizes by enlarging \(\kappa\). Choose the outer constant \(\kappa\) once and for all large enough that, for every supported size \(r\), an integer side length \(b_r\ge2\) and a translation vector \(v_r\) satisfy \(\mathcal B_r\subseteq v_r+\{0,\ldots,b_r-1\}^d\) and \(b_r\le\kappa r\). All occurrences of \(\kappa\) in the quantitative bound below refer to this uniform choice, including the finitely many small-size exceptions. A mark with anchor \(l\) empties \(l+\mathcal B_r\). For every \(t\in\mathbb Z\), \(l\in\mathbb Z^d\) and \(r\ge2\), the mark counts are independent Poisson variables with means \(\rho\nu_r\); the chain has the update and initial-layer convention of Section 2.2. Write \(\Lambda=\sum_r|\mathcal B_r|\nu_r\) and assume \(0