\documentclass[11pt,a4paper]{article}
\usepackage[margin=25mm]{geometry}
\usepackage{hyperref}

\usepackage{amsmath,amssymb,amsfonts}
\usepackage{booktabs,longtable,array,graphicx,calc}
\usepackage{upquote,xcolor}
\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
\providecommand{\passthrough}[1]{#1}
\allowdisplaybreaks[4]
\setlength{\emergencystretch}{3em}
\hypersetup{hidelinks}
\newif\ifmarkchanges
\newcommand{\changed}[1]{\ifmarkchanges{\color{red}#1}\else#1\fi}
\begin{document}
\section*{Online Resource 1. Effective threshold supplement}\label{online-resource-1.-effective-threshold-supplement}

\textbf{Article:} Finite-preparation response under heavy-tailed catastrophes: exact amplitudes and separation from Poisson coverage

\textbf{Preprint manuscript:} revised 27 October 2026\\
\textbf{Author:} HUANG Man\\
\textbf{Affiliation:} Independent Researcher, Shanghai, China\\
\textbf{Corresponding author:} huangman@technocentrics.com\\
\textbf{ORCID:} https://orcid.org/0009-0003-3809-5105

\section*{S0. Scope and notation}\label{s0.-scope-and-notation}

This supplement proves Theorem 4.6 of \emph{Finite-preparation response under heavy-tailed catastrophes: exact amplitudes and separation from Poisson coverage}. References to Sections 2--8, Appendices A--B and Lemmas 6.3--6.4 refer to the accompanying manuscript. Bracketed references use the manuscript bibliography. All inputs are deterministic law and protocol data; time is integer valued. The normalized law is denoted by \(\nu\) throughout this supplement.

The quantitative argument has the following dependencies. Each row records a probability estimate or deterministic approximation; the final scalar checks evaluate its sufficient conditions.

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}}
  >{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}}@{}}
\caption{Dependencies of the effective sign certificate.}\tabularnewline
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Estimate
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Source
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Role in the final bound
\end{minipage} \\
\midrule\noalign{}
\endfirsthead
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Estimate
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Source
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Role in the final bound
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Short-background history and rate strings & S2; manuscript Lemmas 6.1--6.2 & Zero-root discrepancy, young target-root cancellation and gate variance \\
Exact origin integration and attachment debt & S3; manuscript Lemmas 5.2 and A.6 & Actual ancestor span, first-row correction and birth-field trimming \\
Signed kernel and attachment moments & S4 & Centered noise, discarded tails and the retained open-gate covariance \\
Local large kernel and capped small kernel & S5 & Slope, curvature and small-span error payments \\
Complete-path root remainder and gate projection & S6; manuscript Lemmas 6.3--6.4 & All-count root remainder and unresolved-openness error \\
Adaptive small-kernel estimate at gamma one & S7 & Stronger one-sided certificate for the displayed examples \\
Root cancellation, tail replacement and quadrature & S8 & Common target cancellation, effective-tail error and ideal history signal \\
Intensity budget and decreasing time envelopes & S9 & A sufficient threshold valid at every later integer time \\
\end{longtable}

The sixteen evaluations in S11 and Online Resource 2 check the scalar inequalities at their stated fixed intensities. The all-future conclusion uses the decreasing error envelopes proved in S6 and S9.

\section*{S1. Effective tail data and constants}\label{s1.-effective-tail-data-and-constants}

Use \(\nu=\widehat\nu\), so \(\chi=1\) and \(b=\rho\chi\) in original coordinates. The supplied tail bound is \[
|\nu_r-cr^{-p}|\le cr^{-p}\omega(r)\quad(r\ge R_*),\qquad p>3.
\] Here \(\omega\ge0\) is a specified nonincreasing envelope tending to zero, with a supplied integer modulus \(\Omega\) satisfying \[
r\ge\Omega(\varepsilon)\ \Longrightarrow\ \omega(r)\le\varepsilon,
\qquad \Omega(\varepsilon)\ge R_*.
\] The finite prefix and tail data admit certified evaluation. Set \(c_+=c[1+\omega(R_*)]\). This hypothesis permits positive or negative oscillating corrections within the envelope. It requires an effective approach to a constant power-law coefficient.

Choose a finite admissible tuple \(\pi=(\theta,\alpha,\delta_B,\delta)\) such that \[
0<\theta\le1,\qquad\theta<\gamma,\qquad
\alpha>\gamma,\qquad
0<\delta_B<\theta/\alpha,\qquad
0<\delta\le\min\{1/64,\delta_B/128\}.
\] Write \(M\) for the number of common preparation stages, and set \[
\begin{gathered}
E=\gamma(\gamma+1)Q_\gamma[h]>0,\quad B=B_*,\quad C=H+2,\quad D=D_p,\quad S=cDE,\\
d=\theta-\alpha\delta_B>0,\quad\sigma=\alpha-\gamma>0,\\
C_A=\max\{1,(R_*+1)^\gamma,2c_+(1+\gamma^{-1})\},\quad s=\lceil\gamma+1\rceil,\\
K_h=\frac{8BC_A s!}{(3/4)^{s+1}},\quad K_R=2\,64^\gamma(K_h+B),\quad C_R=(64B)^2K_R4^\gamma,\\
m_\chi=\left\lceil\max\{R_*,2,(2c_+/\gamma)^{1/\gamma}\}\right\rceil,\quad D_\theta=1+\frac{2C_A\theta}{\gamma-\theta},\\
K_B=\left\lceil\max\left\{2,\frac{m_\chi+1}{\delta_B},
\left[D_\theta\alpha^\theta\left(\frac{2\theta}{ed}\right)^\theta\right]^{2/d}\right\}\right\rceil .
\end{gathered}
\] The root, history and quadrature constants are \[
\begin{aligned}
L_\delta&=BCc_+\left[\frac{(\delta/2)^{2-p}}{p-2}+\frac{C(\delta/2)^{1-p}}{p-1}\right],\\
P_\delta&=\frac{BCc_+}{p-2}(\delta/2)^{2-p},\qquad
Q_\delta=\frac{4BCc_+}{p-2}(\delta/2)^{2-p},\\
K_2&=3P_\delta^2+(4+8\,2^\gamma)K_RL_\delta,\\
L_{\rm id}&=c_+\left\{2BC(9/10)^{1-p}+\frac{4CMB^2}{p-1}(4/5)^{1-p}\right.\\
&\hspace{12mm}\left.+\frac{B[C(\gamma+1)(9/10)^{-\gamma-2}+2(M+1)(9/10)^{-\gamma-1}]}{(p-1)(p-2)}\right\}.
\end{aligned}
\] Using \(c_+\) in \(L_{\rm id}\) is a conservative bound for the hybrid long roots of weight \(cr^{-p}\). The constant \(c_\gamma\) is from Theorem 4.4. Certified upper constants and a certified positive lower \(S\) can be used in all budgets and denominators.

For the short background, put \[
\begin{gathered}
A_\chi=\frac{c_+}{\gamma}(\delta/2)^{-\gamma},\\
M_*=\sum_{2\le r<R_*}\nu_r\frac{r(r+1)(r+2)}6,\\
A_M=
\begin{cases}
M_*+\frac{c_+}2[1+(1-\gamma)^{-1}],&0<\gamma<1,\\
M_*+\frac{c_+}2,&\gamma=1,\\
M_*+\frac{c_+}2[1+(\gamma-1)^{-1}],&\gamma>1,
\end{cases}\\
\ell_\gamma(T)=
\begin{cases}
T^{-\gamma},&0<\gamma<1,\\
(1+\log T)/T,&\gamma=1,\\
T^{-1},&\gamma>1.
\end{cases}
\end{gathered}
\] For \(m=\lfloor\delta T\rfloor\ge R_*\) and \(T\ge2/\delta\), \(\chi_{\rm tail}(m)\le A_\chi T^{-\gamma}\) and \(M_{3,m}/T\le A_M\ell_\gamma(T)\). The latter follows from \(r(r+1)(r+2)/6\le r^3/2\) and the integral bounds for \(\sum r^{-\gamma}\).

\section*{S2. Spatial tails and history control}\label{s2.-spatial-tails-and-history-control}

\subsection*{S2.1. Quantitative cutoff}\label{s2.1.-quantitative-cutoff}

For every \(k\ge1\), \(A_k\le C_Ak^{-\gamma}\). Below the declared tail onset use \(A_k\le\chi=1\); above it compare the power-law sum with its integral. In the branching comparison of Appendix A the success-count bound is \(\Pr(J=j)\le(8Bb)^j\) and the cost tail is at most \(C_Ak^{-\gamma}\). Conditional on \(J=j\), a sum of \(j\) costs reaching \(k\) has a cost at least \(k/j\). Consequently \[
h_k\le C_Ak^{-\gamma}
\sum_{j\ge1}(8Bb)^j j^{\gamma+1}
\le K_hb k^{-\gamma}.
\] The last inequality follows from \(j^s\le s!\binom{j+s-1}{s}\) and \(8Bb\le1/4\). This bound is uniform in length cutoffs.

The forcing-moment tail satisfies \(\chi-\chi_m\le c_+m^{-\gamma}/\gamma\) for \(m\ge R_*\). Thus \(\chi_m\ge1/2\) once \(m\ge m_\chi\). The truncated branching cost \(W_m\le m\) has \[
\Pr(W_m\ge j)\le2C_Aj^{-\gamma},
\qquad \mathbb EW_m^\theta\le D_\theta .
\] For \(m=\lfloor\delta_Bk\rfloor\) and \(s_k=\alpha\log k/k\), use \(e^x-1\le x^\theta e^x\) to obtain \[
\mathbb Ee^{s_kW_m}-1
\le D_\theta\alpha^\theta(\log k)^\theta k^{-d}\le1
\quad(k\ge K_B).
\] The definition of \(K_B\) uses \((\log k)^\theta\le(2\theta/(ed))^\theta k^{d/2}\). The tree generating-function bound is then at most two, giving \[
h_k^{(\lfloor\delta_Bk\rfloor)}\le2k^{-\alpha}
\quad(k\ge K_B).
\] By coupling it also holds for every smaller cutoff.

\subsection*{S2.2. Finite-error history contraction}\label{s2.2.-finite-error-history-contraction}

For all cutoffs and every \(n\ge2\), the thin-ancestor probability in (24) is bounded by \[
\eta_{\rm con}=64Bb.
\] To prove this, select one site of the input before exposing the future. If its family is empty but the entire input has a nonempty family, pay its eventual extinction probability, at most \(\beta_+\le2Bb\). Otherwise use the single selected family.

After its first killing row the actual ancestor process is an ordinary process over \(8n-1\) independent rows followed by a final expansion. For reflected propagation, take the two \(n\)-site integer ports \([-5n,-4n)\) and \([-3n,-2n)\). If each bounding half-line edge has lost at most \(n\) sites relative to its no-disaster displacement, both ports lie between the edges. Lemma 5.2 bounds the mean lost displacement of either edge by \(8nm_+/u_+\). Markov's inequality therefore pays at most \(16m_+/u_+\) for failure of the two edge bounds. This uses the marginal boundary drift, without conditioning it on the full past.

The finite-history full-source baseline has at most \(n\beta_+\) expected holes in either port. Fewer than \(\lceil n/2\rceil\) open sites requires at least \(\lfloor n/2\rfloor+1\ge n/2\) holes. Markov's inequality pays at most \(2\beta_+\) per port. On nonempty survival, prefix exchange identifies the seed field between its extreme descendants with the full-source baseline. Thus \[
\Pr(0<|U_{8n}|<\lceil n/2\rceil)
\le16m_+/u_++5\beta_+<64Bb.
\] The estimates hold under all deletions of marks.

Apply (28) with this constant contraction. For a time \(t\ge64\), its block target is at least \(t/64\) and its remaining time at least \(t/2\). The budget gives \(\eta_{\rm con}2^\gamma\le1/2\). Strong induction, with \(R_m(t)\le\beta_+\le2Bb\) for smaller times, yields \[
R_m(t)\le K_Rb t^{-\gamma}\qquad(t\ge1).
\] For \(m=\lfloor\delta T\rfloor\) and \(T\ge\max(256,128K_B)\), apply the same recurrence twice. Its target sizes are at least \(T/64,T/128\), and the final remaining time is at least \(T/4\). Both targets exceed \(K_B\) and their cutoff ratios are below \(\delta_B\). Hence \[
R_m(T)\le4\,128^\alpha T^{-\alpha}
+C_Rb^3T^{-\gamma}.
\] The cubic residual is sufficient for a quadratic sign margin.

\subsection*{S2.3. Deterministic rate strings}\label{s2.3.-deterministic-rate-strings}

Fix the short cutoff \(m\), and write \[
p_+(C)=p_{Bb,m}(C),\qquad
f_+(C)=p_+(C)\mathbf1_{\{C\ne\varnothing\}},
\qquad p_+(\varnothing)=1.
\] The stationary phase is that of the actual normalized cutoff law at the maximum rate \(Bb\). Assume the phase and contraction restrictions already imposed in S9.2: \[
Bb\le1/64,\qquad 64Bb\,2^\gamma\le1/2.
\] For an arbitrary deterministic string \(0\le a_i\le Bb\), let \(U_h(C)\) be the actual kill-then-expand ancestor chain. Then \[
\boxed{
R_{\rm var}(h):=
\sup_{\substack{C\ {\rm finite}\\0\le a_1,\ldots,a_h\le Bb}}
\mathbb E_C f_+(U_h)
\le b\min\{2B,K_Rh^{-\gamma}\},
\qquad h\ge1,
}
\] with exactly the constant \(K_R=2\,64^\gamma(K_h+B)\) of S1.

Here is a proof uniform in the rate string and cutoff. Couple one row of rate \(a\le Bb\) below a maximum-rate row. Its ancestor set contains the maximum-rate ancestor set. All-hole probabilities \(p_+\) decrease with target inclusion, including the empty-target convention. Stationary harmonicity therefore gives \[
\mathbb E_a p_+(G_D C)\le p_+(C).
\] For nonempty \(C\), deleting the empty-state contribution on the left proves \(\mathbb E_a f_+(G_D C)\le f_+(C)\). For empty \(C\) both sides are zero. Thus \(f_+\) is superharmonic under every row rate in the string. Also \(f_+(C)\le\beta_+\le2Bb\).

The fixed thin-ancestor bound in S2.2 extends to these strings through its component events. Choose one site of a finite nonempty input before revealing the next \(8q\) rows, \(q\ge2\). Extinction of this selected family while the full input survives costs at most \(\beta_+\) by coupling to the maximum-rate homogeneous chain.

On survival of the selected family, its first killing row did not remove the selected site. The remaining \(8q-1\) rows form an ordinary process from that site, followed by the final expansion of Section 6.1. Couple its two half-line edges and full-source field to the maximum-rate homogeneous versions, starting at the same location. Deleting marks decreases each half-line's loss relative to its disaster-free endpoint and decreases the full-source hole set. Hence endpoint and deterministic-port good events for the maximum-rate field imply the corresponding events for the actual string. Prefix exchange is used only on survival of the actual selected family.

For the two ports of S2.2, the maximum-rate marginal drift bound pays at most \(16m_+/u_+\) for the two endpoint failures. The two deterministic port-density failures cost at most \(4\beta_+\). Together with selected-family extinction, this gives \[
\Pr_C(0<|U_{8q}|<\lceil q/2\rceil)
\le16m_+/u_++5\beta_+<64Bb.
\] All drift and density estimates here concern the maximum-rate homogeneous process. The argument does not require monotonicity of the thin-population event itself.

Split the ancestor set after \(8q\) rows into empty, large and small nonempty outcomes, and use superharmonicity on the large branch. The remaining deterministic rows are independent, so \[
R_{\rm var}(t+8q)
\le h_{+,m}(\lceil q/2\rceil)+64Bb\,R_{\rm var}(t).
\] For \(L\ge64\), take \(q=\lfloor L/16\rfloor\) and \(t=L-8q\). Then \(\lceil q/2\rceil\ge L/64\) and \(L/2\le t<L\). Compression gives \(h_{+,m}(k)\le K_hbk^{-\gamma}\), and strong induction gives \[
R_{\rm var}(L)
\le[64^\gamma K_h+64Bb\,2^\gamma K_R]bL^{-\gamma}
\le K_RbL^{-\gamma}.
\] The bound \(2Bb\) covers the smaller times because \(K_R\ge2B64^\gamma\). This proves the displayed estimate.

\section*{S3. Ancestor representation and drift debt}\label{s3.-ancestor-representation-and-drift-debt}

\subsection*{S3.1. Exact origin integration}\label{s3.1.-exact-origin-integration}

Fix a root birth row \(t\) and the terminal site \((0,t+n)\). The baseline contains only marks of length at most \(m\). Let \(X_t\) be its open set immediately after the birth row's disasters. Starting at the terminal singleton, iterate the reverse updates over the subsequent \(n\) rows: \[
G_D(U)=(U\setminus D)\cup[(U\setminus D)-1].
\] The resulting ancestor set \(U\) reads only rows after \(t\) and is independent of \(X_t\). Set \[
A=U\cap X_t,\qquad O=\mathbf1_{\{A\ne\varnothing\}},\qquad
D_A=\max A-\min A\quad\text{on }\{O=1\}.
\] Thus \(O\) is exactly terminal baseline openness and \(0\le D_A\le n\) on that event.

An interval \([q,q+r-1]\) inserted after the birth row closes the terminal site exactly when \(A\) is nonempty and contained in that interval. Its permitted integer origins satisfy \(\max A-r+1\le q\le\min A\). Consequently, pathwise, \[
\sum_{q\in\mathbb Z}\mathbf1_{\{\text{insertion closes the terminal site}\}}
=O(r-D_A)_+.
\] Origins outside the deterministic backward cone contribute zero. The root expansion still counts all marks in that deterministic cone; it does not select marks by the random ancestor set.

For nonnegative length weights \(w_r\) with finite first moment, define \[
\Psi_m(x)=\sum_{r>m}w_r(r-x)_+.
\] Tonelli's theorem gives the exact origin-and-length integrated benefit \(\mathbb E[O\Psi_m(D_A)]\). This retains both the terminal open gate and absorption after ancestor extinction. For the hybrid process in S8.3, use \(w_r=cr^{-p}\); its short background and all attachment laws remain those of the actual normalized law. No full hybrid normalization is used.

\subsection*{S3.2. Uniform attachment moments and mean corrections}\label{s3.2.-uniform-attachment-moments-and-mean-corrections}

Use the normalized law and tail data of S1. Let \(x=Bb\le1/32\). The phase inequality implies \[
\beta_+\le16x/15,\qquad u_+\ge15/16,\qquad m_a\le16a/15.
\] Indeed \(\beta_+\le x/(1-2x)\), and \(e^{aS_1}\le(1-2a/3)^{-1}\le48/47<16/15\). Therefore each half-line's accumulated extra debt in S3.3, is at most \[
\frac{4096}{3375}CB^2b^2T.
\] The mean corrections obey \[
0\le m_a-a\le(32/47)a^2,\qquad
0\le m_a-m_{a,m}\le(48/47)a\chi_{\rm tail}(m).
\] The cumulative forcing-tail correction is retained explicitly through \(\chi_{\rm tail}(m)\) in S3.3 and \(A_\chi T^{-\gamma}\) in S9.

\subsection*{S3.3. Centered attachments and trimming}\label{s3.3.-centered-attachments-and-trimming}

For deterministic row intensities at most \(Bb\), apply Lemma A.6 at a deterministic source or checkpoint and then the Palm calculation in Lemma 5.2. The full-source open density is at least \(u_+\), so the fresh attachment at the actual half-line edge obeys \[
m_{a,m}\le\mathbb E[J_t\mid L_t=x]\le m_{a,m}/u_+.
\] The conditioning is on the displayed edge value; the next row is independent. Reflection gives the other edge. This supplies the uniform drift input used below.

Assume \(Bb\le1/64\). Write \(a_1,\ldots,a_n\) for the future forward intensities after root birth, and put \[
s=n-2\sum_{i=1}^n a_i.
\] For \(n=0\), \(D_A=s=0\) on openness, and every error below is zero.

For \(n\ge1\), the first reverse update kills the terminal singleton before expanding it. On openness this killing does not occur. The remaining \(n-1\) reverse rows form an ordinary single-source process, followed by one final expansion. Couple its two extremes to the corresponding two half-line processes. On single-source survival, prefix exchange identifies the extremes and the full-source field between them. For each half-line let \(J'_{L,i},J'_{R,i}\) be its fresh-row attachments, with old exterior holes erased. Each side has independent increments across rows with means \(m_{a_i,m}\); the two sides can be dependent.

Let \(V_L,V_R\) be the excess attachment debts, and let \(\tau_L,\tau_R\) be the inward trimming when \(U\) is intersected with \(X_t\). Define these trimming terms to be zero off openness. Then, on \(\{O=1\}\), \[
\begin{gathered}
D_A=s+\mu-N-V,\qquad
\mu=2a_n-2\sum_{i<n}(m_{a_i,m}-a_i),\\
N=\sum_{i<n}
[(J'_{L,i}-m_{a_i,m})+(J'_{R,i}-m_{a_i,m})],\\
V=V_L+V_R+\tau_L+\tau_R\ge0,\qquad \mathbb EN=0.
\end{gathered}
\] The term \(2a_n\) records the first-kill/final-expansion convention.

Let \(C_k\), \(k=n-1\), be the ordinary reverse full-source open field. Continue each auxiliary half-line through all \(k\) rows, including outcomes where the finite single-source family has died. Write \(e_R,e_L\) for the right and left edges of these half-lines before the final expansion. On survival, prefix exchange identifies the single-source family with \(C_k\) between these edges. Its final expansion contains \(C_k\) there and also contains \(C_k-1\) on the corresponding interval translated by \(-1\). In particular, the expanded right endpoint is \(e_R\) and the expanded left endpoint is \(e_L-1\).

On \(\{O=1\}\), the right trim is at most the number of consecutive sites, read leftward from \(e_R\), in \(C_k^c\cup X_t^c\). The left trim is at most the number read rightward from \(e_L-1\) in \((C_k-1)^c\cup X_t^c\). If the first jointly open site lies beyond the opposite endpoint, the corresponding run still exceeds the trim. Thus these are pathwise upper bounds on openness. They are nonnegative and defined also off openness, so the openness and finite-family-survival indicators can be dropped after this domination.

Conditional on \(e_R=z\), the coordinate \(z\) is open in \(C_k\). Clamp that coordinate open before averaging a bounded run over the independent birth field \(X_t\). The resulting function is increasing in the strict-interior holes of \(C_k\), exactly as in Lemma A.6(iv). The reflected argument applies at \(e_L-1\), using the translated field \(C_k-1\). Since \(\Pr(z\in C_k)\ge u_+\), Lemma A.6 bounds each edge-conditioned run by \(1/u_+\) times its unconditioned deterministic-position mean. The half-line comparison is applied to each marginal; independence between the two half-lines is unnecessary.

To bound this unconditioned mean, align the propagation directions of the independent past and reverse full-source fields by deterministic rowwise spatial translations, and pad the shorter history at its beginning with zero-intensity rows. The translations preserve their spatially stationary terminal laws and their independence. If \(E\) is the common eroder, then

\[
E(H\cup K)\supset E(H)\cup E(K).
\]

Induction therefore dominates their union of holes by a full-source process whose row intensity is at most \(\lambda=2Bb\), with the same length cutoff \(m\).

Write \(\chi_m=\sum_{r\le m}\nu_r r(r+1)/2\le1\) and let \(u_{\lambda,m}\) be the stationary open density of this dominating process. The branching comparison of A.4 gives

\[
\mathbb E Q\le
\frac{\lambda M_{3,m}}{u_{\lambda,m}-2\lambda\chi_m},
\qquad
M_{3,m}=\sum_{r\le m}\nu_r\frac{r(r+1)(r+2)}6.
\]

Indeed, its cost has mean \(M_{3,m}/\chi_m\), while its expected number of successful vertices is \((\lambda\chi_m/u_{\lambda,m})/(1-2\lambda\chi_m/u_{\lambda,m})\). For \(Bb\le1/64\), the small-intensity density estimate gives \(u_{\lambda,m}\ge1-2\lambda\chi_m\), hence \(u_{\lambda,m}-2\lambda\chi_m\ge1-4\lambda\chi_m\ge7/8\). In particular \(\mathbb E Q\le2\lambda M_{3,m}=4BbM_{3,m}\). Finite histories and lower deterministic row intensities are dominated by the stationary maximum-intensity field. Summing the two edge-run bounds yields

\[
\boxed{\mathbb E[O(\tau_L+\tau_R)]
\le\frac{8Bb}{u_+}M_{3,m}.}
\]

If \(\chi_m=0\), there are no retained marks and the run is zero.

The uniform boundary comparison, mean corrections and trimming bound give \[
\begin{aligned}
|\mu|+\mathbb E[OV]\le{}&
\left(\frac{8192}{3375}+\frac{64}{47}\right)CB^2b^2T
+2Bb\\
&+\frac{96}{47}CBbT\,\chi_{\rm tail}(m)
+\frac{128}{15}BbM_{3,m}.
\end{aligned}
\] Indeed, each half-line's excess debt has mean at most \((u_+^{-1}-1)\sum m_{a_i,m}\). The two mean corrections use \(m_a-a\le32a^2/47\) and \(m_a-m_{a,m}\le48a\chi_{\rm tail}(m)/47\). The half-lines start with empty exteriors; their initial-field cost is the displayed trimming term. The static coefficient satisfies \[
\frac{8192}{3375}+\frac{64}{47}
=\frac{601024}{158625}<4.
\]

\section*{S4. Signed kernels and attachment tails}\label{s4.-signed-kernels-and-attachment-tails}

\subsection*{S4.1. Taylor identity with the open gate}\label{s4.1.-taylor-identity-with-the-open-gate}

Let \[
\lambda_m=\sum_{r>m}w_r,\qquad
k_m(s)=\sum_{r>m,\ r>s}w_r,\qquad
\kappa_m=\sup_{r>m}w_r.
\] For every real \(z\), convexity and counting integer kinks give \[
0\le R_s(z):=\Psi_m(s-z)-\Psi_m(s)-k_m(s)z
\le\kappa_m(z^2/2+|z|).
\] For centered square-integrable \(Z\) and any gate \(O\), \[
\mathbb E[O\{\Psi_m(s-Z)-\Psi_m(s)\}]
=-k_m(s)\mathbb E[(1-O)Z]+\mathbb E[OR_s(Z)].
\] Thus the unconditional linear term cancels exactly. The retained covariance is bounded by \(\sqrt{\Pr(O=0)\mathbb EZ^2}\); independence of the gate is unnecessary. For the actual complete-ancestor gate, Lemma 6.4 gives the conditional-variance bound without changing the centered attachment variable.

For a deterministic \(K\ge2\), truncate each fresh attachment at \(K\). Write \(N=Z+R-A_K\), with \(R\ge0\), \(A_K=\mathbb ER\) and \(\mathbb EZ=0\). Suppose the full attachment laws have certified bounds \[
\mathbb E(J_a-K)_+\le a\,d_K,\qquad
\mathbb E(J_a\wedge K)^2\le a\,v_K.
\] Independence within each side and the variance inequality across the two sides give \[
\mathbb EZ^2\le4CBbT\,v_K,\qquad
A_K\le2CBbT\,d_K.
\] Since \(\Psi_m\) is decreasing and \(\lambda_m\)-Lipschitz, the centered discarded tail costs only \(\lambda_m A_K\) in either signed direction. The shift \(\mu\) costs \(\lambda_m|\mu|\), while the nonnegative debt contributes at most \(\lambda_m\mathbb E[OV]\). It follows that \[
\begin{aligned}
\left|\mathbb E[O\Psi_m(D_A)]
-\Pr(O=1)\Psi_m(s)\right|
\le{}&\lambda_m\left[
|\mu|+\mathbb E[OV]+A_K+
\sqrt{\Pr(O=0)\mathbb EZ^2}\right]\\
&+\kappa_m\left[\mathbb EZ^2/2+\sqrt{\mathbb EZ^2}\right].
\end{aligned}
\] This is a terminal integrated-kernel estimate; it uses no maximal-in-time noise bound.

The first-row correction is essential: for one future row and a fully open birth field, openness gives \(D_A=1\) exactly, whereas \(s=1-2a\). Hence \(\mathbb E[O(D_A-s)]=2a\Pr(O=1)\). Separately, if \(J\) is Bernoulli with mean \(a\), \(Z=J-a\) and \(O=1-J\), then \(\mathbb EZ=0\) but \(\mathbb E[OZ]=-a(1-a)\). Centering alone therefore does not remove the gate covariance.

\subsection*{S4.2. Attachment tail and truncated moments}\label{s4.2.-attachment-tail-and-truncated-moments}

Let \(J_a\) denote the fixed-boundary attachment length in Lemma 5.1. Put \[
M_i=e^{aB_i}-1,\qquad M(z)=\sum_{i\ge0}M_i z^i,\qquad
Q(z)=\sum_{j\ge0}q_jz^j=[1+(1-z)M(z)]^{-1}.
\] The renewal decomposition in Lemma 5.1 gives \(\Pr(J_a\ge n)=\sum_{j\ge n}q_j\), so \(Q\) is the probability generating function of \(J_a\) and \(\mathbb EJ_a=m_a=M(1)\). Hence \[
\frac{1-Q(z)}{1-z}=M(z)Q(z),\qquad
\Pr(J_a\ge n+1)=\sum_{i+j=n}M_iq_j.
\] For \(k\ge1\), split the nonnegative convolution tail according to \(i\ge k\) or \(j\ge k\). Markov's inequality and \(\sum_{i\ge k}B_i=A_{k+1}\) give \[
\begin{aligned}
D_a(2k):=\mathbb E(J_a-2k)_+
&=\sum_{n\ge2k}\Pr(J_a\ge n+1)\\
&\le\sum_{i\ge k}M_i+m_a\Pr(J_a\ge k)\\
&\le a e^{aS_1}A_{k+1}+\frac{m_a^2}{k}\\
&\le2aC_Ak^{-\gamma}+\frac{4a^2}{k}.
\end{aligned}
\] Here \(e^{aS_1}\le2\) and \(m_a\le2a\) follow from the normalized small-intensity budget. Thinning makes these full-law bounds valid for the truncated attachments as well.

For \(\gamma=1\) this gives, for \(k\ge1\), \[
D_a(2k)\le(2aC_A+4a^2)/k.
\] For integers \(n\ge4\) take \(k=\lfloor n/4\rfloor\ge n/7\). Since \(n-2k\ge n/2\), \[
\Pr(J_a\ge n)\le
\frac{28aC_A+56a^2}{n^2}\le32aC_A n^{-2}.
\] For \(n\le3\) use \(\mathbb EJ_a\le2a\). The point-tail bound therefore holds for all \(n\ge1\). For every integer \(T\ge2\), taking \(k=\lfloor T/2\rfloor\ge T/3\) directly in the mean-tail bound also gives \[
D_a(T)\le(6aC_A+12a^2)/T\le8aC_A/T.
\] Consequently \[
\mathbb E(J_a\wedge T)^2
\le64aC_A(1+\log T).
\] This does not require a finite untruncated second moment.

For \(0<\gamma<1\), let \[
H_\gamma=(4\,7^\gamma+2)C_A,\qquad
d_\gamma=(2\,3^\gamma+1)C_A.
\] The same integer split of the mean-tail estimate proves \[
\Pr(J_a\ge n)\le H_\gamma a n^{-1-\gamma},\qquad
D_a(K)\le d_\gamma aK^{-\gamma}\quad(K\ge2).
\] Here the \(a^2n^{-2}\) contribution is absorbed using \(a\le1/32\). The tail-sum formula then gives \[
\mathbb E(J_a\wedge K)^2
\le\frac{2H_\gamma a}{1-\gamma}K^{1-\gamma}.
\] For \(\gamma>1\), the enveloped power tail gives the finite quantity \[
W=\sum_{r\ge2}\nu_r\frac{r(r-1)(r+1)}6,\qquad V=3+4W.
\] The supplied tail bound gives an effective remainder \(c_+K^{-(\gamma-1)}/[6(\gamma-1)]\) for \(W\) beyond \(K\ge R_*\). An upper bound for \(V\) therefore suffices. Differentiating the nonnegative attachment generating function of Lemma 5.1 gives \[
\mathbb EJ_a^2=m_a+2M'(1)+2m_a^2,\qquad
M'(1)\le2aW.
\] Hence \(\mathbb EJ_a^2\le aV\). For finite caps, valid constants in S4.1 are \[
d_K=8C_A/K,\quad v_K=64C_A(1+\log K)\quad(\gamma=1).
\] For every \(\gamma>0\), an even cap \(K=2k\) permits \(d_K=2C_Ak^{-\gamma}+1/(8k)\) and \(v_K=2K\). For \(\gamma>1\) the untruncated option has \(d_K=0\), \(v_K=V\).

\section*{S5. Local and capped ancestor kernels}\label{s5.-local-and-capped-ancestor-kernels}

\subsection*{S5.1. Kernel decomposition}\label{s5.1.-kernel-decomposition}

Use the normalized law, fixed protocol and ancestor representation of S3. Set \(m=\lfloor\delta T\rfloor\) and \(\ell=\lfloor T/4\rfloor\). For \(T\ge256\) and \(\delta\le1/64\), \(m<\ell\) and \(\ell\ge T/8\). Throughout the hybrid argument the long-root weights are \(w_r=cr^{-p}\) and the short background uses the actual normalized law.

For a root of age \(n\ge\lceil T/2\rceil\), its ideal span \(s=n-2\sum_{i=1}^n a_i\) satisfies \(s\ge31n/32>\ell\). The exact kernel splits as \[
\Psi_m(x)=\Psi_\ell(x)+h_{m,\ell}(x),\qquad
h_{m,\ell}(x)=\sum_{m<r\le\ell}cr^{-p}(r-x)_+.
\] On terminal openness, \(D_A\ge0\), so \[
0\le h_{m,\ell}(D_A)
\le\Psi_m(0)\mathbf1_{\{D_A<\ell\}},
\qquad h_{m,\ell}(s)=0.
\] Consequently the signed large-kernel estimate of S4.1 can use \(\Psi_\ell\) globally, while the omitted short-root kernel is paid only on the actual small-span event. This split does not assume that the noise or debt stays within a prescribed neighborhood.

The small-kernel mass obeys \(4BC T^{p-2}\Psi_m(0)\le Q_\delta\), with \(Q_\delta\) from S1. S5.2 supplies the large-kernel slope and curvature constants used in the final certificate.

\subsection*{S5.2. Local slope and curvature}\label{s5.2.-local-slope-and-curvature}

Retain \(m=\lfloor\delta T\rfloor\), \(\ell=\lfloor T/4\rfloor\), \(T\ge256\), and the hybrid weights \(cr^{-p}\). The first retained large-kernel kink is \(\ell+1>T/4\), so \[
\kappa_\ell=c(\ell+1)^{-p}\le c(T/4)^{-p}.
\] Convexity on unit cells centered at the integers gives \[
\sum_{r>\ell}r^{-p}
\le\int_{\ell+1/2}^{\infty}x^{-p}\,dx.
\] Since \(\ell+1/2\ge T/4-1/2\ge127T/512\), define \[
G^\sharp=\frac{4BCc_+}{p-1}(127/512)^{1-p},
\qquad H^\sharp=4BCc_+4^p.
\] Use these constants for the large-kernel estimates. Set \(Q_\delta=4BCc_+(\delta/2)^{2-p}/(p-2)\).

\subsection*{S5.3. Direct cap estimate}\label{s5.3.-direct-cap-estimate}

For each remaining root age \(n\ge\lceil T/2\rceil\), put \[
\bar s=n-2\sum_{i<n}m_{a_i,m}=s+\mu.
\] Since \(m_{a,m}\le16a/15\) and \(Bb\le1/64\), \(\bar s\ge29n/30\ge29T/60\). The excess debts and trimming in S3.3 separately satisfy \[
\mathbb E[OV]\le\frac{8192}{3375}CB^2b^2T+\frac{128}{15}BbM_{3,m}.
\] Indeed the two half-lines have excess-attachment mean at most \(2(u_+^{-1}-1)\sum_i m_{a_i,m}\), and the remaining term is the weighted trimming bound in S3.3. Thus use \[
D_A=\bar s-Z-R+A_K-V,\qquad
\bar s\ge29T/60,\qquad R,A_K,V\ge0.
\] The representation is asserted on the actual open event \(O=1\), where \(D_A\ge0\). For \(0<q\le15/29\), let \[
f_q(t)=\min\{1,(t-1+q)_+/q\},\qquad t\ge0.
\] Then \[
f_q(t)\le\frac{841}{840}\min\{t^2,1\}.
\] For \(t\ge1\) this follows directly. On \(1-q\le t\le1\), maximize \((t-1+q)/(qt^2)\): the maximum is one when \(q\le1/2\), and is \([4q(1-q)]^{-1}\le841/840\) otherwise. Below \(1-q\) the ramp vanishes.

Also, for \(x,y\ge0\), \[
\min\{(x+y)^2,1\}\le x^2+2y.
\] When \(x+y\le1\), use \(2x+y\le2\). When \(x+y\ge1\) and \(x\le1\), use \(y\ge1-x\); \(x\ge1\) is immediate.

Put \(u=R+V\). For every \(m<r\le\ell\), \(r/\bar s\le15/29\), and the nonnegative compensator has a favorable sign. The actual span cap gives \[
\begin{aligned}
(r-D_A)_+
&\le\min\{r,(Z_++u-\bar s+r)_+\}\\
&\le r\frac{841}{840}
\left(\frac{Z^2}{\bar s^2}+\frac{2u}{\bar s}\right)\\
&\le r\left(\frac{30Z^2}{7T^2}+\frac{29u}{7T}\right).
\end{aligned}
\] Summing over kink lengths and averaging, \[
\boxed{
\mathbb E[O h_{m,\ell}(D_A)]
\le\Psi_m(0)\left[
\frac{30}{7T^2}\mathbb EZ^2+
\frac{29}{7T}\{\mathbb E[OV]+\mathbb ER\}\right].
}
\] The bound uses the weighted debt estimate of S3.3 and permits dependent half-lines and gates. Substituting S3.3--S4.1 gives \[
\begin{aligned}
\mathbb E[O h_{m,\ell}(D_A)]
\le\Psi_m(0)\bigg[
&\frac{120}{7}CBb\,\frac{v_K}{T}
+\frac{58}{7}CBb\,d_K\\
&+\frac{237568}{23625}CB^2b^2
+\frac{3712}{105}Bb\,\frac{M_{3,m}}T
\bigg].
\end{aligned}
\]

\section*{S6. Complete-path remainder and open-gate covariance}\label{s6.-complete-path-remainder-and-open-gate-covariance}

Lemma 6.3 of the manuscript proves the all-count long-root remainder bound and supplies the coefficient \(3P_\delta^2\). Its complete paths are selected from the unchanged short background before long roots are exposed. Lemma 6.4 proves the gate covariance estimate using the rate-string bound in S2.3. The following refinements and scalar envelopes are used in the effective certificate.

\subsection*{S6.1. Signed path-pair refinement}\label{s6.1.-signed-path-pair-refinement}

On a terminal-open short background \(\mathcal B\), the mean number of roots hitting both extremal paths is \[
I_{\mathcal B}
=\int L_eR_e\,\mu(de)
=\sum_t a_t\sum_{r>m}w_r(r-D_{A_t})_+,
\qquad 0\le I_{\mathcal B}\le\mathsf P.
\] This is the origin-integrated ancestor-span kernel of S3.1. Conditional on \(\mathcal B\), the both-hit, left-only and right-only roots form three disjoint Poisson classes, of means \(I_{\mathcal B}\), \(\mathsf P-I_{\mathcal B}\) and \(\mathsf P-I_{\mathcal B}\). Their independence is conditional on paths selected from the short background alone.

On \(B_\zeta\ge1\), the remainder is exactly \(1-B_\zeta\), with expectation contribution \(1-e^{-I_{\mathcal B}}-I_{\mathcal B}\). On \(B_\zeta=0\), a positive remainder requires at least one left-only and at least one right-only root. Hence \[
\begin{aligned}
1-e^{-I_{\mathcal B}}-I_{\mathcal B}
&\le\mathbb E[\mathcal E\mid\mathcal B]\\
&\le1-e^{-I_{\mathcal B}}-I_{\mathcal B}
+e^{-I_{\mathcal B}}
\bigl(1-e^{-(\mathsf P-I_{\mathcal B})}\bigr)^2.
\end{aligned}
\] These inequalities retain the terminal-open gate when averaged over backgrounds. They give the signed split between redundant singleton closures and cooperative pairs. In particular, \[
-\tfrac12I_{\mathcal B}^2
\le\mathbb E[\mathcal E\mid\mathcal B]
\le(\mathsf P-I_{\mathcal B})^2.
\] For a scalar finite-time bound, set \(\varphi(x)=x-1+e^{-x}\). The arm remainder lies in \[
[-\varphi(\mathsf P),\ (1-e^{-\mathsf P})^2].
\] Thus the adverse three-arm remainder is bounded by \[
\varphi(\mathsf P_+)+2(1-e^{-\mathsf P_b})^2
+\varphi(\mathsf P_-).
\] The quadratic bound in Lemma 6.3 follows from \(\varphi(x)\le x^2/2\) and \(1-e^{-x}\le x\). Specified row rates can be used in the actual \(\mathsf P_j\) instead of their common envelope.

Spatial overlap of the inserted roots is not required for cooperation. For example, start fully open, take six updates with no short disasters, and insert the two length-three intervals \([-5,-3]\) and \([-2,0]\) in the first row. At that row the complete terminal-path vertices are \([-5,0]\). Each insertion alone leaves a path, while their union removes every path. Both roots exceed the short cutoff \(m=2\). This configuration realizes the positive pair term without either root being individually effective.

\subsection*{S6.2. Covariance envelopes and scalar times}\label{s6.2.-covariance-envelopes-and-scalar-times}

Use \(q=(1-\gamma)/2\) and the envelope \(\overline L_\gamma\) from Lemma 6.4. The integrated covariance payment is \[
\mathcal C_{{\rm anc},K}(T)=
G^\sharp\min\left\{
\sqrt{\frac{8CB^2v_K}{T}},
\frac{2\sqrt{BK_Rv_K}}{T}\overline L_\gamma(CT)
\right\}.
\] The first branch is the general gate bound of S4.1. The second follows by multiplying Lemma 6.4 by the local kernel slope, root birth rates, at most \(CT\) birth rows and the absolute arm weights. The other local error terms are specified in S9. The second branch has orders \[
T^{-(1+\gamma)/2}\quad(0<\gamma<1),\qquad
(1+\log T)/T\quad(\gamma=1),\qquad
T^{-1}\quad(\gamma>1).
\] These are sufficient error bounds for the actual ancestor gate.

Define \[
A_{\rm cov}=\frac{128G^\sharp\sqrt{BK_Rv_K}}S.
\] A closed sufficient time for the second branch to cost at most \(S/64\) is \[
N_{\rm dyad}=
\begin{cases}
\left[A_{\rm cov}(2C)^q/(2^q-1)\right]^{2/(1+\gamma)},
&0<\gamma<1,\\
2a_*\log(4Ca_*),\quad a_*=\max\{1,A_{\rm cov}/\log2\},
&\gamma=1,\\
A_{\rm cov}/(1-2^q),&\gamma>1.
\end{cases}
\] At \(\gamma=1\) the defining scalar inequality is \(T\ge A_{\rm cov}[1+\log_2(CT)]\). For the displayed choice, write \(L=\log(4Ca_*)>1\). Then \(\log(2C N_{\rm dyad})=L+\log L\le2L\), which proves the inequality. Each continuous covariance envelope decreases for \(T\ge2\), so an upward rounding preserves every later-time bound. A directly verified scalar time can replace the closed expression.

Let \[
N_{\rm gate}=\min\left\{
\max\{2,32768CB^2v_K(G^\sharp/S)^2\},
\max\{2,N_{\rm dyad}\}\right\}.
\] \#\# S7. One-sided small-kernel control at \(\gamma=1\)

The two summands \(\Psi_\ell\) and \(h_{m,\ell}\) can use different deterministic attachment caps: their errors are added, and their noises are never multiplied. Use a fixed cap \(K_L\) for \(\Psi_\ell\) and the ancestor-gate estimate. For the small kernel choose, at each deterministic observation time, \[
K_S(T)=\lfloor T/64\rfloor,\qquad T\ge256.
\] Use its centered sum \(Z_S\), discarded tail \(R_S\) and compensator \(A_S\) in the exact representation. On \(Z_S\le7T/60\), \(\bar s-Z_S\ge11T/30>\ell\). For \(s_0\ge r\) and \(u\ge0\), \[
\min\{r,(u-s_0+r)_+\}\le ru/s_0.
\] For \(u\le s_0\), multiply the positive ramp by \(s_0\) and use \((s_0-r)(u-s_0)\le0\); for \(u\ge s_0\) the cap proves the inequality. Together with \(D_A\ge0\) on openness, this gives \[
\boxed{
\mathbb E[O h_{m,\ell}(D_A)]
\le\Psi_m(0)\left[
\Pr(Z_S>7T/60)
+\frac{30}{11T}\{\mathbb E[OV]+\mathbb ER_S\}\right].
}
\] The gamma-one attachment bounds imply \[
d_{K_S}\le1024C_A/T,\qquad
v_{K_S}\le64C_A(1+\log T).
\] In the complete reverse-row filtration of Lemma 6.4, the centered sum has row increments bounded above by \(2K_S\le T/32\) and total predictable variance at most \[
V_*=256CBC_A bT(1+\log T).
\]

The one-sided martingale bound follows from conditional exponential moments. For \(\lambda\ge0\) and \(x\le c\), \[
e^{\lambda x}\le1+\lambda x+
\frac{e^{\lambda c}-1-\lambda c}{c^2}x^2.
\] Conditional centering, iteration and exponential Markov give \[
\Pr(Z_S>x)\le
\exp\{-V_*h(cx/V_*)/c^2\}
\le(eV_*/(cx))^{x/c},
\] where \(h(u)=(1+u)\log(1+u)-u\). Take \(c=T/32\) and \(x=7T/60\), so \(x/c=56/15>3\). With \[
A_{\rm B}=\frac{1474560}{7}CBC_A,\qquad
q_T=A_{\rm B}b\frac{1+\log T}{T},
\] \(e<3\) implies \[
\boxed{\Pr(Z_S>7T/60)\le q_T^3
\quad\text{provided }q_T\le1.}
\] The condition on \(q_T\) is part of the certificate.

Multiply the kernel bound by root rates and sum as in S9.1. After division by \(b^2T^{-\gamma}\), its small-kernel contribution is at most \[
\boxed{
Q_\delta\left[
\frac{16384}{2475}CB^2b
+\frac{256}{11}BA_M\frac{1+\log T}{T}
+\frac{61440}{11}CBC_A\frac1T
+A_{\rm B}^3b^2\frac{(1+\log T)^3}{T^3}
\right].
}
\] This removes the small-kernel global-variance payment. The remaining logarithmic trimming term comes from \(M_{3,m}\).

\section*{S8. Root expansion, history cancellation and quadrature}\label{s8.-root-expansion-history-cancellation-and-quadrature}

\subsection*{S8.1. Long-root expansion}\label{s8.1.-long-root-expansion}

For each observation time \(T\), retain the actual law at lengths \(r\le m=\lfloor\delta T\rfloor\) and replace longer weights by \(cr^{-p}\). This hybrid keeps the short background and its normalization unchanged. The number of all long candidates in an arm's deterministic cone has mean at most \(bL_\delta T^{-\gamma}\). Expanding around the short background as in Section 7 and applying Lemma 6.3 to the complete remainder gives a weighted error at most \(3P_\delta^2b^2T^{-2\gamma}\).

The three no-root backgrounds differ from the target stationary cutoff field by at most the history error of S2.2. Their weighted contrast costs at most \[
16\,128^\alpha T^{-\alpha}+4C_Rb^3T^{-\gamma}.
\] In root terms, replacing baseline-open marginals by the common target marginal costs at most \(4K_RL_\delta b^2T^{-2\gamma}\). The span representation in S3.1 sums integer origins exactly. The signed large-kernel and capped small-kernel errors are collected in S9.

\subsection*{S8.2. Cancellation of young target roots}\label{s8.2.-cancellation-of-young-target-roots}

A target-period root of age \(0\le n\le\lfloor T/2\rfloor\) is born at time \(t=T-n\ge T/2\) after the switch. Conditional on its future ancestor set \(U\), define \(\pi_{j,t}(C)\) as the probability that the finite target \(C\) is all holes in arm \(j\)'s birth field. For an inserted interval \(I\), its conditional benefit is \[
\pi_{j,t}(U\setminus I)-\pi_{j,t}(U).
\] This formula includes disconnected targets and the convention \(\pi_{j,t}(\varnothing)=1\).

At the switch, each arm and the stationary target cutoff field can be bracketed by the all-open field and the stationary \(Bb\) cutoff field. The uniform finite-target definition in (25) therefore gives, for every finite \(C\), \[
|\pi_{j,t}(C)-p_{b,m}(C)|
\le R_{b,Bb,m}(t).
\] The future \(U\) is independent of the birth field, so the bound applies conditionally to both \(U\) and \(U\setminus I\). Each root benefit differs from its stationary target counterpart by at most \[
2R_{b,Bb,m}(t)\le2^{\gamma+1}K_RbT^{-\gamma}.
\] For \(n=0\) the same argument uses \(U=\{0\}\). If \(U\) is empty both terms are one and the benefit is zero.

Target root intensities and future rows are identical across the arms. Hence the stationary target benefit cancels pointwise under weights \(1,-2,1\). The deterministic candidate mean for these roots is at most \(bL_\delta T^{-\gamma}\). Their weighted residual is therefore at most \[
8\,2^\gamma K_RL_\delta b^2T^{-2\gamma}.
\] This uses the coarse history bound, uniform in the cutoff and finite input set. It requires no new cutoff condition at \(T/2\). Every root remaining in the localized kernel estimate has age greater than \(T/2\).

\subsection*{S8.3. Tail replacement}\label{s8.3.-tail-replacement}

For each fixed \(T\), construct a hybrid marked process. Retain the actual law for every length \(r\le m\), and use \(cr^{-p}\) only for \(r>m\). The no-root background is identical in the actual and hybrid processes. Its moment, attachment and phase estimates are therefore those of the actual cutoff law. The full hybrid law need not have moment one, and no normalized full-law estimate is applied to it.

For a row cone of width \(w\), a length-\(r\) mark has \(r+w-1\) origins intersecting the cone. Couple actual and hybrid long marks through their common intensities and independent remainders. Since \(\omega(r)\le\omega(m)\) for \(r>m\), the mean of mismatched relevant marks in each arm is at most \(bL_\delta\omega(m)T^{-\gamma}\). Thus the weighted terminal contrast differs by at most \[
4bL_\delta\omega(\delta T/2)T^{-\gamma}
\] once \(m\ge\delta T/2\). This is a finite-cone bound, uniform for every later integer \(T\).

\subsection*{S8.4. Ideal shrink and quadrature}\label{s8.4.-ideal-shrink-and-quadrature}

For a preparation root at age \(j\), the ideal shrink is \[
S_j=T+j-2b\left[T+\sum_{\ell=0}^{j-1}f(\ell/T)\right].
\] Put \(D_f(z)=1+z-2b[1+\int_0^z f]\). The cumulative stage quadrature gives \(|S_j-TD_f(j/T)|\le4bMB\). For \(T\ge10M\) active ideal lengths are at least \((4/5)T\), so the short cutoff contributes zero. The function \(r^{-p}(r-S_j)_+\) has total variation at most \(2S_j^{1-p}\). Its sum-to-integral error is paid by the first term of \(L_{\rm id}\). The cumulative-shrink error is paid by its second term. On each age stage, the derivative of \(D_f^{-\gamma-1}\) is bounded by \((\gamma+1)(9/10)^{-\gamma-2}\); the final term pays both stage boundaries and the age Riemann sum. Thus the per-arm ideal old-root expression differs by at most \(L_{\rm id}bT^{-\gamma-1}\) from \[
cDT^{-\gamma}\gamma u_{b,m}b
\int_0^H f(z)D_f(z)^{-\gamma-1}\,dz.
\] The target-root ideal terms among the remaining older roots cancel exactly, since their intensities, ages and ideal drifts are identical across arms. Young roots have already been paid in S8.2.

Taylor's theorem as in Section 8.4, with the explicit drift \(1-2a\), gives a three-arm ideal coefficient at least \(2u_{b,m}Eb^2\). Since \(u_{b,m}\ge15/16\), its positive contribution is at least \(15Sb^2T^{-\gamma}/8\).

\section*{S9. Final budget and complete threshold}\label{s9.-final-budget-and-complete-threshold}

\subsection*{\texorpdfstring{S9.1. Direct capped-kernel residual for every \(\gamma>0\)}{S9.1. Direct capped-kernel residual for every \textbackslash gamma\textgreater0}}\label{s9.1.-direct-capped-kernel-residual-for-every-gamma0}

Define \[
\begin{gathered}
G_d^\sharp=G_M^\sharp=G^\sharp+\frac{29}{7}Q_\delta,\quad H_v^\sharp=H^\sharp+\frac{60}{7}Q_\delta,\\
K_{3,\rm cap}=4C_R+4CB^2G^\sharp+\frac{237568}{23625}CB^2Q_\delta.
\end{gathered}
\] Choose a fixed cap \(K\) satisfying \(2CBG_d^\sharp d_K\le S/64\) with the moment bounds of S4.2. The untruncated finite-variance option is allowed. Let \(\mathcal C_{{\rm anc},K}^\sharp\) be the S6.2 covariance payment with the displayed \(G^\sharp\). The localized residual is \[
\begin{aligned}
\mathcal R_{\rm cap}(T,b)={}&\frac{2BG^\sharp}{T}+\frac{96}{47}CBG^\sharp A_\chi T^{-\gamma}
+\frac{128}{15}BA_MG_M^\sharp\ell_\gamma(T)\\
&+2CBG_d^\sharp d_K+\mathcal C_{{\rm anc},K}^\sharp(T)
+\frac{2CBH_v^\sharp v_K}{T}+\frac{2H^\sharp\sqrt{CBv_K}}{\sqrt b\,T^{3/2}}.
\end{aligned}
\] After division by \(T^{-\gamma}\), S3--S6 bound the one-root error by \((K_{3,\rm cap}-4C_R)b^3+b^2\mathcal R_{\rm cap}\). For its seven slots a sufficient scalar time is \[
\begin{aligned}
\mathcal N_{\rm cap}=\max\Bigg\{&256,\frac{128BG^\sharp}{S},
\left(\frac{6144CBG^\sharp A_\chi}{47S}\right)^{1/\gamma},\mathcal L_\gamma,N_{\rm gate},\\
&\frac{128CBH_v^\sharp v_K}{S},
\left(\frac{16384(H^\sharp)^2CBv_K}{S^2b}\right)^{1/3}\Bigg\}.
\end{aligned}
\] Here \(N_{\rm gate}\) is S6.2's value with the displayed \(G^\sharp\). Put \(A_{\rm tr}=(8192/15)BA_MG_M^\sharp/S\). The trimming scalar time is \[
\mathcal L_\gamma=
\begin{cases}
A_{\rm tr}^{1/\gamma},&0<\gamma<1,\\
2A'[1+\log(2A')],\quad A'=\max(1,A_{\rm tr}),&\gamma=1,\\
A_{\rm tr},&\gamma>1.
\end{cases}
\] The fixed discarded-tail condition and these times give \(\mathcal R_{\rm cap}\le7S/64\). Each time-dependent envelope decreases for \(T\ge256\). A smaller certified integer satisfying the same seven inequalities is admissible.

\subsection*{S9.2. Intensity budget and threshold map}\label{s9.2.-intensity-budget-and-threshold-map}

For the specified cubic constant \(K_3\) and local scalar time \(\mathcal N\), choose \(0<\eta<81/64\) and define \[
b_0(K_3)=\min\left\{\frac1{64B},\frac{2^{-\gamma}}{128B},
\frac{E}{2c_\gamma B^3[1-(1+H)^{-\gamma}]},\frac{\eta S}{K_3}\right\}.
\] For every fixed \(0<b\le b_0(K_3)\), put \(\Delta_b=81S/64-K_3b>0\) and \[
\boxed{
\begin{aligned}
T_0(b)=\mathcal T(K_3,\mathcal N;b)=\Bigg\lceil\max\Bigg\{&256,128K_B,\frac{2(R_*+1)}\delta,10M,\mathcal N,\\
&\left(\frac{64\,128^\alpha}{\Delta_b b^2}\right)^{1/\sigma},
\left(\frac{4K_2}{\Delta_b}\right)^{1/\gamma},
\frac{16L_{\rm id}}{\Delta_b b},
\frac{2[\Omega(\Delta_b b/(16L_\delta))+1]}\delta
\Bigg\}\Bigg\rceil .
\end{aligned}}
\] The threshold \(T_0(b)\) depends on the fixed intensity \(b\) and the chosen admissible tuple. The direct certificate uses \(K_3=K_{3,\rm cap}\), \(\mathcal N=\mathcal N_{\rm cap}\). Exact tails use \(\omega=0\) and \(\Omega(\varepsilon)=R_*\). Nonexact tails with \(\omega(r)=D_{\rm err}r^{-\zeta}\) may use \(\Omega(\varepsilon)=\lceil\max\{R_*,(D_{\rm err}/\varepsilon)^{1/\zeta}\}\rceil\).

\subsection*{\texorpdfstring{S9.3. Stronger certificate at \(\gamma=1\)}{S9.3. Stronger certificate at \textbackslash gamma=1}}\label{s9.3.-stronger-certificate-at-gamma1}

For this certificate define \[
K_{3,\rm B}=4C_R+4CB^2G^\sharp+
\frac{16384}{2475}CB^2Q_\delta,\qquad
G_M^{\rm B}=G^\sharp+\frac{30}{11}Q_\delta.
\] Choose the fixed large-kernel cap by \(2CBG^\sharp d_{K_L}\le S/64\). At \(\gamma=1\) the explicit choice \[
K_L\ge\max\{2,1024CBC_AG^\sharp/S\},\qquad
d_{K_L}=8C_A/K_L,\quad
v_{K_L}=64C_A(1+\log K_L)
\] suffices. Let \(\mathcal C_{\rm anc}^{\sharp,L}\) be S6.2's covariance payment with \(G^\sharp,v_{K_L}\) and define \[
\begin{aligned}
J_{\rm B}(T,b)&=\frac{A_0}{T}
+Q_\delta A_{\rm B}^3b^2\frac{(1+\log T)^3}{T^3},\\
A_0&=2CBH^\sharp v_{K_L}
+\frac{61440}{11}CBC_AQ_\delta .
\end{aligned}
\] The complete local residual is \[
\begin{aligned}
\mathcal R_{\rm B}(T,b)={}&
\frac{2BG^\sharp}{T}
+\frac{96}{47}CBG^\sharp A_\chi T^{-1}
+\frac{128}{15}BA_MG_M^{\rm B}\frac{1+\log T}{T}\\
&+2CBG^\sharp d_{K_L}
+\mathcal C_{\rm anc}^{\sharp,L}(T)
+J_{\rm B}(T,b)
+\frac{2H^\sharp\sqrt{CBv_{K_L}}}{\sqrt b\,T^{3/2}}.
\end{aligned}
\] It has seven payment slots. The joint slot \(J_{\rm B}\) pays both the large-kernel curvature and the small-kernel tail.

Here are closed sufficient scalar times, alongside the directly checkable seven inequalities. Set \[
F_{\log}(x)=2x_*[1+\log(2x_*)],\qquad x_*=\max\{1,x\}.
\] Then \(T\ge F_{\log}(x)\) implies \(x(1+\log T)/T\le1\) for every later \(T\). Define \[
\begin{aligned}
N_{\rm adm}&=F_{\log}(A_{\rm B}b),\\
N_{\rm joint}&=\max\left\{
\frac{128A_0}{S},
F_{\log}\!\left[
\left(\frac{128Q_\delta A_{\rm B}^3b^2}{S}\right)^{1/3}
\right]\right\},\\
N_{\rm trim}&=F_{\log}\!\left[
\frac{8192}{15}\frac{BA_MG_M^{\rm B}}S\right].
\end{aligned}
\] For \(N_{\rm joint}\), the linear and cubic-logarithmic terms each cost at most \(S/128\). Let \(N_{\rm gate}^{\sharp,L}\) be S6.2's covariance scalar time with \(G^\sharp,v_{K_L}\). Take \[
\begin{aligned}
\mathcal N_{\rm B}=\max\Bigg\{&
256,\frac{128BG^\sharp}{S},
\frac{6144CBG^\sharp A_\chi}{47S},
N_{\rm trim},N_{\rm gate}^{\sharp,L},N_{\rm joint},N_{\rm adm},\\
&\left(\frac{16384(H^\sharp)^2CBv_{K_L}}{S^2b}\right)^{1/3}
\Bigg\}.
\end{aligned}
\] The fixed-cap tail condition and these times imply \(\mathcal R_{\rm B}\le7S/64\). A smaller integer satisfying the actual seven inequalities and \(q_T\le1\) also suffices. Every displayed continuous error envelope decreases for \(T\ge256\). The adaptive \(K_S(T)\) is chosen separately at each deterministic time; the proof requires a bound on expectations at each time, with no simultaneous-event assertion.

Use \(K_3=K_{3,\rm B}\) and \(\mathcal N=\mathcal N_{\rm B}\) in S9.2. The large and small kernels use separate deterministic caps. Their errors are added; no product of the two centered noises is used.

\subsection*{S9.4. Completion of the proof}\label{s9.4.-completion-of-the-proof}

The ideal contribution from S8.4 is at least \(15Sb^2T^{-\gamma}/8\). S2, S8 and the selected local certificate bound the total adverse error after division by \(T^{-\gamma}\) by \[
K_3b^3+16\,128^\alpha T^{-\sigma}+K_2b^2T^{-\gamma}
+4L_{\rm id}b/T+4bL_\delta\omega(\delta T/2)+b^2\mathcal R(T,b).
\] The four finite-time terms between the cubic error and the local residual each cost at most \(\Delta_b b^2/4\) at \(T\ge T_0(b)=\mathcal T(K_3,\mathcal N;b)\). The local residual is at most \(7Sb^2/64\). Therefore \[
\Xi_h(\rho,T)\ge b^2T^{-\gamma}
\left[\frac{15}{8}S-K_3b-\left(\frac{81}{64}S-K_3b\right)-\frac7{64}S\right]
=\frac12Sb^2T^{-\gamma}>0.
\] All envelopes remain bounded at every later integer time. This proves Theorem 4.6. \(\square\)

\section*{S10. Nonuniformity at fixed summary parameters}\label{s10.-nonuniformity-at-fixed-summary-parameters}

Fix \(p>3\), \(\chi>0\) and \(0<\rho\chi\le10^{-3}\). For any prescribed integer \(N\ge1\), put \(R=16N\) and \[
v=\frac{2\chi}{R(R+1)},\qquad
\nu^{\rm atom}_r=v\mathbf1_{\{r=R\}}.
\] Consider the constant three-arm protocol with \(H=1,T=N\). Its preparation cone parameters are \(k_i=2N-i\), \(1\le i\le N\). Put \[
A_k=\sum_{r\ge k+1}(r-k)\nu_r,\qquad
B(k,h)=\sum_{\ell=1}^{k}\sum_{j=1}^{h}
\mathbf1_{\{\ell+j\ge k+1\}}\tau_\ell\tau_j
\quad(k\ge h).
\] The two-event counting of Section 8.5 gives \[
[\rho^2]W=
2\sum_iB(k_i,k_i)+4\sum_{i<j}B(k_i,k_j)
-\left(\sum_iA_{k_i}\right)^2.
\] For the atom, \(A_k=v(R-k)\) and \(B(k,h)\le v^2kh\). With \(\sum k_i=(3N^2-N)/2\), this coefficient is at most \[
-\frac{823}{4}v^2N^4
\le-\frac{823}{73984}\chi^2<-\chi^2/100.
\] The largest relevant four-multiplier cone mass is \(v(4NR+5N^2-2N)\le69\chi/128\). The third-difference estimate in Section 8.5 bounds the cubic remainder by \(\chi^3\rho^3\). Thus \(W^{\rm atom}\le-0.009\chi^2\rho^2\).

Let \(\chi_p=\sum_{r\ge2}r^{-p}r(r+1)/2\) and \(\nu^{\rm pow}_r=(\chi/\chi_p)r^{-p}\). Both component laws have moment \(\chi\). Set \[
\varepsilon=\frac{\rho\chi}{10000N^2},\qquad
\nu=(1-\varepsilon)\nu^{\rm atom}+\varepsilon\nu^{\rm pow}.
\] This has the same moment and a positive exact power-law tail beyond \(R\). Since \(S_1\le2\chi/3\) and \(S_0\le\chi/3\), the differing mark measures in each arm's cone have total mass below \(6\varepsilon\chi N^2\). Poisson coupling gives a weighted contrast change at most \(24\rho\varepsilon\chi N^2=0.0024\chi^2\rho^2\). Therefore \(W<-0.0066\chi^2\rho^2<0\) at the prescribed \(N\). This proves Proposition 4.7. The construction varies the law with \(N\); it concerns general histories and does not establish the corresponding assertion restricted to equal-dose histories. \(\square\)

\section*{S11. Evaluated complete certificates}\label{s11.-evaluated-complete-certificates}

For the normalized pure law use \(2<c<4\), \(C_A=16\) and \(S_{\rm lo}=1/54\). For the normalized law \(r^{-4}(1+1/r)\), normalization gives \(1<c<2\), \(\omega(r)=1/r\), \(c_+\le3\), \(C_A=12\) and \(S_{\rm lo}=1/108\). In both cases use the equal-dose profiles, \(B=C=3\), \(M=2\), \(E=1/18\) and \(\eta=5/4\).

Choose \(\theta=19/20\), \(\alpha\in\{16/10,\ldots,50/10\}\), \(\delta_B\in\{6/100,\ldots,45/100\}\) and \(\delta=\delta_B/128\), retaining \(d=\theta-\alpha\delta_B\ge1/10\). Only tuples whose own budget admits the specified intensity are evaluated. The table below lists rows that satisfy all seven local inequalities, four finite-time margin inequalities, structural time bounds and the full intensity budget of S9.3.

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.1579}}
  >{\raggedleft\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2105}}
  >{\raggedleft\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2105}}
  >{\raggedleft\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2105}}
  >{\raggedleft\arraybackslash}p{(\linewidth - 8\tabcolsep) * \real{0.2105}}@{}}
\caption{Complete parameter choices and certified upper bounds for \(T_0(b)\) at the listed fixed intensities for the two normalized laws.}\tabularnewline
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Normalized law
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
\(b\)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
\(\alpha\)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
\(\delta_B\)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Certified \(T_0(b)\) upper bound
\end{minipage} \\
\midrule\noalign{}
\endfirsthead
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Normalized law
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
\(b\)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
\(\alpha\)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
\(\delta_B\)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedleft
Certified \(T_0(b)\) upper bound
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
\(r^{-4}\) & \(10^{-30}\) & \(3\) & \(3/50\) & \(10^{36}\) \\
\(r^{-4}\) & \(3\times10^{-16}\) & \(29/10\) & \(3/50\) & \(10^{22}\) \\
\(r^{-4}\) & \(5\times10^{-16}\) & \(3\) & \(2/25\) & \(10^{21}\) \\
\(r^{-4}\) & \(2\times10^{-15}\) & \(29/10\) & \(2/25\) & \(10^{21}\) \\
\(r^{-4}\) & \(3\times10^{-15}\) & \(29/10\) & \(2/25\) & \(10^{21}\) \\
\(r^{-4}\) & \(10^{-14}\) & \(29/10\) & \(9/50\) & \(10^{20}\) \\
\(r^{-4}\) & \(2\times10^{-14}\) & \(29/10\) & \(9/50\) & \(10^{20}\) \\
\(r^{-4}\) & \(2.5\times10^{-14}\) & \(14/5\) & \(9/100\) & \(10^{21}\) \\
\(r^{-4}\) & \(5\times10^{-14}\) & \(14/5\) & \(1/10\) & \(10^{21}\) \\
\(r^{-4}\) & \(10^{-13}\) & \(14/5\) & \(7/50\) & \(10^{21}\) \\
\(r^{-4}\) & \(1.3\times10^{-13}\) & \(14/5\) & \(9/50\) & \(10^{21}\) \\
\(r^{-4}(1+1/r)\) & \(3\times10^{-16}\) & \(23/10\) & \(27/100\) & \(10^{32}\) \\
\(r^{-4}(1+1/r)\) & \(10^{-15}\) & \(11/5\) & \(21/100\) & \(10^{32}\) \\
\(r^{-4}(1+1/r)\) & \(10^{-14}\) & \(11/5\) & \(21/100\) & \(10^{31}\) \\
\(r^{-4}(1+1/r)\) & \(3\times10^{-14}\) & \(11/5\) & \(3/10\) & \(10^{30}\) \\
\(r^{-4}(1+1/r)\) & \(5\times10^{-14}\) & \(11/5\) & \(7/25\) & \(10^{30}\) \\
\end{longtable}

At \(T=10^q\), the scalar checks use \(1+\log T<1+3q\) and \(1+\log_2(3T)<3+10q/3\). The covariance condition is certified by \[
T^2\ge\frac{16384(G^\sharp)^2BK_Rv_{K_L}}{S_{\rm lo}^2}\left(3+\frac{10q}{3}\right)^2.
\] The additional one-sided and joint-slot tests are \[A_{\rm B}b(1+3q)\le T,\qquad
\frac{A_0}{T}+\frac{Q_\delta A_{\rm B}^3b^2(1+3q)^3}{T^3}\le S_{\rm lo}/64.\] The parameter tuples and complete rational outputs are identified by the reproducibility index in Online Resource 2. At \(\delta_B=2/5\), \(\alpha=9/5\), the sufficient budget has the lower bound \[
b_0\ge\frac{112661065}{695667801466437894144}>1.619\times10^{-13}.
\] For each listed fixed \(b\), the continuous error envelopes in S9 prove the all-future assertion from its scalar-certified upper bound for \(T_0(b)\). The time upper bounds need not equal the first positive observation time.

The profile discretization is the one specified in the manuscript. The sign certificate covers every later integer time; for the two equal halves of the example, exact equality of cumulative discrete doses holds at even observation times.

\end{document}