Proof. For fixed \(Z_h\), let \(\mathcal Z_\epsilon[Z_h]\) be the partition function of the remaining particles with their usual mutual exclusion and with the additional restriction that they avoid all the balls \(B(x_j,\epsilon)\). Directly from the ensemble, \[ F_{h,0}^{\epsilon}(Z_h)= \mathbf 1_{\mathcal D_h^\epsilon}(Z_h)\prod_{j=1}^h f_0(z_j) \frac{\mathcal Z_\epsilon[Z_h]}{\mathcal Z_\epsilon}. \tag{23}\] The ratio is at most one, proving the factorial upper bound. When \(h=1\), it is the probability under the original ensemble that the ball \(B(x_1,\epsilon)\) contains no particle center. The factorial upper bound with one particle and the union bound show that this probability is at least \(1-\mu b_\epsilon\). This proves (17). Integrating the factorial upper bound gives the first part of (18); expressing \(N^p\) as a finite positive linear combination of \((N)_j\), \(j\le p\), gives its second part.

We give the connected expansion and its convergence explicitly. Write \(\chi_{ij}=\mathbf 1_{|x_i-x_j|\le\epsilon}\) and set \[\mathcal U_1=1, \qquad \mathcal U_n(Z_n)= \sum_{H\text{ connected on }[n]} \prod_{\{i,j\}\in H}(-\chi_{ij})\quad(n\ge2).\] The Penrose partition scheme (Penrose 1967), in the form proved in (Bodineau et al. 2023b, Proposition 2.3.3), gives the following tree bound. \[ |\mathcal U_n(Z_n)|\le \sum_{\mathcal T\text{ tree on }[n]} \prod_{\{i,j\}\in\mathcal T}\chi_{ij}. \tag{24}\] To see this, root a connected labelled graph at vertex \(1\), assign its graph-distance levels, and choose the least-labelled neighbor in the preceding level as the parent of each other vertex. For a fixed resulting tree \(\mathcal T\), the graphs that give this tree form the interval \(\mathcal T\subset H\subset R(\mathcal T)\). Here \(R(\mathcal T)\) contains all same-level edges and every edge from a vertex to a preceding-level vertex whose label is at least the label of its chosen parent. No other edge is permitted. Summing this interval yields \[\prod_{e\in\mathcal T}(-\chi_e) \prod_{e\in R(\mathcal T)\setminus\mathcal T}(1-\chi_e).\] These intervals partition all connected graphs, and every factor in the second product lies in \([0,1]\), proving (24).

Integrating a tree successively from its leaves, and keeping one free vertex, gives \[ \int f_0^{\otimes n}|\mathcal U_n| \le n^{n-2}b_\epsilon^{n-1}\qquad(n\ge2). \tag{25}\] For a bounded measurable activity multiplier \(a\), the formal connected-graph identity is therefore an absolutely convergent identity whenever \(e\mu b_\epsilon\|a\|_\infty<1\): \[ \log\mathcal Z_\epsilon[a] =\sum_{n\ge1}\frac{\mu^n}{n!} \int\mathcal U_n(Z_n)\prod_{j=1}^n f_0(z_j)a(z_j)\,\mathrm dZ_n. \tag{26}\] For completeness, first multiply the activity by a complex scalar \(w\). The identity follows coefficient by coefficient near \(w=0\) from the finite exponential formula for connected components. The tree bound makes the right side absolutely convergent on the stated disk, so exponentiation and analytic continuation prove the identity there. This also fixes the branch of the logarithm. The same bounds justify every fixed number of derivatives in bounded source multipliers.

Apply (26) to \(a=1+u\). The logarithm of \(\mathbb E\prod_i(1+u(z_i))\) has coefficients \(\mu^h c_{h,0}^{\epsilon}/h!\) by (16). Differentiating the absolutely convergent series gives \[ c_{h,0}^{\epsilon}(Z_h) =\prod_{j=1}^h f_0(z_j) \sum_{n\ge0}\frac{\mu^n}{n!} \int f_0^{\otimes n}(Z_{h+1:h+n}) \mathcal U_{h+n}(Z_{h+n})\,\mathrm dZ_{h+1:h+n}. \tag{27}\] We next give bounds uniform in the number \(h\) of fixed roots. From any tree on the \(h+n\) vertices, delete \(h-1\) edges so that each remaining component contains exactly one fixed root. Such a deletion is obtained by repeatedly cutting an edge on a path between two roots. Dropping the deleted constraints and integrating the auxiliary leaves then costs at most \(b_\epsilon^n\). For \(n\ge1\), \[ \frac{(h+n)^{h+n-2}}{n!} \le (eh)^h e^{2n}. \tag{28}\] Indeed, \(n!\ge(n/e)^n\), \((1+h/n)^n\le e^h\), and \((h+n)^h\le h^h e^n\). The \(n=0\) count is also bounded by \((Ch)^{Ch}\). It follows that (27) is absolutely convergent for \(\epsilon\le\epsilon_0\), with this \(\epsilon_0\) independent of \(h\).

Let \(q\) be the number of connected components of the graph on the fixed roots with edges \(|x_i-x_j|\le\epsilon\). If \(q\ge2\), every contributing full tree must attach each such component to an auxiliary vertex. An auxiliary vertex can be adjacent to at most \(27\) different root components: choose one adjacent root from each; their mutual distances exceed \(\epsilon\), and their disjoint balls of radius \(\epsilon/2\) lie in a ball of radius \(3\epsilon/2\). Thus a contributing tree with \(n\) auxiliary vertices has \(q\le27n\). Using (28) and \(d_\epsilon=O(\epsilon)\), we obtain \[\frac{|c_{h,0}^{\epsilon}(Z_h)|}{\prod_j f_0(z_j)} \le (Ch)^{Ch}\epsilon^{q/27} \le (Ch)^{Ch}\epsilon^{(q-1)/81}\qquad(q\ge2),\] with the quotient interpreted through (27) when some \(f_0(z_j)=0\). For \(q=1\) the same argument without the restriction on \(n\) gives \((Ch)^{Ch}\). There exists a tree of the roots using edges inside each of the \(q\) components and exactly \(q-1\) edges between components. Its product in (19) is at least \(\epsilon^{(q-1)/81}\). This proves the coarse link estimate.

For fixed \(h,D\), truncate (27) at an integer \(n_0\) with \(n_0+1\ge D(h-1)\). Its tail is bounded by \[C_{D,h}\epsilon^{D(h-1)}\prod_j f_0(z_j).\] In any contributing term with \(n\le n_0\), all fixed roots are connected by paths of at most \(h+n_0-1\) links of length at most \(\epsilon\). Thus their diameter is at most \((h+n_0)\epsilon\). On that set the tree sum in (20) is at least one when \(C_{D,h}\) is chosen large enough; everywhere it is at least \(\epsilon^{D(h-1)}\). This proves (20). Integrating all roots in (27) and using (25) instead gives \[\|c_{h,0}^{\epsilon}\|_1 \le b_\epsilon^{h-1} \sum_{n\ge0}\frac{(h+n)^{h+n-2}}{n!}d_\epsilon^n \le C_h\epsilon^{3(h-1)},\] as asserted.

For a real random variable \(X\) with exponential moments near the origin, write \[\kappa_k(X)=\left.\frac{\mathrm d^k}{\mathrm du^k}\log\mathbb Ee^{uX}\right|_{u=0}, \qquad k\ge1,\] for its ordinary cumulants. To prove the initial central limit theorem, let \(\phi\) be bounded and real and put \(S_\phi=\sum_i\phi(z_i)\). Substitute \(a(z)=e^{u\phi(z)}\) into (26) and differentiate \(k\) times at \(u=0\). The resulting ordinary cumulant is \[\kappa_k(S_\phi)= \sum_{n\ge1}\frac{\mu^n}{n!} \int f_0^{\otimes n}\mathcal U_n \left(\sum_{j=1}^n\phi(z_j)\right)^k.\] For each fixed \(k\), (25) gives \[ \kappa_k(S_\phi)=\mu\int f_0\phi^k+O_{k,\phi}(\mu\epsilon). \tag{29}\] In fact the absolute error is bounded by \(\mu\|\phi\|_\infty^k\sum_{n\ge2} n^{n+k-2}d_\epsilon^{n-1}/n!\), which is \(O_{k,\phi}(\mu\epsilon)\). After exact centering and division by \(\sqrt\mu\), the second cumulant tends to \(\int f_0\phi^2\) and every higher cumulant tends to zero. The moment–cumulant identity gives every Gaussian moment. To see directly that this implies scalar convergence, compare the characteristic functions of the normalized variable and the Gaussian with variance \(\sigma^2=\int f_0\phi^2\), using their Taylor polynomials of degree \(2n-1\). After taking \(\limsup_{\epsilon\to0}\), the sum of their two remainder bounds is at most \(2|u|^{2n}\sigma^{2n}/(2^n n!)\), which tends to zero for each real \(u\). Applying scalar convergence to each linear combination of the tests gives joint convergence. This argument also includes zero variance.

Finally, the \(n=1\) auxiliary term of (27) with \(h=1\) gives, uniformly in \(z\), \[g_0(z)=f_0(z)\left[ 1-\mu\int_{|x-x_*|\le\epsilon}\varrho_0(x_*)\,\mathrm dx_* +O(\epsilon^2)\right].\] The bounded gradient of \(\varrho_0\) implies \[\int_{|x-x_*|\le\epsilon}\varrho_0(x_*)\,\mathrm dx_* =\frac{4\pi}{3}\epsilon^3\varrho_0(x)+O(\epsilon^4).\] Since \(\sqrt\mu=\epsilon^{-1}\), integration against \(\phi\) proves (22). ◻

Proof. Let \(Q^+\) denote the positive gain part of \(Q\), and define \(P_tg=Q^+(g,f_t)\), keeping the second factor fixed. Write \(\mathsf T_t g(x,v)=g(x-tv,v)\) for density transport, and \[(\mathcal Vg)_t=\int_0^t\mathsf T_{t-s}P_sg_s\,\mathrm ds.\] Positivity and the transport Duhamel formula imply \[ f_t\le\sum_{j=0}^{n-1} (\mathcal V^j\mathsf T_{\cdot}f_0)_t +(\mathcal V^nf)_t. \tag{31}\] We remove the remainder without presupposing spatial integrability. On the Gaussian decay spaces \(Y_a=\{g:\|g\|_{Y_a}=\sup e^{a|v|^2}|g(x,v)|<\infty\}\), energy conservation at a collision gives, for \(0<b<a\le2\beta\), \[|P_tg(x,v)|\le CM\|g\|_{Y_a}(1+|v|)e^{-a|v|^2}, \qquad \|P_tg\|_{Y_b} \le CM\bigl(1+(a-b)^{-1/2}\bigr)\|g\|_{Y_a}.\] Constants are uniform when \(b\) is bounded away from zero. Distribute the exponent interval from \(\beta\) to \(\beta/2\) equally among the \(n\) factors in the remainder. Transport preserves these norms and the time simplex has volume \(T^n/n!\), so \[\sup_{t\le T}\| (\mathcal V^nf)_t\|_{Y_{\beta/2}} \le M\frac{(CT)^n}{n!} \bigl(1+\sqrt{2n/\beta}\bigr)^n\longrightarrow0.\] Thus \(f\) is bounded pointwise by the full positive gain series.

Let \(X_q=L^1(e^{q|v|^2}\mathrm dx\mathrm dv)\). The pre/postcollision substitution, followed by energy conservation and the bound on \(f_t\), gives \[\|P_tg\|_{X_q} \le CM\int |g(x,v)|(1+|v|)e^{q|v|^2}\,\mathrm dx\,\mathrm dv \le CM\bigl(1+\delta^{-1/2}\bigr)\|g\|_{X_{q+\delta}},\] uniformly for exponents in a compact subinterval of \((0,2\beta)\). For example, in the first inequality use \(e^{q|v'|^2}\le e^{q|v|^2+q|v_*|^2}\) and integrate the second factor against its Gaussian bound. The initial Bol norm implies \(f_0\in X_{q_0}\) for every \(q_0<2\beta\). Given \(q<q_0\), distribute \(q_0-q\) among the factors of the positive gain series. The same convergent bound with \(n^{n/2}/n!\) proves that its \(X_q\) norm is uniformly finite for \(t\le T\). Integrating the pointwise majorant proves (30). ◻

Proof. For \(a\) in a compact subinterval of \([0,2\beta)\), energy conservation and the Gaussian bound give \[ |K_th(x,v)|\le CM\|h\|_a(1+|v|)e^{a|v|^2}, \qquad \|K_th\|_b\le CM\bigl(1+(b-a)^{-1/2}\bigr)\|h\|_a. \tag{35}\] Indeed \(|\Delta_h|\le4\|h\|_a e^{a(|v|^2+|v_*|^2)}\) and \(\int B(v-v_*,\omega)\mathrm d\omega=\pi|v-v_*|\). The operator also maps \(E_a^0\) into \(E_b^0\). To check the spatial tail, replace \(\|h\|_a\) in the first bound by \(\eta_h(x)=\sup_v e^{-a|v|^2}|h(x,v)|\), which tends to zero as \(|x|\to\infty\). The exponent loss controls the velocity tail. Classical continuity of \(f\), local dominated convergence, and these tail bounds show that \(t\mapsto K_th\) is continuous in \(E_b^0\). Free transport is a strongly continuous isometry on each \(E_a^0\).

Iterate (33). Giving each of the \(n\) collision factors the exponent allowance \((b-a)/n\) bounds the \(n\)th time-ordered term by \[ \frac{(CM(t-r))^n}{n!} \bigl(1+\sqrt{n/(b-a)}\bigr)^n\|h\|_a. \tag{36}\] The series converges for every finite \(T\). Its terms are Bochner integrals on the stated spaces. Splitting a time-ordered simplex at an intermediate time proves composition; the same estimate applied to an iterated homogeneous remainder proves uniqueness, leaving a positive exponent allowance between its two endpoint spaces.

For a compactly supported smooth \(\phi\), the function \(A_s\phi\) has compact spatial support, grows at most linearly in \(|v|\), and is continuous in \(s\) in every \(E_a^0\) with \(a>0\). Differentiate the series at its upper endpoint \(t\). Differentiating its final transport factor gives \(U(r,t)(v\cdot\nabla_x\phi)\), and the upper faces of the simplices give \(U(r,t)K_t\phi\). Both differentiated series converge with the bound (36), using a small extra exponent allowance for \(K_t\phi\). Hence \(\partial_tU(r,t)\phi=U(r,t)A_t\phi\), which proves (34). ◻

Proof. For \(b<\beta\), energy conservation and 5 give \[ |C_t(h,k)|\le CM\|h\|_b\|k\|_b \int f_t(x,v)(1+|v|)e^{2b|v|^2}\,\mathrm dx\,\mathrm dv \le C_b\|h\|_b\|k\|_b. \tag{38}\] Choose the moment exponent slightly above \(2b\) to absorb the factor \(1+|v|\). The initial quadratic form has the corresponding bound.

Let \(I_0\) and \(I\) be independent isonormal Gaussian maps on, respectively, \(L^2(f_0\mathrm dz)\) and \(L^2(\nu)\), where \[\mathrm d\nu=\frac12 B(v-v_*,\omega) f_r(x,v)f_r(x,v_*) \,\mathrm dr\,\mathrm dx\,\mathrm dv\,\mathrm dv_*\,\mathrm d\omega.\] Such maps can be constructed by choosing orthonormal bases and independent standard normal coefficients; their defining series converge in \(L^2\). Define \[ \zeta_t(h)=I_0\bigl(U(0,t)h\bigr) +I\bigl(\mathbf 1_{r\le t}\Delta_{U(r,t)h}\bigr), \qquad W_t(h)=I\bigl(\mathbf 1_{r\le t}\Delta_h\bigr). \tag{39}\] For \(h\in E_a^0\), propagate to any exponent \(b\in(a,\beta)\) and apply (38). This proves that the maps are well defined and that \[ \sup_{t\le T}\|\zeta_t(h)\|_{L^2(\Omega)}\le C_a\|h\|_a. \tag{40}\] They are jointly Gaussian and linear, and their covariance is (37). Orthogonality of disjoint time intervals in \(L^2(\nu)\) gives the independent increments of \(W\); independence of \(I_0\) and \(I\) gives the stated initial independence.

We verify the weak equation, including the possibly unbounded drift test. For \(\phi\in C_c^\infty\), use (34) and the boundedness of the two isonormal maps to interchange them with Bochner integrals. All the triangular kernels involved have finite squared norm by (38). One obtains \[\begin{split} \int_0^t\zeta_s(A_s\phi)\,\mathrm ds ={}&I_0\bigl(U(0,t)\phi-\phi\bigr)\\ &+I\bigl(\mathbf 1_{r\le t} \Delta_{U(r,t)\phi-\phi}\bigr). \end{split}\] Together with (39), this is precisely \[\zeta_t(\phi)=\zeta_0(\phi) +\int_0^t\zeta_s(A_s\phi)\,\mathrm ds+W_t(\phi).\]

For uniqueness, work on the probability space of any candidate solution. Construct the backward mild field there from its own initial field and noise: the deterministic integrals are mean-square limits of compact tests and time-step functions, with the bounds just proved. Subtract this field from the candidate and call the difference \(D_t\). It has zero initial field and noise. Its compact-test weak equation and uniform second-moment bound give mean-square continuity. Riemann sums then permit the time-dependent tests \(\mathsf S_{t-s}\phi\), whose supports lie in one compact set for \(0\le s\le t\). They give the transport mild identity \[D_t(\phi)=\int_0^t D_s\bigl(K_s\mathsf S_{t-s}\phi\bigr)\,\mathrm ds.\] It extends by density and (35) to exponent scales with a positive allowance. For \(0\le a<b<\beta\), iteration \(n\) times and (40) bound its \(L^2\) norm by \[C_b\|\phi\|_a\frac{(CT)^n}{n!} \bigl(1+\sqrt{n/(b-a)}\bigr)^n\longrightarrow0.\] This proves uniqueness in the stated class and therefore uniqueness in law of the constructed Gaussian solution.

This construction is a Gaussian generalized field in the explicit mean-square test-space sense of (40); it does not assert membership in the Banach dual of \(E_a^0\) sample by sample. If desired, its restriction to \(C_c^\infty(\mathbb R^6)\) has a random distribution version. One direct construction expands locally in Fourier functions multiplied by a compact cutoff. Their variances are uniformly bounded by (40) with \(a=0\); the expected squared local \(H^{-s}\) norm is therefore finite for \(s>3\). A countable exhaustion by cubes yields consistent local distributions. No path-space assertion is needed here. ◻

Proof. Apply (Deng et al. 2025, Proposition A.1) with decreasing Gaussian exponent \(\beta_*(t)=2\beta-\beta t/T\). The weighted uniform hypothesis follows from (6), and its two initial summable spatial norms are (5). Since \(\beta_*(t)\ge\beta\), its conclusion implies (43). This is the same application as (Deng et al. 2025, Equation (1.29)). ◻

Source statements and the envelope adaptations. For (a), (Deng et al. 2025, Proposition 7.2) counts the root-labeled unlabeled histories by \(C^b|\log\epsilon|^{C_*\rho}\). (Deng et al. 2025, Proposition 9.7, Equation (9.51)) retains a positive geometric gain proportional to \(\rho\). After the regular cutting operations, (Deng et al. 2025, Equation (9.53)) bounds each full edge integral by \[\tau^{b/9}\epsilon^{-2h+2\upsilon/5+c_*\rho} |\log\epsilon|^{C_*\rho}.\] Multiplication by \(\mu^{-h}=\epsilon^{2h}\) cancels the negative root power. Incorporating the counting and the cutting alternatives, weakening \(1/9\) to \(1/10\), and discarding the extra positive power gives (50). These are the estimates in the proof of (Deng et al. 2025, Proposition 6.2, Section 9.2), rather than just its final displayed conclusion.

The only use there of a birth density is the decomposition in (Deng et al. 2025, Equations (9.58)–(9.63)): write its absolute value as a sum of spatially localized Gaussian envelopes, whose coefficients have bounded sum. Indeed, from (1) choose a summable sequence \((m_a)\) bounding the weighted supremum on the unit balls; the bounded overlap of those balls gives a decomposition dominated by \(C\sum_a m_a\mathbf 1_{|x-a|\le1}e^{-\beta'|v|^2}\), at any permitted exponent \(\beta'>0\). Apply the positive source estimates to each product of such envelopes and sum their coefficients. This proves the stated extension to arbitrary bounded inputs, including absolute values of signed inputs. Multipliers between zero and one are kept in the support throughout, or dropped only when a positive upper bound is being taken.

For (b), the one-root integral is a function of a fixed endpoint by (Deng et al. 2025, Equation (14.2)). The C/O involution of (Deng et al. 2025, Proposition 14.2) cancels the noncausal trees in the complete signed sum. The remaining causal trees satisfy the Gaussian Bol estimate (Deng et al. 2025, Proposition 14.3, Equation (14.8)). The comparison to the short-time Boltzmann expansion, using 8, is the proof of (Deng et al. 2025, Proposition 6.1, Section 14). To replace its preceding input by \(g\), expand each birth factor as \(f_{(i-1)\tau}+(g-f_{(i-1)\tau})\). Each residual factor gains \(\epsilon^{\theta_{i-1}}\) by (51); the source multilinear tree bound sums all choices. Since \(\theta_{i-1}>\theta_i\), the residual and the finite-diameter comparison errors are \(o(\epsilon^{\theta_i})\), proving (52) for small \(\epsilon\).

For the positive subleading terms the explicit source bound is (Deng et al. 2025, Equation (14.28)), in \(\mathrm{Bol},\widetilde\beta_i\), with a time power per atom and gain \(\epsilon^{\upsilon/3+c_*\rho}\) in the cyclic and extra-encounter cases. Points 1–2 of Section 14.3 retain the Gaussian weight at the fixed root and the spatial Bol sum; Point 4 regularizes an O-atom adjacent to that fixed root without altering its integral. Thus this is stronger than an integrated mass estimate. The tree-tail branch of Point 5 uses only that the atom number is of cutoff order; replacing \(\Lambda_i\) by \(\Lambda_i/3\) leaves an exponentially small tail. Graph enumeration, the bounded number of selected-edge choices, and fixed exponential factors are absorbed by the time power after taking \(L\) large. This proves (b) with the asserted norm and size range. For the partial-slab assertion, restrict every contact time to \(((i-1)\tau,(i-1)\tau+\sigma)\). After translating the slab to start at zero, the fixed top end is evaluated at \((x-\sigma v,v)\). Repeat the root Gaussian and spatial localization argument of (Deng et al. 2025, sec. 14.3, Points 1–2) at this endpoint: all spatial shifts have duration at most \(\tau\), and the same Gaussian exponent margin remains available. The resulting fixed-root Bol bounds are therefore uniform for \(\sigma\le\tau\). Positivity enlarges the contact-time domain only with the intercept held fixed; the preceding localization supplies the bound in the physical endpoint norm. The signed tree involution preserves this restriction, and the Boltzmann expansion and its finite-diameter comparison use the same shortened time domain. Their upper estimates retain the original \(\tau\) bound, uniformly in \(\sigma\), so the preceding argument proves (53) with unchanged budgets. When \(\sigma=0\), only the zero-atom tree remains, giving \(g\); (51) and \(\theta_{i-1}>\theta_i\) give the claimed bound there too. ◻

Proof. For (i), use the error version of the cutting inequality in (Deng et al. 2025, Equation (9.51)), retaining its deficit proportional to \(h\). This is an adaptation of the source proof: its formal error class also requires a large terminal component. The finale of (Deng et al. 2025, sec. 13.5, p. 169) treats the missing top continuation in two cases. When \(\rho\) is at most a fixed multiple of \(C_*h\), the source’s UP cutting algorithm produces at most \(h\) elementary pieces carrying a free-component activity cost, called bad pieces in that proof. The permitted root deficit pays for them. For larger \(\rho\), the same cutting argument as for dependent histories applies; the continuation condition used in its layer selection concerns a strictly lower layer, where that condition is still available. Thus the cutting inequality with its deficit does not require a large terminal component. The subsequent positive integrations give the loss \(\epsilon^{-C_*h}\) in (Deng et al. 2025, Equation (9.82)). Terminal size is used only later, when the time tail absorbs that deficit. Keeping the deficit gives precisely (i).

For (ii), choose one line of the terminal component as root. Its normalization multiplies the original integral by \(\mu^{-1}\); multiplication by \(\mu\) restores it. The stated enlarged terminal class must be split into two cases for the source application. If the component has at most \(\Lambda_i\) lines and excess at most \(\Gamma\), its root and the stated lower continuation make it an ordinary admissible dependent history. If either cutoff is exceeded, it belongs to the enlarged terminal class in (Deng et al. 2025, Definition 3.20(4)), and the same root operation is (Deng et al. 2025, Equation (15.4)). A sum over the choice of line costs a cutoff polynomial.

For (iii), take any fixed nonnegative \(q_0\) with \(\int q_0\,\mathrm dz=1\) and finite Gaussian Bol norm. For every \(x\), \[\int q_0(x',v')\ell_\epsilon(x,x')\,\mathrm dx'\mathrm dv' \ge\epsilon^\upsilon.\] Insert this integral once for each required continuation. The activity \(\mu\) of a dummy input cancels its additional root factor \(\mu^{-1}\). Its straight line satisfies continuation in every slab, and the new links are pendant edges, preserving the initial forest. The positive kernel inequality proves the claimed cost independently of whether \(f_0\) vanishes anywhere. The envelope extension in 10 permits \(q_0\) as the dummy input. Added roots and the finitely many existing roots used for their records fit the prescribed slack. No record is changed by this multiplication. ◻

Proof. The collision components of the full truncated history give one summand equal to one. Its restrictions to those components are autonomous: no accepted collision joins different components, and the rule at an internal contact depends only on the prefix component involved there. Conversely, suppose the factors in a summand equal one. Juxtapose the corresponding autonomous histories and examine their incoming contacts in chronological order. The internal accepted contacts obey the full rule. Every cross-contact fails that rule by \(\chi(C,D)=1\). Thus the juxtaposed history obeys the full truncated dynamics. Uniqueness identifies it with the full history and identifies its collision components with \(\mathcal C\). This proves both assertions. Rejected contacts are not inserted in a prefix collision graph in this argument. ◻

Proof. Choose a total order on pairs. Start with the empty selected forest and all pair factors unprocessed. At each step choose the first unprocessed pair joining distinct forest components, at least one of which meets \(R\). Expand its factor as \(1-a_e\). The positive branch removes the factor; the negative branch selects \(e\) and merges the two components. Never expand a factor internal to a current forest component. Stop a branch as soon as a component reaches \(M\) vertices. If there is no eligible pair, complete the branch.

A selected edge always joins two distinct components. Hence selected edges form a forest, and every nontrivial selected component is rooted. An unrooted vertex cannot join another unrooted vertex. At a stop, both components just merged had fewer than \(M\) vertices, which gives the stated size range and shows that there is exactly one large component. The selected forest determines its branch: replay the deterministic rule, choosing the negative branch precisely on its edges. This proves the multiplicity assertion.

For a completed branch, expand the retained internal factors. Each full graph thereby produced has a unique exploration branch, obtained by replaying the same rule and selecting precisely those eligible edges present in the graph. Its components meeting \(R\) are precisely the reached components of the branch. Grouping by these components gives the products of (57); the remaining vertices retain their compatibility product. If all these components have size less than \(M\), this replay cannot stop prematurely. This proves the main term in (58). In particular, after grouping, priorities involving vertices outside a block do not affect its coefficient \(u(B)\).

For a stopped branch, take absolute values and drop every unused factor except those internal to unreached vertices. All dropped factors are in \([0,1]\). The branch multiplicity just proved gives the asserted forest bound.

For completeness, the ordinary tree inequality can be seen without a stopping rule. Fix a total edge order on \(B\) and associate to each connected graph its unique minimum spanning tree. For a tree \(T\), let \(Q(T)\) be the set of additional edges whose order is later than every edge on the corresponding \(T\)-path. The graphs with minimum spanning tree \(T\) are exactly \(T\subseteq G\subseteq T\cup Q(T)\). Therefore \[ u(B)=(-1)^{|B|-1}\sum_T \prod_{e\in T}a_e\prod_{e\in Q(T)}(1-a_e). \tag{60}\] Dropping the second product proves (59). ◻

Proof. Apply (56) and 13 inside (61). For a completed branch, fix the reached particle labels \(V\supset S\), of size \(s+j\). The cluster partitions on the unreached labels sum to one by (55), pointwise before integration against \(p_n\). No independence of the input is used. Choosing the reached labels gives \[ \frac{1}{(n-s)!}\binom{n-s}{j} =\frac{1}{j!(n-s-j)!}. \tag{66}\] Summing the remaining labels gives \(\mu^{s+j}F_{s+j}\); the output normalization \(\mu^{-s}\) leaves exactly \(\mu^j/j!\).

For the stopped terms, retain the mutual compatibility of unreached clusters and discard the other unused factors and the branch-selection restrictions. Positivity of the input and the multiplicity bound give \(|R_s(H)|\le(\mathcal E_sF)(|H|)\) for every bounded measurable \(H\). The same spectator summation and (66) apply. There are at most \(s\) rooted components, only one of which has up to \(2M-2\) clusters. Their total number of clusters is at most \((s+1)(M-1)\), proving (64).

These bounds also justify the Janossy summations: the dominating expression involves only finitely many finite factorial measures. Finally, the spectator cancellation and compatibility factors do not use records, and root-output maps carry them without alteration except for the prescribed updates. This proves the record version. ◻

Proof. Apply (60) to the single retained block. Refine each autonomous C-cluster according to its complete ordered accepted collision history. Its internal indicator is at most the indicator requiring those prescribed C contacts. Every chosen cluster-tree edge requires the existence of a prospective accepted incoming encounter between two isolated histories. Each history has finitely many flight segments: a cluster of \(q\) particles has at most \(q-1+\Gamma\) accepted collisions. A fixed pair of straight segments has at most one incoming contact. Partition the encounter event by the first admissible segment pair in a fixed order. This writes it as a sum of disjoint encounter indicators, each with an additional factor in \([0,1]\). Ordering the resulting contacts along particle lines gives particle-line molecules. All indicators not requiring the displayed encounters can be kept as an additional weight between zero and one.

The C clusters and the selected O forest determine the connected retained block. Its sign is \((-1)^{\#\mathrm O}\). Its global cycle excess is the sum of the C-cluster excesses. Thus all cyclic terms and all terms with \(n\ge\Lambda_i/3\) have the first two dominations. Their cluster counts are below \(M_i<\Lambda_i\), and their C-cluster caps are the original \(\Lambda_i,\Gamma\), so they belong to the indicated DHM class.

Consider a remaining tree molecule. Every prescribed C component is a tree on fewer than \(\Lambda_i/3\) particles. At every prescribed contact its acceptance test is below both thresholds. If there are no other incoming encounters on this prescribed history, its autonomous C-clusters are exactly the displayed ones, every chosen O contact is the unique contact selected on that cluster pair, and every unused factor in (60) equals one. The added weight is therefore one. Consequently a discrepancy from the unrestricted tree is supported on a further incoming encounter.

This last assertion also covers an internal history that fails the autonomous-cluster indicator: take the first undesignated accepted contact. Until that contact the autonomous and prescribed histories coincide, so the prescribed history has the required additional incoming encounter. The same argument applies to a discarded unused compatibility factor or to a nonfirst choice of a cluster-pair contact. Inserting an O atom at that encounter changes no velocity and hence does not change the prescribed output map. It supplies a positive dominating history integral. There are at most a fixed power of \(n\) segment-pair choices and order insertions, which is a fixed cutoff power.

Conversely, every small unrestricted particle tree is admitted by the retained block rule when it has no further encounters: its C components have size at most \(n\), their excess is zero, and there are at most \(n<M_i\) clusters. The C flags determine those clusters and the O flags their spanning tree. The contact ordering determines the segment placements. Thus it occurs with precisely the displayed sign, without an omitted symmetry factor. Equivalently, keep all fresh particle labels throughout the argument: all the refinements partition indicators and leave \(1/(n-1)!\) unchanged. This proves the lemma. ◻

Proof. 3 gives \(\|g_0-f_0\|_{\mathrm{Bol},\beta_0}\le C\epsilon\), which satisfies the initial smallness hypothesis for \(\theta_0<1\). Suppose the assertion holds at the preceding interface. Apply 15. The fixed-endpoint part of 10 bounds its three error families, including their cutoff-polynomial coefficients, by \(o(\epsilon^{\theta_i})\) in \(\mathrm{Bol},\beta_i\). That same part compares the signed small-tree sum with \(f_{i\tau}\), under the preceding \(\mathrm{Bol},\beta_{i-1}\) error hypothesis, with the remaining margin in \(\epsilon^{\theta_i}\). This proves the first estimate by induction; the uniform Bol bound follows from the Bol bound for \(f\).

This use is a fixed-endpoint Bol estimate, not an inference from an integrated \(L^1\) estimate. The source representation underlying the one-root item of 10 fixes the root endpoint before integrating the other lines. It therefore also provides the asserted density representatives and preserves the summable spatial cell norm. Restricting all contact times to a partial slab gives the uniform partial-slab version of that item. The same bridge and induction then apply for every \(\sigma\le\tau\). ◻

Proof. At time zero expand the factorial hierarchy into exact initial cumulant blocks. Put in \(H\) every label in a nontrivial block and every label in \(R\). An unmarked singleton contributes \(g_0\) on the complement. A marked singleton remains in \(H\) and carries its deterministic initial record. This proves the initial case.

Insert the preceding-interface expansion into (63). Let \(H_-\) be its dependent input set. Separate the blocks that have one new root and are disjoint from \(H_-\). Their inputs are independent \(g_{i-1}\) factors; summing each such block is exactly (68). Since \(R\subseteq H_-\), these blocks carry no required record. Keep all the other blocks and call their new root set \(H\). This gives (71) and the stated rule. Each required record label lies in \(H_-\) at every step, so its own particle line is continued through all interfaces.

Fresh labels assigned to different blocks are disjoint and distributed with their multinomial coefficients. Identity (66) therefore permits the separate sums used here. All sums are finite; no conditional convergence or factorization of the actual input law is involved.

Finally, forgetting records commutes with each root-output kernel in (62), because its weight uses only the unmarked histories. It also commutes with the initial one-particle updates. Thus it commutes with their composition. In the regrouped expression, the blocks freed from the requirement of carrying a record are summed using the same one-root definition. This proves the asserted diagram interpretation of projection. ◻

Proof. First expand each \(g_i\) using its finite definition all the way to time zero, only for this algebraic argument. Each slab block and each initial cumulant block has a weight depending only on its labels. The restrictions on its size and excess are componentwise. Every global component contains a designated root, since each slab block meets a root in the interface above it. On distributing \(j\) fresh labels among component sizes \(j_1,\ldots,j_q\), the coefficient is \[\frac1{j!}\frac{j!}{j_1!\cdots j_q!} =\prod_{a=1}^q\frac1{j_a!}.\] All integrations and all remaining restrictions consequently factor. Grouping the components proves (73). Ordinary finite partition inversion gives its cumulant assertion. A one-root resummation contracts a history attached to the rest through one particle line at its top. It cannot split or merge global components, so the same assertion holds after those resummations.

For the counting statement, the only labels in a slab that are not fresh are the designated roots or labels continued through an interface. Their number is bounded when \(\rho\) and the observation list are fixed, proving the first assertion in (74); the second follows from \(n!/(n-d)!\le n^d\). Choosing or matching boundedly many such labels costs a fixed power of total size. The same is true of the initial link vertices and the segments carrying a fixed list of observations. There are \(2^b\) C/O flag choices. Once an ordered C/O graph is fixed, its C components determine the autonomous collision clusters and its selected O forest determines their blocks. There is no further arbitrary partition of all ordinary lines. The forest exploration has at most one branch per selected forest. Thus no additional factorial or order-dependent exponential base is introduced by these operations. At unbounded complexity the same argument bounds each distinguished choice by a power of the maximum slab size; its exponent is controlled by the crossing and initial-link counts. This is the stated cutoff power, consistent with the molecule count in 10. ◻

Proof. Write \(\Delta_i=F_i-\widehat F_i\). By 14, \[ \Delta_i=R_i+\mathcal L_i\Delta_{i-1},\qquad |R_i|\le\mathcal E_iF_{i-1}. \tag{76}\] The initial difference is zero. In the positive stopped bound substitute \(F_{i-1}=\widehat F_{i-1}+\Delta_{i-1}\), take absolute values, and expand the retained part as in 17. Use the initial tree domination and 16. Refining autonomous histories into C graphs and selecting one incoming contact for every incompatibility gives exactly the positive C/O histories of the coarse package, with their previous-interface continuations. The only top-slab difference is a stopped component with between \(M_i\) and \(2M_i-2<\Lambda_i\) clusters. Hence the same coarse family contains it, and its atom count is at least \(M_i-1\).

Here are the bookkeeping losses in this application. The independent inputs from the lower-layer expansion have already been resummed into \(g_{i-1}\) (and earlier \(g\)’s). We do not resum any component of the positive stopped forest in the top slab. Every top component that fails the continuation condition is covered by the retained root-deficit bound in 11(i), at total cost \(\epsilon^{-C_*|S|}\). To satisfy the bottom continuation condition on a required record line that has no physical initial link, use 11. Equivalently, add an independent, unit-mass, positive Gaussian Bol input on a straight line continued from zero to the top, and connect it to that line by (72). Since this kernel is at least \(\epsilon^\upsilon\), insertion costs at most \(\epsilon^{-\upsilon}\); the new activity and top-root normalization cancel. Only boundedly many such lines are needed. The \(A_i/2\) slack accommodates them. They are used only in this upper bound.

The remaining label and initial-block factors are the cutoff powers in 18. The coarse bound therefore gives, after summing complexities and atoms, \[ \|\mathcal E_i\widehat F_{i-1}\|_{\mathrm{TV}} \le \epsilon^{-C_*|S|-C_{m}} \sum_{b\ge M_i-1}(C\tau^{c_0})^b \sum_{\rho\ge0}|\log\epsilon|^{C_*\rho}\epsilon^{c_*\rho} \le e^{-c\Lambda_i}. \tag{77}\] In this display \(c_0,c_*>0\) are the fixed coarse exponents; absorbed constants can depend on \(L\) and the fixed observation list. Choose \(L\) so that \(C\tau^{c_0}<1\). For small \(\epsilon\) the complexity series is geometric. Finally \(\Lambda_i=A_i^{30}\) and \(A_i\ge|\log\epsilon|\) imply \(A_i\log(1/\epsilon)=o(\Lambda_i)\), so the displayed top losses are absorbed in the exponential size gain.

All input orders are at most \((|S|+1)\Lambda_i^2/2<A_{i-1}/2\). The terms involving \(\Delta_{i-1}\), both in the stopped bound and in (76), are bounded by (67). The induction hypothesis and \[A_i\Lambda_i^2\log(1/\epsilon)=o(\Lambda_{i-1}), \qquad A_{i-1}=\Lambda_i^{30},\] absorb these terms into an error smaller than \(e^{-c\Lambda_i}\). This proves the induction. Restricting all times in the last slab to a subinterval preserves every upper bound just used, proving the partial-slab assertion. ◻

Proof. The first assertion combines the preceding propositions and lemmas. For the second, apply factorial-cumulant inversion to the actual and retained hierarchies. Formula (73) gives precisely the connected sum for the latter. Every term in the difference of the two finite partition polynomials has a factor \(\Delta_{L,B}\). The actual normalized factorial measures have mass at most one: initially this follows from the insertion bound in 3, and the dynamics preserves \(N\). The retained measures therefore have variation at most \(1+e^{-c\Lambda_L}\) at fixed orders by 19. This proves the stated cumulant error.

For the third assertion use the coarse bound on each resummed core, with the same fixed record and top-root losses used in (77). Independent top singleton factors are bounded by 16. For sufficiently small \(\epsilon\), \(|\log\epsilon|^{C_*\rho}\epsilon^{c_*\rho}\le\epsilon^{c_*\rho/2}\). After summing the geometric atom factor, the sum with \(\rho>K\) is at most \[C_{h,m}\epsilon^{-C_*h-C_m}\epsilon^{c_*K/2}.\] Increase \(K\) to make this at most \(\epsilon^P\). Adding the boundedly many dummy lines changes the complexity by a fixed amount and is absorbed by the same choice. Remove these artificial lines before applying any low-complexity asymptotics.

All root-order and observation choices in this reasoning occur in the cutoff powers of the complexity estimate or in the fixed polynomial factors of (74). The exponential atom bases come from the fixed graph-counting and Bol bounds, with \(\Gamma\) fixed first. Thus the required choice of \(L\) is independent of each subsequently fixed order. This completes the proof. ◻

Proof. Here \(n\le3\Lambda_i<A_{i-1}/2\) for small \(\epsilon\), so substitute the preceding-interface expansion from 20 in (78). Every lower slab component contains a root leading to the slab above, and the final set of those lines enters the connected witness. This proves the attachment assertion before discarding any initial link. The contribution of the preceding variation error is bounded by \[\sum_{n\le3\Lambda_i}\frac{\mu^n}{n!} e^{-c\Lambda_{i-1}} \le e^{C\Lambda_i\log(1/\epsilon)-c\Lambda_{i-1}},\] which has the claimed smallness. Arbitrary witness restrictions are indicators and are kept in the positive integrals. Designating an artificial top root places the remaining diagrams in the stated terminal-error class when there is a dependent continuation below; removing its normalization costs \(\mu\). If there is no such continuation, the diagram has only its terminal slab with independent inputs, and its complexity is at most \(1+2\Gamma\). The same geometric complexity summation as in the proof of 20, with this additional fixed power, proves the last assertion. For \(i=1\) there is no preceding variation error; the identical argument starts directly from the initial cumulant expansion. Nothing here compares true and pasted evolutions. ◻

Proof. Choose the birth velocities first. Initially every line has an independent translation of its free affine trajectory. An initial forest link determines the difference between the translations of its two groups by a vector in the stated ball. Orient the forest toward one representative per group. The changes from individual translations to representatives and these differences have determinant one.

Proceed in increasing contact time. Inductively, the histories within each group have been constructed, including all velocity changes before the next merger, and are known up to a common translation of that group. For a merger between groups with translations \(X\) and \(X_*\), the contact condition on the indicated straight segments is \[X-X_*=-d(t)-\epsilon\omega,\] where \(d(t)\) is the known relative displacement with the translations removed, and \(\omega\) is opposite to the outward geometric relative normal. Thus incoming contact has \((v-v_*)\cdot\omega>0\), as in (2). On these segments \(\dot d(t)=v-v_*\). The determinant of the map from time and surface coordinates on \(S^2\) to \(X-X_*\) is \(\epsilon^2|(v-v_*)\cdot\omega|\). The incoming convention selects the sign in (85). A scattering contact applies the elastic velocity reflection; an O contact retains the velocities. The two groups now have one common free translation.

If the next specified contact is redundant, its relative displacement and velocities on the indicated segments are already fixed. A straight relative trajectory has at most one incoming intersection with the sphere of radius \(\epsilon\). More precisely, for its relative flight \(r(t)=d+tu\) on the common segment interval \(I\), \[\int_I\delta(|r(t)|-\epsilon) [-u\cdot r(t)/\epsilon]_+\,\mathrm dt =\mathbf 1_{\{\text{an incoming intersection occurs in }I\}}.\] At a transversal incoming crossing, the flux cancels the absolute derivative of \(|r(t)|-\epsilon\). The outgoing delta factors then make the prescribed C or O substitution. Thus its time and normal, when the prescribed intersection exists, are determined. If it does not exist, the branch has weight zero. This includes segments starting inside the sphere: their outgoing exit is not an incoming intersection. Subsequent translations of the entire group leave this test and its resulting velocity reflection unchanged.

Every step therefore uses only coordinates fixed at earlier steps. Its derivatives with respect to those earlier coordinates may be nonzero, but the full determinant is triangular and is the product of the displayed diagonal determinants. There is no new unknown relative translation at a redundant step. Contacts at different birth interfaces are treated identically: the affine origins of the new lines differ, but the derivative of their current displacement is still their current velocity. At the end retain one representative translation per global component. On an incoming transversal branch the reverse construction recovers every relative translation, so the multiplicity is one.

For a finite graph the excluded segment boundaries, equality of two specified times, and zero incoming flux have measure zero in merger coordinates unless they belong to a redundant-contact requirement. Such a requirement is retained as an indicator; no change of variables is performed at it. Equivalently, one may first work on compact subsets of the transversal branches and then exhaust them. This also justifies the formula for nonnegative history integrals without smooth densities. ◻

Proof. In slab \(i\), let \(s_i\) be the number of labels arriving as roots from above, including the final designated roots in the highest slab. The finite factorial expansion integrates \(j_i=n_i-s_i\) new labels with coefficient \(1/j_i!\). Since \(s_i\le K\), \[\frac1{j_i!}=\frac{n_i!}{(n_i-s_i)!}\frac1{n_i!} \le\frac{(1+n_i)^{s_i}}{n_i!}.\] The choices of continued labels, dependent subsets, and labels in initial links all involve boundedly many labels; their number is a fixed power of \(1+n\).

The rooted compatibility expansion in 20 represents each connected cluster coefficient by spanning forests. Its deterministic exploration gives at most one branch per selected forest. After selecting an incoming encounter for each chosen forest edge, the C components are exactly the isolated collision clusters; the O forest specifies their blocks. Consequently these partitions are recovered from the C/O graph and do not furnish a further partition factor. The retained unexpanded independent factors and the continued labels specify the lower-slab attachment. This is also why taking a tree bound for a compatibility coefficient does not remove the fresh-label denominator.

There are at most \(r\le C_K\) redundant contacts. Their ordered pair labels cost at most \((1+n)^{2r}\), and inserting them among the merger contacts costs at most \((1+n+r)^r\). Initial forests and the choices of the bounded exceptional labels have the same polynomial form. Each ordinary contact has at most two C/O flags, which gives an absolute exponential base. A chronological order already determines the order of contacts on every line; it has no separate permutation factor. These observations give (87). The bounded exceptional counts alter \(C_K,d_K\), not the base multiplying each ordinary merger. ◻

Proof. We give the energy and time estimates jointly, to avoid putting a separate logarithmic velocity loss at each vertex. Let \(E_0=\sum_{p=1}^m|v_p^{\mathrm{birth}}|^2\). Elastic collisions conserve the energy of each colliding pair, O contacts do not change it, and a line ending at an interface discards its energy. Introducing a birth adds the energy already included in \(E_0\). Hence the sum of squared velocities of all lines active at any time is at most \(E_0\).

On the bin \(E_0\in[P/2,P]\), with the bin \(E_0<1\) included by using \(P=1\), the one-step sum of incoming speed factors over all choices of a merger pair in slab \(i\) satisfies \[ \sum_{p<q}|v_p-v_q| \le (n_i-1)\sum_p|v_p| \le n_i^{3/2}\sqrt P. \tag{89}\] This estimate is applied to a discrete tree of successive pair choices at fixed merger times, normals, and birth velocities. At a node, all earlier velocities and all intervening specified redundant contacts are already determined by 23. The sum of the weights of its children is bounded by (89); an unrealizable child contributes zero. Induction over that discrete tree bounds the sum of the products of fluxes by \[ \prod_i(n_i^{3/2}\sqrt P)^{a_i}. \tag{90}\] Integrating the normals adds \(C^{\sum_i a_i}\). The merger times are globally ordered within each slab, even if some redundant times are inserted between them. Dropping further constraints leaves time volume at most \(\prod_i\tau^{a_i}/a_i!\).

There remains a common translation for each component. For any fixed birth velocities and merger coordinates, write the spatial arguments of its inputs as \(X+d_p\). Integrate one chosen input in \(X\) and use the spatial supremum of every other input. The Bol norms in (79) give a factor at most \(C^m\exp(-bE_0)\). This bound is independent of the displacements \(d_p\), which can be large. To prescribe an observation position instead, solve \(X=x-d_{\mathrm{obs}}\) and use a spatial supremum for every input. The Jacobian is one and the same Gaussian bound holds uniformly in \(x\).

We spell out the velocity-volume step. By reserving half of the Gaussian suppression, for a suitable \(b_1>0\), \[\mathbf 1_{E_0\in[P/2,P]}e^{-2b_1E_0} \le e^{-b_1P/2}e^{-b_1E_0}.\] The integral of the last factor over the \(3m\) birth velocity coordinates is \((\pi/b_1)^{3m/2}\). On the lowest bin \(E_0<1\), omit the dyadic suppression factor and use \(P^{a/2}=1\); integrating the reserved Gaussian still costs only \(C^m\). Thus no uncounted power of the energy-bin volume remains. If \(a=\sum_i a_i\), summing the dyadic energy bins yields \[ \sum_{P=1,2,4,\ldots} e^{-b_1P/2}P^{a/2} \le C^{a+1}(1+a)^{a/2}. \tag{91}\] For example, comparison with the integral of \(e^{-b_1u/4}(1+u)^{a/2}\) proves this by the gamma integral and Stirling’s inequality. Its constant depends on \(b_1\), not on \(K\).

Combining these estimates with 24, the remaining quantity is bounded by a fixed polynomial in \(1+n\) times \[ C^{n+a}\tau^a(1+a)^{a/2} \prod_i\frac{n_i^{3a_i/2}}{a_i!n_i!}. \tag{92}\] Set \(\delta_i=n_i-a_i\). By (83), \(\delta_i\ge0\) and \(\sum_i\delta_i\le C_K\). The inequalities \(n_i!/a_i!\le(1+n_i)^{\delta_i}\) and \(n_i!\ge(n_i/e)^{n_i}\) imply \[\frac{n_i^{3a_i/2}}{a_i!n_i!} \le e^{2n_i}(1+n_i)^{\delta_i}n_i^{-n_i/2},\] where zero counts contribute one. Also \(\tau^a\le\tau^n\max(1,\tau^{-1})^{C_K}\), and \((1+a)^{a/2}\le C(1+n)^{C_K}n^{n/2}\), after adjusting the bounded-size cases. Finally convexity of \(u\log u\) gives \[ \prod_i n_i^{n_i}\ge(n/L)^n. \tag{93}\] Substitution in (92) proves (88).

At a fixed time the sum of squared designated velocities is at most \(E_0\). At \(q_0\) record times their total is at most \(q_0E_0\). A weight with exponent smaller than \(b_1/(2\max(1,q_0))\) can therefore be absorbed in the Gaussian factor reserved above. This proves the weighted assertions. The exponent is chosen so that the fraction of birth Gaussian reserved for all record times together is fixed; the exponential base therefore does not grow with the number of observations. The powers supplied by initial physical links and erased links are those already computed in (86) and do not affect the summation argument. ◻

Proof. For clarity first set \(\epsilon=0\) in the formal trajectories. At each prescribed C contact the velocities are transformed by the linear reflection of the pair velocity in its contact direction; at an O contact they are unchanged. Every segment velocity is therefore polynomial in the birth velocities and in the coordinates of the sphere directions. A segment displacement is polynomial also in the prescribed times. The parameters lie on the connected real analytic manifold given by the product of their Euclidean spaces, the spheres, and an open time simplex. Consequently a polynomial expression that is not identically zero has a null zero set. It is legitimate to use grazing or ineffective reflections to test nonidentity: the reflection formulas extend analytically to those parameters, although the physical flux there may vanish.

There are finitely many pairs of lines and finitely many pairs of their straight segments. On a common time interval write their relative displacement as \(d+tu\). A coincidence on this interval requires \[ d\times u=0. \tag{94}\] We check the only cases in which this condition could be forced. If the two lines belong to different forest components, variation of their free relative translation disproves the identity (94) immediately. The assertions used for the first redundant contact and for a connected full tree concern lines in the same component, as treated next.

Suppose first that the lines are not neighbors in the prescribed tree. Make all reflections ineffective. If the unique joining path is \(p_0,p_1,\ldots,p_k\), \(k\ge2\), with its edge contact times \(s_1,\ldots,s_k\), then solving the contact equations gives the endpoint displacement as a sum of terms \((v_{p_{j-1}}-v_{p_j})s_j\), plus the endpoint free-flight term. For \(1\le j<k\), the independent velocity \(v_{p_j}\) occurs with coefficient \(s_{j+1}-s_j\), up to an immaterial common sign. These two times are distinct. Set the endpoint relative velocity nonzero and vary this intermediate velocity in a direction not parallel to it. This disproves the identity (94). Branches off the path and birth interfaces do not alter this test: ineffective reflections leave their velocities unchanged, and birth times only alter the affine origins solved for by the contact equations.

Suppose next that the two lines are neighbors and have their prescribed meeting at time \(a\). On segments after \(a\), if neither endpoint has subsequently changed velocity, their flight is the single separating relative flight from that meeting. It cannot give a second incoming crossing, either at zero diameter or at positive diameter. This includes an O contact: its relative flight passes through the sphere and exits once, but does not enter it a second time without a velocity change.

If an endpoint has a subsequent scattering at time \(b\), take the first effective such scattering in testing nonidentity and make all other reflections ineffective. Its third particle is distinct from the original pair, since another edge between that pair would form a cycle. With all other reflections ineffective, this third particle’s current velocity equals its independent birth velocity, even if it has a prescribed earlier history. One can choose this velocity and the scattering normal so that the relative velocities of the original pair before and after \(b\) are two nonparallel vectors \(u_1,u_2\). Their displacement on the later segments then contains \[(b-a)u_1+(t-b)u_2.\] Its cross product with \(u_2\) is nonzero, since \(b\ne a\). For an explicit choice, keep the untouched endpoint at velocity zero, take the other endpoint’s velocities to be \(e_1\) before and \(e_2\) after the scattering, and use a third-particle velocity \(-e_1\) with normal \((e_1-e_2)/\sqrt2\). The equal-mass reflection gives the stated outgoing endpoint velocity. This again disproves (94). For a segment before the neighboring pair’s prescribed meeting, choose an intervening scattering time \(b<a\) with the same change from \(e_1\) to \(e_2\). Before \(b\), the relative displacement required to meet at \(a\) is \[(t-b)e_1-(a-b)e_2,\] which is not parallel to the segment relative velocity \(e_1\). With no intervening reflection the earlier relative flight has only its prescribed incoming crossing. This proves the earlier-segment case without any condition on time-reversed feasibility.

These arguments use the entire tree and therefore also exclude an unintended earlier contact between groups whose relative translation is fixed only by a later prescribed merger. For a first required redundant contact, they need only the joining tree in the prefix: the two lines are already in the same group, and all their relative parameters are fixed there. Adding parameters belonging to the rest of the graph preserves a null exceptional set by Fubini’s theorem.

Zero relative velocities and grazing prescribed contacts have null parameter sets. Indeed the incoming relative velocity at a merger is the difference of velocities from two disjoint previous groups; it is not an identically zero linear function of their independent birth velocities. The normal is a free sphere parameter. Specified time coincidences, including equality with an interface or an observation time, have null time measure. A third line at a prescribed vertex has no forced coincidence: at least one of its paths to the two incident lines is a nonneighbor path, to which the intermediate-velocity argument applies. Endpoints of line lifetimes at interfaces are treated by the same finite-segment test. We discard this finite union of null sets.

It remains to justify that an absent zero-diameter coincidence excludes positive-diameter contacts, particularly contacts converging to a prescribed vertex. For fixed merger times, directions, and birth velocities the formal tree velocities do not depend on \(\epsilon\). Write each line as \(x_p(t)=X_p+Y_p(t)\), with \(Y_p\) determined by its velocities. The edge equation is \[X_p-X_q=-\epsilon\omega_{pq}-Y_p(s_{pq})+Y_q(s_{pq}).\] Solving these equations along the forest shows that, after fixing its representative translations, \[ x_p^\epsilon(t)=x_p^0(t)+\epsilon d_p, \tag{95}\] where each \(d_p\) is a fixed finite sum of merger normals. Thus convergence is uniform on every active line interval.

Away from small neighborhoods of the finitely many prescribed vertices, every pair of closed straight-segment pieces either has a strictly positive minimal separation or is the already treated separating flight of a neighboring pair. For sufficiently small diameter no new contact can occur there. Near a prescribed vertex, a different pair would require an additional zero-offset pair coincidence, already excluded by the finite-segment argument. This includes a third-particle coincidence at that vertex and a disjoint pair coinciding elsewhere at the same time. The incident pair has no other velocity change in a sufficiently small fixed neighborhood, because all event times are distinct. It has only the one prescribed incoming crossing on that neighborhood. In particular two incoming crossings cannot collapse onto that vertex. The same compactness argument applies to interfaces and observation times. This proves (ii) and (iii), including contacts whose times vary with \(\epsilon\). ◻

Proof. For a finite tree choose its free representative translation to be the position \(x\) of the designated observation. Every birth position is then \(x+d_p\), and every recorded position has the same form, with displacements independent of \(x\). The derivative in the representative translation is the identity. This constructs (96) as an actual integral formula, not just as an almost-everywhere equivalence class. The fixed-position and integrated versions of 25 give both asserted variation bounds.

After truncation in size and velocity, the integrand against a continuous record test is continuous in \(x\), by continuity of the limiting inputs and the observation updates. It has the same integrable majorant uniformly for \(x\) in a compact set. The velocity tails and the graph-size tails converge uniformly in \(x\), by the fixed-position estimate. This proves the stated continuity. Gaussian growth in the test is handled by reserving a slightly stronger Gaussian weight first.

Let the designated variables in two different components have positions \(x,x_*\). Their joint integral before joining uses two independent translations and the product measure. Setting \(x_*=x\) gives the claimed conditional product. The bound \(B(v-v_*,\omega)\le |v|+|v_*|\), with a slight Gaussian exponent loss, makes its variation finite: use the uniform conditional bound for one component and the integrated bound for the other. The same estimate dominates a compactly supported smooth approximate identity in \(x-x_*\), uniformly in its width. On bounded velocity sets conditional weak continuity implies convergence of the product measures against continuous tests; products of continuous single-component tests suffice first, and their finite sums approximate a continuous test on each compact set. The Gaussian and spatial variation bounds then remove the compact truncations. This proves the approximate-identity characterization and its agreement with merger coordinates. Forgetting records commutes with every displayed integral. ◻

Proof. We first prove that an unmarked dependent one-root component cannot be a tree with trivial initial cumulants. In its highest occupied slab, a one-root component has a single connected block. If all lower inputs of this block are independent, it is precisely a term in the definition of \(g_i\), and is regrouped into that independent factor. If a connected lower dependent component attaches to the block through two lines, the upper connection between these lines and their lower connection form a global cycle. For a global tree, every lower dependent component therefore attaches through at most one line. At least one such component must exist for the upper one to remain dependent. Following it downward repeats the same argument on a smaller number of slabs. At time zero there is no dependent singleton, and a dependent nonsingleton requires a nontrivial initial cumulant. This contradiction proves the claim by downward induction. Required records are the only reason to expand an otherwise independent singleton history further.

Now cut a global forest at an interface and contract each of its upper components to a vertex. Suppose there are \(u\) such vertices. A lower connected component with \(k\ge2\) attachments cannot touch the same upper component twice, since that creates a cycle. The bipartite incidence graph of these lower components and the upper vertices is a forest. If there are \(v\) lower vertices of degree at least two and \(d\) incidence edges, then \(d\le u+v-1\) in each nonempty connected part, while \(2v\le d\). In particular \[ d\le 2u. \tag{97}\] A lower component attached at a single line either is absorbed into an independent input, by the first paragraph, or continues a required record. The latter contribute at most the fixed number of recorded lines at that interface. Empty continued lines are included in this count.

Starting with the fixed number of final roots and iterating (97) through \(L\) slabs therefore bounds the total number of interface roots and recorded singleton continuations. There are no cycle or initial-link contributions to complexity in a leading tree. This gives \(K_0\). The argument applies separately to disconnected expressions and to projections with fewer required records.

Regrouping an unmarked one-root component is the defining finite sum for its independent factor, with its original fresh-label coefficient. Its components attach at one line only. Thus undoing it neither changes a connection between different remaining components nor introduces a constraint between them. The exact finite identity in 20 shows that forgetting records and regrouping commute before a limit. Absolute summability from 25 gives the same identity for the limiting trees. Their weights and projections are therefore preserved. ◻

Proof. Temporarily expand all independent factors in the highest slab by their one-root definitions for that slab, retaining their lower inputs in the canonical regrouped form. Bound the total number of lines in each resulting limiting tree by \(N\). For sufficiently small \(\epsilon\), every particle cap \(\Lambda_i\) and retained cluster cap \(M_i\) exceeds \(N\). The convergence already proved therefore supplies every such leading tree with its original factorial weight. We perform the following finite algebra on this limiting tree class, preserving the inherited bound on the combined size after a split. This is not a claim that arbitrary finite-\(\epsilon\) capped classes are closed under joining.

Deleting an edge of a global tree makes exactly two components. Each contains one incident line of the deleted contact, which is now designated. If a component in the highest slab loses its old designated root, it contains the newly designated incident line or another original root. Thus it still satisfies the rooted-block rule. All lower attachments are unchanged. They can be put in the canonical retained form by the singleton regrouping of 28.

Conversely, the two components are disjoint before joining. Adding their incident edge creates a tree, and its highest block meets the original root. The inverse singleton regroupings restore precisely its retained representation. This is a bijection at the level of the temporarily expanded labelled trees. In particular it is not a factorization of the actual particle law at an interface.

For the coefficients, choosing an additional incident particle among \(j\) fresh labels changes \(1/j!\) into \(1/(j-1)!\). If the remaining fresh labels are divided into sets of sizes \(j_1,j_2\), their distribution contributes the binomial coefficient satisfying \[\binom{j_1+j_2}{j_1}\frac1{(j_1+j_2)!} =\frac1{j_1!j_2!}.\] Apply this identity in each slab. A contact between two already designated particles uses their single unordered pair; it has no fresh-label factor. The sign of the deleted contact is its C/O sign, and all other signs multiply over the two components. This proves the coefficient assertion.

The finite identities retain the combined-size restriction just specified. Letting \(N\) increase removes that restriction and passes the sums through the deletion and joining integrals by [lem:sharp-summability,lem:sharp-spatial-traces]. The uniform conditional Gaussian bound controls the velocity flux introduced by the join, with a fixed exponent loss. Record projection commutes with the finite hierarchy and with the one-root regrouping, as established above. Hence a temporarily designated partner for which no past record appears in the integrand requires only the projected component measure. Approximation locally uniformly with a stricter Gaussian exponent extends the identity to the stated test class. ◻

Proof of 22. We first justify weighted control of the retained high-complexity tail. Choose the common birth exponent \(b>0\) in (79) independently of \(L\), and choose \(L\) large enough for the coarse package with the fixed smaller exponent \(3b/4\) as well. For a fixed observation list let \(J\) be the number of designated endpoint and record velocity slots. On every original retained history, conservation of energy gives \[\sum_{\text{these slots}}|v|^2\le J E_0.\] Choose the weight exponent \(\alpha>0\) with \(\alpha J\le b/4\). Its Gaussian weight is dominated by \(e^{bE_0/4}\), which can be absorbed by replacing every absolute birth input \(|g_i(x,v)|\) or \(f_0(x,v)\) by that input times \(e^{b|v|^2/4}\). The resulting inputs have a common \(\mathrm{Bol},3b/4\) bound. The positive coarse estimate in 10(a) requires this envelope, not closeness to the Boltzmann solution. Apply it to the tilted inputs, with the same initial-link and fixed-record bounds used in 20. Dummy inputs, if used for that bound, may be tilted too; their additional birth energy only enlarges the majorant. The same geometric complexity sum makes the weighted retained tail smaller than any assigned power of \(\epsilon\), after choosing \(K\) for the fixed order and accuracy. The Gaussian fraction reserved here is fixed, so its atom base does not depend on \(J,h,K\). Only the observable weight exponent decreases with the record list. This argument concerns retained diagrams, not a microscopic expansion remainder.

Use 20 with an accuracy \(\epsilon^M\) where \(M>2(h-1)\), and with a complexity cutoff large enough both for that accuracy and for 28. The expansion error and the high-complexity tail are negligible after multiplication by \(\mu^{h-1}\). They need only their unweighted variation bound, because the observation tests here are bounded.

For the remaining connected diagrams, (86) and 25 prove the uniform bound at scale \(\mu^{1-h}\). Physical initial links gain \(\epsilon\); nonphysical initial tails gain any assigned power after choosing \(D\) as above. Cycles are negligible by 26, finite-size dominated convergence, and the uniform size tails. The diagrams that remain are the complete class of leading trees, by 28. Their fluxes, signs, input limits, and continuous record tests were evaluated above. The same summable majorant proves the asserted Gaussian variation bounds for their retained sums and their limits. In particular it gives the one-root and two-root limits needed in the next section. All exponential bases used in this proof are independent of \(K\) and of the fixed order \(h\); dependence on those quantities occurs only in front constants and polynomial degrees. Thus the choice of \(\Gamma,L\) remains the single choice made before the moment order was specified. ◻

Proof. We derive the identities from the tree integrals, after the limit, rather than taking a microscopic contact trace. Start with smooth bounded marked tests, bounded on the allowed mark interval. Expand the independent one-root factors of the highest slab by their definition for that slab, retaining the canonical lower inputs, and then truncate the resulting trees to finitely many vertices and bounded total birth energy. The size restriction thus includes the newly exposed lines, as in 29. Away from the finitely many observation times, the test at the upper endpoint depends on current root phase points and already accumulated marks.

Vary that upper endpoint, or equivalently integrate in the time of the last contact of the highest slab. The scattering and non-scattering choices at a last contact that meets no tested root cancel: no later contact or update uses either line, so their tests are unchanged, while the two choices have the same incoming flux and opposite signs. The change when a tested root is incident is its scattered value minus its free value. Deleting that last edge splits a global tree into two components. By 29, these are exactly the two component measures before contact, multiplied at the common position in the sense just specified. This lemma also supplies the inverse joining operation and its fresh-label coefficients. At a finite total-size truncation, the pair of component histories retains the inherited restriction on their combined size. The unrestricted product of the two component measures appears only after absolute summability removes that auxiliary restriction.

For clarity, the root and label counts can be checked as follows.

1. With one original root, its partner has been chosen from the fresh labels; selecting it changes \(1/j!\) into \(1/(j-1)!\). Dividing the remaining labels between the two component trees uses \[\frac1{(j_1+j_2)!}\binom{j_1+j_2}{j_1} =\frac1{j_1!j_2!}\] in each slab. Thus the contact contribution is \(\int BFF_*(\Phi(\xi')-\Phi(\xi))\). Particle interchange, including \(\omega\mapsto-\omega\), symmetrizes this into the collision term in (104).

2. With two original roots, a contact between the original pair is one unordered pair and introduces no fresh-label factor. Deleting it gives two one-root means. Its action on the product test is precisely the last integral in (105). There is no factor \(1/2\) in this unsymmetrized expression for the two designated roots.

3. Suppose the contact is on the first original root and a new partner. The second original root lies in one of the two components obtained by deleting the contact. If it lies with the first root, the contribution is \(J\) on that component and \(F\) on the partner component, with change \(\Phi(\xi')-\Phi(\xi)\). If it lies with the new partner, interchange the two incident variables; the \(J\) component then carries the change of the test on the other incident line, \(\Phi(\xi_*')-\Phi(\xi_*)\). Adding the two placements gives \(J(H_t\Phi\otimes\Psi)\). The same argument at the second original root gives \(J(\Phi\otimes H_t\Psi)\).

If the temporarily designated new partner has no record in one of these integrands, its component is simply projected on the needed variables. Conversely, when its mark occurs after an interchange, the full recorded component is used. The projection identity in [thm:marked-expansion,lem:sharp-tree-completeness,lem:sharp-tree-splitting] identifies these as projections of the same \(F,J\); it does not postulate factorization of the microscopic law at an interface.

Free flight gives the \(D\) terms. One can formulate the calculation without differentiating a sum: for an interval \([a,t]\) contained in one slab and containing no update, freely transport the final test from \(t\) to the last-contact time \(s\). The no-contact term is the measure at \(a\), tested against this free transport, and the last-contact terms are the integrals of the expressions above over \(a<s<t\). For \(J\) use free transport in both test slots. These are exactly the transport mild versions of (104) and (105).

The finite re-expansion, deletion, and joining were bijections of labelled trees with their factorial weights. Passing to the full sums and the unbounded velocity domain is justified by [lem:sharp-summability,lem:sharp-spatial-traces,lem:sharp-tree-splitting]. In particular the last-contact products are integrable in time and space with a fixed Gaussian exponent allowance. The same bounds justify locally uniform approximation of continuous tests with smaller Gaussian growth. They also extend the identities to bounded tests such as \(y\), by cutting off space and velocity first and using the integrated variation bounds.

At a slab interface with no contact at its endpoint, the empty upper slab has the input hierarchy as its no-contact term. The finite marked expansion has this identity before regrouping, and the absolutely convergent tree sums preserve it. Hence there is no jump at an artificial interface, and the mild identities concatenate across all slabs. Actual observation updates are the deterministic pushforwards of the finite marked hierarchy; bounded continuous record convergence gives (106). ◻

Proof. Put \(S^\epsilon=\sum_i y_i(T)\). The bound \(|S^\epsilon|\le M_{\mathrm{obs}}N\) and the partition function imply that \(S^\epsilon\) has exponential moments for each fixed \(\epsilon\). For the normalized factorial cumulants at the final marked time, the exact factorial-to-ordinary identity is \[ \kappa_k(S^\epsilon)= \sum_{\pi\in\mathcal P([k])}\mu^{|\pi|} \int c_{|\pi|}^{\epsilon}(\mathrm d\xi_1\cdots\mathrm d\xi_{|\pi|}) \prod_{B\in\pi} y_B^{|B|}. \tag{114}\] Here one associates the blocks of \(\pi\), in any fixed canonical order, with the distinct particle slots in the integral. For a direct derivation, write \(\exp(\lambda S^\epsilon)=\prod_i[1+(\exp(\lambda y_i)-1)]\), insert the factorial generating series, and take its logarithm. Each factor \(\exp(\lambda y)-1\) must receive a nonempty set of the \(k\) differentiations at zero. Distributing those sets gives exactly the partitions in (114); their ordering cancels the factorial denominator of the factorial series. This is also a finite formal identity through degree \(k\), so no uniform radius of convergence in \(\epsilon\) is being assumed.

All the powers of \(y\) in (114) are bounded on the common mark interval; they can be extended to bounded continuous functions on the full mark line. By 22, every term on its right is \(O_k(\mu)\). For the centered normalized variable \[X^\epsilon=(S^\epsilon-\mathbb ES^\epsilon)/\sqrt\mu\] the first cumulant is zero. The second is \[\kappa_2(X^\epsilon) =c_1^\epsilon(y^2)+\mu c_2^\epsilon(y\otimes y) \longrightarrow F_T(y^2)+J_T(y\otimes y)=G_T(y,y).\] For \(k>2\), \[ \kappa_k(X^\epsilon)=O_k(\mu^{1-k/2})\longrightarrow0. \tag{115}\] The statement that these estimates hold on one dynamics is material: the bases determining \(\Gamma,L\) in [thm:marked-expansion,thm:sharp-cumulants] are independent of the fixed factorial order. Choosing a larger auxiliary complexity threshold or a larger \(D\) only changes the error representation used to prove (115); it does not change the random variable \(X^\epsilon\).

The finite moment–cumulant identities now give every centered Gaussian moment with variance \(\sigma^2=G_T(y,y)\). Here the usual Gaussian moment-determinacy argument can be made explicit. Truncate the characteristic functions of \(X^\epsilon\) and of a centered Gaussian \(Z\) with variance \(\sigma^2\) at degree \(2n-1\). Taylor’s integral remainder on the real axis is bounded by \(|u x|^{2n}/(2n)!\), so moment convergence gives \[\limsup_{\epsilon\to0} \left|\mathbb Ee^{iuX^\epsilon}-\mathbb Ee^{iuZ}\right| \le \frac{2|u|^{2n}\mathbb EZ^{2n}}{(2n)!} =\frac{2|u|^{2n}\sigma^{2n}}{2^n n!}.\] Letting \(n\to\infty\) proves characteristic-function convergence for every real \(u\), including when \(\sigma=0\). It implies the asserted weak convergence, with variance identified by (113). ◻

Proof. All atom and birth counts are bounded by a fixed power \(B_\epsilon=(1+|\log\epsilon|)^H\), by the layer cutoffs and (119). A crude count of all graphs, time orders, and coefficient choices in this finite range is at most \(\exp(C_*B_\epsilon^2\log(2+B_\epsilon))\). For example, label all births, list at most \(B_\epsilon\) ordered contacts by their two labels and C/O type, and then list their slabs and interface gluings. Discarding factorial denominators and allowing all such lists gives this upper bound. Bounded complexity initial coefficients cost at most an additional fixed power of \(B_\epsilon\).

The two-incoming-flight identity requires no velocity localization. Indeed, for fixed relative intercept \(x\) and velocity \(v\), a transversal root of \(|x+tv|=\epsilon\) has derivative \(v\cdot(x+tv)/\epsilon\). Thus \[\int \delta(|x+tv|-\epsilon) [-v\cdot(x+tv)/\epsilon]_+\,\mathrm dt\leq1:\] a straight flight has at most one incoming crossing of a ball, and the flux cancels the scalar coarea Jacobian. The outgoing-position and outgoing-velocity delta factors then make a deterministic substitution with unit mass. Zero-speed and grazing cases carry no transversal incoming flux; use the same almost-everywhere convention, or exhaust the strictly transversal set. This proves the unrestricted incoming-contact bound directly, before localization.

For a fixed fully ordered graph, let \(h_j\) be the absolute input at birth \(j\): it is either \(|g_{i_j}|\) or an \(f_0\) factor from an initial cumulant bound. Both have the common Gaussian envelope from [prop:one-root,ass:kinetic]. Integrate the original coarea measure forward, starting with all birth phase points. At each contact its two incoming affine lines have already been determined. The unrestricted identity just proved integrates the contact time and outgoing variables to an indicator at most one and a deterministic update. Births at later interfaces are inserted at their assigned times; they can equivalently be integrated in advance and held inactive until then. Thus, after dropping all encounter restrictions and initial links, a graph with \(n\) births has mass at most \[\mu^n\int\prod_{j=1}^n h_j(x_j,v_j) \mathbf 1_{\{\sum_j|v_j|^2>P_\epsilon\}} \prod_j\mathrm dx_j\,\mathrm dv_j \leq \mu^n C^n e^{-cP_\epsilon}.\] The last inequality uses half of the uniform birth Gaussian for the energy tail and integrates the other half. This forward bound is an estimate of the original positive measure, so it involves no local recollision gain. Summing the crude finite count, and using \(n\leq B_\epsilon\), gives an upper bound \[\exp\{C_*B_\epsilon^2\log(2+B_\epsilon) +2B_\epsilon|\log\epsilon|-cP_\epsilon\}.\] Choose \(H_*>2H+2\). The result is smaller than every fixed power of \(\epsilon\). ◻

Proof. Part (i) is the incoming coarea formula (Deng et al. 2025, Proposition 9.1, Equations (9.1)–(9.6)); the two-end kernel counts at most one incoming crossing between the two fixed flights. The powers of \(\mu\) also follow directly from (48). The type-A assertion follows from (120) and its defining cut. Part (ii) is (Deng et al. 2025, Proposition 9.2(1), Equations (9.14)–(9.15)), with precisely the restrictions on distinct and serial edges written there. Part (iii) is (Deng et al. 2025, Proposition 9.3(2), Equation (9.23)); its range \(0<\upsilon<1/(6d)\) contains \(1/81\) for \(d=3\). The estimates are bounds for a positive kernel with a supremum over all fixed ends, which gives the last assertion and permits their successive conditional use. ◻

Source identification. The transversal definition and closure properties are (Deng et al. 2025, Definition 15.3 and Proposition 15.4). The chain and its transition cutting sequence are the construction in the proof of (Deng et al. 2025, Proposition 15.1, case (1)). The displayed bound \(G\), including its realizability hypothesis, is (Deng et al. 2025, Proposition 15.2); for \(X^\pm(A)\) its entering boundary consists of the stated connection bonds. That proposition uses the hard-sphere collision bound of Burago, Ferleger and Kononenko (Burago et al. 1998), adapted to extended dynamics in (Deng et al. 2025, Appendix A). Part [item:cutoff-trans] is (Deng et al. 2025, Proposition 15.6). The two sequences in [item:cutoff-main] are (Deng et al. 2025, Proposition 15.10, (15.7)–(15.8)). We deliberately weaken the first count in (15.7) when retaining only good singletons. Its proof has at most \(r(A)\) type-two or type-A pieces inside \(A\), covering at most \(2r(A)\) vertices, and one free seed. The remaining weakly degenerate vertices or later-cut members of weak pairs give good singletons. Discarding these exceptional vertices and weakening the count gives the stated \(y/10-3(r(A)+1)\). For the second sequence, the type-B pairs lie in the remaining part of \(A\), after the weakly degenerate atoms and weak pairs have been removed. Both the time separation and all the velocity separations required by 34(iii) therefore hold there. The integer \(y\) is the size of the set \(Y_1\) in that proposition’s parameter partition. Thus its name and internal classification are unnecessary for the operational statement here. The uniform local estimates and conditioning convention are 34. We use none of the final exponents in DHM’s truncation-error proposition. ◻

Proof. If a remaining vertex is a successor of the exposed set, choose one minimal among the unexposed successors. It is adjacent to the exposed set, and adding it preserves order convexity. Every fixed incident edge is on its incoming side: fixed edges on both sides would place this vertex between two already exposed vertices. There are one or two such edges. If no unexposed successor is available, choose dually a maximal unexposed predecessor. It has one or two fixed outgoing edges. Continue until the components touching the exposed set are exhausted. Adjacency guarantees that one of these choices is available whenever such a component remains. The local formulas for both permitted singleton types apply, including to a nonscattering contact. Given top ends can be regarded as fictitious exposed vertices later than all contacts on their lines. For this argument one may use literal cuts in the uncontracted four-edge coarea representation (46)–(48). Tonelli’s theorem identifies each split bond variable with the given boundary value of the next piece. The activity exponents add because a split bond is replaced by one integrated free end and one fixed end. The final singleton pieces are regular: their fixed ends are on the same time side, so no two of them are serial. We do not assert that every intermediate complement of these literal cuts is regular under the serial-segment cutting convention described above. ◻

Proof. Reverse time within the local calculation if necessary. In the type-A arrangement the second contact has both outgoing edges available, and its fixed incoming edge is not the shared bond. Integrate the first contact as a type-three singleton and the second as a type-two singleton. The free first-contact parameters are \((t_1,\omega,w)\). Denote the two given velocities by \(u,v\), and the intervening bond velocity by \(z\).

For the free pair in (i), use the two-contact tree parameters: one free collision position, the two velocities at the first contact, the new incident velocity at the second contact, two normals, and two times. The activity prefactor is \(\mu\). Elastic reversal preserves velocity Lebesgue measure and the flux, so this parametrization also covers a prospective fixed edge chosen on the outgoing side of the first contact. Conditional on the first contact, the second given velocity is the independent incident velocity of the third particle. Its permitted set in (123) has volume \(C\epsilon^{12/5}\); the second time has interval length at most \(2\epsilon^{9/10}\). All remaining localized volumes and fluxes cost at most \(CR^C\). This proves (124). By primitivity, 37 finishes the rest of the connected graph without another free singleton.

For (ii), first suppose \(t_2-t_1>\epsilon^{9/10}\). Let \(q_1(t)\), \(q_2(t)\) be the positions on the given affine lines. The two contact equations imply \[ (t_2-t_1)(z-v)=q_2(t_1)-q_1(t_1)+O(\epsilon). \tag{125}\] The \(O(\epsilon)\) error includes both contact offsets and is uniform in the normals. At fixed \(t_1\), the vector \(z-v\) consequently lies within distance \(C\epsilon^{1/10}\) of the line \(\mathbb R(q_2(t_1)-q_1(t_1))\). If this displacement is zero it instead lies in a ball of that radius, which is a stronger restriction.

Put \(h=C\epsilon^{1/10}\). Depending on which edge is fixed at the first contact, the bond velocity is one of \[w,\qquad P_{\omega^\perp}w+(u\cdot\omega)\omega,\qquad u+s\omega,\quad s=(w-u)\cdot\omega ,\] up to signs that do not change these estimates. For the first map, the preimage of an \(h\)-tube has three-dimensional volume at most \(CR^C h^2\). For the second, a plane intersects a bounded part of the tube in area at most \(CR^C h\), and the unused longitudinal coordinate costs only \(CR\). For the third, integrating the two tangential coordinates of \(w\) costs \(CR^2\), while the image of \(\mathrm ds\,\mathrm d\omega\) has density at most \(C|z-u|^{-2}\). Inside \(|z-u|\leq\sqrt h\) its mass is at most \(C\sqrt h\). Outside that ball, the density is at most \(Ch^{-1}\) and the tube volume is at most \(CRh^2\). Thus its total mass is at most \(CR^C\sqrt h\). These bounds, with the first flux bounded by \(CR\) and the type-two kernel by one, give \(CR^C\epsilon^{1/20}\).

Next suppose \(t_2-t_1\leq\epsilon^{9/10}\) and \(|u-v|>\epsilon^{4/5}\). Equation (125) gives \[|q_2(t_1)-q_1(t_1)|\leq CR\epsilon^{9/10}.\] The difference of the given affine lines has velocity \(v-u\); the set of such \(t_1\) is an interval of length at most \(CR\epsilon^{1/10}\). Integrating this restricted first time, and bounding all other local factors by \(CR^C\), gives a stronger estimate.

Finally suppose (123) holds. The pair is then nonprimitive. Choose another directed chain between its vertices. At least one intermediate vertex of this chain has already been cut. Indeed, if all were still uncut, the first vertex would have two uncut outgoing bonds and the second two uncut incoming bonds. Their fixed edges would be on the opposite outer sides, which is precisely not the type-A arrangement. A time \(t_*\) strictly between \(t_1,t_2\) is therefore part of the previously exposed data. The first free collision time lies within \(\epsilon^{9/10}\) of \(t_*\). Cutting the pair into a type-three and a type-two singleton gives the claimed estimate. There are only polynomially many choices of such intermediate vertices in a fixed graph; a deterministic choice suffices. ◻

Proof. First take the union over all strongly close primitive pairs and all prospective type-A fixed-edge choices. There are at most a constant times \(b\) candidate bonds and a bounded number of edge choices per bond. A fixed priority partitions this union. Cut the selected pair freely and use 37. This is the first alternative. On the complement every type-A pair has the gain in 38.

Here is an explicit finite hierarchy for the second alternative. Choose \[D=\lceil4/\delta\rceil,\qquad B=100/\delta,\qquad R_0=(10^5+100/\delta)(1+2B),\qquad \kappa_1=2,\] and choose increasing integers recursively so that \[\kappa_{j+1}> \kappa_j+40\{1+G(\lceil R_0\kappa_j\rceil)\}, \qquad 1\leq j<D .\] Use an increasing envelope of \(G\) if needed at its two smallest arguments. Take \(\Gamma_0\geq\kappa_D\). All these choices precede \(L\).

Apply the transversal chain of 36. If it has at least \(D\) transitions, its transition sequence has \(g\geq D\), \(s_4=1\), and \(\delta D-2\geq2>6/5\). Otherwise the chain reaches \(M\) in at most \(D\) sets. Since \(r(A_1)=0<\kappa_1\) and \(r(M)>\Gamma\geq\kappa_D\), there is a first \(j\) such that \[r(A_j)<\kappa_j,\qquad r(A_{j+1})\geq\kappa_{j+1}.\] Put \(A=A_j\). The growth bound shows that one of \(X^\pm(A)\) has size greater than \(G(\lceil R_0\kappa_j\rceil)\). Its number \(m\) of connections to \(A\) therefore satisfies \(m>R_0\kappa_j\), by 36[item:cutoff-bfk]. The retained realizability of the witness is used at exactly this invocation of the collision-count bound.

For the chosen side put \(z=|X_0^\pm(A)|\), \(a=r(A)\), and use the partition and integer \(y\) of 36[item:cutoff-main]. If \(z\geq B\kappa_j\), use the transversal sequence: \[\delta g-2s_4 \geq \delta\{(B/2-1)\kappa_j-1\}-2>6/5 .\] If \(y\geq B\kappa_j\), use its first main sequence: \[\delta g-2s_4 \geq \delta\{(B/10-3)\kappa_j-3\}-2>6/5 .\] Otherwise \(S=a+z+y\leq(1+2B)\kappa_j\). In the second main sequence, every counted type-A or type-B pair has a gain. Hence \[\delta g-2s_4 \geq \frac{\delta}{10}(m-10^5S)-2S > 8(1+2B)\kappa_j>6/5 .\] If more uniformly good singletons are produced than needed, retain only the number of their gains needed for this inequality.

For completeness, a degree count bounds the remaining exceptional pieces. Write \(s_j\) for singleton counts and \(p_{33},p_{44}\) for pair counts. Counting each bond once, except that an internal pair bond contributes at both endpoints when available degrees are summed, gives \[ s_2+p_{33}=r(M)+s_4+p_{44}-1 . \tag{128}\] One way to verify this identity is to use \(b=s_2+s_3+s_4+2p_{33}+2p_{44}\) and \(n=b+1-r(M)\). The sum of available degrees is both \[2s_2+3s_3+4s_4+6p_{33}+8p_{44} \quad\text{and}\quad 2b+n+p_{33}+p_{44}.\] This proves (128). The constructed \(s_4\) is bounded, \(r(M)\leq2\Gamma\), and \(p_{44}\) is zero except for the single primitive seed. Thus \(s_2,p_{33}\) are bounded. Splitting a nonprimitive type-A pair as in 38 changes these counts by a bounded amount. The number of retained good singletons may also be bounded in terms of the same constants. The subcase count follows from 36 and the additional polynomial choices. ◻

Proof. First put all ordinary variables in bounded boxes. Fix their entire vector \(\vartheta\). Their deterministic updates still occur in their original positions in the cutting sequence. The exceptional kernels remain in their original order, with \(\vartheta\) treated as exterior parameters. Integrating these kernels backwards from the last exceptional step bounds the resulting total mass by \(\prod_\ell H_\ell\). Intervening deterministic substitutions have norm at most one on bounded functions. Nonnegative kernel Fubini therefore represents \(\nu\) as the product measure in \(\vartheta\) times these remaining conditional kernels. Majorizing the indicator of \(E\) by \(\mathbf 1_{\mathcal D}(\vartheta)\) proves (130). Exhaustion removes the bounded boxes.

In particular, this calculation does not condition an exceptional kernel on the global energy event. That event is retained until its ordinary projection is majorized. Fixing a future ordinary parameter does not alter any exceptional kernel or require an averaged estimate to become pointwise. Uniformity in the actual earlier-cut data is sufficient.

There is also a rectangle formulation useful for the coarea measures. Restrict the ordinary relative speeds away from zero and bound their product on the original feasible support. After division by those speeds, an ordinary local kernel in a rectangle of its \((t,\omega,w)\) coordinates has mass bounded by the rectangle’s product volume. In a product rectangle for all ordinary steps, successive conditional estimates give that volume times \(\prod_\ell H_\ell\). Cover a compact containing set \(\mathcal D\) by finite unions of product rectangles with volume decreasing to its outer measure, using finitely many sphere charts. Summing the rectangle bounds and taking the infimum gives (130). The containing sets used below are compact and measurable. Finally let the speed restriction decrease to zero; a zero-speed incoming contact has zero flux. This proves the same statement directly by exhaustion of transversal coarea histories. ◻

Proof. Work on the original feasible histories, before dropping any collision constraints. Let \(u\) be the vector of all birth velocities. At each exceptional atom replace its two outgoing velocities by independent additional inputs, and write \(a\) for their concatenation. There are at most \(6s\) scalar coordinates in \(a\). On the energy support each of its three-dimensional entries has norm at most \(\sqrt P\).

Run the resulting velocity computation forward in the graph’s chronological order. Ordinary scattering applies an orthogonal matrix to two particle velocities; an ordinary nonscattering contact applies the identity. At an exceptional atom, erase the dependence of its two outgoing coordinates on \(u\), and insert the prescribed values from \(a\). Birth coordinates can be present from the start as inactive coordinates, and departed coordinates can be retained as inactive ones. The part depending linearly on \(u\) is thus propagated by orthogonal matrices and coordinate projections. Its operator norm is at most one at every time.

Every ordinary parameter \(w_j\) is one of the physical edge velocities immediately before or after its contact. This includes a parameter on the outgoing side when the cut parametrization runs backwards. Consequently \[ W=A(\omega)u+d(\omega,a),\qquad \|A(\omega)\|_{\mathrm{HS}}^2\leq3q . \tag{132}\] Repeated observation of one velocity does not require orthogonality between different rows; the stated sum of their squared norms is all that is used. These maps do not depend on numerical contact times. Different total extensions of the causal order produce the same maps, since incomparable contacts act on disjoint particle coordinates and commute.

Take a Euclidean net of the possible \(a\)’s of mesh \(c_s(1+q)^{-1/2}\). It has cardinality at most \(C_s\{(1+P)(1+q)\}^{Cs}\). A change of size \(\zeta\) in this entire vector changes any one chronological velocity state by at most \(\sqrt s\,\zeta\): propagate the reset errors and use Cauchy–Schwarz. Its effect on the \(q\) observations is at most \(C\sqrt{qs}\,\zeta\). Choose \(c_s\) so that the simultaneous error is at most one. The allowed \(W\)’s are contained in the union, over net points \(a_0\), of \[ d(\omega,a_0)+A(\omega)B_{3n}(\sqrt P)+B_{3q}(1). \tag{133}\]

For \(m=3q\) the Minkowski sum in (133) is contained in \[\sqrt2\,(PAA^{\mathsf T}+\mathrm{Id}_m)^{1/2}B_m(1).\] Indeed its support function in direction \(\xi\) satisfies \[\sqrt P|A^{\mathsf T}\xi|+|\xi| \leq \sqrt{2(P|A^{\mathsf T}\xi|^2+|\xi|^2)}.\] Combine the trace bound in (132) with the arithmetic–geometric mean inequality for eigenvalues and the ball-volume estimate \[\operatorname{vol}(B_m(1))\leq(C/\sqrt m)^m.\] This gives \[2^{m/2}\operatorname{vol}(B_m(1)) \det(\mathrm{Id}_m+PAA^{\mathsf T})^{1/2} \leq \left(C\sqrt{(1+P)/q}\right)^{3q} \leq \left(C\sqrt{1+P/q}\right)^{3q}.\] Sum over the net. The identity-matrix enlargement handles zero or small singular values without any inverse Jacobian. For varying normals, the sets in (133) form a finite union of continuous images of compact parameter sets. They provide a compact measurable containing set for use in 40. ◻

Proof. Let \(e_i\) be the number of linear extensions of this partial order. Apart from equality hyperplanes, its time-order region is the disjoint union of \(e_i\) simplexes, so \(|\mathcal T_i|=e_i\tau^{q_i}/q_i!\).

An antichain of ordinary vertices occupies disjoint incoming pairs on a common causal slice. To see this, place all strict ancestors of the antichain first in a total extension of the full graph. No two vertices in the antichain share a particle line. Their incoming states are present simultaneously on the resulting slice. Conservation at all original contacts, including exceptional contacts, bounds the sum of their pair energies by the total birth energy \(P\). Retaining unborn and departed velocities as inactive coordinates makes the same argument valid in the presence of interfaces.

For a fixed feasible velocity assignment write \(a_v=|v_v-v_{*v}|\). Every antichain \(\mathcal A\) therefore satisfies \[\sum_{v\in\mathcal A}a_v \leq\sqrt{2|\mathcal A|P}\leq\sqrt{2q_iP}.\] In the recursive enumeration of linear extensions, the available minimal vertices are an antichain at each step. Summing the products of their weights successively gives \[e_i\prod_v a_v\leq(2q_iP)^{q_i/2}.\] This bound holds for every feasible assignment, so also for the supremum of its product. Multiplication by \(\tau^{q_i}/q_i!\) proves (134). In particular, no assumption that the maximizing assignment is independent of the times is necessary. ◻

Proof. By [prop:one-root,lem:kinetic-envelopes], choose \(\beta_*>0\) and a common Gaussian Bol bound for all birth inputs, with \(\beta_*\) independent of \(L\). For each birth factor use its spatial-cell decomposition \[|g_i(x,v)|\leq C\sum_{a\in\mathbb Z^3}m_{i,a} \mathbf 1_{\{|x-a|\leq C\}}e^{-\beta_*|v|^2}, \qquad \sum_a m_{i,a}\leq C .\] The same form applies at time zero. Sum over the birth-cell indices at the end; their coefficients have total at most \(C^n=C^{b+O_K(1)}\).

Decompose total birth energy into \([0,1]\) and the dyadic bins \([P/2,P]\), \(P\geq1\). On a bin, the Gaussian factors give \(C^{b+O_K(1)}e^{-cP}\). Fixed birth cells and bounded energy localize each physical particle’s positions within a polynomially bounded distance of its birth cell over \([0,T]\). The same is true of its affine intercept. Thus each exceptional kernel has the position localization required by its stated uniform estimate, with an arbitrary but fixed cell-dependent center.

For the ordinary steps use the relative speed rather than the smaller normal flux as \(a_j\) in 40. Its quotient density is at most one. Pull out the product of the per-slab suprema in 42, on the original feasible support. Use the containing velocity sets of 41, and keep the full ordinary time orders. All gain-producing exceptional restrictions remain attached to their kernels. Other future restrictions may now be dropped when taking their uniform mass bounds. The ordinary sphere integration contributes \((4\pi)^q\). After absorbing the bounded exceptional-net prefactor into the logarithmic power, the bin is bounded by \[\epsilon^\alpha|\log\epsilon|^{C_*} C^q e^{-cP}(1+P/q)^{3q/2} \prod_i\frac{\tau^{q_i}(2q_iP)^{q_i/2}}{q_i!}.\] When \(q\geq1\), Stirling’s inequality and convexity give \[q_i!\geq(q_i/e)^{q_i},\qquad \prod_iq_i^{q_i}\geq(q/L)^q ,\] with zero counts omitted. The last bound becomes \[ \epsilon^\alpha|\log\epsilon|^{C_*} (C\tau\sqrt L)^q e^{-cP} (P/q)^{q/2}(1+P/q)^{3q/2}. \tag{136}\] For \(x=P/q\), \[\sup_{x\geq0}x^{1/2}(1+x)^{3/2}e^{-cx/2}<\infty .\] Consequently the final three factors in (136) are bounded by \(C^qe^{-cP/2}\). Summing the dyadic bins costs only another absolute constant. The lowest bin is treated by the same bound with \(P=1\). If \(q=0\), integrate the bounded number of exceptional pieces directly; no time factor is asserted for them. This proves (135). ◻

Proof of 32. Use the terminal witness reduction and choose \(K\) as in (119). Apply 33. For the remaining graphs all velocities are polylogarithmically bounded. Decomposing the birth positions into cells as in 43 also gives the required local position regions: along each original particle line the physical travel distance is at most \(T\sqrt{P_\epsilon}\); its affine intercepts are within another \(T\sqrt{P_\epsilon}\). Their centers are fixed by the birth cells and may be arbitrary in the local estimates.

First consider witnesses with \(\Gamma<r(M_{\mathrm{top}})\leq2\Gamma\). Cut the terminal component freely and apply 39. The terminal atom set is order-convex: every lower atom lies in an earlier slab, while a directed path has increasing times. Straight continuations beyond the stopping contact add no atoms. Finish all lower components from their now fixed terminal data by 37. These lower components create no extra type-four singleton: by 21, each is attached to the terminal part through particle lines, without using initial links. The total cycle and crossing bounds give (129) for the entire cut sequence.

In the primitive branch the product of exceptional masses is at most \(\epsilon^{13/10}|\log\epsilon|^{C_*}\). In every other branch it is at most \(\epsilon^{6/5}|\log\epsilon|^{C_*}\), by (127); a type-four singleton costs \(\mu=\epsilon^{-2}\). The positions, exceptional variables, and intermediate-time choices only enlarge the fixed logarithmic exponent. Therefore 43 bounds either branch by \[\epsilon^{6/5}|\log\epsilon|^{C_*} (C\tau\sqrt L)^{b-O_K(1)}.\]

We explain explicitly the uniformity needed to sum these bounds over graphs. For bounded \(\rho\), the unlabelled root and interface counting in [thm:dhm-package,cor:terminal-witness] costs \(C_1^b\) times a fixed power of the cutoffs. The pure-collision cutting partitions have the same form by 36. The additional selected pairs, crossing labels, and initial coefficients have boundedly many choices of locations, and therefore cost only fixed-degree polynomials in the graph size. The exponential bases depend on \(\Gamma\) and the uniform kinetic envelopes; they do not depend on \(K\) or \(L\). The \(K\)-dependence is confined to the fixed polynomial and logarithmic exponents. To see explicitly why assigning slabs does not add a factor \(L^b\), cut the bounded number of interface bonds and delete the bounded number of cycle bonds. The remaining atom forests have degree at most four. Ordered-tree encodings and the finite local C/O and serial-edge choices cost \(C^b\); restoring bonds and marked ends costs \((1+b)^{O(K+\Gamma)}\). Each resulting component lies in one slab, so its slab choices cost only \(L^{O(K+\Gamma+1)}\), a front constant. Equivalently, first choose the boundedly many occupied slab indices, then count the positive size compositions by \(2^b\). Empty continued lines are bounded by the root and crossing counts. Choose \(L\) so large that \[C_1 C\tau\sqrt L=C_1CT/\sqrt L<1/2 .\] The \(O_K(1)\) missing ordinary factors contribute a constant depending on \(K,L\), and the geometric series in \(b\) converges. The sum of this first class is \[ O\bigl(\epsilon^{6/5}|\log\epsilon|^{C_*}\bigr). \tag{137}\]

For the other witnesses use \(n(M_{\mathrm{top}})>\Lambda_i\) and \(r(M_{\mathrm{top}})\leq2\Gamma\). Choose one terminal vertex as a free type-four seed and finish by 37; complete the lower part in the same manner. There is one activity loss \(\mu\), and no gain is required. Degree counting again leaves \(b-O_K(1)\) ordinary vertices. Moreover \(b\geq b(M_{\mathrm{top}}) =n(M_{\mathrm{top}})-1+r(M_{\mathrm{top}}) \geq\Lambda_i-1\). The same summation now gives \[C_*\epsilon^{-2}|\log\epsilon|^{C_*} \sum_{b\geq\Lambda_i-1}2^{-b},\] which is smaller than every fixed power of \(\epsilon\), since \(\Lambda_i\geq|\log\epsilon|^{30}\).

Finally sum over the finitely many slabs and add the coarse high-complexity, high-energy, and expansion remainders. For sufficiently small \(\epsilon\), (137) is bounded by \(C\epsilon^{1+1/20}\). This proves (116).

The order of choices is worth recording. The local exponent \(\delta\) and the finite hierarchy in 39 fix \(\Gamma_0\) using only dimensional constants. Choose \(\Gamma\), then choose \(L\) for the imported estimates and the common exponential base. Only afterwards choose the complexity threshold \(K\), localization powers, and the sufficiently small upper bound on \(\epsilon\). No observation number or moment order enters the first two choices. ◻