Proof. Taking two particles in (8) gives \(\phi(r)\geq-2B\) for \(r>0\). In the nonsingular case, approximation by distinct configurations extends stability to coincident positions. The vector field is locally Lipschitz on its domain and hence has a unique maximal local solution. Differentiation gives energy conservation on its existence interval. For \(n=|R|\), \[ \frac12\sum_{i\in R}|v_i(t)|^2 \leq\mathcal H_R(Z_R(0))+Bn. \tag{10}\] For \(n\geq2\), bounding every pair except one below gives \[ \Phi_\varepsilon(x_i(t)-x_j(t)) \leq\mathcal H_R(Z_R(0)) +2B\left(\binom n2-1\right). \tag{11}\] In the singular case this bounds every pair distance away from zero. Equation (10) prevents escape to spatial infinity in finite time. Thus on each bounded time interval the solution remains in a compact subset of the vector field’s domain and extends beyond any proposed finite endpoint, forward or backward. In the nonsingular case one can also use the globally Lipschitz gradient of the compactly supported \(C^2\) potential directly. The cases \(n=0,1\) are immediate.

The vector field has zero divergence in \((x_i,v_i)_{i\in R}\): \(\dot x_i\) is independent of \(x_i\) and \(\dot v_i\) is independent of \(v_i\). Its flow Jacobian therefore solves \(\dot J=0\) with \(J(0)=1\). ◻

Proof. Insert \(r\) particles into (15) at time zero. Their mutual exclusions give \(\mathbf1_{{\cal D}_R^\varepsilon}\). The remaining partition sum is \(\mathcal Z_\varepsilon\) times the probability that the original cloud misses \(\bigcup_iB_i\). This proves (17). Initial exclusion already gives external isolation.

On \(\mathcal D_R^\varepsilon\) every subset is separated, so substitution into (18) gives the exact identity \[ E_R^\varepsilon(0,Z_R) =(-1)^{|R|}f_0^{\otimes R}(Z_R) \mathbb P\{N(B_i)\geq1\text{ for every }i\in R\}. \tag{21}\] Choose a maximal subcollection of centers with mutual distance greater than \(2\varepsilon\). Its balls \(B_i\) are disjoint up to null boundaries. A ball of radius \(2\varepsilon\) contains at most \(125\) original centers: their disjoint balls of radius \(\varepsilon/2\) fit inside a ball of radius \(5\varepsilon/2\). Hence the chosen subcollection has size \(m\geq |R|/125\). Disjointness ensures that occupants are distinct, and the factorial upper bound gives \[\begin{align*} \mathbb P\{N(B_i)\geq1\text{ for every chosen }i\} &\leq\mathbb E\prod_{\text{chosen }i}N(B_i)\\ &=\mu^m\int_{\prod_i(B_i\times\mathbb R^3)} F_m^\varepsilon(0)\,dZ_m\\ &\leq\bigl(\mu\|\rho_0\|_\infty|B(0,\varepsilon)|\bigr)^m \leq(C\varepsilon)^m. \end{align*}\] For \(C\varepsilon\leq1\) this proves (19).

The loss from mutual exclusion of \(r\) independent inserted positions is at most \(Cr^2\varepsilon^3\), by a union bound and bounded \(\rho_0\). For each inserted position, \[\mathbb P\{N(B_i)\geq1\}\leq\mathbb E N(B_i) =\mu\int_{B_i\times\mathbb R^3}F_1^\varepsilon(0,z)\,dz \leq C\varepsilon.\] A second union bound proves the first estimate in (20). The integral of (15) is \(\mu^{-r}\mathbb E(N)_r\), where \((N)_r=N(N-1)\cdots(N-r+1)\). It is finite by the initial factorial bound and conserved since \(N\) is conserved. ◻

Proof. Relative energy is \[\tfrac14|q'|^2+\phi(|q|)=\tfrac14|g|^2.\] Since \(\phi\geq-2B\), we have \(|q'|^2\leq M^2+8B\) throughout the encounter. Consider \[(n,g,s)\longmapsto(q(s),q'(s)), \quad n\cdot g<0,\quad |g|\leq M,\quad0<s<\tau_*(n,g).\] Its volume element is \(|n\cdot g|\,d\sigma(n)\,dg\,ds\). At \(s=0\), append the velocity vector field to five tangent vectors of the entrance section. Its normal component in position space is \(n\cdot g\). Phase-volume preservation propagates that determinant.

The map is injective. Two representations of an interior state lie on the same orbit by uniqueness. A preceding outward crossing cannot be followed by an inward crossing: outside the ball the path is a straight line with increasing distance from the origin. If the orbit repeated an interior state it would be periodic, so it could not have a past entrance from outside. Hence there is at most one entrance and elapsed time.

Apply change of variables on compact pieces of the entrance section and bounded elapsed-time intervals, then exhaust the domain. The image lies in \(B(0,1)\times B(0,\sqrt{M^2+8B})\). Its finite volume proves (23). Markov’s inequality gives the tail and excludes infinite duration outside a null set. The same flow-box determinant on the transverse exit section proves flux preservation. Tangential section data carry zero flux. ◻

Proof. Let \((n_+,g_{\rm out})\) be the exit state of the relative orbit with entrance \((n,g)\). Since \(\Phi\) is even, \(-q(\tau_*-s)\) has entrance \((-n_+,g_{\rm out})\) and exit \((-n,g)\). Thus \((n,g)\mapsto(-n_+,g_{\rm out})\) is an involution of the incoming section outside a null set. It preserves flux by Lemma 5 and the reflection \(n_+\mapsto-n_+\). The center velocity is unchanged, and \((v,v_*)\leftrightarrow(c,g)\) has absolute determinant one. Equation (22) proves invariance of \(d\Lambda\).

Particle exchange also preserves \(d\Lambda\), so the two test-function terms in either pair of (24) give equal integrals. This cancels the factor \(1/2\). The scattering involution transfers the remaining outgoing test to the incoming variable, giving (25). Energy conservation gives \(|v'|^2+|v_*'|^2=|v|^2+|v_*|^2\). The finite disk area and Gaussian integration justify absolute convergence. No scattering-angle density or choice of one branch is introduced. ◻

Proof. Put \(G=|g|\), \(\rho=|b|\), \(J=G\rho\), and \(E=G^2/4\). The cases \(G=0\), \(J=0\), and \(\rho=1\) are null in incoming flux. For \(J>0\), the effective radial potential is \[W_J(r)=\phi(r)+\frac{J^2}{4r^2}.\] An incoming turning radius \(y\) satisfies \(E=W_J(y)\). A nonsimple turning point also requires \(W_J'(y)=0\). At each fixed \(J\), the critical values of the \(C^2\) function \(W_J\) on \((0,1)\) are Lebesgue null, by the one-dimensional Sard theorem on a countable compact exhaustion. The radial incoming flux is \[ G^3\rho\,dG\,d\rho=2J\,dE\,dJ. \tag{27}\] Fubini excludes nonsimple turning energies up to zero flux. Lemma 5 has already excluded infinite duration.

At a simple incoming turning radius \(y\in(0,1)\), \(W_J'(y)<0\) and \(E>W_J(r)\) for all \(r>y\). Set \(a=J^{-2}\) and \[A_y(r)=y^{-2}-r^{-2},\qquad C_y(r)=4(\phi(y)-\phi(r)),\qquad H_{y,a}(r)=A_y(r)+aC_y(r).\] The half angular sweep is \[ {\cal A}_y(a)=\int_y^\infty\frac{dr}{r^2\sqrt{H_{y,a}(r)}}. \tag{28}\] Indeed \(\dot r^2=J^2H_{y,a}(r)\) and the angular speed is \(J/r^2\). For admissible \(a\), one has \(H_{y,a}(r)>0\) for \(r>y\), \(\partial_rH_{y,a}(y)>0\), and \(\lim_{r\to\infty}H_{y,a}(r)>0\). The ratios \(C_y/A_y\) and \(H_{y,a}/A_y\) extend continuously to \([y,\infty]\). At \(y\) use the ratios of first derivatives; at infinity use finite range. The second ratio is strictly positive on this compact interval, so it has a positive minimum, while \(C_y/H_{y,a}\) is bounded. These properties also show that the admissible set of \(a\) is open.

For \(a'\) near \(a\), the binomial expansion in \[H_{y,a'}^{-1/2} =H_{y,a}^{-1/2} \left(1+(a'-a)\frac{C_y}{H_{y,a}}\right)^{-1/2}\] converges uniformly after multiplication by the integrable majorant \(r^{-2}H_{y,a}^{-1/2}\). Thus \(\mathcal A_y\) is real analytic in the parameter \(a\), although \(\phi\) is only \(C^2\). Twice differentiating gives \[ {\cal A}_y''(a)=\frac34 \int_y^\infty\frac{C_y(r)^2\,dr}{r^2H_{y,a}(r)^{5/2}}. \tag{29}\] Near \(y\) the integrand is \(O((r-y)^{-1/2})\), since \(C_y(r)=O(r-y)\) and \(H_{y,a}\) has a simple root. It is also integrable at infinity. It is strictly positive unless \(\phi(r)=\phi(y)\) for every \(r\geq y\). Finite range makes this constant zero; the encountered orbit is then a straight force-free passage.

For a non-free encounter \(\mathcal A_y\) is strictly convex. The deflection angle is \(\pi-2\mathcal A_y\) modulo \(2\pi\). Therefore \(g_{\rm out}=g\) or \(g_{\rm out}=-g\) requires \(\mathcal A_y\in(\pi/2)\mathbb Z\). Each such level has at most two preimages on an admissible interval. There are countably many levels and countably many open intervals. On a simple turning branch, \((y,J)\mapsto(E,J)\), \(E=W_J(y)\), is a local \(C^1\) diffeomorphism. A countable cover by these charts, Fubini in \((y,a)\), and (27) prove flux nullness. These exceptional equalities are exactly the two failures of (26).

For the compact-set assertion, exhaust the simple-scattering non-free radial parameter domain by compact subsets avoiding \(G=0\), \(J=0\), \(\rho=1\), and the zero-increment levels. This covers all non-free data outside a null set. On each such compact subset, continuous dependence for the \(C^1\) ODE flow and transversality of the finite exit give a continuous exit time and scattering map. Thus durations are bounded and all the listed nonzero quantities have positive minima. The relative flux measure on \(G\leq M\) is finite, so an exhaustion member loses at most \(\eta\) flux. Lifting with all rotations makes the set rotation-invariant.

Finally, write \((g,b)=(GRe_1,\rho Re_2)\), \(R\in SO(3)\). At fixed \((G,\rho)\) the orientation factor in incoming flux is a constant multiple of Haar measure on \(SO(3)\). Central-force rotation equivariance maps each nonzero increment to \(R\) times a fixed vector. Its direction has uniform measure on \(S^2\). This is a marginal statement and leaves the joint pair correlated. ◻

Proof. Set \[\gamma(t)=\frac{3\beta}{2}-\frac{\beta t}{2T},\qquad c=\frac{\beta}{2T},\qquad M(v)=e^{-2\beta|v|^2},\] and \((S(r)h)(x,v)=h(x-rv,v)\). Notice \(\beta\leq\gamma(t)\leq3\beta/2\). Define a positive linear operator for \(h\geq0\) by \[ \begin{split} ({\cal B}h)(x,v)=\int_{\mathbb R^3}|v-v_*|\int_{|b|<1} \bigl[&M(v_*')h(x,v')+M(v')h(x,v_*')\\ &+M(v_*)h(x,v)+M(v)h(x,v_*)\bigr]\,d^2b\,dv_* . \end{split} \tag{32}\] The disk is in \((v-v_*)^\perp\).

We first prove that for \(0\leq s<t\leq T\), \(r=t-s\), \[\begin{align*} \|S(r){\cal B}h\|_{\mathrm{Bol},\gamma(t)} &\leq Cr^{-1/2}\|h\|_{\mathrm{Bol},\gamma(s)}, \tag{33}\\ \|S(r){\cal B}h\|_{\infty,\gamma(t)} &\leq Cr^{-1/2}\|h\|_{\infty,\gamma(s)}. \tag{34}\end{align*}\] Let \(H_s(y)=\sup_u e^{\gamma(s)|u|^2}h(y,u)\). For each product in (32), energy conservation, or its identity version for a loss product, gives \[M(u_*)h(y,u) \leq H_s(y)e^{-\gamma(s)(|v|^2+|v_*|^2)}.\] After multiplication by \(e^{\gamma(t)|v|^2}\) this is at most \[H_s(y)e^{-\beta|v_*|^2} e^{-cr(|v|^2+|v_*|^2)}.\] Since \[|v-v_*|e^{-cr(|v|^2+|v_*|^2)/2}\leq C_c r^{-1/2},\] the disk area \(\pi\) and the Gaussian integral in \(v_*\) imply \[ e^{\gamma(t)|v|^2}(S(r){\cal B}h)(x,v) \leq Cr^{-1/2}e^{-cr|v|^2/2}H_s(x-rv). \tag{35}\] This proves (34).

For the summed spatial estimate put \(a_k=\sup_{|y-k|\leq1}H_s(y)\). If \(m\leq r|v|<m+1\) and \(|x-k|\leq1\), then \(|x-rv-k|\leq m+2\), whence \[H_s(x-rv)\leq \sum_{\substack{\ell\in\mathbb Z^3\\|\ell-k|\leq m+3}}a_\ell.\] Indeed lattice-centered unit balls cover \(\mathbb R^3\). For \(r\leq T\), the damping factor on that shell is at most \(e^{-cm^2/(2T)}\). Bound the supremum over \(v\) by summation over \(m\geq0\) and then sum over \(k\). Each \(a_\ell\) occurs at most \(C(m+4)^3\) times. Convergence of \(\sum_{m\geq0}e^{-cm^2/(2T)}(m+4)^3\) proves (33). The same shell argument, now using \[e^{\gamma(t)|v|^2}|S(t)h_0(x,v)| \leq e^{-ct|v|^2} \sup_u e^{\gamma(0)|u|^2}|h_0(x-tv,u)|,\] gives \[ \sup_{0\leq t\leq T}\|S(t)h_0\|_{\mathrm{Bol},\gamma(t)} \leq C\|h_0\|_{\mathrm{Bol},\gamma(0)}. \tag{36}\] At \(t=0\) this is immediate.

Define \[({\cal V}h)(t)=A\int_0^t S(t-s){\cal B}h(s)\,ds.\] The characteristic equation, \(f\geq0\), and \(f\leq AM\) imply \[ 0\leq f(t)\leq S(t)f_0+({\cal V}f)(t). \tag{37}\] Its \(n\)-fold iteration is the finite inequality \[f\leq\sum_{j=0}^{n-1}{\cal V}^j(S(\cdot)f_0)+{\cal V}^n f.\] No finiteness of the unknown summed spatial norm has been assumed. Using (34) and \(\|f(t)\|_{\infty,\gamma(t)}\leq A\) yields \[ \|({\cal V}^n f)(t)\|_{\infty,\gamma(t)} \leq A\frac{(CA\Gamma(1/2))^nt^{n/2}} {\Gamma(1+n/2)}. \tag{38}\] This tends to zero uniformly for \(t\leq T\). The time-convolution formula follows inductively from \[\int_0^t(t-s)^{-1/2}s^{n/2}\,ds =t^{(n+1)/2} \frac{\Gamma(1/2)\Gamma(1+n/2)} {\Gamma(1+(n+1)/2)}.\] Equations (33) and (36) bound the summed spatial norm of the \(j\)th series term by \[C B_0\frac{(CA\Gamma(1/2))^jT^{j/2}}{\Gamma(1+j/2)}.\] The sum is finite. Letting \(n\to\infty\) in the pointwise comparison and then taking the summed spatial norm proves the required bound for \(f\) at exponent \(\gamma(t)\).

To handle the derivative without assuming its envelope, fix a coordinate unit vector \(e\) and \(0<|\delta|\leq1\), and put \[d_\delta(t,x,v) =\frac{f(t,x+\delta e,v)-f(t,x,v)}{\delta}.\] Use \(a'b'-ab=(a'-a)b'+a(b'-b)\) for both gain and loss products in (25). Each undifferenced factor is a spatial translate of \(f\), bounded by \(AM\), so \[|d_\delta(t)| \leq S(t)|d_\delta(0)|+{\cal V}|d_\delta|(t).\] For fixed \(\delta\), \(\|d_\delta(t)\|_{\infty,\gamma(t)}\leq2A/|\delta|\). The remainder argument (38) therefore applies and vanishes as \(n\to\infty\), before taking \(\delta\to0\). The fundamental theorem of calculus gives \[|d_\delta(0,x,v)| \leq\int_0^1|\partial_e f_0(x+\theta\delta e,v)|\,d\theta.\] Translations of length at most one change the summed spatial supremum norm by at most an absolute factor, since each translated unit ball is covered by a fixed number of lattice-centered unit balls and each covering ball occurs a bounded number of times. Hence \(\|d_\delta(0)\|_{\mathrm{Bol},\gamma(0)}\leq CB_0\), uniformly in \(\delta\). The same series estimate bounds \(\|d_\delta(t)\|_{\mathrm{Bol},\gamma(t)}\) uniformly in \(t,\delta\). Pointwise convergence to \(\partial_e f\), followed by Fatou’s inequality for the sum of local suprema, gives the derivative bound. Sum over the three coordinate directions and use \(\gamma(t)\geq\beta\) to obtain (30). ◻

Proof. First decompose the full flow according to its unique union of contact components meeting \(R\). The flow observation preceding Definition 11 gives \[ \mathsf H^I_{R,S} =\sum_{\substack{R\subset A\subset S\\ A\text{ union of blocks of }\pi_0(S)}} c_I^R(A)\mathsf H^I_{R,A}\Xi_I(\{A\};S\setminus A). \tag{47}\] Precisely one term has its two component indicators nonzero: the actual union of rooted contact components. Endpoint isolation of \(R\) against the rest of \(S\) can then be checked inside \(A\), because \(A\) and its complement avoid throughout the cell.

Here is the finite identity that expands the last factor. Fix \(\mathcal X\), and consider a collection \(\mathcal B\) of disjoint nonempty subsets of \(V\), each a union of the fixed initial polymers. Put \(U_{\mathcal B}=\bigcup\mathcal B\) and \[d_I(\mathcal B) =\prod_{B\in\mathcal B}c_I(B) \prod_{\{B,C\}\subset\mathcal B}\alpha_I(B,C).\] Then \[ \Xi_I(\mathcal X;V) =\sum_{\mathcal B} (-1)^{|\mathcal B|}d_I(\mathcal B) \prod_{B\in\mathcal B}\tau_I(\mathcal X,B) \Xi_I(\mathcal B;V\setminus U_{\mathcal B}). \tag{48}\] The term with \(\mathcal B=\varnothing\) equals one. To prove (48), expand the product (39) over subsets of its actual components. A given collection \(\mathcal B\) consists of components of \(V\) exactly when its members are connected, mutually avoid, and avoid the remaining \(V\setminus U_{\mathcal B}\) flow. These conditions are respectively \(\prod c_I(B)\), the pair product in \(d_I\), and the last \(\Xi_I\) in (48). On their support the required flows agree. This proves the formula, including all signs.

Apply (48) repeatedly to (47). Every nonempty selection strictly decreases the number of remaining labels, so the process terminates. At termination its coefficient is exactly (41)–(42); the unselected labels have no remaining restriction. Grouping by the union of selected initial polymers proves (45). The initial partition of such a union is the restriction of \(\pi_0(S)\), so its coefficient is exactly the kernel defined in (43), not a kernel depending on unselected coordinates. Finite Boolean inversion proves (46).

There are at most \(2^m\) choices of \(A\setminus R\), at most \(m^m\) partitions of the remaining labels, and at most \(m^m\) assignments of their blocks to successive nonempty generations. This proves the count, with the cases \(m=0,1\) understood separately. ◻

Proof. Apply (45) to each ordered tuple of roots in a realization of the cloud at \(a\). A selected union of its initial polymers is exactly an externally isolated displayed subset at \(a\). Consequently the expectation of each selected union uses \(\widetilde F_{r+m}(a)\). Its \(m\) extra labels were unordered, whereas the factorial sum orders them, giving the divisor \(m!\). The scaling of the original \(r\) roots is \(\mu^{-r}\), and that of the new factorial measure is \(\mu^{-(r+m)}\), giving \(\mu^m\).

For a bounded test \(\psi\), Boolean inversion bounds a coefficient in total variation by \(2^{|\mathcal U|}\|\psi\|_\infty\). Summing this over all selected polymer subsets in a fixed cloud costs at most \(3^N\|\psi\|_\infty\), and the root tuple count costs at most \(N^r\). The integrable bound \(\mu^{-r}N^r3^N\|\psi\|_\infty\) justifies the stated grouped limit. Expanding every large-order coefficient into all its diagrams before taking absolute values would give a different, unproved convergence assertion and is unnecessary. ◻

Proof. Cross avoidance makes the joint root flow equal to the two separate root flows. Its rooted-component and endpoint-isolation indicators therefore split. Initial-polymer admissibility also splits, since all cross initial distances are greater than \(\varepsilon\). Within a merged generation, all additional cross avoidance factors are one. A detached component in \(S_i\) can touch the preceding generation only through that generation’s \(S_i\) part. Hence its touch indicator is exactly its original touch indicator. The number of detached groups is additive, so the signs multiply. The output Dirac measures split with the root flow. These facts give (50).

For the inverse map, a connected detached group cannot contain labels from both families: a connected graph across that partition would have a cross contact, contrary to \(\eta_I=1\). The same avoidance splits the root group’s flow by uniqueness. A nonempty generation in one family must have a predecessor in that same family unless it is the first generation: otherwise its required touch indicator vanishes. There can therefore be no gap followed by a later nonempty generation in one family. Removing its terminal empty batches produces its unique local generation list. This proves the asserted bijection. Formula (51) is the ordinary ordered-label count. ◻

Proof. Use (47). Stop a summand immediately if its root group reaches the threshold. Otherwise apply the weighted product expansion with \(d=k-|A|\), extract its selected components, and continue its nonempty completed selections by the same rule with the updated remaining threshold. Empty selections terminate a complete diagram. The explicit finite identities (52) and (53) prove the resulting equality at every stage. A diagram of total size less than \(k\) never triggers the stopping rule and appears with exactly its original coefficient. Every other branch stops in one of the two stated ways.

Weights are positive before their displayed inclusion-exclusion sign, and the unexpanded factors are in \([0,1]\), proving the variation bound. All selected groups are unions of the original initial polymers. Their union is consequently externally isolated in the original cloud at the initial cut; removing the last factor does not remove this initial-polymer selection condition. Taking factorial expectations therefore gives the asserted input measures and normalization. On the stated trial-component bound each component of the root group contains a root and has at most \(D\) labels, proving \(|A|\leq |R|D\); the other size bound follows from the one-component overshoot.

These expected finite stopped identities are well defined for the grand ensemble without a tail theorem. Prior to the last selected component there are fewer than \(k\) selected labels, and therefore at most \(k\) selected groups. The root-large branch has one group. At fixed \(k\) the number of stopped descriptions in an \(N\)-particle cloud is bounded by \(C_k^{N+1}\): assign each label to one of at most \(k+2\) group or remainder slots, and then order the bounded number of selected groups and generations. The exponential moments of \(N\), with its root-tuple polynomial factor, justify expectation and the finite-\(N\) limit. This argument establishes finiteness only; it supplies no useful uniform bound as \(\varepsilon\to0\). ◻

Proof. On \(\{\chi_\pi^\varepsilon=1\}\) every restricted pattern \(\pi_A\) has indicator one. Substitute (56) on the right of (57). For \(D\subset S(\pi)\), the coefficient of \(f_{S(\pi)\setminus D}\widetilde F_{B(\pi)\cup D}^\varepsilon\) is \[\sum_{D\subset A\subset S(\pi)}(-1)^{|A\setminus D|} =(1-1)^{|S(\pi)\setminus D|}.\] Only \(D=S(\pi)\) survives. Off that pattern both sides vanish. ◻

Proof. At a fixed array exactly one full pattern occurs. Its singleton labels are exactly the labels which may belong to a set \(D\) with \(\iota_D^R=1\). Deleting these labels leaves the unique restricted pattern. Summing (56) and (57) therefore gives (59)–(60). For disjoint \(D,A\) one can also check inversion directly from \[\iota_D^R\,\iota_A^{R\setminus D}=\iota_{D\cup A}^R\] and the binomial cancellation in Proposition 20. Because pattern supports are disjoint, the sum of the patternwise absolute values is the absolute value of their assembled density. Taking absolute values in the collapsed formulas, dropping \(\iota_D^R\leq1\), and integrating each deleted factor using \(\int f(t)=1\) proves (61). ◻

Proof. We give the majorant and the limiting argument separately. Put \(p=n+1\) and \(\mathcal E=|v|^2+\sum_{i=1}^n|w_i|^2\). At every insertion sphere the pair potential is zero. Elasticity consequently gives \[ \sum_{k=1}^p|v_k(u)|^2=\mathcal E. \tag{66}\] Intermediate pair speeds are bounded by \(C(\mathcal E+p)^{1/2}\) by stability. These bounds hold for both signs. At a birth, summing the flux over possible parents, with unused parent choices assigned zero, gives \[ \sum_{a\ \mathrm{available}}\int_{\mathbb S^2} ((w-v_a)\cdot\omega)_+\,d\omega \leq Cp\bigl(1+|w|^2+\mathcal E/p\bigr)^{1/2}. \tag{67}\] The inequality follows from \(\sum_a|v_a|\leq\sqrt{p\sum_a|v_a|^2}\). It may be used successively, since its right side no longer depends on the chosen parents or the angular variables. The product satisfies \[ \prod_{i=1}^n\bigl(1+|w_i|^2+\mathcal E/p\bigr)^{1/2} \leq C^p(1+\mathcal E/p)^{p/2}. \tag{68}\] Indeed the arithmetic–geometric mean inequality bounds the product by the \(n\)th power of its arithmetic mean, and \(\sum_i|w_i|^2\leq\mathcal E\).

Let \(M\) bound both the uniform Gaussian norm and the summed spatial Gaussian norm of \(f(u)\) at exponent \(\beta\). Use the spatial envelope on the continued label \(1\) and the uniform bound on all other bottom labels. Every displacement of that label has length at most \(Cb(\mathcal E+p)^{1/2}\). Translation of its summability boxes therefore costs at most \(C(1+b\sqrt{\mathcal E+p})^3\): any unit box enlarged by that radius is covered by this number of unit boxes. To handle the velocity supremum without interchanging it with the spatial sum, first split the root velocities into shells \(m\leq|v|<m+1\). On each shell the displacement radius is at most \(Cb\sqrt{(m+1)^2+|W|^2+p}\), independently of the root velocity. Sum the spatial boxes with this enlarged radius and then sum over \(m\), reserving a Gaussian factor \(e^{-c m^2}\). This proves the same bound with a convergent Gaussian shell sum in place of a root supremum. Reserve the fixed Gaussian margin \(\beta-\beta_0\). For a fixed \(\delta>0\) smaller than that margin, \[ \sup_v e^{-\delta|v|^2} \int_{\mathbb R^{3n}} e^{-\delta\sum_i|w_i|^2} (1+\mathcal E/p)^{p/2}(1+b\sqrt{\mathcal E+p})^3\,dW \leq C^p. \tag{69}\] To verify this, use \(1+y\leq C_\delta e^{\delta y}\) with \(y=\mathcal E/p\), choose its exponent smaller than the remaining Gaussian exponent, and absorb the fixed cubic polynomial with another fixed part of that exponent. The resulting Gaussian integral is \(C^n\); its root supremum is finite. Its Gaussian velocity-shell version, needed in the preceding spatial estimate, has the same \(C^p\) bound. Adjusting \(C\) includes the finitely many cases \(p\leq3\).

The ordered simplex has volume \((t-u)^n/n!\), and \(p^n/n!\leq e^{2p}\). Equations (66)–(69), including the \(2^n\) sign choices, prove the absolute estimate \[ \left\|\sum_{a,\sigma}\int \mathcal S_n^{\varepsilon,J} f_p(u,Z_p^\varepsilon(u))\,d\lambda_n\right\|_{\mathrm{Bol},\beta_0} \leq C_0(C_1b)^n, \tag{70}\] where the sum inside the norm is understood with absolute integrands. Choose \(b_*\) so that \(C_1b_*<1/2\). The identical estimate holds for instantaneous scattering trees.

Fix now \(n\) and truncate the velocity and relative insertion parameters to a compact set. The Gaussian majorant just proved makes the omitted part arbitrarily small, uniformly in \(u,t\) and \(\varepsilon\). Remove incoming parameters whose unscaled residence time exceeds \(D\), whose relative speed is too small, or which belong to the null exceptional sets of Proposition 8. Lemma 5 controls the residence-time cutoff. The omitted flux tends to zero as \(D\to\infty\) and the other cutoffs are removed. On the retained set a true or free passage has length \(O(\varepsilon D)\) and displacement \(O(\varepsilon D)\). The set of insertion times lying within \(C\varepsilon D\) of a cell boundary has measure at most \(C_nJ\varepsilon D\); two insertion times in one cell cost \(C_n/J\). Both quantities vanish, taking \(D\) fixed first. Shortened endpoint cells obey the same estimates.

Here is the needed null-set assertion for other meetings. Parameterize each later birth by its velocity relative to its current parent in laboratory axes. This is a triangular translation of the \(w_i\) variables with Jacobian one. By Galilean invariance, later relative increments do not depend on an earlier common velocity boost. At the first divergence of two distinct rays, keep the relative speed and the scalar scattering parameters fixed, and rotate the relative incident velocity and impact configuration together. The separating relative velocity has nonzero length and a spherical direction with absolutely continuous area measure. On any subsequent interval between scheduled births, the relative position of the two rays has the form \[ (s-s_*)\,\xi+\Gamma(s), \tag{71}\] where \(s_*\) is their divergence time, \(\xi\) is that relative velocity, and \(\Gamma\) is a fixed affine function after the later relative increments and times have been fixed. There are finitely many such intervals. For \(s\ne s_*\), equality to zero confines the direction of \(\xi\) to a finite union of curves on a sphere, which has area zero. The interval immediately after divergence is free of other scheduled births and its nonzero relative speed separates the two rays. For a true scattering both outgoing rays must be included when identifying this divergence; their directions are not asserted to be independent. This argument also covers the free sign. Fubini’s theorem therefore excludes every unprescribed meeting for fixed \(n\). Relative positions and this exceptional set are independent of the root position and root velocity, by translation and Galilean invariance. Thus the compact exceptional estimates are uniform for the root supremum in the spatial norm.

Away from these exceptions, the microscopic paths converge to the instantaneous paths, with the same velocities outside the short pair arcs. All input positions differ by \(O_n(\varepsilon D)\), while their input velocities agree exactly. The outgoing segments between a contact and its upper cell boundary have length \(O(b/J)\) and are also checked for unwanted contacts. Their length tends to zero, so the same null-set argument covers their avoidance indicators. Proposition 9 and the spatial mean-value inequality give convergence of the products of input densities in the summed spatial norm, uniformly in \(u,t\). Dominated convergence in the compact parameters, followed by removal of the compact cutoffs, proves convergence of every fixed term. For uniformity in the layer length, use backward elapsed times in the fixed simplex \(0<\tau_1<\cdots<\tau_n<b\) and restrict by \(\tau_n<t-u\). Extend the bottom rays freely to elapsed time \(b\) in the null-set test. This tests a larger finite time interval and still has zero exceptional measure. The estimates for short arcs and endpoint cells are uniform in \(t-u\). If \(t-u\to0\), the terms with \(n\geq1\) are directly bounded by \(C_n(t-u)^n\). These observations justify the asserted supremum over all shortened layers.

Finally the limiting insertion integrates over the outgoing hemisphere with weight \((g\cdot\omega)_+\). For the plus sign its bottom velocities are the inverse scattering velocities; for the minus sign they are the unaltered velocities. The flux-preserving scattering change of variables in Lemma 7 and the sphere-to-disk identity (22) identify their difference with precisely \(Q_\Phi\), with ordinary disk area. Iterating the mild kinetic equation on \([u,t]\) gives the instantaneous tree sum. Its remainder tends to zero by (70), using the same Gaussian bound for the terminal \(f\); this also proves uniqueness on this short interval in the class at issue. First make the common geometric tail small, then use fixed-term convergence. This proves (65) without any statement about a microscopic large-history remainder. ◻

Proof. Retain a tag saying whether each centered singleton root belongs to the microscopic marginal in \(\mathsf M_{t_j}\) or to its negative \(f(t_j)\) factor. In either case expand a simple candidate family all the way to the lower cut and label every line it displays. Split the domain of every diagram into the event of no cross contact with this family and its complement.

On the no-cross-contact event, Proposition 15 splits the cell generation diagram into its restriction to that family and its restriction to the other families. Explicitly, the root subsystem flow splits; a detached connected group cannot meet two families; each generation is the disjoint union of its family generations. Deleting empty terminal generations in each restriction and padding them when merging are inverse operations. The signs are the products of the signs of the two restrictions, and the within-generation avoidance tests split. This statement uses the trajectories in the displayed signed diagram, including the detached ones. Avoidance merely in the full interacting system would not justify any of these operations.

The one-label rule in each occupied cell is exactly the plus/minus insertion used in (63). Empty cells propagate freely. Thus the family restriction is one of the terms of (64), with identical times, sphere normals, velocities, indicators, and input factors on the two tagged sides. The inverse transform at the lower cut supplies exactly the independent \(f\) factors on its leaves. Restoring the full pattern indicators makes the deletion and reinsertion operations inverse also at the cuts. In particular, a nonsingleton polymer cannot enter this pairing.

If \(m\) extra labels split into \(r\) in this family and \(m-r\) in the rest, the label choice contributes \(\binom mr\) and \[\frac{\mu^m}{m!}\binom mr =\frac{\mu^r}{r!}\frac{\mu^{m-r}}{(m-r)!}.\] Applying this in the occupied cells proves equality of the measures and all factorial coefficients. Change only the root tag. The centering sign reverses and every other factor stays the same. The first eligible root in a fixed order defines an involution: eligibility depends on the displayed paths and lower retained slots, which the tag change preserves. The paired contributions sum to zero.

Perform the pairing coefficient by coefficient for \(p<K_j\), counting all formal lifetimes and reusing the upper root label when its tag changes. Both the resulting physical block and each individual reference tree then have size less than \(K_j\). Hence both tagged terms occur in the separately truncated expansion, and both belong to \(\mathsf A_j\). The pairing preserves \(p\) and is an exact involution on this retained part. The complementary terms are precisely the already defined finite operator \(B_j\), with the size bound (75); they are not identified with the physical stopped remainder. ◻

Proof. Use the disjoint ownership of Definition 24 in each layer. Every original singleton birth has its parent’s owner. If a family has no marked member, its births are single and every passage is isolated, complete, and separated at its required cuts. Every cross-owner contact would mark its endpoint in that family, so none exists. In a joint root force group the cross potentials therefore vanish along these trajectories, and uniqueness splits off this family’s flow. A connected detached group cannot straddle this family and another one without a cross-owner contact. Its generation diagram consequently splits by Proposition 15. This proves that an unmarked family is precisely eligible for the simple reference construction; it is not an assumption about the components of the joint root group. A centered singleton root of this family must therefore either have a residual factor, or have a retained lower singleton descendant: otherwise its family would cancel by Proposition 23. A root in a nonsingleton block is already marked.

For each uncharged centered root choose one retained lower descendant, using the first one in a fixed order. Different unmarked families have disjoint labels, so these choices are distinct. Draw an arrow from that root occurrence to the chosen root entering the next lower expanded layer, or end its path at the chosen initial factor if the lower cut is time zero. The counted vertices are precisely the entering-layer root occurrences in \(H\). Root occurrences charged to marked families have no arrow; assign each such root one marked member of its family. This assignment is injective within a layer because the families are disjoint. Residual roots and uncentered restarting roots also terminate paths. Every other vertex has one downward continuation, and every vertex receives at most one arrow. Hence the graph is a disjoint union of downward paths. Every path has at most \(L\) counted root occurrences and ends at one of the specified charges.

At most \(h_0\) ends are restarting roots, at most \(r\) are residual roots, and at most \(i\) are initial roots. At any layer the marked family ends inject into the marked lives. One life occurs in at most \(L\) layers, so there are at most \(Ld\) such ends. Multiplying the number of ends by the maximal path length proves (78). The last inequality follows immediately. ◻

Proof. Between two consecutive mesh times the augmented map is a product of Hamiltonian maps and identity maps. Each factor is invertible and volume preserving by Proposition 2. Injection merely changes which of the already present coordinates is active; departure changes which coordinate is frozen. Both operations are identity maps on the augmented space. A fixed prescription therefore gives a finite composition of volume preserving maps. Coincident singular positions at a switch form a null set; apply the argument first on compact sets avoiding them and then exhaust the domain.

For the energy identity, keep the Hamiltonian of every force group. At the earlier endpoint of a cell its distinct force groups are unions of distinct whole initial polymers. Their cross potentials are zero. Thus passing from the preceding force partition to this partition changes no interaction energy. At the later endpoint the root group \(A\) splits into its surviving labels \(R\) and \(A\setminus R\). The indicator \(e_I^R(A)\) makes their cross potentials zero. The other force groups depart in their entirety. Store the Hamiltonian of each departing group at that time. At a layer cut the singleton sources are inserted on the restored polymer pattern, so their cross potentials with each other and with all existing groups in the new force system are zero. A nonsingleton polymer is never split by this operation.

At the top of a terminal witness all final groups are stored intact, including their distinguished anchors. No split into \(R\) and \(A\setminus R\) is made there, and no endpoint isolation is used. The mixed tuple still has one terminal velocity for every label. This proves the same identity for the positive witness variant.

There is a relevant distinction at a late cell endpoint. A detached group may geometrically overlap a different-generation group or the root group. Those groups had separate Hamiltonians throughout the cell. Their geometric overlap introduces no term into either Hamiltonian, and they are not merged when the detached group departs. Consequently it causes no missing cross-energy term in this calculation.

Starting with the empty active system, conservation on open cells and the preceding switch identities give, at every time \(t\), \[ \frac12\sum_{i:\,\text{source}_i\leq t}|v_i^{\rm source}|^2 =\sum_{\substack{A:\,A\text{ current at }t}}\mathcal H_A(t) +\sum_{\substack{D:\,D\text{ departed by }t}}\mathcal H_D(\text{departure}_D). \tag{84}\] The current and departed groups partition the injected labels. Stability on each group therefore subtracts at most \(B\) times the number of those labels, not \(B\) times that number at every cell. Adding the kinetic energy of future sources to both sides proves (82). Splitting the source exponential into two equal factors and using the first inequality gives (83). ◻

Proof. At the late endpoint initially tie all root labels into one anchored translation component. This is a convention about fixed coordinates, not an assertion that the roots interact. Process cross-component endpoint overlaps, choosing a pair by a fixed label order. An overlap fixes the relative translation by an offset of length at most \(\varepsilon\). Ties between two already anchored pieces fix no new coordinate and are ignored. Repeat until every remaining floating component is separated from all other components at that endpoint.

Every component carries the original force partition restricted to its labels. Translate the component as a whole. Its restricted dynamics commutes with that translation. Starting at the late endpoint, evolve all these components backward. Until their first intercomponent geometric contact, they exert no cross force and these restricted motions equal the original prescribed motions. At that contact fix one relative component translation, merge the two translation components, and continue with the restricted original force partition on their union. Physical interactions are included when the original partition includes them; a virtual contact only merges translation coordinates. Interactions already internal to a component need not be enumerated. Because the prescribed contact graph links every extra label to roots, the algorithm eventually anchors every extra coordinate.

For two floating components write their paths as \(a_A+q_i^A(t)\) and \(a_B+q_j^B(t)\), where all internal coordinates are fixed. At the selected contact, \[a_B-a_A=\varepsilon\omega+q_i^A(t)-q_j^B(t).\] The derivative in time is the relative velocity; the other two derivatives span the tangent plane of the sphere of radius \(\varepsilon\). Their determinant is (87). A contact with the anchored component uses the same formula with its position already fixed. Endpoint pins are linear changes from a relative translation to an offset and have determinant one.

At each stage keep the previously chosen internal and pin coordinates unchanged. The next change acts only on one still free relative translation. Thus the changes are triangular in merger order. Large derivatives with respect to earlier pin coordinates enter only off-diagonal blocks. This proves the product assertion, including for singular potentials by compact exhaustion of the regular flow domains.

There are at most \(m\) mergers that remove an extra translation degree of freedom. Choosing their pairs, representatives, orders, and endpoint/contact types costs at most \((Cp)^{Cm}\). The extra-label partitions and generations have the same type of bound by Proposition 13. Roots have already been anchored together and require no partition sum. On each original first-contact domain the construction recovers the starting configuration. For an upper bound the area formula permits summing the images of all these charts. It does not require a uniform bound on repeated physical contacts. Tangencies lie in zero-Jacobian portions of the contact maps, and deterministic tie rules suffice at multiple choices.

When there is one extra label, before its first contact with its own factor that label is free and the old roots follow their isolated root flow. This proves (88). An earlier contact with a different tensor factor exerts no force and hence does not invalidate this original-factor chart. Its indicators remain part of the term. The force group and original factor are determined by the chosen original parent and the full/virtual choice.

Finally, on the valid energy shell every velocity at a selected contact has magnitude at most \(E_r^{1/2}\). A contact pin therefore integrates to at most \(C\varepsilon^2 E_r^{1/2}\Delta\), whereas an endpoint pin costs \(C\varepsilon^3\). After \(\mu^m=\varepsilon^{-2m}\), (89) follows. The energy bound is used on the original chart domain before other restrictions are dropped; no energy bound on an artificially reconstructed off-domain trajectory is presumed. ◻

Proof. For a set of at most \(q\) old labels with squared-velocity sum at most \(E\), direct Cauchy–Schwarz gives \[ \sum_a|v_a-w|^2\leq2q|w|^2+2E,\qquad \sum_a|v_a-w| \leq Cq\left(1+|w|^2+\frac E q\right)^{1/2}. \tag{97}\] The root state at an ordinary pin is determined before its parent and new velocity are chosen. On a fixed source-energy shell, retain the adapted gate that this root state has squared-velocity sum at most \(E_r\). Every original term in the shell satisfies that gate. Sum the last parent first using (97), then the preceding one, and continue. The resulting factors are independent of the parent states. This proves the ordinary-pin product in (93). Remaining future admissibility indicators can now be dropped. The majorant in \(V\) is the fixed Gaussian of Lemma 27, so these integrations introduce no additional per-layer exponent loss.

For globally ordinary cells the first contact times are chronologically ordered. Summing their disjoint cell boxes is an integral over a subset of the corresponding ordered simplex. Hence it supplies \(b^{n}/n!\), not \(J^n\) times this quantity. A maximal stretch of empty cells simply composes the same prescribed root flows. It has one deterministic map and no independent empty-cell choice.

If a cell has \(m\geq2\) additions, use (89), without a factorial improvement in its internal contact times. For a total of \(m_*\) such participants there are at most \(m_*/2\) occupied cells. Their location choices cost at most \(J^{m_*/2}\) in a layer. The time factors supply \((b/J)^{m_*}\), since \(\varepsilon J\leq b\) eventually. Distributing the participants among those cells and choosing their pin/generation data costs \(p^{C m_*}\). A missing ordered-time factorial for these participants is restored with \(n_j!/(n_j-m_j)!\leq p^{m_j}\). On the energy shell, their speed factors cost at most \(p^{m_*/2}(r+2)^{m_*/2}\) times a fixed exponential. All these polynomial powers are absorbed in \(d\) in (94).

A singleton endpoint pin has factor \(C\varepsilon\) after its activity normalization, instead of a contact-time integral. Its cell choice costs at most \(J\). Restoring the time simplex and any parent factor loses at most a further fixed power of \(p\) per such pin. This gives the second term in (94). The unordered-extra factorials of Corollary 14 cancel permutations of new names; at a singleton no independent factor assignment remains after choosing its original parent. At multiple cells all remaining assignments are among the polynomial costs just counted.

The energy and spatial bounds were proved in (83)–(92). Combining them with these changes of variables proves (93). Notice that \(\chi_\varepsilon\to0\) uniformly for \(p\leq P^A\). No rarity cutoff has entered its exponent.

For the coarse consequence replace every parent capacity by \(p\) and use the global ordered simplex. For \(n=p-h_0\leq p\), \[p^n\frac{T^n}{n!}\leq e^p\max(1,T)^p.\] The source shell gives \(\sum_i|w_i|^2\leq E_r\) on its original domain. Alternatively integrating the product Gaussian moments gives the same bound \(C^p(r+2)^{Cp}\) for the speed product and the spatial factor, since \(h_0\leq p\) and \(h_0\log(1+p/h_0)\leq Cp\). Thus the shell sum is bounded by \[C^p\sum_{r\geq0}e^{-c rp}(r+2)^{Cp}\leq C^p.\] All retained-slot binomials are at most \(2^{\sum_jq_j}\leq2^{(L+1)p}\), and all other fixed layer choices cost a further \(C_L^p\). This proves (95). Starting the shell sum at \(r\geq R-1\) proves (96). Polynomial costs of special cells have already been paid by their mesh factors before this shell sum or a later constant-cutoff estimate is used. ◻

Proof. Choose \(R\) using (96). Given an old-root state with squared-velocity sum at most \(R_1p\), at most \(R_1p/U^2\) of its parent slots have speed above \(U\). The conditional reference probability of choosing one is therefore at most \(R_1/U^2\). A large added absolute velocity has Gaussian probability at most \(Ce^{-cW^2}\). The ball \(|w-v|<\gamma\) has reference probability at most \(C\gamma^3\), uniformly in \(v\).

On the remaining compact set let \(G_*=U+W\). In the incoming disk variables, relative velocities and impact parameters have reference density bounded by a constant with respect to \(dg\,d^2\zeta\). On \(\gamma\leq|g|\leq G_*\) this measure is equivalent to incoming flux. Proposition 8 therefore gives a function \(\omega_\Phi(\kappa;\gamma,G_*)\to0\) as \(\kappa\downarrow0\) such that the second mark has conditional probability at most \(C\omega_\Phi\). The bound is uniform over \(|v|\leq U\): translate \(w\) to \(g=w-v\), bound its Gaussian density above, and enlarge to the fixed relative-velocity annulus. Force-free passages are crossings and are not marked for their zero parent increment.

Lemma 5 gives \[\int_{|g|\leq G_*}\int_{|\zeta|<1} |g|\,\mathbf1_{\{\tau(g,\zeta)>D\}}\,d^2\zeta\,dg \leq C_{\Phi,G_*}/D.\] Dividing by \(|g|\geq\gamma\) shows that the duration mark has conditional probability at most \(C_{\Phi,G_*}\gamma^{-1}\varepsilon^{.04}\). This is the duration of the isolated continuation, not an assumption that an actual many-particle encounter is isolated. An interruption will be an encounter defect in Section [sec:exposure].

Take \(U,W\) large and then \(\gamma,\kappa\) small to make the sum of the first two reference-mark probabilities as small as needed. Equation (99), \(\binom{n_*}{k}\leq2^p\), and the finite exponential normalization give the required bound for an extensive number of these marks. For fixed \(\delta\) any prescribed exponential is obtained by making their one-step probability sufficiently small. Multiple cells and endpoint pins have instead the factor \(\chi_\varepsilon^{m_*+o_*}\) from Proposition 31, which proves their assertion.

For the time marks use unordered uniform times before sorting in each layer. A fixed time lies within \(\varepsilon^{.95}\) of a cell boundary with probability at most \(CJ\varepsilon^{.95}/b\). If \(k\) birth times have a neighbor at distance below \(2\varepsilon^{.9}\), their closeness graph on the line has a matching or a disjoint consecutive-pair selection of size at least a fixed multiple of \(k\). For a prescribed disjoint pair selection the probability is at most \((C_L\varepsilon^{.9})^{ck}\). There are at most \(p^{Ck}\) selections. Times in distinct layers which are this close lie near their common boundary and can be charged there. Since \(p\leq P^A\) and \(J\) is a power of \(P\), these bounds tend to zero with a positive power of \(\varepsilon\) per fixed fraction of marked times. Apply the same global Cauchy–Schwarz estimate to the fluxes. A finite union bound, with thresholds reduced by the number of categories, completes the proof. ◻

Proof. Use the augmented phase coordinates at \(t_*\) from Lemma 27 and first fix their velocities. Root each component of the forest at its least label. Replace its positions by one common translation and the edge differences along that rooted tree. The linear map has determinant one. The \(r\) relative vectors range in balls of radius \(\rho\), at cost \((C\rho^3)^r\).

Each current translation component carries the original time-dependent force partitions restricted to its labels, as well as the frozen coordinates outside their lifetimes. Begin by merging any remaining cross-component overlaps at the pivot, using endpoint offsets in \(B(0,\varepsilon)\). Next scan all geometric contacts of the extended paths, both forward and backward from \(t_*\). This includes contacts between different physical force systems and contacts of inactive frozen coordinates. Choose the first remaining contact in increasing distance \(|t-t_*|\), with a deterministic rule for ties, fix its relative translation, and merge the two translation components. Recompute the merged component’s restricted prescribed motion from the pivot in both directions. Continue until no cross-component contact remains anywhere in \([0,T]\).

The reconstruction has a useful induction. Inside the already explored two-sided time window, distinct translation components have had no geometric contact. Hence they exert no physical force, and their separately computed motions agree with the full prescribed motion. A newly chosen earliest contact lies on the boundary of an extension of that window. Merging its two components changes no motion inside the old window. This statement holds even when successive mergers occur on opposite sides of the pivot. Interactions beyond the known window are recomputed with the original force partitions after the merger. A virtual merger never adds a force.

The contact Jacobian is again (87); for a frozen coordinate its position derivative is zero. Its integral over all possible times is bounded by \(C_T\varepsilon^2\sqrt{R_1p}\). Piecewise smooth time intervals are summed as one time integral. There is no independent endpoint choice at each mesh time: extended positions are continuous, so an overlap which becomes physically relevant at a switch was either already a pivot overlap or was preceded by a geometric sphere contact. This is why all extended geometric contacts, rather than only currently interacting pairs, were included in the scan.

The triangular determinant and area-formula argument of Lemma 29 applies verbatim. At a fixed-time kink use the adjacent smooth time intervals; the endpoint itself has zero three-dimensional contact-parameter measure unless it is covered by an already counted overlap. Tangent and repeated contacts need no separate counting. Compact exhaustion avoids singular physical configurations. There are at most \(p-1\) mergers, with at most \(p^{Cp}\) choices of pairs, directions, orders, and contact/overlap types.

After the first \(r\) forest links, if the final number of components is \(k\), there have been exactly \(p-r-k\) further pins. At termination there are no remaining geometric contacts between these components. Every required contact belongs to the scanned set, so each final component contains a top and \(k\leq h_0\). Every extra pivot overlap costs \(\varepsilon^3\leq\varepsilon^2\); every other pin has the \(C_T\sqrt{R_1p}\varepsilon^2\) bound just proved.

It remains to integrate the \(k\) free translations. In a final component all of its source positions have the form \(a+d_i\) and its source velocities are independent of \(a\). Choose one source label in that component. Integrate its \(g_i(a+d_i)\) using its \(L^1\) norm and bound the other spatial envelopes by their \(L^\infty\) norms. Both norms are at most \(C M\) by (80). On the original stratum the restricted component flows equal the full flows; extending these restricted functions after discarding the stratum indicators preserves their translation property. Thus all free translations cost \(C^p\). The mixed Gaussian integrates to \(C^p\), and the speed powers cost \(p^{Cp}\). This proves the sharper stratum bound and hence (100). ◻

Proof. The proximity graph of \(m\) crowded labels has a forest of at least \(m/2\) edges: each nontrivial connected component with \(s\) vertices contributes \(s-1\geq s/2\). Summing possible forests costs at most \(p^{Cp}\). Take a grid of spacing at most \(\varepsilon\) in \([0,T]\). On \(S\leq Rp\), the displacement of an extended path to its closest grid point is at most \(C_R\sqrt p\,\varepsilon\leq\rho/2\) for small \(\varepsilon\). Thus an active-path crowding event gives the same forest with radius \(2\rho\) at a grid point, even across a lifetime boundary. The grid costs at most \(C_T\varepsilon^{-1}\).

Use the sharper form of Lemma 33. On a stratum with \(k\leq h_0\), multiplying by \(\mu^{p-h_0}=\varepsilon^{-2(p-h_0)}\) leaves the exponent \[3(.85)r+2(p-r-k)-2(p-h_0) =.55r+2(h_0-k)\geq .55r.\] Since \(r\geq\delta p/2\), this gives (101) for a fixed prescription up to a \(p^{Cp}\) factor. There are only \(O_L(p)\) nontrivial cell and retained-slot choices: empty stretches introduce no new prescription. Even the crude count \((LJ)^{Cp}p^{C_Lp}\) of all their locations and finite partitions is \(P^{C_{A,Q,L}p}\). This proves the asserted summed estimate. When \(\delta p\geq8\), the grid loss leaves an exponent at least \(.15\delta p\); the remaining logarithmic powers are absorbed by \(.05\delta p\) for small \(\varepsilon\). ◻

Proof. First remove \(S>Rp\) with Proposition 31. Apply Proposition 34 with a sufficiently small fixed crowding fraction \(\theta\), to be chosen below. On its complement, at every ordinary pin at most \(\theta p\) of the known old paths have another known path within \(\varepsilon^{.85}\). Their known states at the original first-contact time equal the corresponding states of the prescribed term before the insertion. Exclude the new label itself in this count.

In the reference measure retain the adapted gate that the known old-path crowding count is at most \(\theta p\). It contains every term under consideration. Uniform parent selection now gives conditional probability at most \(\theta\) of choosing a crowded old parent. Equation (99) bounds the integral with at least \(\delta p\) such choices by \(C_{L,R}^p2^{p/2}\theta^{\delta p/2}\). Choose \(\theta\) small enough to make this \(e^{-(M_0+1)p}\). The extensive crowding estimate with this fixed \(\theta\) is smaller than the same quantity once the history size and \(P\) are sufficiently large. This proves the claim. Neither step presupposes a good event for the exact cloud. ◻

Proof. There is exactly one added label across all expansions in the cell under consideration. Let \(\sigma\) be its original backward pin time, and \(a\) its original parent. The original first-pin condition says that on its prelaunch interval the new label is free and has no contact with any root of its own factor. The current-cell root paths of all other factors are also independent of its free velocity \(w\) and position. These paths are known after the root states at the later cell endpoint are fixed.

Write \(v=v_a(\sigma)\) and \(g=w-v\). With \(u>0\) denoting elapsed time from \(\sigma\) in the prelaunch direction, the new ray is \[x_a(\sigma)+\varepsilon n+u(v+g).\] The original parent is separated by \(\varepsilon^{.85}\) from every other already anchored path at time \(\sigma\), outside the original crowding exception. Hence, with polynomial speed bounds, a premature contact is impossible for \(u<\tau_\varepsilon\). For fixed \(\sigma,n,|g|\), the remaining angular measure is bounded by a constant times spherical area. Indeed, \(w\mapsto g\) is a translation, the Gaussian density in \(w\) is bounded above on the retained compact range, and the incoming-hemisphere restriction and bounded flux weight preserve this upper bound. Uniformity of the original angular law is not needed. The proof of Lemma 38, with time run in this prelaunch direction, bounds the contact measure against each fixed other-factor root curve by \(p^C\varepsilon^{.01}\). The union over target lives costs only a polynomial factor. An endpoint contact is included in the same estimate.

Use the product Gaussian/normal reference measure in the Cauchy–Schwarz parent comparison of Proposition 31, retaining all first-pin tests as indicators. Fix the entire ordered time vector and the discrete original parent choices before this argument. Index the remaining coordinate blocks in backward cell order, including an entire multi-addition block whenever necessary, and let \(\mathcal F_i\) contain all fixed times and choices, the top coordinates, and the first \(i\) blocks. A complete path or prelaunch segment from a previously processed block is then fixed. The current cell’s roots and the other factors’ current-cell root paths are \(\mathcal F_{i-1}\)-measurable. There is no other new label in a designated singleton cell. Its absolute velocity and normal are fresh coordinates of block \(i\).

Define \(A_i\) using only these paths and block \(i\): its original parent is separated at the pin, the paths on the tested prelaunch interval have the required polynomial speed bound, the relative velocity has the retained compact bounds, and the free new ray meets an already anchored path of another factor before its original pin. In particular we do not include any test against a future pin or condition on the eventual set of marks. The separation and exterior-speed requirements are \(\mathcal F_{i-1}\)-measurable. On an original good history the global energy bound implies these local speed requirements; outside that domain we may simply omit the history. The angular calculation gives, uniformly in all past values, \[A_i\in\mathcal F_i,\qquad \mathbb E_{\mathrm{ref}}[\mathbf1_{A_i}\mid\mathcal F_{i-1}] \le\rho_p:=p^C\varepsilon^{.01}.\] This is an adapted assertion, not a separate conditional bound with every other block held fixed. For any fixed set \(S\) of \(k\) designated indices, let \(i\) be its largest element. The product of all other designated indicators is \(\mathcal F_{i-1}\)-measurable. Conditional expectation removes \(\mathbf1_{A_i}\) at cost \(\rho_p\). Iterating gives \[\mathbb E_{\mathrm{ref}}\prod_{i\in S}\mathbf1_{A_i}\le\rho_p^k.\] All original-domain indicators not used in \(A_i\) can be dropped for this upper bound, without changing the normalized reference kernels. Finally sum over possible fixed sets \(S\); this is a union bound, not conditioning on which indices turn out to be marked. The square-flux comparison contributes \(C_L^p\) and at worst halves the positive exponent. Choices of target lives cost \(p^{Ck}\); choices of the \(k\) cells cost at most \(2^p\) when summed. This proves (103). Since \(p\) is a fixed power of \(P\), an extensive \(k\) supplies the claimed suppression. ◻

Proof. For a fixed forest shape with \(n\) internal vertices, the number of increasing numberings is \[ \frac{n!}{\prod_{v\in\mathcal T}s(v)}. \tag{104}\] Indeed, assign the smallest available number to each tree root, distribute the remaining numbers between its two child subtrees with the appropriate binomial coefficient, and repeat. The resulting factorials cancel at every internal edge and give (104); the same multinomial calculation distributes numbers between forest components. A plane binary forest with \(h_0\) roots has at most \(C^{n+h_0}\) shapes. Ordered times in an interval of length \(T\) have volume \(T^n/n!\). Dropping the layer and lifetime restrictions only increases this volume. Consequently, if \(\alpha n/8\) vertices have \(s(v)>M\), summation over their shapes and numberings has bound \[ C_T^{n+h_0} M^{-\alpha n/8}. \tag{105}\] The Cauchy–Schwarz comparison with uniform parent counting in Proposition 31 can replace the last exponent by a fixed positive fraction of it, which still allows arbitrary exponential suppression by increasing \(M\). Quantitatively, choose \(\theta\le\min\{1/4,\alpha/8\}\), so that \(n=p-h_0\ge p/2\). Let \(\zeta>0\) be the fixed exponent left by the parent comparison. Choose \(M\) so large that \[\frac{\zeta\alpha}{16}\log M \ge A+\log C_*+2,\] where \(C_*^p\) bounds the shape and fixed-layer constants in the comparison. Then the large-subtree exclusion is exponentially small uniformly for every admissible \(p\), without requiring \(h_0/p\) to tend to zero.

By Lemma 36, include the few remaining premature births among the irregular vertices. This count still uses the original forest; no global-parent multiplicity has been introduced. Irregular vertices can be cut out before this count. Alternatively order the pins inside an irregular cell consistently with attachment to the anchored pool and refine them to binary vertices. The additional orderings are bounded by the factorial in that cell’s number of labels. The multi-addition estimates in Proposition 32 absorb these factors. Cutting \(r\) vertices creates at most \(2r\) extra roots, so the shape count remains exponential. Here the irregular count concerns added insertion vertices and the old lives targeted by their primary pins, not all old lives that the added group may subsequently meet. A cell with \(m\) added labels has at most \(m\) primary attachment edges, even if its motion later touches many more old labels. Those secondary contacts remain defects on the corresponding regular candidate trees; they are not removed as \(O(m)\) contaminated participants. Choose the allowed irregular fraction only after \(M\) is fixed.

A vertex of a subtree with at most \(M\) internal vertices has at most \(M\) ancestors whose subtrees also have at most \(M\) vertices. Thus \(r\) irregular vertices contaminate at most \(Mr\) candidate launch vertices. Distinct new-particle lives have distinct birth vertices. Excluding the \(h_0\) top lives and the contaminated candidates leaves a fixed fraction of the \(\alpha p\) affected lives. Their candidate subtrees are nested or disjoint; each contains at most \(M\) candidate birth vertices. Choose an inclusion maximal candidate, discard its at most \(M\) nested candidates, and repeat. This produces a disjoint family with at least a \(1/M\) fraction of the remaining candidates. There is one additional mark needed before this greedy selection. Form the global pins only as combinatorial data on the current configuration. For each rerouted singleton birth mark its new global parent life, and for each irregular multi-pin mark the lives to which its pins attach. There are at most a constant times the number of irregular participants such marks. Count a target at its actual attachment segment; a life persists through a cut along the same continuation path, so no new mark is created merely by crossing a layer. An original bounded subtree containing a marked life is among at most \(M\) candidate ancestors of that life, with another constant to include the life itself. Remove these candidates also. For example, take \(h_0\le\alpha p/8\) and choose the irregular participant bound so that all child and target marks together eliminate at most \(\alpha p/8\) bounded candidates; a bound \(r\le c\alpha p/(M+1)\) with a sufficiently small absolute \(c\) suffices. Together with the \(\alpha p/8\) large candidates, this leaves at least \(\alpha p/2\) affected candidates. Greedy selection therefore retains at least \(\alpha p/(2(M+1))\) disjoint trees, up to integer rounding. Now an outside global pin cannot attach to a selected tree: an unchanged singleton attachment would already be its original descendant, a rerouted attachment has a marked target, and a multi-pin has a marked target. No vertex within a selected tree is rerouted. Thus both genealogies agree there. This target marking is necessary; marking only the rerouted child’s original ancestors would not exclude attachment to an unrelated hidden parent.

The marks and the selected antichain can depend on the phase coordinates. No probability estimate is obtained by conditioning on this selection. In the later exposure bound we sum over its possible labels and choices at cost \(p^{Cp}\), paid by an extensive geometric gain. Replacing a selected passage subtree by its new-child part preserves disjointness and its designated affected life. Choose the irregular fraction after \(M\), as specified above, and increase the requested accuracy of its finitely many exclusions by a fixed constant to sum them. Finally take \(p_0\ge 8(M+1)/\alpha\) to absorb integer rounding; for example the number of selected trees is then at least \(\alpha p/(4(M+1))\). All these choices are fixed before \(\varepsilon\downarrow0\). ◻

Proof. On \([a,2a]\cap[\tau,T]\), take a mesh of spacing \(\delta/(2(V+r_1))\). A hit implies that a mesh point satisfies the same inequality with \(2\delta\) in place of \(\delta\). There are at most \(C(1+a(V+r_1)/\delta)\) points. At a fixed time the allowed directions are empty or contained in a spherical cap with area at most \(C\delta^2/(r_0^2a^2)\). The mesh union therefore has measure at most \(C(V+r_1)\delta/(r_0^2a)\). Sum this geometric bound over \(a=2^k\tau\) to obtain (109).

Apply the estimate to (108), with \(\delta=C_M\varepsilon^{.96}\) enlarged to include the interaction radius, and \(\tau=\tau_\varepsilon=\varepsilon^{.95}/4\). On the earlier side of a physical launch time use elapsed time \(a=t_0-t\) and replace the spherical direction by its negative; on the prelaunch side use \(a=t-t_0\). The quotient \(\delta/\tau\) is \(O(\varepsilon^{.01})\). Polynomial speed and finite choices of descendant rays cost a factor \(p^C\). A lifetime segment tested only after \(\tau_\varepsilon\) needs no launch-separation condition: extend its independent comparison curve to \([\tau_\varepsilon,T]\) without increasing its Lipschitz constant and apply (109). The following initial-interval argument concerns only segments that are tested before \(\tau_\varepsilon\). The launch parent is separated from all other anchored paths by \(\varepsilon^{.85}\), except for the already excluded crowded-parent choices. Since \(\tau p^C=o(\varepsilon^{.85})\), no hit with these other exterior paths occurs at smaller elapsed time. For a passage tree the unchanged launch parent remains visible and is not separated from the new child at launch. Their prescribed passage is exempt. Separation from all other paths and the absence of another birth or a cut in the initial \(2\tau_\varepsilon\) interval leave that parent free there. The no-cut assertion follows from (102) and the original boundary gate; consequently no component from another cell can be installed during this interval. The nonzero relative straight ray cannot return after its prescribed passage. For later times that parent is included among the exterior curves in (109). For internal contacts, use the last common divergence and use the retained exclusion of other births within \(2\varepsilon^{.9}>2\tau_\varepsilon\) of it. The initial scattering itself is prescribed and finishes before this interval ends. The next free outgoing relative ray cannot return to its partner without a later change of velocity. A union over the at most \(M^2\) relevant divergence/ray pairs completes the proof. ◻

Proof. The unconditioned charts. Fix the cells, force partitions, lifetimes, and labels of the translation pins. Inactive labels may be extended freely. The augmented map from independent input coordinates to the current/departed coordinates preserves volume by Lemma 27. At a cell, keep the end coordinates of all its roots anchored and decompose the added coordinates into velocities, internal relative positions, and one common translation for each floating group. Grow the groups backward using the common first-contact convention. Until a merger, each group’s motion is independent of the other group’s common translation. If the contact is at \(t\) between labels \(i\) and \(j\), its relative translation is \[ r=\varepsilon n+X_i(t)-X_j(t),\qquad |\det D_{(t,n)}r| =\varepsilon^2|n\cdot(V_i(t)-V_j(t))|. \tag{112}\] Internal coordinates in this formula are held fixed. Once a relative translation is fixed, regard its groups as a single coordinate group, retaining the prescribed forces, including absence of force across a virtual contact. Ordering the internal merger edges before their enclosing edges makes the full change of variables triangular. Its Jacobian is the product of (112) and the endpoint-overlap volume factors. No derivative of a many-particle scattering map is introduced.

Earliest-contact and no-prior-hit tests are kept as indicators. Fix a deterministic ordering of candidate contact pairs and resolve simultaneous first contacts by that ordering; retain the resulting strata rather than asserting that all ties are null. The area formula applied to the translation map in (112) gives zero spatial-translation measure to its nontransverse images. Endpoint overlaps are represented separately by their volume pins. For each discrete merger forest and tie stratum the inverse reconstruction is unique outside the nontransverse images: the pin data recover its relative translations, and the prescribed flows recover the coordinates. There are at most \((pJ)^{Cp}\) merger/partner choices. Singleton cells have one ordinary parent choice, in the original coordinates. In the global coordinates there can be additional choices even for an irregular reroute; they are included in the present \((pJ)^{Cp}\) bound. This coarse multiplicity is used only in the present geometric-gain estimate. The factors \(\varepsilon^2\) cancel the activity normalization. Multi-pin internal volumes and endpoint factors retain the bounds of Lemma 29.

For regular singleton vertices belonging to the selected trees make (106), followed by (107). Keep the absolute terminal velocities of every other floating component among the known coordinates \(\xi\); do not replace them by relative velocities depending on a future anchor. The parent state is known before that vertex, so these substitutions remain triangular across cells. Partition the source positions into summability boxes. The energy bound restricts all velocities and displacements to polynomial ranges; source boxes contribute the weights \(\prod_i a_{i,k_i}\) of (110). Every integrated top position is within a polynomial displacement of its own source box, giving a polynomial volume per top. The sum of the source-box weights is exactly \(\mathcal A_p\). On each box choice, bound every remaining flux and compact radial density by a polynomial in \(p\), and enlarge coordinate domains to product ranges. Let \(d|\mathfrak I_{\boldsymbol k}|\) be the absolute chart measure of the activity-normalized integral of Section [sec:history-estimates], restricted to these source boxes. It has the domination \[ d|\mathfrak I_{\boldsymbol k}|\le \Bigl(\prod_i a_{i,k_i}\Bigr)(pJ)^{Cp}\,d\nu_0(\xi) \prod_{\ell=1}^k d\nu_\ell(\theta_\ell). \tag{113}\] The reference measures are finite product measures. Put the entire ordered global-chart time vector in \(\xi\), including all selected launch times, internal birth times, removals, and lifetime cuts. It is fixed before exposure: a visible parent must be removed at a known time, even while the output velocities of its genuine launch remain hidden. The block \(\theta_\ell\) contains the relative velocities, radial impact parameters, and rotations in tree \(\ell\). The variable \(\xi\) contains all other chart coordinates and the fixed genuine/passage types. At a rerouted vertex, the original required contact time is derived from the reconstructed flow and retained only through its indicator; it is not an additional independent time. Original and global pin times coincide on the selected trees. The tube estimate is uniform in the fixed time vector, including all retained spacing restrictions. Normalize their polynomially bounded total masses and absorb them in \((pJ)^{Cp}\). Equation (113) is applied to the entire joint integral. We never divide by a conditional mass of an admissible sector. In particular, a rare sector is not asserted to have bounded normalized conditional density.

Visible, dormant, and private states. Fix a list specifying the order of exposures, the one or two selected trees revealed at each exposure, and the partner lives or internal divergences and cell/ray segments involved. There are at most \((pJ)^{Cp}\) such discrete choices. Each selected tree is revealed once. The list does not fix the exposure times; these will be recovered successively from the blocks it reveals. We first describe the state and its updates between exposures.

Within a cell the state has three parts: anchored visible labels with known phase points; selected trees that have already launched and remain hidden; and components of added labels that have not yet merged with any rooted component. Call the last components dormant. Every dormant component is omitted from the bath, whether or not it contains a selected label. In particular, a nonselected singleton or a multi-label addition is not represented in the bath by an absolute prelaunch path whose later anchor has yet to be determined.

For a dormant component \(C\) with known parameters, choose one terminal position as origin. Its terminal velocities, internal endpoint offsets, and already specified internal merger pins determine relative paths \(q_i^C(t)\) and velocities \(v_i^C(t)\), up to one common translation. Use the original force partition restricted to \(C\) to compute these paths; a virtual internal merger changes the coordinate component, not the force partition. These are functions of \(\xi\) and already revealed blocks only. A selected regular addition is a singleton in its cell. Its as yet unknown relative velocity is kept in its hidden block and is used only when its known parent state is available at launch. Internal additions on an already hidden selected tree are handled privately in that tree, using its block. They are not part of the visible bath. Its private paths follow the scheduled isolated binary scatterings and free passages, with all later relative parameters fixed in laboratory axes. On the enlarged product domain these paths continue through unlisted hypothetical contacts without adding forces. They therefore retain the affine representation (108). Agreement with the prescribed whole-cell dynamics is required only up to the tree’s first defect on the retained domain.

Read the fixed global merger times in backward order. If two dormant components merge, the contact normal and time fix their relative translation by (112). Install that relative translation inside their union and continue the restricted dynamics of the union, retaining one free common translation. It remains dormant. No absolute position or force from it is used in the visible evolution. Its internal mergers precede its first rooted merger in this construction.

Suppose next that \(C\) first merges with the rooted pool at fixed time \(t\), through its label \(i\) and a visible label \(j\). With the normal oriented from \(j\) to \(i\), set \[ a_C=x_j(t)+\varepsilon n-q_i^C(t). \tag{114}\] Install every label \(\ell\in C\) at \((a_C+q_\ell^C(t),v_\ell^C(t))\) and thereafter evolve the enlarged rooted system with its original prescribed forces. The common translation uses only the visible anchor and the known relative state. Endpoint rooting uses the corresponding overlap displacement in place of \(\varepsilon n\). For a selected launch this rule first computes its parent point and velocity independently of its hidden block. A genuine launch then hides both outputs, including the continuing parent; a passage launch hides only the new child. No velocity of an unrevealed selected output is needed to continue the remaining visible system.

The selection in Lemma 37 rules out an outside primary pin whose first rooted target belongs to a selected hidden tree. This includes rerouted singleton targets, multi-pin targets, and endpoint targets. An unchanged singleton pin into a hidden branch is its original descendant and uses that tree’s private block. The launch antichain therefore ensures that the parent of a different selected tree is visible when that tree launches.

At a listed exposure, reveal the indicated block or pair of blocks, install the current private states, and resume the original prescribed force systems. A virtual contact reveals the block without changing a force. An internal defect reveals its tree and resumes its prescribed internal system. Future additions whose data have just been revealed remain dormant until their own merger times; revealing a block does not insert all its future labels immediately. No coordinate change at the random exposure time is made. The independent variables remain the global whole-cell coordinates in (113).

At a cell boundary carry the visible root states to the next cell, retain the private states of continuing hidden lines, and initialize that cell’s new additions as dormant. A line that terminates is removed according to its prescribed lifetime. These updates use the global time vector already fixed in \(\xi\), including the removal time of a parent hidden by a genuine launch.

Recovering the listed exposure times. Suppose the first \(m-1\) entries have been replayed. The visible evolution now uses only \(\xi\) and those revealed blocks. To replay entry \(m\), supply the one or two blocks named in that entry and construct their private paths from their launch states. Starting after the preceding replayed exposure, find the first eligible contact of the listed type on its listed segments, in backward chronological order with a fixed order for ties. At that time make the revelation update just described. If there is no such contact, the list contributes zero. The same tie order permits successive entries at the same physical time. In this search we neither compare with a contact involving another still-hidden block nor test that such a block has had no earlier hit. These restrictions from the original first-exposure event are dropped.

The replay retains a revealed tree’s private path before revelation and its resumed path afterward. They join at the installed phase point. Thus a later test may use the whole past of a previously revealed life, even when that life was hidden at the later-tested tree’s launch. Both this past and the replayed revelation time depend only on the previously revealed blocks. In particular, revelation is not a new spatial placement.

Definition on the product domain and agreement. The enlarged product ranges need not satisfy the original energy and isolation tests. Choose a polynomial speed limit \(V_p\) strictly larger than the actual speed bound on \(\mathcal K\le Rp\). Stop each visible simulation at its first violation of that limit or at failure of a validity condition for its known finite flow. Apply the same known-data rule to a dormant relative flow needed for an installation. Keep stopped positions constant afterward, with zero velocity for any subsequent launch. If a prescribed visible anchor is unavailable, the list contributes zero and its replay ends. These rules use only visible and already revealed data. In particular, neither future contact compatibility nor a dormant or hidden no-prior-hit test is used to continue the simulation. Its visible portions are \(V_p\)-Lipschitz for every value of the product coordinates.

This exterior stopping rule is not a bound obtained by inspecting a hidden tree. Its private scheduled paths instead inherit the known launch-parent velocity and add at most \(M\) compactly bounded relative increments and isolated scattering arcs. Their speeds therefore have a polynomial bound, uniform in the hidden rotations. The common Galilean drift cancels in the arc-to-ray error, leaving the bound \(C_M\varepsilon^{.96}\) in (108). Unlisted contacts do not alter those paths. Their private pasts and the resumed visible portions thus give polynomially Lipschitz exterior curves. The affine representation and tube estimate remain available on the enlarged domain, without asserting an energy inequality for its artificial trajectories.

We verify agreement on the retained chart stratum with the specified actual exposure list. Before a component’s first rooted merger it has no contact with an anchored path. An earlier contact with a selected hidden path would itself be an earlier rooted merger, and its outside hidden target would have been excluded. Omitting a dormant component therefore changes no rooted motion before its scheduled installation. This remains true within an original force group, since finite range makes the cross force zero before contact. A dormant–dormant merger recovers the relative states up to one common translation, and (114) recovers that translation at a rooted merger.

Before the next actual first defect, the hidden private continuations are their original isolated binary paths. Thus the next listed contact found by the replay occurs at its actual time and installs its actual phase points. Uniqueness then gives agreement of the resumed visible flow. Induction over the list proves agreement at every revelation; the energy bound ensures that the visible stop is not reached before any tested exposure. No bound on the number of internal bath collisions is used. Connectivity roots every added component by the earlier cell boundary on this stratum. A hidden–visible contact at a cut is an exposure or an excluded cut irregularity, and the intact-polymer condition prevents a hidden force from being transferred silently through a cut. Hence the same induction passes through all cells and macro cuts.

We have defined curves on the entire product domain and proved that they contain every hit event from the retained stratum. Outside that stratum they need only have the stated dependence on revealed blocks and the uniform speed bounds. No hidden coordinate has been conditioned on or renormalized in this construction.

First-hit blocks. A hit of two hidden trees reveals them both and uses one estimate; every other exposure reveals one tree. Thus the fixed list has at least \(k/2\) entries. Before its next test the bath and launch anchors depend only on \(\xi\) and previously revealed blocks. Fixing the nonangular parameters of the next block gives (108) against an independent exterior.

For a one-tree test of \(B\), the initial-interval hypothesis of Lemma 38 also needs verification after earlier exposures. Fix \(\xi\) and the previously revealed blocks. Their replayed paths, including the private pasts retained at their revelations, use no parameter of \(B\). Before its own launch the continuing parent of \(B\) remains visible. On the retained domain these curves agree with the original paths; on the enlarged domain their speed bounds were supplied above.

The launch cell of \(B\) has exactly one addition across all factors, and its boundary gate excludes a cut in the initial \(2\tau_\varepsilon\) interval. Thus every exterior life tested before elapsed time \(\tau_\varepsilon\) was already displayed at \(B\)’s launch; an earlier tree’s revelation during this interval merely makes its independent private past available. Extend the tested life back to that launch using the paths just constructed. Except for the unchanged parent of a passage, retain the gate that this extension starts farther than \(\varepsilon^{.85}\) from \(B\)’s launch parent, and declare the enlarged early test empty if the gate fails. The gate depends only on \(\xi\) and previously revealed blocks, and it holds on the selected good domain by separated original parents and the agreement induction. The initial offset is \(O(\varepsilon)\) and \(V_p\tau_\varepsilon=o(\varepsilon^{.85})\), so the gate rules out an early hit.

For the unchanged passage parent, first consider the retained domain. At launch it is separated from every other old displayed path by \(\varepsilon^{.85}\). The original polynomial speed bound and the absence of another birth or cut in the initial \(2\tau_\varepsilon\) interval preclude a third-path force there. After the prescribed passage the nonzero free relative ray cannot return to that parent. Thus an original first defect against the unchanged parent cannot occur at elapsed time below \(\tau_\varepsilon\). On the product domain we consequently make this parent test only at elapsed times at least \(\tau_\varepsilon\); no freeness of its artificial stopped path is asserted. The tube estimate then applies without an initial-interval gate.

Exterior segments born later are tested only after \(\tau_\varepsilon\); the same globally singleton cell and boundary gate exclude an earlier installation. For all tests at elapsed times at least \(\tau_\varepsilon\), apply the tube bound without a launch-separation gate. These extensions and gates are independent of \(B\)’s tested rotation.

For a two-tree test, call \(A\) the earlier-launched tree and \(B\) the later-launched tree in backward chronology; the retained spacing excludes equal launch times. Fix all of \(A\)’s block first and integrate \(B\)’s launch rotation. Both launch anchors were computed without either hidden block. Retain the gate that \(B\)’s launch parent is at least \(\rho=\varepsilon^{.85}\) from the already anchored paths of \(A\) at that launch. The gate depends on \(A\)’s fixed block and the visible anchor of \(B\), not on \(B\)’s launch rotation. If it fails, define this enlarged pair test to be empty. On the original selected good domain it holds by parent separation. During elapsed time \(u<\tau=\tau_\varepsilon\) after \(B\)’s launch, the initial offsets and displacements are bounded by \(C\varepsilon+Cp^C\tau=o(\rho)\). The retained time spacing and boundary gate leave no other birth or cut in this interval. Thus no \(A\)–\(B\) hit is possible there; \(B\)’s own prescribed pair is not an \(A\) path. For \(u\ge\tau\), apply Lemma 38 to \(B\) against the fixed paths of \(A\). The bound is uniform in \(A\)’s block, so its subsequent integration preserves it. Internal defects use their divergence parameters as in that lemma. The uniform bound for each one- or two-tree test is \(\rho_p=p^C\varepsilon^c\).

For a fixed list, let \(B_m\) denote the one- or two-tree block revealed by its \(m\)th entry, and let \(A_m\) require that its replay finds the listed contact and satisfies the gates used in the corresponding test. If the replay has already failed, set \(A_m=\varnothing\). The construction and the uniform one-test bound give \[A_m=A_m(\xi,\theta_{B_1},\ldots,\theta_{B_m}), \qquad \int \mathbf1_{A_m}\,d\nu_{B_m}\le\rho_p,\] The blocks \(B_m\) are disjoint, and the retained event for this list is contained in \(\bigcap_m A_m\). The preceding no-hit requirements on unexposed blocks have been dropped, without renormalizing their measures. Integrate in reverse block order in (113). Each last remaining restriction gives the factor \(\rho_p\), while all earlier restrictions are independent of that block. The resulting factor is \(\rho_p^{\#\{B_m\}}\). Summation over the guessed lists and the source-box weights, whose sum is \(\mathcal A_p\), proves (111). ◻

Proof. We divide by the explicit source factor \(\mathcal A_p\); if that factor is zero the integral is zero. First make the high-energy, multi-addition, endpoint, noncompact, long scattering, close-time, and crowded-parent exclusions of Propositions 32 and 34. Apply Lemma 36 to premature contacts in the original coordinates. Next choose \(M\) and the irregular fraction in Lemma 37. A positive fraction of the affected lives then yields \(k\ge c_{\alpha,M}p\) regular disjoint trees. Choose \(\theta\) and \(p_0\) from Lemma 37, decreasing \(\theta\) if the finite-participant transfer below requires a smaller affected fraction. Every exclusion is requested at a sufficiently increased fixed accuracy so that their sum is at most \(e^{-Ap}\). Proposition 39 gives (115). Since \(p\) and \(J\) are fixed powers of \(P\), their logarithms are \(O(\log P)=o(P)\), and hence the expression in (115) is smaller than \(e^{-Ap}\) for all sufficiently small \(\varepsilon\), uniformly in the indicated range of \(p\): its logarithm per label is at most \(C_L\log P-c_L|\log\varepsilon|\). For a fixed number \(s\) of top roots, this applies to every \(p\ge m\) once \[m\ge\max\{p_0,\lceil s/\theta\rceil\}.\] Thus it supplies a uniform tail in \(p\), rather than a statement requiring an \(\varepsilon\)-dependent limit of \(s/p\).

The affected lives used by the algebraic centering argument transfer to this geometric count with a fixed finite-participant loss. An unchanged algebraic extra contact marks its participating geometric lives. If a birth is rerouted to an earlier virtual global pin, mark its new child and the old and new parents, at most three lives. The later original required contact is then a geometric defect of that child unless it is an already charged irregularity. A rerouted genuine contact cannot disappear by calling it a different primary birth: each added label has exactly one primary geometric pin. This accounts for all discrepancies between the two forests. The charging statement of Proposition 25 may therefore be used with a reduced positive fraction.

No estimate asserts that an already exposed bath has few collisions or no bound cluster. Such a bath is allowed in Lemma 38 through its speed bound. Its first influence on each fresh regular tree costs a geometric factor; primary births into a small crowded bath are included in the parent-choice exclusions. This is why the argument treats interference with clusters without deleting clusters from the actual dynamics. ◻

Proof. Work throughout in the original charts, without the coarse global-chart multiplicity. The ordinary parent/time bound of Proposition 31 gives a finite constant depending on \(p_*\) and \(T\) after summing cell locations. Empty cells introduce no extra choice. Its Gaussian bound allows velocity truncation at a fixed large constant, with uniformly small error. The summability-box decomposition may be retained throughout; its weights sum to (110), so no spatial truncation depending on the observation time is required.

Fix next \(\eta>0\). Delete births whose time is within \(\eta\) of another birth and parameters with incoming relative speed or a nonzero output increment smaller than \(\eta\). The time error tends to zero with \(\eta\) by the finite-dimensional simplex measure. The parameter error tends to zero by the flux-null statement of Proposition 8 and continuity from above of finite measure. There is no claim of an algebraic rate in \(\eta\). A scattering lasting more than \(\varepsilon^{.96}\) has vanishing flux measure. Births within that distance of a cell endpoint have total time measure at most \(C_{p_*}J\varepsilon^{.96}\to0\). The multi-addition bounds give a vanishing factor in a cell containing at least two additions, while an overlap pin of an added label costs \(\mu\varepsilon^3=\varepsilon\) in place of a normalized flux. A close pair of already integrated top roots instead costs an unnormalized spatial volume \(O(\varepsilon^3)\). Endpoint polymers are handled by a spatial spanning forest of these pins. Thus no isolation of a top array has been assumed in this deletion.

First test additional force-bearing encounters. Before the first such encounter every factor is a finite collection of isolated binary trees, with completed scattering arcs replaced, at a position error \(\delta_\varepsilon=C_{p_*,\eta}\varepsilon^{.96}\), by their elastic broken rays. Purely virtual contacts do not change this reconstruction and are not excluded at this stage. The exact prescribed dynamics after the first force-bearing event is unrestricted. Only its volume and Gaussian majorants are used. There are finitely many ray pairs, divergence vertices, and birth/cut incidences to test.

Consider first a force-bearing contact between two families issuing from different top roots. Keep the internal collision data of both families fixed and float one family by a common spatial translation \(r\). Before their first contact, their motions are independent and translation equivariant. The relative position is \(r+G(t)\) with \(G\) uniformly Lipschitz on the compact parameter set. The translations producing a hit within radius \(\delta\) have volume at most \[ C\bigl(\delta^3+T\,\operatorname{Lip}(G)\delta^2\bigr). \tag{116}\] Indeed, sample the time interval at spacing \(\delta/(1+\operatorname{Lip}(G))\) and cover each sampled point by a ball of radius \(2\delta\). Formula (116) is uniform in the locations of the source boxes. Integrating their envelope weights proves that root–root and cross-family contacts have vanishing integral. The same calculation controls two top roots already close at an endpoint. This argument uses the integrated top translations; it asserts no pointwise smallness at fixed top positions.

For two rays within one family, use their last common divergence. Its nonzero relative output direction is spherical, even though the two individual output velocities can be correlated. Fix all relative data below that divergence in laboratory axes. Their relative position then has the form used in Lemma 38, with elapsed time at least \(\eta\) whenever another descendant has been born. Before that time the prescribed pair is a straight outgoing relative ray and cannot return. Taking \(\delta\) larger than \(\varepsilon+2\delta_\varepsilon\) gives a vanishing bound. This proves smallness of internal force-bearing recollisions without restricting what happens after the first one. Remove these vanishing physical-defect sets. The complete remaining factor paths are then their prescribed binary paths and free virtual rays throughout their lifetimes. We may now test all virtual contacts on these complete paths; they introduce no new dynamical dependence. Cross-family virtual contacts obey (116), and internal virtual contacts obey the same divergence-direction argument.

The same two tests control a parent being within \(\varepsilon^{.85}\) of a third path at a scheduled birth: use the spatial translation for different top families and the divergence direction for one family. The prescribed new child is exempt; all other divergences are separated in time by the retained \(\eta\). Hence the near-parent exclusions needed for finite singleton geometry have vanishing measure without invoking the extensive crowding estimate. For a premature cross-factor contact of an added particle’s full-cell prelaunch ray, use its fresh incoming direction in the original chart, exactly as in Lemma 36; the comparison is with already anchored paths, and no global reparenting is necessary. The unchanged parent of a passage is treated as in the proof of Lemma 38. Thus full-cell ghost segments are included in the finite event list.

These estimates exhaust a first extra contact or interference of the retained binary histories. They also cover the short displacement and cut errors, after the already removed duration and endpoint-time sets. For fixed \(p_*\), the number of combinatorial cases is finite and the time sums are the original simplex integrals. Take \(\varepsilon\downarrow0\) with the velocity and \(\eta\) cutoffs fixed, then let \(\eta\downarrow0\) and the velocity cutoff tend to infinity. The resulting upper bounds depend on the observation time only through its upper bound \(T\); shortened cells only reduce their integration domains. This proves uniformity. Finally, replacing one source norm in the finite majorant by a vanishing norm gives the last assertion directly. ◻

Proof. The coefficient of \(z^{H'}\) in \(\prod_j(1+z)^{q_j}=(1+z)^Q\) is the sum of the products on the left of (117) over all allocations of \(H'\). The first inequality follows. The second follows from \(\binom Qk\leq(eQ/k)^k\). A label contributes an available slot at its first input cut, and an additional available slot each time it is retained across another cut. This gives the asserted bound on \(Q\). A bounded number of choices per retained slot contributes \(\exp(C H)\). Bounded choices at first-input slots instead cost \(C^p\); these belong to the baseline constant in the history estimate, not to its small retained-slot entropy. The last two assertions follow from the displayed inequalities, using \(\binom Qk\leq2^Q\) in the general case. ◻

Proof. Use the source sum and augmented squared-speed budget of Lemma 27: \[S=\sum_{\text{independent inputs }a}|v_a|^2, \qquad \mathcal E=S+2Bp.\] By Lemma 27, every active or mixed sum of squared speeds is at most \(\mathcal E\). The same assertion bounds the sum of squares of the velocities of the independently added free rays. The source sum \(S\) is not replaced by terminal kinetic energy: a departing interacting component may have positive potential energy. Retaining its full Hamiltonian gives instead the Gaussian comparison \[ e^{-\beta _0S} \leq e^{\beta _0Bp} e^{-(\beta _0/2)\sum_a|V_a|^2} e^{-(\beta _0/2)S}, \tag{120}\] where \(V\) is the vector of mixed terminal velocities. Further splitting into a fixed number of portions is allowed. All these exponents are fixed independently of the layer subdivision.

The source energy depends on the parent prescription. We therefore first partition its domain into shells \[rp\leq S<(r+1)p,\qquad r=0,1,\ldots.\] On shell \(r\) use the deterministic upper budget \(E_r=(r+1+2B)p\). Parent choices that cannot belong to this shell are killed. The parent-sum bound of Proposition 31 then applies with this common budget; it does not require moving a prescription-dependent energy through a sum over parents. At a birth in layer \(j\), if \(u\leq q_j\) paths are present, its summed flux is bounded by \[C u\left(1+|w|+\sqrt{E_r/u}\right) \leq Cq_j\left(1+|w|+\sqrt{E_r/q_j}\right).\] Let \(m\) count multiple-insertion participants and endpoint pins, and let \(\mathcal O\) be the ordinary births. The nonordinary shell-speed factor in Proposition 31 is \((r+2)^{m/2}\). Pad the speed product with \(h_0\) unit factors. AM–GM and \(\sum_jn_j/q_j\leq L\) give \[\begin{align*} &(r+2)^{m/2} \prod_{i\in\mathcal O} \left(1+|w_i|^2+\frac{E_r}{q_{j(i)}}\right)^{1/2} \\ &\hspace{12mm}\leq \left(C L(r+2)+\frac{|V|^2}{p}\right)^{p/2}. \tag{121}\end{align*}\] Indeed the sum of the \(p\) squared factors is at most \(p+m(r+1)+|V|^2+LE_r\). This form of the estimate remains valid after the parent majorization has enlarged the velocity integral to the full product Gaussian; it does not incorrectly retain a source-shell constraint on that enlarged integration domain.

For a fixed product Gaussian \(\gamma\), \[\int\left(1+\frac{|V|^2}{p}\right)^{p/2}\,d\gamma(V) \leq C^p.\] To check this, reserve half of its Gaussian exponent and bound \(e^{-c u}(1+u/p)^{p/2}\) by \(C^p\) for \(u\geq0\); the remaining Gaussian integrates with an exponential-in-\(p\) normalization. Applying this to (121) gives \([C\sqrt{L(r+2)}]^p\). A fixed portion of the source-energy factor in (120) gives \(e^{-crp}\). Thus the sum of the shell bounds is at most \[ C^p\sum_{r\geq0}e^{-crp} \bigl(C L(r+2)\bigr)^{p/2} \leq (C\sqrt L)^p. \tag{122}\] For the last inequality, bound \(e^{-cr/2}\sqrt{1+C(r+1)}\) uniformly in \(r\) and sum the remaining geometric series. Reserve an additional fixed Gaussian portion before this calculation if spatial factors are present.

In each layer the ordered-time volume is \(b^{n_j}/n_j!\). Moreover, \[\frac{q_j^{n_j}}{n_j!}\leq e^{n_j+h_j},\] because \(n!\geq(n/e)^n\) and \(n\log(1+h/n)\leq h\). Hence the parent and time product is at most \(b^n e^{n+H}\). Combining this with (122), and using \(n=p-h_0\), proves (119). ◻

Proof. First choose \(L\) so that \(b=T/L\) is within the short interval of Proposition 22, is at most one, and satisfies \[C T/\sqrt L<e^{-4a-20}.\] Here \(C\) is the constant in Lemma 43, including the independent source-envelope constants. Its independence of \(L\) is essential. Next choose \(\delta>0\) small enough that (118), (125), and the factor \(e^H\) together cost at most \(e^{ap/4}\) whenever \(H\leq\delta p\). Choose \(\eta_L\) still smaller so that the upper-root factor in (119) and any upper-root localization also have logarithm at most \(ap/4\) for \(h_0\leq\eta_Lp\). The finite occupancy factors in (123) are absorbed by increasing \(p_L\). The histories with few crossings, \(H\leq\delta p\), then obey (126) with room to spare.

For the complementary histories apply Proposition 25, after reducing \(\eta_L\) to at most \(\delta/(2L)\). If \(d\) is the number of distinct marked lives, \(r\) the number of residual factors, and \(i\) the number of retained initial singleton inputs, then \[d+r+i\geq\theta_Lp,\qquad \theta_L=\frac{\delta}{2L^2}>0.\] The fixed-factor transfer between algebraic families and the geometric pin forest in Remark 26 can be absorbed by reducing \(\theta_L\). A new geometric first pin is still a single birth. If it reroutes a birth to another displayed family, the original required later encounter is a defect; it is not counted as another birth.

The unrestricted finite-term bound is \(C_L^p\) by Proposition 31. If at least \(\theta_Lp/3\) charges are residual or initial factors, their additional weight is at most \[\gamma_\varepsilon^{\theta_Lp/3},\qquad \gamma_\varepsilon= \max\{C\varepsilon^c,C\eta_\varepsilon\} \longrightarrow0,\] where \(\eta_\varepsilon\) is the supremum of the fixed summed spatial Gaussian residual norms in Proposition 22, taking the maximum over the finitely many choices \(K_j\). Its common exponent is not weakened through the layers. This contribution beats any fixed \(C_L^p\) once \(\varepsilon\) is small. No rate for the residual is needed.

Otherwise an extensive number of lives is marked. The tunable parameter, tree, and multiple-insertion discards of Proposition 32 and Lemma 37, followed by Proposition 40, give a bound of the form \[ C_L^p e^{-A_1p} +P^{C_Lp}\varepsilon^{c_Lp}. \tag{127}\] The desired \(A_1\) is chosen after \(L\) and \(\theta_L\). Choose it larger than all the fixed exponential costs plus \(4a+20\). Reduce \(\eta_L\) to the top-root fraction supplied by Proposition 40 for affected fraction \(\theta_L/3\) and this accuracy, and increase \(p_L\) to its fixed size threshold. These choices leave the preceding few-crossing estimates valid. The second term is also at most \(e^{-(4a+20)p}\) for small \(\varepsilon\), because \(p\leq P^A\) and \(\log P=o(|\log\varepsilon|)\). The expensive exposure guesses occur only in this geometric-gain term, not in the term controlled solely by residual smallness. This proves (126) in the many-crossing case.

The argument used only the finite-history estimates and the finite centered identities. An admissible mixed upper block is allowed because the charge estimate imposes no condition on its \(h_0\) upper roots. Signed completed cells, reference families, and a positive actual prefix are all admissible finite terms; taking their absolute weights does not change their force rules or contact constraints. The upper physical/reference choices cost only \(2^{h_0}\), with bounded alternatives absorbed in \(C^p\). The original parent determines a birth’s factor, so these choices introduce no extra parent power. Ordinary births across all simultaneous factors retain the ordered-time bound; multiple insertions have the mesh factors already estimated. Below this one upper layer every retained input is an ordinary centered slot, so Proposition 25 applies with precisely the same allowance \(Lh_0\). The source, spatial, and parent bounds, and hence both crossing cases, remain unchanged.

Finally fix the total number of labels. The path argument in Proposition 25 makes every surviving centered upper root lead to an initial small factor, a vanishing residual, or a marked finite history. For the last case, first restrict velocities and scattering parameters to compact regular sets. Corollary 41 makes its integral tend to zero. The complement has arbitrarily small integral by the Gaussian and flux bounds, so the compact restriction may then be removed. This finite-size argument does not mistake a fixed small-parameter cutoff for a quantity tending to zero. Ordered time domination and shortened cells make the argument uniform in the endpoint time. ◻

Proof. Throughout this proof the threshold is the same number \(D\). Take the first violating cell in the preceding prefix. There cannot be a previously unaccounted giant polymer at its start. A giant polymer at a layer boundary was contained in the preceding actual cell component, and hence would have violated the same threshold earlier. Initially the exclusion makes all polymers singletons.

Before the violating cell, every current polymer has an actual ancestry of size at most \(D\). Regard these polymers as atoms, with their ancestry sets as weights. In a violating trial system follow the increasing contact-component graph until a union of weights first exceeds \(D\). Before this merger all participating components have weight at most \(D\). At a binary merger their union has size at most \(2D\). If several contacts occur together, select connected pre-contact components one at a time, stopping when the union first exceeds \(D\). The last added component has weight at most \(D\), so the same bound holds. Thus ties require no probabilistic convention. Before the selected merger these components have had no contacts with the other trial components; their trajectories up to that time are consequently their own isolated trajectories.

Let \(U\) be the union of the selected current polymers and let \(W\) be the union of their actual ancestries. Then \[D<|W|\leq2D,\qquad U\subseteq W.\] Trace the members of \(U\) backwards through their actual whole-cell components to the beginning of layer \(i\). The union of these histories has exactly the required ancestry labels in \(W\). An ancestor not in \(U\) terminates at its prescribed earlier cut. In particular, it is not kept artificially active in the last trial cell. The last cell runs only the isolated flow on \(U\). At every such removal a whole close component departs; otherwise a contact across that cut would have included the departing label in the next required polymer.

Extend this last isolated trial flow to the fixed end of its cell. It already has the required connected-contact event before the selected merger, even if its subsequent path differs from the original larger trial system. All coordinate changes can therefore be made at fixed times; no stopping-time Jacobian is introduced. Choose a label \(a\in U\) as the upper distinguished label. The trial contact graph connects \(U\) to this label, and the actual ancestry graphs connect all remaining labels in \(W\) to \(U\) backwards, cell by cell. All of \(U\) terminates at this final endpoint. Its distinguished label need not be isolated from \(U\setminus\{a\}\) there; in particular, this construction does not exclude a cluster persisting to the end of the cell. Keep the full terminal Hamiltonian before applying stability, and integrate all terminal coordinates. Connectivity supplies \(p-1\) translation pins, with one spatial anchor and hence \(h_0=1\); endpoint close components use the endpoint pins of Lemma 29. This gives the asserted positive history. Keeping the cruder union of at most \(2D\) histories of size at most \(D\) would give the stated safe bound \(2D^2\). ◻

Proof. For a fixed term, product Hamiltonian transport preserves phase volume, marginalization decreases total variation, and every contact or isolation indicator has absolute value at most one. Tensoring with an independent input has bounded integrated norm. The factorial normalization costs at most \(\mu^M\). With at most \(M\) labels, the finite subset, generation, cut, and parent descriptions cost at most \((C_L M J)^{C_LM}\). This proves (129), since \(\log M+\log J=O(\log P)=o(P)\). The centering and inverse transforms have norm at most \(2^M\) by (61); this factor is absorbed by (129). Pattern summation introduces no additional partition count, by Corollary 21.

For (130), first work on an \(N\)-particle cloud. The exact root observable is bounded by its root-tuple count. Every completed below-\(K\) term involves fewer than \(K\) labels. Its absolute value, summed over their choices, is thus bounded by \((C_LKJ)^{C_LK}(1+N)^K\), times normalization factors no larger than \(\mu^K\). No bound for the number of terms with an unbounded last component is used here: we are bounding the exact observable minus the completed small terms on \(\mathcal A\).

The initial factorial bound gives \(\mathbb E(N)_\ell\leq\mu^\ell\), and \(N\) is conserved. Expanding ordinary moments in factorial moments therefore gives, for integer \(r\geq1\), \[\mathbb E(1+N)^r\leq \bigl(Cr(1+\mu)\bigr)^r.\] Cauchy–Schwarz with \(r=2K\), followed by \(\log\mu=2|\log\varepsilon|\), proves (130). These finite polynomial moments also justify truncating first to \(N\leq N_0\) and then taking \(N_0\to\infty\) in all the capped identities. The original unconditioned ensemble is used throughout; no truncated ensemble is recentered in place of it. ◻

Proof. The finite-cloud stopping and witness constructions are performed first. Integrate their finite identities with \(N\leq N_0\) and then let \(N_0\to\infty\), using Lemma 47 and the finite expectations in Proposition 16. Only after this passage do we apply the centered identities to the original ensemble and its initial insertion formula. In particular, no centering bound for a law conditioned on \(N\leq N_0\) is asserted.

We induct on \(j\) in chronological order, with \(R_k\) already controlling both remainder types whenever \(k<j\). First estimate the finite reference cutoff mismatch. By (75), every term of \(B_j\) has \[K_j\leq p<(U_j+1)K_j,\qquad h_0\leq U_j.\] This cap is below \(W_j\). Treat this finite product of signed physical terms, reference families, and residual upper slots as an uncentered mixed upper block. Expand its true input \(E(t_{j-1})\) by (76) only through the strictly earlier layers \(j-1,\ldots,1\). Apply the usual simple-family cancellations at those lower cuts. Every completed history still contains at least \(K_j\) labels, and its starting top count is at most \(U_j=o(K_j)\). Therefore Proposition 44 bounds their total norm by \(C_L e^{-32K_j}\). Every interruption in a layer \(k<j\) contains the complete centered remainder \(\mathsf Z_k\) from (134). Its upper block fits \(U_k\) by (132), so Lemma 47 bounds it by \(e^{C_LU_kP}R_k\). We have proved \[ V_j\leq C_L e^{-32K_j} +C_L\sum_{k<j}e^{C_LU_kP}R_k. \tag{138}\] No good-cloud event or physical overshoot is used in this estimate. In particular, neither \(R_j\) nor a bound on \(\mathcal B_j(D_j)\) occurs on its right side: \(B_j\) is already a finite upper block, and no layer-\(j\) expansion is inserted below it.

To prove the probability bound for \(\mathcal B_j(D_j)\), apply Lemma 46 with the fixed threshold \(D_j\) throughout the preceding prefix. A witness whose last cell is in layer \(i\leq j\) has more than \(D_j\) labels and at most \(2D_j^2\). Drop its first-occurrence restrictions, as in (128). Expand its start marginal through layers \(1,\ldots,i-1\), using the exact finite centered recurrence (76), including both remainder terms. Every completed term has at least \(D_j\) labels and one integrated uncentered upper root. Its total integral is therefore at most \(C e^{-32D_j}\) by Proposition 44. This is a statement about finite terms with independent inputs; it does not presume a bound for the probability being proved.

Every interrupted lower term occurs in a layer \(k<i\). Its upper block is bounded by \(U_k\) by construction. Its contribution is at most \(e^{C_LU_kP}R_k\) by Lemma 47, and \(R_k\) is already bounded by the induction hypothesis. Thus \[ \mathbb P\bigl(\mathcal B_j(D_j)\bigr) \leq \mu LJ\left( C_L e^{-32D_j} +C_L\sum_{k<j}e^{C_LU_kP}R_k\right). \tag{139}\] For \(j=1\) the sum is empty and all input estimates are initial ones. For \(k<j\), (133) makes \(e^{C_LU_kP}R_k\leq e^{-7K_k}\) for small \(\varepsilon\). The same schedule makes each \(K_k\) exceed every fixed multiple of \(D_j+P+\log J\). Since \(\log\mu=2|\log\varepsilon|\), the right side of (139) is at most \(e^{-8D_j}\).

Now estimate the physical remainder norm \(Q_j\). On \(\mathcal B_j(D_j)\), discard the exact observable minus the completed below-\(K_j\) terms before estimating an overshoot. Lemma 47 bounds this part by \[ e^{C_LK_jP} \mathbb P\bigl(\mathcal B_j(D_j)\bigr)^{1/2}. \tag{140}\] On the complementary cloud, all trial components have at most \(D_j\) labels and every earlier actual history within the layer has at most \(D_j\) labels. In the first stopped cell, Proposition 16 therefore bounds its selected union by \[\max\{U_jD_j,K_j-1+D_j\}.\] Its positive continuation through the earlier cells of the layer adds at most a factor \(D_j\) in label count. Together with the already completed upper part of that layer, this is less than \(W_j\) in (131). Every such history still has at least \(K_j\) labels: the positive continuation never removes a label counted by the stopping rule. Its upper root count is at most \(U_j\), and \(U_j/K_j\to0\).

The completed signed cells above the stopped cell and the positive actual prefix below it form an admissible mixed upper block. Drop the remaining good-set restrictions and expand its input through the strictly earlier layers, using the full centered recurrence there. Completed terms are bounded by \(C_L e^{-32K_j}\) using Proposition 44; all their labels are below the fixed polynomial cap. Earlier interruptions again cost at most \(e^{C_LU_kP}R_k\). Consequently \[ Q_j\leq C_L e^{-32K_j} +e^{C_LK_jP} \mathbb P\bigl(\mathcal B_j(D_j)\bigr)^{1/2} +C_L\sum_{k<j}e^{C_LU_kP}R_k. \tag{141}\] Polynomially many final-cell and stopping choices are absorbed by the exponential margins in this inequality.

Finally combine (138), (141), and (136). Since \(U_j=o(K_j)\), the factor \(2^{U_j}\) changes \(e^{-32K_j}\) to at most \(e^{-31K_j}\) for small \(\varepsilon\). Its multiplication of the other terms is absorbed by increasing \(C_L\) in their raw amplifications. Thus the common scalar obeys \[ R_j\leq C_L e^{-31K_j} +e^{C_LK_jP} \mathbb P\bigl(\mathcal B_j(D_j)\bigr)^{1/2} +C_L\sum_{k<j}e^{C_LU_kP}R_k. \tag{142}\] By the probability bound just proved and (133), its middle term is at most \(e^{-3D_j}\). Every earlier term is at most \(e^{-7K_k}\), and the earlier \(K_k\) dominate \(K_j\). The right side is at most \(e^{-8K_j}\) for small \(\varepsilon\).

This completes the induction. Notice its order: the reference cutoff mismatch and the probability of a layer-\(j\) witness used only remainders with index less than \(j\); the physical layer-\(j\) remainder then used that newly proved probability. Centering its physical remainder and adding the separately estimated finite mismatch produced \(\mathsf Z_j\) and the common bound \(R_j\). No estimate assumed its own layer’s remainder. The larger threshold \(D_i\) assigned to a preceding layer was never used to infer that a polymer exceeding \(D_j\) was absent. The same-\(D_j\) first-witness construction is what justifies that inference. ◻

Proof of Theorem 1 and Corollary 2. Fix \(s\). For sufficiently small \(\varepsilon\), it is at most \(U_L\). Expand a retained centered singleton array of this order with the exact finite identity [cent:finite-layers]. Proposition 48 and (129) remove both the physical and the reference cutoff remainders: the total error is bounded by a finite sum of \(e^{C_LU_jP}R_j\), which tends to zero.

For the completed terms, first fix an integer \(m\) larger than the finite constants required in Proposition 44. The sum of all terms with more than \(m\) labels is at most \(C_{s,L}e^{-m}\), uniformly for small \(\varepsilon\). The finitely sized part, with at most \(m\) labels, tends to zero by the last assertion of that proposition. Here finiteness refers to the number of labels and tree shapes; the contact times are integrated over their ordered simplices. Letting first \(\varepsilon\to0\) and then \(m\to\infty\) proves integrated convergence of every fixed retained centered singleton array.

These estimates are uniform over deterministic observation times. Use the fixed full mesh of \([0,T]\) and truncate only its last cell at the observation time. Full-cell trial-component and ancestry caps imply the corresponding caps on every prefix of that cell. All ordered-time estimates use intervals of length at most \(b\) and remain valid after shortening. The residual and finite-size convergence are uniform for these endpoints by Proposition 22. At \(t=0\) the expansion is empty and Proposition 4 supplies the conclusion directly. Thus no positive lower bound on the observation time is needed.

Let \[\mathcal D_s^\varepsilon= \{Z_s:|x_i-x_j|>\varepsilon\text{ for every }i\ne j\}, \qquad G_s^\varepsilon= \mathbf1_{\mathcal D_s^\varepsilon} \widetilde F_s^\varepsilon.\] The inverse centered identity in Proposition 20, with the pattern restrictions restored, gives \[ \sup_{0\leq t\leq T} \|G_s^\varepsilon(t)-f(t)^{\otimes s}\|_1\longrightarrow0. \tag{143}\] Indeed every nonempty retained subset contributes one of the centered arrays just bounded. Integrating the other independent \(f\) slots costs at most their mass one; the separation indicators only decrease this bound. The missing independent mass outside \(\mathcal D_s^\varepsilon\) is \(O_s(\varepsilon^3)\), uniformly in time, because the spatial density of \(f\) is bounded by its uniform Gaussian envelope.

We have \(0\leq G_s^\varepsilon\leq F_s^\varepsilon\). Since \(N\) is conserved, \[m_s^\varepsilon:=\int F_s^\varepsilon(t) =\mu^{-s}\mathbb E(N)_s =\int F_s^\varepsilon(0)\longrightarrow1\] by Proposition 4. Hence \[\begin{align*} \|F_s^\varepsilon-f^{\otimes s}\|_1 &\leq\|G_s^\varepsilon-f^{\otimes s}\|_1 +\int(F_s^\varepsilon-G_s^\varepsilon)\\ &\leq2\|G_s^\varepsilon-f^{\otimes s}\|_1 +|m_s^\varepsilon-1|. \end{align*}\] Together with (143), this proves (6). Thus close configurations, including any bound clusters formed by the actual dynamics, are removed by their nonnegative mass. They have never been excluded from the microscopic evolution.

Finally put \[A_\varepsilon(t)=\mu^{-1}\sum_{i=1}^N a(z_i(t)), \qquad a_f(t)=\int a(z)f(t,z)\,dz.\] Factorial normalization gives \[\mathbb E A_\varepsilon(t)=\int aF_1^\varepsilon(t), \qquad \mathbb E A_\varepsilon(t)^2 =\int a(z_1)a(z_2)F_2^\varepsilon(t) +\mu^{-1}\int a(z)^2F_1^\varepsilon(t,z)\,dz.\] It follows that \[\begin{align*} \mathbb E|A_\varepsilon(t)-a_f(t)|^2 \leq\|a\|_\infty^2\bigl(& \|F_2^\varepsilon(t)-f(t)^{\otimes2}\|_1 +2\|F_1^\varepsilon(t)-f(t)\|_1 +\mu^{-1}m_1^\varepsilon\bigr). \end{align*}\] The right side tends to zero uniformly over deterministic \(t\). Chebyshev’s inequality proves (7). This conclusion is the stated uniformity of probabilities at deterministic times; it makes no assertion about a supremum inside a sample path. ◻