Proof. Put \(g=p-\xi\). Reflection sends \(g\) to \(g-2(g\cdot\omega)\omega\). On either hemisphere of the normal sphere, the change from the polar angle of \(\omega\) to the reflected direction shows that \(|g\cdot\omega|\,\,\mathrm d\omega\) pushes forward to \((|g|/4)\,\,\mathrm d\sigma\). The two hemispheres therefore give \((|g|/2)\,\,\mathrm d\sigma\), where \(\sigma\in S^2\) is the outgoing relative direction. Equivalently, for every nonnegative test function \(\psi\), the one-output measure is \[ \frac{|p-\xi|}{2}\int_{S^2} \psi\left(\frac{p+\xi}{2} +\frac{|p-\xi|}{2}\sigma\right)\,\mathrm d\sigma. \tag{10}\] Its mass is \(2\pi|p-\xi|\). The affine map in this integral multiplies surface area by \(|p-\xi|^2/4\), so the density on its image sphere is \(2/|p-\xi|\).

The same sphere is the zero set of \(f(v)=(v-p)\cdot(\xi-v)\). On that sphere, \(|\nabla_v f(v)|=|p-\xi|\). The coarea formula consequently identifies the measure in (10) with \(2\delta(f(v))\,\,\mathrm dv\). Integration against the two incoming densities and Tonelli’s theorem give (9). The usual pre/post-collisional change of variables identifies this incoming-pair expression with the gain in (1). Coincident incoming velocities have zero collision rate. The same identities can first be integrated against a velocity measure \(c(\,\mathrm dp)\), proving the asserted extension. ◻

Proof. Set \(r=|\bar v|\), \(n=\bar v-\bar p\), and \(e_n=n/|n|\). Except on a set of transverse measure zero, \(n\ne0\). For fixed \(p_3,\xi_3\), the delta in the transverse variables is line measure divided by \(|n|\), on a line whose signed distance from zero is \[A-B(\xi_3-v_3),\qquad A=e_n\cdot\bar v,\quad B=\frac{v_3-p_3}{|n|}.\] The regular part of its integral over \(|\xi-v|\leq R\) is at most \(2R/|n|\). The singular part is bounded by its integral over the full line, namely \[ \frac{1}{|n|}\int_{\mathbb R} \bigl((A-B(\xi_3-v_3))^2+s^2\bigr)^{-q/2}\,\,\mathrm ds =\frac{a_q}{|n|}|A-B(\xi_3-v_3)|^{1-q}, \tag{13}\] where \(a_q=\int_{\mathbb R}(1+s^2)^{-q/2}\,\,\mathrm ds<\infty\) because \(q>1\). Infinite values on a zero-distance slice are harmless in the nonnegative iterated integral.

For every interval \(J\subset[-R,R]\) of length at most \(\ell\), \[ \int_J |A-Bx|^{1-q}\,\,\mathrm dx \leq C_q R^{q-1}\ell^{2-q}|A|^{1-q}. \tag{14}\] If \(2R|B|\leq|A|\), the integrand is at most \(C_q|A|^{1-q}\), and the assertion follows from \(\ell\leq2R\). Otherwise, integration about the possible root gives an upper bound \(C_q|B|^{1-q}\ell^{2-q}\); since \(1-q<0\) and \(|B|>|A|/(2R)\), this proves (14). When \(A=0\) the displayed upper bound is understood in the extended sense.

We next prove \[ \int\frac{1+|e_n\cdot e_{\bar v}|^{1-q}}{|n|}\, \lambda(\,\mathrm d\bar p)\leq\frac{C_q}{r}. \tag{15}\] It suffices to consider a disk of radius \(d\). If \(d\leq r/2\), then \(|n|\geq r/2\) and \(e_n\cdot e_{\bar v}\geq1/3\), which proves the bound. If \(d>r/2\), use polar coordinates for \(n\). Its range lies in \(|n|<r+d<3d\). The factor \(|n|^{-1}\) cancels the radial Jacobian, while \[\int_0^{2\pi}\bigl(1+|\cos\theta|^{1-q}\bigr)\,\,\mathrm d\theta<\infty\] because \(q<2\). Division by the disk area gives \(C_q/d\leq2C_q/r\). The point mass at zero satisfies the same estimate directly. Finally, the Gaussian is a probability mixture of these disks: for \(d>0\) its mixture density in the disk radius \(a\) is \[\frac{2a^3}{d^4}e^{-a^2/d^2}\,\,\mathrm da,\] whose integral is one. This proves the Gaussian assertion as well.

Integrating (13) in \(\xi_3\), applying (14), and then (15), bounds the expression in (12) by \[C_q R\ell\frac{r^{-1}}{1+r^{-q}} +C_qR^{q-1}\ell^{2-q}\frac{r^{-q}}{1+r^{-q}}.\] Both fractions are bounded for \(q>1\). These bounds are independent of \(p_3\), so integration against the arbitrary probability \(\rho\) preserves them. Taking the entire axial range of length \(2R\) also gives (11). ◻

Proof. By positivity it suffices to treat an elementary sheet. Put \(k=\xi-v\) and \(w=p+k\). On the collision plane, \[ |p|^2+|\xi|^2=|v|^2+|w|^2, \qquad \frac{H(\xi)}{H(v)} =e^{|p|^2-|w|^2} \frac{\langle v\rangle^m}{\langle\xi\rangle^m} \frac{W(\xi)}{W(v)}. \tag{20}\] The same identity implies \(\langle v\rangle\leq\langle p\rangle\langle\xi\rangle\). Since \(\langle p\rangle^m\leq C_m e^{|p|^2}\), the kernel is bounded by \[ C_m e^{2|p|^2}c(\,\mathrm dp)\, e^{-|w|^2}\delta\bigl((v-p)\cdot k\bigr) \frac{W(v+k)}{W(v)}\,\,\mathrm dk. \tag{21}\]

The tilted cold measure retains a product majorant. For \(0<d\leq1/8\), \[ e^{2|\bar p|^2}\mathcal G_d(\,\mathrm d\bar p) \leq4e^{-46|\bar p|^2}\mathcal G_{2d}(\,\mathrm d\bar p). \tag{22}\] This follows by dividing the two Gaussian densities: their ratio is \(4\exp[-(3/(4d^2)-2)|\bar p|^2]\), and \(3/(4d^2)-2\geq46\). The assertion at \(d=0\) has the evident interpretation. The axial tilted measure \(\widetilde\rho(\,\mathrm da)=e^{2a^2}\rho(\,\mathrm da)\) has mass at most \(L\), and satisfies \[ \widetilde\rho\{|a|>u\}\leq Le^{-6u^2},\qquad u\geq0. \tag{23}\] Consequently \(e^{2|p|^2}c(\,\mathrm dp)\) is bounded by \(4L\) times a product of a centered transverse Gaussian and an axial probability. Moreover, its restriction to \(|p|>u\) is bounded by \(CLe^{-cu^2}\) times a sum of at most two such product probabilities. To see the last assertion, split into \(|\bar p|>u/\sqrt2\) and \(|p_3|>u/\sqrt2\), and apply (22) and (23), respectively. The axial probabilities obtained by normalization may have large support; Lemma 4 permits this.

On the shell in (19), either \(|p|>R/4\) or \(|w|>R/2\), because \(|p|+|k|\leq2|p|+|w|\). In the first case the preceding tilted-measure tail contributes \(CLe^{-cR^2}\). In the second case \(e^{-|w|^2}\leq e^{-R^2/4}\). In either case the remaining integral has \(|k|\leq2R\) and is bounded by \(C_qR^2\) using Lemma 4. This proves (19). The base region \(|p|+|k|\leq2\) is bounded by the same geometric lemma and the total tilted mass. Summing the dyadic shells proves (18) and also the finiteness of (17). ◻

Proof. The dyadic estimate (19) proves (24). Write \(k=\xi-v\). We may consequently work on \(|p|+|k|\leq R\) and remove that restriction at the end of each argument. On this region the exponential and polynomial factors in (20) are bounded by a constant depending on \(R\).

For (25), if \(|\bar v|>2R+1\), then \(|\bar v-\bar p|\geq|\bar v|/2\) and \(|\bar\xi|\geq|\bar v|-R\). The transverse line estimate, followed by the axial integration over a length at most \(2R\), gives \(C_R/|\bar v|\). If \(|v_3|>2R+1\), integrate the delta in \(\xi_3\) instead. Its divisor is \(|v_3-p_3|\geq|v_3|/2\), and \[ \sup_{a\in\mathbb R^2}\int_{|\zeta-a|\leq R} (1+|\zeta|^{-q})\,\,\mathrm d\zeta \leq C_q(R^2+R^{2-q}). \tag{26}\] For completeness, if \(|a|\leq2R\) the disk is contained in the disk of radius \(3R\) about zero; if \(|a|>2R\), then \(|\zeta|>|a|/2>R\) on the disk, giving the same bound. We obtain \(C_R/|v_3|\). At least one of the two outgoing components becomes large with \(|v|\), proving (25). ◻

Proof. The Gaussian majorant is a product measure. Its transverse image is \(\mathcal G_d\), and its axial image is the probability \(\rho=\Phi_*(\pi^{-1/2}e^{-a^2}\,\,\mathrm da)\). There is no need to differentiate or invert \(\Phi\). From (27), \[e^{8|\Phi(a)|^2}\leq e^{16A^2}e^{16\delta^2a^2}.\] Even under the wider axial probability \((2\pi)^{-1/2}e^{-a^2/2}\,\,\mathrm da\), the integral of the right side is \[e^{16A^2}(1-32\delta^2)^{-1/2}\leq2e^{16A^2}.\] We may therefore take \(L_A=2e^{16A^2}\) for both the original and the wider parameter laws. The original image has the required sheet domination.

On the parameter tail, \[\mathbf 1_{|w|>B}e^{-|w|^2} \leq e^{-B^2/2}e^{-|w|^2/2}.\] After normalization the right side is \(2^{3/2}e^{-B^2/2}\) times a Gaussian probability with parameter widths enlarged by \(\sqrt2\). Its transverse velocity width is at most \(\sqrt2/16<1/8\), and its axial image has the exponential moment just proved. Lemma 6 applied to this dominating image gives (28).

For a probability mixture, mix the dominating sheets just constructed; this is again a member of \(\mathcal S_{L_A}\). If the mixing measure has mass \(B_0>0\), normalize it by \(B_0\) and retain that factor in the domination amplitude. If \(B_0=0\), the mixed image is zero. The same argument applied to the wider Gaussian majorants of the component tails proves the asserted factor in (28). Thus the estimates are uniform over families of mixtures whose masses \(B_0\) have a common finite bound independent of the tail cutoff. Masks taking values in \([0,1]\) only decrease the measures, whether imposed before or after the pushforward. ◻

Proof. The integrand is symmetric in \(p,\xi\). On the collision plane, \(|v|^2\leq|p|^2+|\xi|^2\), so at least one incoming speed is at least \(|v|/2\). It suffices, at a factor of two, to retain \(|\xi|\geq|v|/2\). Formula (20) now gives \[\begin{align*} \frac{P(H,H)(v)}{H(v)} &\leq C_m\int_{|\xi|\geq|v|/2} \delta\bigl((v-p)\cdot(\xi-v)\bigr) e^{-|p+\xi-v|^2}\langle p\rangle^{-m} \frac{W(p)W(\xi)}{W(v)}\,\,\mathrm dp\,\,\mathrm d\xi, \tag{31}\end{align*}\] because \(\langle v\rangle^m/\langle\xi\rangle^m\leq2^m\).

Again write \(k=\xi-v\) and \(w=p+k\). On \(R<|p|+|k|\leq2R\), \(R\geq1\), either \(|p|>R/4\) or \(|w|>R/2\). Consequently \[ \langle p\rangle^{-m}e^{-|w|^2} \leq C_mR^{-m}e^{-|w|^2/2}. \tag{32}\] In the second alternative this uses the finite supremum of \(s^m e^{-s^2/2}\); in the first it follows directly from the polynomial factor.

The measure \(W(p)\mathbf 1_{|p|\leq2R}\,\,\mathrm dp\) is dominated by \[\bigl(1+|\bar p|^{-q}\bigr) \mathbf 1_{|\bar p|\leq2R}\,\,\mathrm d\bar p \ \otimes\ \mathbf 1_{|p_3|\leq2R}\,\,\mathrm dp_3.\] Its transverse factor is a positive mixture of centered disks. Indeed, for \(D>0\), \[|\bar p|^{-q}\mathbf 1_{|\bar p|\leq D} =D^{-q}\mathbf 1_{|\bar p|\leq D} +q\int_0^D a^{-q-1}\mathbf 1_{|\bar p|\leq a}\,\,\mathrm da\] almost everywhere. Upon normalizing the disks, the mixture has total mass \(\pi D^2+2\pi D^{2-q}/(2-q)\). The axial factor has mass \(4R\). For \(D=2R\) the resulting product mixture therefore has total mass at most \(C_qR^3\).

Discard \(e^{-|w|^2/2}\) and the incoming-speed restriction in (31). Apply Lemma 4 to each normalized product in this mixture, with \(|k|\leq2R\). In conjunction with (32), this yields the shell estimate \[ \frac{P(H,H)(v)}{H(v)}\bigg|_{R<|p|+|\xi-v|\leq2R} \leq C_{q,m}R^{5-m}. \tag{33}\] The notation on the left denotes the restricted nonnegative integral from (31), including its fixed symmetry factor. The base region is bounded by the same disk-mixture argument with \(R=1\). Summation over dyadic \(R\) proves the result for \(m>5\). ◻

Proof. The case \(B=0\) is immediate. Assume \(B>0\). We first establish \[ \lim_{M\to\infty}\mathop{\mathrm{ess\,sup}}_{|v|>M} \frac{P(H,H)(v)}{H(v)}=0. \tag{35}\] As in the proof of Lemma 9, symmetry reduces the integral to \(|\xi|\geq|v|/2\), up to a factor of two. The shell estimate (33) then removes \(|p|+|\xi-v|>R\) with an error tending to zero uniformly in \(v\) as \(R\to\infty\). On the retained region, the right side of (31) is bounded by a constant times \[\int_{|p|\leq R}W(p) \left[\int_{|\xi-v|\leq R} \delta\bigl((v-p)\cdot(\xi-v)\bigr) \frac{W(\xi)}{W(v)}\,\,\mathrm d\xi\right]\,\mathrm dp.\] If \(|\bar v|>2R+1\), the transverse divisor is at least \(|\bar v|/2\), and \(W(\xi)\) is bounded on the retained region. The transverse line integral and the axial interval of length \(2R\) give an upper bound \(C_R/|\bar v|\) after integration in \(p\). If \(|v_3|>2R+1\), integrating the delta in \(\xi_3\) instead and using (26) gives \(C_R/|v_3|\). Here \(\int_{|p|\leq R}W(p)\,\,\mathrm dp<\infty\), since \(q<2\). At least one output component tends to infinity with \(|v|\). First letting \(|v|\to\infty\) and then \(R\to\infty\) proves (35).

Choose \(M\) so large that \(BP(H,H)\leq\varepsilon H\) on \(|v|>M\). It remains to prove the desired estimate for \(|v|\leq M\). On this output ball, the part of the normalized quadratic gain with \(|p|+|\xi|>L\) tends uniformly to zero as \(L\to\infty\). Indeed, this deletion is symmetric in the two inputs. After the same reduction to \(|\xi|\geq|v|/2\), it implies \[|p|+|\xi-v|>L-M,\] so (33) bounds its cost by a summable tail. The condition \(j\leq BH\) transfers this estimate to \(P(H,j)\).

Next retain both inputs in \(|p|,|\xi|\leq L\). The Gaussian and polynomial factors in the normalized quadratic integrand are then bounded by a constant depending on \(M,L\), leaving \[\delta\bigl((v-p)\cdot(\xi-v)\bigr) \frac{W(p)W(\xi)}{W(v)}.\] For \(0<\delta<1\), the restriction \(|\bar p|<\delta\) is dominated in its \(p\) variable by the product measure \[(1+|\bar p|^{-q})\mathbf 1_{|\bar p|<\delta}\,\,\mathrm d\bar p \otimes\mathbf 1_{|p_3|\leq L}\,\,\mathrm dp_3.\] Its transverse factor is a positive mixture of centered disks, by the layer-cake formula used in Lemma 9, and its total mass is \[2L\left(\pi\delta^2+ \frac{2\pi}{2-q}\delta^{2-q}\right).\] After normalization, (11), with radius at least \(L+M\), therefore bounds the cost of this restriction by \(C_{M,L}(\delta^2+\delta^{2-q})\). Symmetry gives the same bound for \(|\bar\xi|<\delta\). These estimates are uniform even when the output velocity approaches its axis. Again \(j\leq BH\) transfers them to the mixed gain.

We have proved that, for a suitable compact set \[K=\{p:|p|\leq L,\ |\bar p|\geq\delta\},\] the truncations \(H_K=H\mathbf 1_K\) and \(j_K=j\mathbf 1_K\) satisfy \[0\leq P(H,j)-P(H_K,j_K)\leq\tfrac12\varepsilon H \qquad (|v|\leq M),\] uniformly over \(0\leq j\leq BH\). The function \(H_K\) is bounded and supported in the ball of radius \(L\). By symmetry of \(P\), put \(j_K\) outside the plane integral in (9). The area of a plane section of that ball is at most \(\pi L^2\), so \[P(H_K,j_K)(v)\leq C_{L,\delta} \int_K\frac{j_K(p)}{|v-p|}\,\,\mathrm dp.\] For \(0<\rho<1\), the part with \(|v-p|<\rho\) is at most \(C_{B,L,\delta}\rho^2\), because \(j_K\) is bounded. The complement is at most \(C_{L,\delta}\rho^{-1}\int j\). Finally, \(H\) has a positive lower bound on \(|v|\leq M\) off its null axis. Choose \(\rho\) small enough that the near part is at most \(\varepsilon H/2\) there. Absorbing the other fixed factors into \(C_{B,\varepsilon}\) proves (34). The cutoff assertion follows from \(P^b\leq P\) on nonnegative inputs. ◻

Proof. An original integer vector \((k,m,\ell)\) has transverse coordinates \(((m-\sqrt2 k)/\sqrt3,\ell)\). If \(\ell\ne0\), its transverse length is at least one. Otherwise \((k,m)\ne(0,0)\), and \[|m-\sqrt2 k| =\frac{|m^2-2k^2|}{|m+\sqrt2 k|} \ge\frac{1}{C(1+|k|+|m|)}.\] The numerator is a nonzero integer. Since rotation preserves length, this proves the claim. ◻

Proof. Integration over one period of the lattice sum equals integration of each unshifted prototype over \(\mathbb R^3_x\). The transverse area is \(O(s^2)\), while \[\int_{-a}^a S(\gamma z/s)\,\mathrm dz\le 2s/\gamma.\] The velocity factor has integral one and support in \(\{|v|<2\}\). Thus every fixed velocity moment of a sign prototype is \(O(s^2)\). On its positive set the logarithm of the density splits into \[\log((K/s)d_s^{-3}),\qquad \log S(\gamma z/s),\qquad \log J(w).\] The first term is \(O(1+|\log s|)\), and \(\int S|\log S|<\infty\), \(\int J|\log J|<\infty\). Its absolute entropy is therefore \(O(s^2(1+|\log s|))\). The corresponding series over the geometric scales converges. Disjointness proves the asserted integrability and convergence for the cold sum. The background is bounded by \(\mu\), and its velocities satisfy \(|v|\le R_\mu\), giving the datum claims.

For a seed \(b=h^0_{N,T,\lambda}\), scaling \(x\) gives \[\int b\langle v\rangle^k\,\mathrm dx\,\mathrm dv\le C_k s_N^4,\qquad \int b|\log b|\,\mathrm dx\,\mathrm dv \le C s_N^4(1+|\log s_N|).\] These constants are uniform because \(\lambda m_T\) stays in a fixed bounded interval and \(r|\log r|\) is bounded on such an interval. The preexisting truncated density \(f=u^0+\sum_{j\le N}c_i^0\) satisfies \(f\le C s_N^{-31}\). For \(f,b\ge0\), integration of \(1+\log(f+r)\) over \(0<r<b\) gives \[|(f+b)\log(f+b)-f\log f| \le b\{2+\log^+(f+b)\}+b|\log b|.\] Indeed, the positive logarithm is bounded by its terminal value, while \(\int_0^b\log^-(f+r)\,\mathrm dr\le b(1+|\log b|)\). The entropy change caused by the seed is consequently \(O(s_N^4(1+|\log s_N|))\). Its reference terms are controlled by its mass and energy. This proves all convergence assertions.

Removing the hole and the velocity tail from \(\mu\) costs precisely the removed mass in \(\mathcal H_\mu\). The inserted jets have bounded velocities, so their contribution is bounded by the preceding absolute entropy, mass, and energy sums, which tend to zero with \(s_0\). The same is true uniformly for the seeds. The stated order of choices therefore makes the initial relative entropy arbitrarily small. ◻

Proof. The gain estimate \(P(H,H)\le CH\) and \[\|\nu_f^b\|_\infty\le 2\pi b\|f\|_H\int H(p)\,\mathrm dp\] make the gain and loss locally Lipschitz in the weighted norm. For the colored system use the norm of the pointwise sum of the absolute values of the components, and the sum of absolute differences for two solutions. All quadratic estimates close in this norm. The initially disjoint lattice copies have finite total norm, and periodic covariance is preserved by uniqueness. Picard iteration in free coordinates gives the local solution; the loss integrating factor and positive gain iteration give nonnegativity. Repetition on short subintervals gives continuation and the difference estimate. These arguments use the integral equation along flights and require no norm continuity of spatial translation for measurable data. On an interval with bounded norm, the increments along flights are Lipschitz in that norm. Since free transport is isometric, each component’s norm is continuous. Transfers only regroup finitely many scales and preserve the total density.

The weight \(H\) has every polynomial moment and belongs to \(L^{1+\epsilon}(\mathbb R^3)\) for some \(\epsilon>0\). Hence the cutoff equation has integrable gain and loss with all polynomial weights on every bounded interval under consideration. Integrating its path formula and using particle interchange and the elastic substitution yields conservation of the collision invariants. The same calculation with a smooth spatial test gives each local balance, including its initial trace.

Here are details of the entropy calculation at a fixed approximation. Put \(g(v)=e^{-2|v|^2}\) and \(F_\epsilon=F+\epsilon g\). Differentiate \(F_\epsilon\log(F_\epsilon/\rho)-F_\epsilon+\rho\) along flights; its derivative is \(Q^b(F,F)\log(F_\epsilon/\rho)\). Replacing \(Q^b(F,F)\) by \(Q^b(F_\epsilon,F_\epsilon)\) changes its integral by a quantity tending to zero. Indeed, the operator difference is \(O(\epsilon)H\) on the fixed interval, while \[|\log(F_\epsilon/\rho)| \le C\bigl(1+|\log\epsilon|+|v|^2+ \log^+(1/|\bar v|)\bigr).\] The right-hand side is integrable against \(H\), and the resulting error is \(O(\epsilon(1+|\log\epsilon|))\). For the shifted density the usual symmetric calculation is now justified by this integrability. The logarithm of either Maxwellian reference is a collision invariant. Symmetrization therefore gives minus the cutoff dissipation. The regularized entropies converge by dominated convergence using the same bounds, and Fatou’s lemma applies to the nonnegative dissipation. This proves (45). The integrated path formula includes every time section, so the conservation and entropy statements hold at every time, also when colors are transferred. ◻

Proof. Within a fixed copy, the radial margins are larger than the corresponding enlargement radii. Adjacent radial cores have gap \(c_\eta(s_j^2+s_{j+1}^2)\), whereas the sum of their enlargement radii is one quarter of that gap. Opposite stripe patterns at a single scale have gap \(s_j^2/4\), larger than the sum \(c_\eta s_j^2/2\) of their enlargement radii. This proves the assertions for one copy.

For distinct copies let \(S_* =\max\{s_i,s_{i'}\}\). An intersection would force \[|\bar n-\bar n'|\le4S_*,\qquad |n-n'|\le C(1+t)\le C(1+S_*^{-1/2}).\] By (37), the first quantity is at least \(c/(1+|n-n'|)\ge c' S_*^{1/2}\). For sufficiently small \(s_0\) this contradicts \(4S_*\). The choice is uniform over all pairs of scales and all lattice copies. ◻

Proof. Put \(A=(a_s^\sigma)'(\tau)\ge0\). Differentiation of (49) gives \[U_z=\frac A{1+tA}\ge0,\qquad U_t=-UU_z,\qquad D_tU=(v_3-U)U_z,\qquad D_t\tau=\frac{v_3-U}{1+tA}.\] Thus \(D_t|\xi|^2=-2U_z\xi_3^2\). The distance functions are locally Lipschitz along every flight, and \[|D_tD_y|\le|\bar\xi|,\qquad |D_tD_z|\le|\xi_3|.\] With \(r=(D_y,D_z)\), differentiation of the logarithm of the barrier therefore yields \[D_t\log m_i =\frac1L+\frac{|\xi|^2}{Ld_s^2} +\frac{2(2-t/L)U_z\xi_3^2}{d_s^2} +\frac{|r|^2}{8d_s^2Lt^2} -\frac{r\cdot D_tr}{4d_s^2Lt}.\] The last two terms are at least \(-|D_tr|^2/(8d_s^2L)\), by completing a square. Since \(|D_tr|\le|\xi|\) and \(2-t/L\ge1\), this proves the derivative estimate. The large axial derivative \(U_z\) has a favorable sign.

For a flight starting in the core at \(x^0\), write \(v=a_s^\sigma(z_c^0)e_3+d_sw\), \(|w|\le1\). At \(x=x^0+tv\), the increasing function \(p\mapsto p-a_s^\sigma(z_c-tp)\) has derivative at least one, value \(d_sw_3\) at \(p=v_3\), and value zero at \(p=U\). Consequently \[|\xi|\le d_s,\qquad D_y\le td_s|\bar w|,\qquad D_z\le td_s|w_3|.\] The distance exponent in (51) is at most \(t/(8L)\), and its velocity exponent is at most two. A fixed \(C_m\), depending only on \(\|J\|_\infty\), thus gives \(f_i\le m_i\). Since \(L_sd_s=d_0s^{19/2}\ll s^2\), the transverse displacement is less than the enlargement radius. The axial speed is below two, so the flight also stays inside the axial interval of (48).

At fixed \((t,x)\), the majorant is a scalar multiple of a Maxwellian centered at \(Ue_3\). Detailed balance gives \(P(m_i,m_i)=m_i\nu_{m_i}\). Its mass is at most \(CK/s\), and its first moment about \(Ue_3\) is at most \(CKd_s/s\), uniformly for \(t\le L_s\). The inequality \(|v-p|\le|\xi|+|p-Ue_3|\) gives (53).

Finally, along a flight starting on the closed initial spatial support, both distance functions vanish initially and are locally Lipschitz, so their squared-distance-over-time terms tend to zero. Away from that support the barrier tends to zero. These observations give its path trace and allow the differential inequalities to be integrated from zero. Boundary conventions for the masks change only null sets; no spatial indicator is differentiated. ◻

Proof. Reduce \(s_0\) so that \(\epsilon_s\le1\) at every scale. Then \(\mathcal U_i\le2m_i\), and Lemma 15 gives \[P^b(\mathcal U_i,\mathcal U_i) \le4C\frac K{s}(d_s+|\xi|)m_i,\qquad D_t\mathcal U_i\ge\frac{\epsilon_s}{L_s} \left(1+\frac78\frac{|\xi|^2}{d_s^2}\right)m_i.\] Since \(1+z^2\ge c(1+z)\), a fixed large \(C_*\) makes \(D_t\mathcal U_i\ge P^b(\mathcal U_i,\mathcal U_i)\). Dropping the nonnegative loss in the equation for \(c_i\) permits positive gain comparison from the initial trace.

For completeness, this comparison is valid in the measurable solution class of Proposition 13. At a fixed approximation \(m_i\le C_sH\) throughout the interval, and \(z_i=(c_i-\mathcal U_i)_+\) has a pathwise upper integral comparison from zero with source \(2P^b(\mathcal U_i,z_i)+P^b(z_i,z_i)\). The bilinear \(H\)-norm estimate and Gronwall’s inequality give \(\|z_i\|_H=0\). One may first integrate from a positive time and then use the trace established in Lemma 15.

For the second assertion, \(e^{-\kappa(t-t_1)}\mathcal U_i\) is a supersolution for \(D_t+\kappa\): its linear derivative has factor \(e^{-\kappa(t-t_1)}\), whereas its quadratic gain has the smaller factor \(e^{-2\kappa(t-t_1)}\). Its value at \(t_1\) dominates \(c_i(t_1)\). The same comparison proves (55). ◻

Proof. The transverse Jacobian is \(d_s^2\), while differentiation of the implicit equation for \(p_3\) gives \[\frac{\partial p_3}{\partial w_3} =\frac{d_s}{1+t(a_s^\sigma)'(z_c-tp_3)}.\] Multiplying by the density in (39) proves (56). Its coefficient is at most \(C/(s+t)\), since \(0\le S\le1\) and \(\alpha\gamma>0\). A fixed product Gaussian dominates \(J\). Moreover, the monotonicity argument in Lemma 15 gives \(|p_3-U|\le d_s|w_3|\). The Gaussian parameter estimate (Lemma 8) therefore supplies the asserted sheet domination, with constants independent of the axial compression.

The remainder is a Gaussian centered at \(Ue_3\), of width comparable to \(d_s\), whose mass is at most \(C\epsilon_s K/s=C L_sd_s/s^2\). Its transverse Gaussian width is within the sheet class and its axial exponential moment is uniformly bounded because \(|U|\le1+\alpha\). Since \(L_sd_s/s^2=d_0s^{15/2}\) and \(t\le s^{-1/2}\), this also gives the last inequality of (57). Uniform Gaussian moments and tails, together with (54), prove (58). ◻

Proof. Free cores vanish off their enlargements. For a nonmain label either \(D_y\ge(c_\eta/4)s^2\), or \(|z_c|>a+2t+b_{\mathrm{sp}}\), in which case \(D_z\ge b_{\mathrm{sp}}\) because \(|U|<2\). Completing the velocity square in the remainder gives \[\mathcal U_i(t,x,v) \le C s^{-31}e^{-4|v|^2} \exp\left(-\frac{D_y^2+D_z^2}{8d_s^2L_st}\right).\] Here the Gaussian rate is at least eight once the widths are fixed small, and its centers stay bounded. Half the distance exponential is at most \(\exp[-c/(s^{16}L_st)]\). Since \(t\le L_s\), this is bounded by \(e^{-c/s^{15}}\), which absorbs every polynomial in \(1/s\). If also \(t\le1\), it absorbs arbitrary powers of \(t\) and \(s\).

It remains to sum the other half over lattice copies. Uniformly, \[D_y\ge(|y_c|-C)_+,\qquad D_z\ge(|z_c|-C(1+t))_+.\] The Gaussian distance scale \(d_s\sqrt{L_st}\) is bounded, and the central spatial region has diameter \(O(1+L_s)\). The lattice has unit separation, hence at most \(C(R+1)^3\) points in a ball of radius \(R\). Summing in successive shells gives at worst a fixed polynomial loss in \(1/s\). The extracted exponential absorbs this loss and the factor \(s^{-31}\). Finally sum over the geometric scales. This proves both claims, including their stated uniformity. ◻

Proof. For (60), use \(f_i\le m_i\) and the same Gaussian lattice sum as in Lemma 18, without extracting a small distance factor. The prefactor and the number of central copies are polynomial in \(1/s\). Completing the velocity square bounds the remaining velocity factor by \(Ce^{-4|v|^2}\le CH(v)\). For (61), the transverse area at scale \(s_j\) is \(O(s_j^2)\), and its axial length while active is \(O(1+L_{s_j})\). Projection onto the torus does not increase volume. Summation gives \(C\sum_j(s_j^2+s_j^{3/2})\le Cs_0^{3/2}\). ◻

Proof. The distance \(D_0\) is \(1\)-Lipschitz. Fix \(4<\beta'<\beta_0\) and a positive absolute constant \(c\), and consider \[B(t,x,v)=\exp(C_1t)\exp\!\left( -(\beta_0-C_1t)|v|^2-\frac{cD_0(x)^2}{t}\right).\] At almost every point of a free path, \[\begin{align*} D_t\log B &=C_1+C_1|v|^2+\frac{cD_0^2}{t^2} -\frac{2cD_0}{t}\,v\cdot\nabla D_0\\ &\ge C_1+(C_1-c)|v|^2. \end{align*}\] The last inequality is completion of the square in \(D_0/t\). At fixed \((t,x)\), \(B\) is a Maxwellian in velocity. Energy conservation in a collision therefore gives \(P(B,B)=B\nu_B\), and, as long as \(\beta_0-C_1t\ge\beta'>4\) and \(e^{C_1t}\le2\), \[P^b(B,B)\le P(B,B)\le C(1+|v|)B.\] Choose \(C_1\) large enough that the preceding lower bound for \(D_t\log B\) dominates \(C(1+|v|)\), and then choose \(t_1\le1/32\) small enough to maintain both restrictions. These choices use only \(\beta_0\) and the collision normalization.

The trace of \(B\) along a path starting outside the hole dominates \(u^0\). Indeed, if \(D_0(x)=0\), then \(D_0(x+tv)\le t|v|\), so the distance factor tends to one. If \(D_0(x)>0\), its trace is zero, as is \(u^0(x,v)\). The differential inequality can be integrated from a positive time and then passed to this trace. The positive-gain comparison used for Proposition 16, with the nonnegative loss omitted, yields \(u\le B\). This uses only absolute continuity along paths at positive times and boundedness in the \(H\) norm; it does not differentiate the initial spatial mask. Since \(e^{C_1t}\) is bounded, (62) follows.

For \(t\le1/32\), the axial interval defining \(\mathcal O_i(t)\) has half-length \[a+2t+b_{\rm sp}\le\frac18+\frac1{16}+\frac1{16}=\frac14<\frac13.\] Its transverse distance from the corresponding lifted axis is at most \(s_0\). Thus every point of \(\mathcal O(t)\) lies within distance \(s_0\) of the segment defining the hole. Since the hole is its open \(2\delta\) neighborhood, the triangle inequality gives \(D_0\ge2\delta-s_0\ge\delta\) when \(s_0\le\delta\). ◻

Proof. At each \((t,x)\) there is at most one main color. Denote it by \(a\), taking \(a=0\) if no main color is present, and put \(r=c-a\). The main-color measure bound in Lemma 17 and the sheet gain estimates, including Lemma 8, give \[P(a,H)\le\frac{C}{t+s_N}H.\] By (59), for each \(M>0\), \[0\le r\le\rho(t)H,\qquad \rho(t)\le C_Mt^M,\] uniformly for sufficiently small \(s_0\). The bath bound gives \(u\le CH\). The quadratic gain bound of Lemma 9 proves the first estimate in (64), absorbing the bounded terms since \(t+s_N\le2\).

If \(a\ne0\), (63) improves the bath estimate to \(u\le C e^{-c\delta^2/t}H\) at this position. The ordered cross-color sum is bounded by \(2P(a,r)+P(r,r)\), so positivity yields \[\mathcal S\le C\left(\frac{e^{-c\delta^2/t}+\rho(t)}{t+s_N} +\rho(t)+\rho(t)^2\right)H.\] The exponential is bounded by any prescribed power of \(t\), with a constant depending on that power and on \(\delta\). Taking \(M=A\) proves the second estimate. When \(a=0\), only the last two types of terms remain, and the same conclusion follows. ◻

Proof. For the dormant solution, the function \(Y(t)=C_dt^A\) is a supersolution of (65) if \[C_d(A-C_2)\ge C_{A,\delta},\] because \((t+s_N)^{-1}\le t^{-1}\). At each fixed \(N\), comparison begins at the regular initial section \(t=0\), where both functions vanish. Choosing an initial interval on which \(C_dt^A<1/2\) closes the condition \(Z\le1\). The cold and bath bounds are finite in the \(H\) norm at each fixed approximation. Thus the continuation criterion of Proposition 13 also proves existence on that interval and gives (68).

Reduce \(T_0\) until this interval contains \([0,T_0]\) and \(C_dT_0^{A-p-1}<1\). Then, for \(0<t\le T\le T_0\), \[\frac{Z(t)}{T^p(t+s_N)} \le\frac{C_dt^{A-1}}{T^p} \le C_dT^{A-p-1}<1.\] At \(t=0\) the ratio is zero. Hence the dormant solution stays strictly below the cap. At the other endpoint, \(\lambda=L_0\), the ratio at time zero equals \(L_0\|\chi\|_H>1\).

Fix \(T,N\), and the cutoff. Let \(I\) be the set of \(\lambda\in[0,L_0]\) whose solutions exist on \([0,T]\) and remain strictly below the cap there. Continuous dependence in the local theory and continuation at the terminal section make \(I\) relatively open. It contains \(0\) and does not contain \(L_0\). Let \(\lambda_*\) be the right endpoint of the connected component of \(I\) containing zero. Along a sequence \(\lambda_k\uparrow\lambda_*\) within that component, all hot parts are bounded by the cap. The cold and bath estimates bound the remaining colors uniformly in \(k\) in the fixed-approximation norm. The difference estimate in Proposition 13 therefore makes these solutions Cauchy, uniformly on \([0,T]\) in free coordinates. Their limit is the solution with parameter \(\lambda_*\) and satisfies the non-strict cap inequality.

The norm \(Z\) is continuous in time, and the cap denominator is positive, so the displayed ratio has a maximum. If that maximum were less than one, continuous dependence and continuation would place a neighborhood of \(\lambda_*\) in \(I\), contrary to its endpoint property. Its maximum is therefore one. Since \(0\) has a neighborhood in \(I\), while \(L_0\|\chi\|_H>1\) also excludes a neighborhood of \(L_0\), one has \(0<\lambda_*<L_0\). A final reduction of \(T_0\) makes the cap at most one on \([0,T]\) for every \(T\le T_0\). ◻

Proof. On \([t_b,t_1]\), the nonmain bound (59) gives \(r\le\varepsilon(s_0)H\), with \(\varepsilon(s_0)\to0\). The main-color sheet amplitude is bounded by \(C/t_b\). Consequently all terms of \(\mathcal S\) containing a nonmain factor are bounded by \(C_{t_b}\varepsilon(s_0)H\), after enlarging \(\varepsilon\) if needed.

For the remaining main–bath term, define a deterministic majorant \[\mathcal U_{\mathrm{main}}(s,x,v) =\sum_i\mathbf 1_{\mathcal O_i(s)}(x)\mathcal U_i(s,x,v), \qquad U_{\mathrm{bath}}(v)=C e^{-\beta'|v|^2},\] \[\mathcal S_{\mathrm{mb}}(s,x,v) =2P(\mathcal U_{\mathrm{main}}(s,x,\cdot),U_{\mathrm{bath}})(v).\] Here the sum includes all scales, so it bounds every finite selection; at each position it has at most one nonzero summand. By Proposition 16 and (62), \[ 0\le\mathcal S(s,x,v) \le C_{t_b}\varepsilon(s_0)H(v)+\mathcal S_{\mathrm{mb}}(s,x,v), \qquad t_b\le s\le t_1. \tag{70}\] The sheet gain estimates give \(\mathcal S_{\mathrm{mb}}\le C_{t_b}H\). The uniform large-output estimate (25) gives \[ \sup_{s,x,\,|v|>R}\frac{\mathcal S_{\mathrm{mb}}(s,x,v)}{H(v)} \le\omega_{t_b}(R),\qquad \omega_{t_b}(R)\longrightarrow0 \quad(R\to\infty), \tag{71}\] with essential suprema in the velocity variable. To handle velocities near the axis, fix \(q'\in(1,q)\) and write \[H_{q'}(v)=e^{-|v|^2}\langle v\rangle^{-100} (1+|\bar v|^{-q'}).\] Since \(U_{\mathrm{bath}}\le C_{q'}H_{q'}\), the same sheet lemma with exponent \(q'\) implies \(\mathcal S_{\mathrm{mb}}\le C_{t_b,q'}H_{q'}\). Hence, for \(0<r\le1\), \[ \sup_{s,x,\,0<|\bar v|<r}\frac{\mathcal S_{\mathrm{mb}}(s,x,v)}{H(v)} \le C_{t_b,q'}r^{q-q'}. \tag{72}\] Both estimates include the Gaussian-parameter measures in Lemma 17, by Lemma 8.

It remains to consider the compact velocity region \(|v|\le R\), \(|\bar v|\ge r\). The spatial support of \(\mathcal S_{\mathrm{mb}}\) is contained, for every \(s\le t_1\), in the projection of the fixed union \[\bigcup_{n\in\Lambda} \bigl(n+B^2_{s_0}\times[-1/4,1/4]\bigr),\] where \(B^2_{s_0}\) is the transverse disk of radius \(s_0\). A lifted free path with \(|v|\le R\) and duration at most \(t_1\) can meet only \(C_R\) of these cylinders, uniformly in its endpoint: their centers must lie in a fixed bounded enlargement of a segment of length at most \(Rt_1\), and the lattice has a fixed separation. For any one cylinder, its transverse projection is crossed in time at most \(2s_0/|\bar v|\le2s_0/r\). Thus the normalized integral of \(\mathcal S_{\mathrm{mb}}\) along such a path is bounded by \(C_{t_b,R}s_0/r\). Combining the three velocity regions gives \[\left\|\int_{t_b}^t \mathcal S(s,x-(t-s)v,v)\,\mathrm ds\right\|_H \le C_{t_b}\varepsilon(s_0) +t_1\omega_{t_b}(R) +C_{t_b,q'}t_1r^{q-q'} +C_{t_b,R}\frac{s_0}{r}.\] First take \(R\) large, then \(r\) small, and finally \(s_0\) small. This proves (69).

All estimates above concern the common pointwise majorant (70), rather than values chosen separately on individual paths. To pass an almost-everywhere inequality to path integrals, use free coordinates \((s,x_0,v)\mapsto(s,x_0+sv,v)\). This map preserves measure, so Fubini’s theorem gives the inequality at almost every time on almost every free path. Integration on those paths then defines representatives at every terminal section. Since free transport preserves \(\|\cdot\|_H\), the resulting bound is precisely the essential norm in (69) at each fixed terminal time. The construction applies on any existing solution interval, so this lemma does not presuppose continuation to \(t_1\). ◻

Proof. We choose an intermediate time \(t_b\in(0,t_1)\) and then require \(T_0<t_b\). At the cap horizon, both the selected and dormant solutions satisfy \[Z(T)\le T^p(T+s_N)\le2T^p,\] using only \(T,s_N\le1\); no relation between \(T\) and \(s_N\) is needed. For \(T\le t\le t_b\), (65) and \((s+s_N)^{-1}\le s^{-1}\) give, while \(Z\le1\), \[ Z(t)\le2T^{p-C_2}t^{C_2} +\widetilde C_{A,\delta}t^A. \tag{74}\] Indeed multiplication of the scalar comparison by \(t^{-C_2}\) integrates the source to at most \(C_{A,\delta}t^{A-C_2}/(A-C_2)\).

From \(t_b\) onward, retain the source inside the path integral rather than using its pointwise bound. The same hot gain estimates and Lemma 23 give \[Z(t)\le Z(t_b)+E_{t_b}(s_0) +\int_{t_b}^t\frac{C_2}{s}Z(s)\,\mathrm ds.\] Consequently, as long as \(Z\le1\), \[ \sup_{t_b\le t\le t_1}Z(t) \le\left(\frac{t_1}{t_b}\right)^{C_2} \left(2T_0^{p-C_2}t_b^{C_2} +\widetilde C_{A,\delta}t_b^A+E_{t_b}(s_0)\right), \tag{75}\] where the supremum is restricted to the current interval of existence until continuation has been established.

The order of choices now matters. Since \(A>C_2\), choose \(t_b\) so that \[\widetilde C_{A,\delta}t_1^{C_2}t_b^{A-C_2}<\frac\zeta6.\] Since \(p>C_2\), reduce \(T_0<t_b\), within the range of Proposition 22, so that \[2T_0^{p-C_2}t_1^{C_2}<\frac\zeta6, \qquad 2T_0^p<\frac\zeta2.\] Finally take \(\bar s_0\) small enough for all preceding cold and bath estimates and for \[\left(\frac{t_1}{t_b}\right)^{C_2}E_{t_b}(s_0)<\frac\zeta6 \qquad(0<s_0\le\bar s_0).\] The last choice is possible by Lemma 23. Equation (75) is then strictly below \(\zeta/2\). The same first two choices make (74) strictly below \(\zeta/2\) on \([T,t_b]\), and the cap gives the same bound on \([0,T]\).

These estimates close the condition \(Z\le1\) with a strict margin. At each fixed approximation the cold sum and bath have bounded \(H\) norm through \(t_1\), so the finite-evolution continuation criterion excludes a finite endpoint before \(t_1\). This proves existence and (73) uniformly in all the stated indices. The dormant estimate was obtained independently on its initial interval and is preserved under this continuation. ◻

Proof. Contain \(V\) in a ball of radius \(R\). The change of variables \(z=x-\tau v\) gives a factor \(\tau^{-3}\). The image ball meets at most \(C(1+\tau R)^3\) translates of a fixed bounded fundamental cell of \(\Lambda\). Periodicity therefore gives \[\int_V A_s(x-\tau v)\,\mathrm dv \le C\tau^{-3}(1+\tau R)^3\|A_s\|_1,\] which proves (80). In particular, its constant does not grow when \(\tau\) tends to infinity.

For \(L\ge1\) let \[H_L(v)=\min\{H(v),L\}\mathbf 1_{\{|v|\le L\}},\qquad \delta_L=\int(H-H_L)\,\mathrm dv.\] Then \(\delta_L\to0\), and (80) implies, for \(\tau\ge\rho\), \[ \int H(v)A_s(x-\tau v)\,\mathrm dv \le A_*\delta_L+C_{L,\rho}a_*. \tag{83}\] For \(0\le\tau<\rho\), the same integral is at most \(A_*M_H\). Consequently the expression in (81) is at most \[\rho A_*M_H+\kappa^{-1} (A_*\delta_L+C_{L,\rho}a_*),\] and that in (82) is at most \[\rho A_*M_H+t_1(A_*\delta_L+C_{L,\rho}a_*).\] Choose first \(\rho\) small, then \(L\) large, and finally \(a_*\) small. Both upper bounds tend to zero in this order. The same proof works when the time integrals are restricted to an interval of existence. ◻

Proof. Fix two closed velocity balls \(V_1,V_2\) of positive radius with \(\mathop{\mathrm{dist}}(V_1,V_2)>3\), and put \(V=V_1\cup V_2\). All constants until the choice of the entropy tolerance may depend on these fixed balls. Since \(b\ge1\) and \(\mu\) is a Maxwellian, \[P^b(\mu,\mu)(v)=\mu(v)\nu_\mu^b(v) \ge \mu(v)\nu_\mu^1(v)\ge m_0>0 \qquad(v\in V).\] At a fixed \((t,x)\) write \(d=(\mu-F)_+\). Positivity of the gain and \(F\ge\mu-d\) imply \[P^b(F,F)\ge P^b(\mu-d,\mu-d) \ge P^b(\mu,\mu)-2P(\mu,d).\] By the plane formula (9), the Gaussian integral over each plane is bounded, so \[P(\mu,d)(v)\le C\int\frac{d(p)}{|p-v|}\,\mathrm dp.\] For \(|p-v|<r\), use \(d\le\mu\le1\) to bound this integral by \(Cr^2\); for \(|p-v|\ge r\), it is at most \(Cr^{-1}D_F(t,x)\). Choose \(r>0\) once so that the first contribution to \(2P(\mu,d)\) is at most \(m_0/2\). With \(m_*=m_0/2\) and a fixed constant \(C\), we obtain \[ P^b(F,F)(t,x,v)\ge m_*-CD_F(t,x),\qquad \nu_F^b(t,x,v)\le C+CD_F(t,x),\qquad v\in V. \tag{87}\] The second inequality follows directly from \(\min\{b,|v-p|\}\le |v-p|\le C\langle p\rangle\) on \(V\).

Let \(\Delta=t_1/2\), fix \(t\ge t_1\), and write \(X_\tau=x-(t-\tau)v\). The Duhamel formula for the total density, whose value is continuous through retirements, gives \[F(t,x,v)\ge \int_{t-\Delta}^{t-\Delta/2} e^{-\int_s^t\nu_F^b(\tau,X_\tau,v)\,\mathrm d\tau} P^b(F,F)(s,X_s,v)\,\mathrm ds.\] Using (87), \(e^{-a}\ge1-a\) for \(a\ge0\), and the upper bound \(1\) for a survival factor, the integrand is bounded below by \[ m_*e^{-C\Delta} \left(1-C\int_s^t D_F(\tau,X_\tau)\,\mathrm d\tau\right) -CD_F(s,X_s). \tag{88}\] This inequality remains valid when its right side is negative.

Integrate (88) over \(s\) and over either ball \(V_k\). The positive constant contributes \[a_k:=|V_k|(\Delta/2)m_*e^{-C\Delta}>0.\] Choose \(0<\rho<\Delta/2\). For \(t-\tau\ge\rho\), (80) and (84) give \[ \int_{V_k}D_F(\tau,x-(t-\tau)v)\,\mathrm dv \le C_{V_k,\rho}\varepsilon_E. \tag{89}\] The term \(D_F(s,X_s)\) always has \(t-s\ge\Delta/2\). Its integrated cost is therefore \(C_\rho\varepsilon_E\). For the double time integral in (88), the portion with \(t-\tau\ge\rho\) has the same type of bound, whereas the remaining portion costs at most \(C(B)\rho\) by (85). Thus \[\int_{V_k}F(t,x,v)\,\mathrm dv \ge a_k-C_\rho\varepsilon_E-C(B)\rho.\] After \(B\) has been fixed, choose \(\rho\) to make the last term at most \(\min(a_1,a_2)/4\), and then choose \(\varepsilon_E\) to make the middle term at most the same quantity. It follows that \[\int_{V_k}F(t,x,v)\,\mathrm dv\ge a_k/2\qquad(k=1,2).\] For any velocity \(v\), one of the balls has distance at least \(1\) from \(v\): otherwise their mutual distance would be less than \(2\). On that ball \(\min\{b,|v-p|\}\ge1\). Consequently \[\nu_F^b(t,x,v) =2\pi\int\min\{b,|v-p|\}F(t,x,p)\,\mathrm dp \ge\pi\min(a_1,a_2)=:\kappa.\] Only the entropy tolerance, and not this value of \(\kappa\), depended on \(B\). ◻

Proof. Choose the entropy tolerance small enough for 26, so that the damping constant \(\kappa\) and (91)–(93) are available. Define \[ j=(w-\mu)_+,\qquad m_j(t,x)=\int j(t,x,v)\,\mathrm dv. \tag{95}\] Since \(w\le F\) and \(j\le BH\), \[ 0\le m_j\le BM_H, \qquad\|m_j(t)\|_1\le\varepsilon_E. \tag{96}\] Write \(C_\mu=\|\mu\|_H<\infty\) and \(A_B=2C_\mu+B\). The inequality \(w\le\mu+j\), bilinearity, and positivity give \[ P^b(w,w)-P^b(\mu,\mu) \le2P(\mu,j)+P(j,j)\le A_BP(H,j). \tag{97}\] The frequency deficit has the bound \[ (\nu_\mu^b-\nu_F^b)\mu \le2\pi\mu(v)\int|v-p|(\mu(p)-F(p))_+\,\mathrm dp \le CH(v)D_-. \tag{98}\] Here \(\mu(v)\langle v\rangle\le CH(v)\).

Between retirements, subtracting the stationary Maxwellian equation from (77) gives \[(D_t+\nu_F^b)(w-\mu) =P^b(w,w)-P^b(\mu,\mu) +(\nu_\mu^b-\nu_F^b)\mu+2P^b(c,w)+\mathcal S_c.\] The chain rule for the positive part on almost every free flight therefore gives an upper comparison for \(j\) with damping \(\kappa\). Using 10 in (97), and then (98), (91), and (92), shows that for each \(\epsilon>0\) its positive source is at most \(H(v)R(s,x)\), where \[ R(s,x)=\epsilon+C_{B,\epsilon}m_j(s,x)+CD_-(s,x) +C(B)\bigl(\mathbf 1_{\mathcal O(s)}(x)+\theta_0\bigr). \tag{99}\] Constants have been enlarged to absorb \(A_B\). At a retirement time, the increase of \(j\) is at most the increase of \(w\), because \((a+d)_+-a_+\le d\) for \(d\ge0\). Thus its increments are bounded by the same \(J_sH\) as in (93).

We also need to estimate the excess already present in the bath at \(t_1\). Put \(j_u=(u-\mu)_+\) and \(m_{j_u}=\int j_u\,\mathrm dv\) for \(0\le s\le t_1\). The bath starts below \(\mu\), so \(j_u(0)=0\); 20 gives the fixed bound \(j_u\le B_0H\). Since \(u\le F\), \[ \|m_{j_u}(s)\|_\infty\le B_0M_H, \qquad\|m_{j_u}(s)\|_1\le\varepsilon_E. \tag{100}\] The bath equation is \((D_t+\nu_F^b)u=P^b(u,u)\). Subtracting the Maxwellian equation and applying the same two difference estimates as above gives, now discarding the nonnegative damping, a source bounded by \(H(v)R_u(s,x)\), where \[ R_u(s,x)=\epsilon+C_{B_0,\epsilon}m_{j_u}(s,x)+CD_-(s,x). \tag{101}\] The cold density appears only through the frequency deficit, which is controlled by \(D_-\). No positive source involving a cold gain occurs in this bath equation.

At \(t_1\), the hot bound gives \(j(t_1)\le j_u(t_1)+\zeta H\). Continue the same backward free flight through \(t_1\), and write \(X_s=x-(t-s)v\). The two comparisons, together with all intervening retirements, imply \[ \frac{j(t,x,v)}{H(v)} \le\zeta+J_0 +\int_0^{t_1}R_u(s,X_s)\,\mathrm ds +\int_{t_1}^t e^{-\kappa(t-s)}R(s,X_s)\,\mathrm ds. \tag{102}\] We have dropped damping from the initial and retirement contributions; doing so also allows the bath comparison to run along that one flight all the way back to zero.

We claim that the velocity integral of the right side times \(H\) can be made smaller than any prescribed number \(\tau>0\), uniformly in \((t,x)\) on the bootstrap interval. The constant terms contribute at most \[ M_H\left(\zeta+J_0+\epsilon(t_1+\kappa^{-1}) +\kappa^{-1}C(B)\theta_0\right). \tag{103}\] For the remaining terms, all the spatial functions in question have a fixed pointwise upper bound once \(B\) is fixed: \(m_j\le BM_H\), \(m_{j_u}\le B_0M_H\), \(D_-\le\int\langle v\rangle\mu\,\mathrm dv\), and \(\mathbf 1_{\mathcal O(s)}\le1\). Their spatial integrals are respectively bounded by \(\varepsilon_E\), \(\varepsilon_E\), \(\varepsilon_E\), and \(Cs_0^{3/2}\), the last by (61). Apply 25 to each term, with its coefficient in (99) or (101). For clarity, the choices are made in the following order. First choose \(\epsilon\) so that its contribution to (103) is less than \(\tau/4\); all the coefficients \(C_{B,\epsilon}\) and \(C_{B_0,\epsilon}\) are then fixed. For each of the finitely many averaged terms, choose the time deletion length \(\rho\) and the truncation \(H_L\) in the proof of that lemma so that the endpoint and velocity-tail errors, including its coefficient, have total less than \(\tau/4\). Next make \(\varepsilon_E\) and \(s_0\) sufficiently small for the remaining averaged errors to total less than \(\tau/4\). Finally take \(\zeta\) small and, if necessary, reduce \(s_0\) further to make the other terms in (103) less than \(\tau/4\). All these choices are uniform in the endpoint time. Thus \[ m_j(t,x)\le\tau. \tag{104}\] The same calculation on a pre-retirement section includes only the jumps that have already occurred, so it does not require a post-jump bootstrap hypothesis.

It remains to select the particular smallness threshold that controls the gain. Apply 10 once more, now with \(\epsilon'=1/(2A_B)\). Write its constant as \(C_{B,\epsilon'}\) and set \[\tau=\frac{1}{2A_B\max\{1,C_{B,\epsilon'}\}}.\] Choose the parameters in the preceding argument to give (104) for this \(\tau\). Then \[A_BP(H,j)\le A_BH\bigl(\epsilon'+C_{B,\epsilon'}m_j\bigr)\le H.\] Equation (97) and \(P^b(\mu,\mu)\le P(\mu,\mu)\) prove (94). The latter has finite \(H\) norm because \(P(\mu,\mu)=\mu\nu_\mu\) is a Gaussian times a function of at most linear growth. ◻

Proof. On a bootstrap interval let \(Z_w(t)=\|w(t)\|_H\). The Duhamel formula for (77), with all retirement increments included, and (86), (91), (92), and (94), give \[\begin{align*} Z_w(t)\le {}&e^{-\kappa(t-t_1)}B_0 +\frac{C_5+C_4\theta_0}{\kappa}+J_0\\ &+2C_3(1+\theta_0) \int_{t_1}^t e^{-\kappa(t-s)}e^{-\kappa(s-t_1)}Z_w(s)\,\mathrm ds. \end{align*}\] Take \(s_0\) small enough that \(\theta_0,J_0\le1\) and discard the first exponential in the last integrand. We obtain \[ Z_w(t)\le B_0+\frac{C_4+C_5}{\kappa}+1 +4C_3\int_{t_1}^t e^{-\kappa(s-t_1)}Z_w(s)\,\mathrm ds. \tag{106}\] The integral version of Gronwall’s inequality therefore gives \[ Z_w(t)\le \left(B_0+\frac{C_4+C_5}{\kappa}+1\right) \exp\left(\frac{4C_3}{\kappa}\right)=:B_*. \tag{107}\] The constants defining \(B_*\) were fixed before \(B\).

Here is the complete order of parameter choices. Fix the kernel and jet parameters, the narrow width factor, the Maxwellian \(\mu\), and the short time \(t_1\) as in the preceding sections; also fix the early growth exponent \(p\) there. The bath estimate fixes \(B_0\). The proof of 26 fixes \(\kappa\) from \(\mu\), \(t_1\), and the two chosen balls, without imposing its entropy tolerance yet. The cold estimates then fix \(C_3,C_4\) uniformly for sufficiently small \(s_0\), while \(C_5=\|P(\mu,\mu)\|_H+1\) is already fixed. Choose \[B>\max\{B_0+2,B_*+3\}.\] For this \(B\), prescribe the entropy tolerance required by 26, and the possibly smaller entropy tolerance, hot tolerance \(\zeta\), and upper bounds on \(s_0\) required by 27. Include \(\theta_0,J_0\le1\) among these requirements.

To realize the entropy tolerance, take the hole radius \(\delta\) small and the bath velocity cutoff \(R_\mu\) large. Their contributions to the initial relative entropy can then be made arbitrarily small. With these parameters fixed, the jet and seed contributions become arbitrarily small as \(s_0\to0\), as proved in (47) and its derivation. To realize the prescribed \(\zeta\), use 24: choose its auxiliary time \(t_b\) and horizon upper bound \(T_0\) after \(\delta,\zeta\), and then take \(s_0\) sufficiently small for its source estimate. Decrease \(s_0\) further to meet the entropy, separation, cold-gain, and retirement requirements already listed. Reducing \(T_0\) when required by the early estimates preserves all later estimates. These are fixed construction parameters; none depends on \(N\), on the chosen cap solution, or on its horizon \(T\in(0,T_0]\).

For completeness, the strict estimate closes the bootstrap even at retirement times. At \(t_1\) it starts by (78). On an interval with no retirement, the characteristic equations make the weighted norm continuous, and (107) bounds it by \(B_*<B-3\) as long as it is at most \(B\). Thus it cannot first reach \(B\) there. At a retirement time \(L_s\), apply the argument only up to the left-hand section. The cold comparison on that interval and (93) give \[Z_w(L_s+)\le Z_w(L_s-)+J_s\le B_*+1<B-2.\] This starts the bootstrap on the next interval without having assumed its conclusion across the jump. The total density and its entropy bound are unchanged by this relabeling.

Finally, at each fixed approximation only finitely many scales occur. The cold barriers bound the sum of the active colors in \(H\) norm on every finite interval, with constants allowed to depend on that approximation. The uniform bound on \(w\) therefore bounds the total \(H\) norm on every such interval. The cutoff local existence and continuation argument from the construction excludes a finite maximal existence time. There are only finitely many retirements at that approximation, after which \(F=w\). This proves global existence and (105), with all the stated auxiliary estimates. ◻

Proof. Localize to a compact time-space set and a compact velocity neighborhood of \(\mathop{\mathrm{supp}}\psi\), using smooth cutoffs. Denote the localized functions by \(q_n\), extend them by zero, and set \(r_n=(1+\partial_t+v\cdot\nabla_x)q_n\). The equation shows that \(r_n\) is uniformly integrable in \(L^1\) and has support in a fixed bounded set of time and velocity, and also of space in the Euclidean case. Spatial cutoff derivatives cause no difficulty because velocity has been localized. Integration from a time preceding the support gives \[q_n(t,x,v)=\int_0^\infty e^{-s}r_n(t-s,x-sv,v)\,\mathrm ds=:Rr_n(t,x,v).\] The damped inverse \(R\) is an \(L^1\) contraction, since free transport preserves Lebesgue measure.

Truncate \(r_n\) at height \(L\), preserving its sign, to obtain \(r_n^{(L)}\). Uniform integrability implies \[\sup_n\|r_n-r_n^{(L)}\|_1\longrightarrow0 \quad(L\to\infty),\qquad \sup_n\|r_n^{(L)}\|_2<\infty \quad\text{for fixed }L.\] The velocity averages of \(R(r_n-r_n^{(L)})\) are therefore uniformly small in \(L^1\). It remains to prove compactness for a bounded \(L^2\) family of right-hand sides.

Take Fourier transform in \((t,x)\), with Fourier series in the torus case. For \(u_n=Rr_n^{(L)}\), Cauchy–Schwarz yields \[\left|\int\psi(v)\widehat u_n(\sigma,k,v)\,\mathrm dv\right|^2 \le I_\psi(\sigma,k) \int|\widehat r_n^{(L)}(\sigma,k,v)|^2\,\mathrm dv, \qquad I_\psi(\sigma,k)=\int \frac{|\psi(v)|^2}{1+(\sigma+k\cdot v)^2}\,\mathrm dv.\] If \(\mathop{\mathrm{supp}}\psi\subset\{|v|\le R_0\}\), integration in the direction of \(k\) gives \(I_\psi\le C_\psi/|k|\) for \(k\ne0\). When \(|\sigma|>2R_0|k|\), the denominator instead gives \(I_\psi\le C_\psi/(1+|\sigma|^2)\). Thus \(I_\psi\to0\) uniformly as \(|\sigma|+|k|\to\infty\). Plancherel’s theorem shows that the averages have uniformly small high-frequency \(L^2\) tails. Convolution with a fixed smooth approximate identity consequently approximates them uniformly in \(L^2\). On every compact time-space set these convolutions are bounded and equicontinuous, by Cauchy–Schwarz applied to translates of the convolution kernel and its derivatives. They are relatively compact there. This proves local \(L^2\), and hence local \(L^1\), compactness at fixed \(L\); the uniform \(L^1\) error as \(L\to\infty\) completes the proof. The argument applies unchanged to the dual lattice of a rotated torus. ◻

Proof. The supports of the free cores lie in their disjoint spatial enlargements through physical time \(r_N\); no retirement occurs on this interval for large \(N\). The exact Jacobian formula (128), or its version before centering a periodic copy, gives the moment estimate because \(r_N/(r_Nt+s_j)\le1/(t+l_N)\). The sheet gain estimate gives the second bound.

In the upper cold estimate, the main-color remainder contributes at most \(C r_Ns_j^{15/2}H\) to its gain against \(H\). The sum of the nonmain contributions has a vanishing Gaussian majorant after multiplication by \(r_N\). Uniformly over all positions, therefore, \[P(C_{r_N},H)\le P(A_N,H)+o(1)H.\] More precisely this inequality follows by applying the positive gain to the nonnegative upper error in \(C_{r_N}\le A_N+R_N\), where \(P(R_N,H)\le o(1)H\); it does not assert a signed total-variation comparison at all scales. On the present time range \(h_{r_N}\le2H\). Drop every nonnegative loss in (130), use \(P^{b_N}\le P\), and bound the positive bath, hot, and source terms exactly as in (131). This proves (135) with an error independent of \(t,x,v\).

All finite approximations obey their equations in free coordinates. Fubini’s theorem gives the comparisons along almost every free path and on full-measure endpoint sets at every specified terminal time. The countably many approximation indices and lattice translations can be treated simultaneously. This is the form of the comparison used in the next section. ◻

Proof. Use the free-core measure from Lemma 32. More explicitly, if the main retained label at \((r_nt,r_nx)\) is \(i\), set \[A_n(t,x,\,\mathrm dp)=r_n f_i(r_nt,r_nx,p)\,\mathrm dp, \qquad f_i(t,x,p)=c_i^0(x-tp,p),\] and otherwise set \(A_n=0\). The disjointness in Lemma 14 holds on the whole physical interval \([0,r_n]\) for all sufficiently large \(n\), and no color is retired there. Free transport therefore defines \(A_n\) on \([0,1]\), regardless of \(U_n\). Equation (135) gives (137). Its error already includes the bath, the source divided by \(m_{T_n}=T_n^p\), the rescaled hot self-gain, and the difference between actual and free cold cores.

The enlargement of a main transverse annulus remains between its original radial boundaries. Thus, relative to its period point, \(\varrho<|y|<e^\eta\varrho\), where \(\varrho=s/r_n\). The change of velocity variables in (56), multiplied by \(r_n\), is exactly (138); equivalently, this is the parameter formula in (128) before velocity testing. For \(0\le S\le1\), \[\frac{KS}{\varrho+t\alpha\gamma S} \le \frac{C}{t+\varrho}.\] Indeed, multiplying by \(t+\varrho\) bounds the \(\varrho\)-term by \(K\) and the \(t\)-term by \(K/(\alpha\gamma)\). The smallest scale is \(\ell_n\), so the amplitude is at most \(C/(t+\ell_n)\).

The implicit axial map depends only on \(w_3\), and its monotonicity gives \(|p_3|\le1+\alpha+d_s|w_3|\). The smooth compactly supported \(J\) has a fixed product Gaussian majorant. Lemma 8 therefore dominates each core by a sheet, with the preceding amplitude and a fixed axial moment bound. The gain bound in Lemma 6 proves (i). The factor \(4\) in (139) is the product of the linear gain coefficient \(2\) and the coefficient \(2\) in the collision-plane formula of Lemma 3. Lemma 7 now gives (iii); Lemma 8 also controls Gaussian parameter tails before the compressed axial change of variables. For the exact free cores the parameters already lie in \(\mathop{\mathrm{supp}}J\). All these estimates allow measurable masks bounded by one, which can only decrease the nonnegative integrals. ◻

Proof. At any one gain, a removed normalized tail has uniformly small mass by Lemma 34. Bound the other four gains and the input by (141), and sum over the five possible locations of a removed tail. This proves the first assertion. The same argument at the first gain, using its uniform decay for large outgoing velocities, proves (i).

Retain bounded endpoint velocities and bounded increments. Then all velocities are bounded by a fixed \(V\), and every collision position is within distance \(V\) of the endpoint, since the total free-flight time is at most one. Suppose \[d_n:=\mathop{\mathrm{dist}}(x_n,r_n^{-1}\Lambda)\longrightarrow\infty.\] We show that the first kernel becomes uniformly small on these paths. A main core of dilated scale \(\varrho\ge M\) has normalized mass at most \(C/M\), by (138) and the sheet bound. Here \(M\) can be chosen arbitrarily large, independently of \(n\). For a main core with \(\varrho\le M\), let \(q/r_n\), \(q\in\Lambda\), be its period point. Write \((y,z)\) for the first collision position relative to this point. Its transverse coordinate obeys \(|y|\le e^\eta M\), while its distance to this point is at least \(d_n-V\). Consequently \(|z|\to\infty\), uniformly over all encountered cores of these scales. For bounded \(p_3,t\), their parameter amplitudes satisfy \[ \frac{K S(\gamma(z-tp_3)/\varrho)} {\varrho+t\alpha\gamma S(\gamma(z-tp_3)/\varrho)} \le \frac K\varrho S(\gamma(z-tp_3)/\varrho)\longrightarrow0 \tag{144}\] uniformly for \(0<\varrho\le M\). In fact, when \(|z-tp_3|\ge Z\), the right side is at most \(4K\varrho^{-1}e^{-2\gamma Z/\varrho}\), whose supremum tends to zero as \(Z\to\infty\). Applying the sheet bound with this vanishing amplitude proves the assertion for \(\varrho\le M\). The remaining gains have bounded mass. First let \(n\to\infty\), then \(M\to\infty\), and finally remove the initial tail error. This proves (ii).

It follows that a sequence with a positive lower bound for its five-fold integral has bounded velocities and bounded distance from \(r_n^{-1}\Lambda\), after subselection. Choose period points \(q_n/r_n\) realizing a bounded distance, and translate the endpoints by these periods. The resulting positions have a convergent subsequence. Such translations preserve both the kernels and the periodic hot inputs used below.

We record also the geometry along the translated bounded paths. Every physical main enlargement for times \(r_nt\le r_n\) lies in a fixed ball of radius less than \(1/2\) about its period point, for sufficiently small \(s_0,r_n\). Distinct points of \(\Lambda\) are at least distance one apart. A path in a bounded dilated set lies at physical distance \(O(r_n)\) from the chosen origin, so for all large \(n\) it can encounter only that copy. Its encountered scales satisfy \(s/r_n\le C\), by the transverse annular restriction. Thus the largest physical width along these paths is \(O(r_n^{10})\), which tends to zero. ◻

Proof. By (24), we may work on \(|p|+|\xi-v|\leq R\) and remove that restriction at the end of each argument. On this region the exponential and polynomial factors in (20) are bounded by a constant depending on \(R\).

For (145), on \(|\bar\xi|\geq\delta\) the factor \(W(\xi)\) is bounded by \(1+\delta^{-q}\). The regular line estimate and (15) therefore give \[C_{R,\delta}\frac{|\bar v|^{-1}}{1+|\bar v|^{-q}} \leq C_{R,\delta}|\bar v|^{q-1} \qquad(0<|\bar v|\leq1).\] For (146), integrate in \(\xi_3\) first, using \(|v_3-p_3|^{-1}\leq\delta^{-1}\) and (26). The bound is \(C_{R,\delta}/W(v)\leq C_{R,\delta}|\bar v|^q\). These are uniform in the axial law. For (147), discard the bounded exponential and polynomial factors and apply (12). This also proves (148) after the tail removal.

It remains to prove (149). Fix the truncation radius \(R\). The part \(|\bar p|>r_0/4\) is uniformly negligible as the transverse widths tend to zero. Indeed, \[\mathbf 1_{|\bar p|>r_0/4}\mathcal G_d(\,\mathrm d\bar p) \leq2e^{-r_0^2/(32d^2)}\mathcal G_{\sqrt2d}(\,\mathrm d\bar p),\] and Lemma 6 applies to the wider Gaussian for all sufficiently small \(d\). Now retain \(|\bar p|\leq r_0/4\). Write \(n=\bar v-\bar p\), \(A=(n/|n|)\cdot\bar v\), and \(B=(v_3-p_3)/|n|\), as in Lemma 4. Then \[|n|\geq\tfrac34r_0,\qquad A\geq\tfrac35r_0.\] If \(|v_3-p_3|\leq r_0^2/(8R)\), then for \(|\xi_3-v_3|\leq R\), \[|B(\xi_3-v_3)|\leq\tfrac16r_0.\] The transverse collision line stays at distance greater than \(r_0/3\) from zero, and cannot meet \(|\bar\xi|<\delta\) when \(\delta<r_0/3\). In the complementary case, \(|v_3-p_3|^{-1}\leq8R/r_0^2\). Integrating first in \(\xi_3\) bounds the small-input contribution by \[C_{R,r_0}\int_{|\bar\xi|<\delta} (1+|\bar\xi|^{-q})\,\,\mathrm d\bar\xi \leq C_{R,r_0}(\delta^2+\delta^{2-q}).\] First let the widths tend to zero, then \(\delta\) tend to zero, and finally remove the truncation. This proves the last assertion. ◻

Proof. The profile satisfies \[\partial_z u_{\varrho,\sigma} =\frac{(\alpha\gamma/\varrho)S} {1+(t\alpha\gamma/\varrho)S}\ge0, \qquad \partial_tu_{\varrho,\sigma} +u_{\varrho,\sigma}\partial_zu_{\varrho,\sigma}=0.\] For fixed \(t_1,t_2,z_1\), the function \(F(w)=w-u_{\varrho,\sigma}(t_2,z_1-\Delta w)\) has derivative \(F'(w)\ge1\). Its root is \(u_{\varrho,\sigma}(t_1,z_1)\): the defining implicit equation shows that this velocity is unchanged along the straight segment between the two points. Integrating \(F'\ge1\) proves (150).

Fix \(0<\delta<\alpha\). If \(u_{\varrho,\sigma}\) is within \(\delta\) of an endpoint speed, there is nothing to prove. Otherwise \[\left|\frac{u_{\varrho,\sigma}-\sigma}{\alpha}\right| \le1-\frac{\delta}{\alpha}, \qquad |z-tu_{\varrho,\sigma}| \le\frac{\varrho}{\gamma} \operatorname{arctanh}(1-\delta/\alpha).\] It follows that \[|u_{\varrho,\sigma}-z/t| \le\frac{\varrho}{\gamma a} \operatorname{arctanh}(1-\delta/\alpha).\] First choose \(\delta\), then \(\varrho\) small. This proves the uniform assertion in (ii). Finally, the function \[p\longmapsto p-\sigma-\alpha\tanh\bigl(\gamma(z-tp)/\varrho\bigr)\] has derivative at least one. Its values at \(u_{\varrho,\sigma}\) and at the perturbed \(p_3\) are \(0\) and \(d_sw_3\). This proves (151). ◻

Proof. Remove normalized parameter and increment tails at an arbitrarily small cost, using Lemma 34. On the retained two-gain paths, all velocities and positions lie in fixed compact sets, the times lie in \([a,1]\), and every active scale satisfies \(0<\varrho\le M\). In particular the largest possible physical width is \(d_0(r_nM)^{10}\to0\).

Write \(t_1,x_1=(y_1,z_1)\) for the first collision point, and \(\xi\) for its input velocity. The next point has axial coordinate \(z_2=z_1-(t_1-t_2)\xi_3\). All estimates below are uniform when \(t_1,x_1\) vary in their retained compact sets. At the first gain, Lemma 36 supplies a common modulus \(\omega(\rho)\downarrow0\) for the mass of any axial interval of length \(\rho\); the interval’s center is arbitrary. It also says that for each fixed \(b>0\), the first-kernel mass on \(\{|\bar\xi|>b\}\) tends to zero as \(\bar v_n\to0\).

Here is an order of choices that handles all scales at the second point. Fix an error \(\epsilon>0\), including the fixed factors from the two time integrations and the bound \(C/a\). Choose \(\delta>0\) so that the sum of the first-kernel masses of the \(\delta\)-neighborhoods of the five numbers \[ z_1/t_1,\quad 1-\alpha,\quad1+\alpha,\quad -1-\alpha,\quad-1+\alpha \tag{152}\] is at most \(\epsilon\). This is possible uniformly because there are five intervals and their lengths are \(2\delta\). Lemma 38 then gives \(\varrho_0>0\), chosen uniformly on the retained compact sets, so that every profile with \(\varrho<\varrho_0\) is within \(\delta/4\) of its corresponding three collapsed speeds.

The geometric mesh has a uniformly bounded number \(J_0\) of scales in \([\varrho_0,M]\). Choose \(b_0>0\) so small that the first-kernel mass of the union of the \(b_0\)-neighborhoods of \[\{u_{\varrho,\sigma}(t_1,z_1): \varrho\in[\varrho_0,M]\text{ in the current mesh},\ \sigma=\pm1\}\] is at most \(\epsilon\). For example it suffices that \(2J_0\omega(2b_0)\le\epsilon\), after incorporating the fixed amplitude bound into \(\omega\). This list may depend on \(n\); its cardinality bound does not. Its centers may depend on \(t_1,z_1\), which are held fixed when the first kernel is integrated. Thus neither dependence weakens the interval bound.

Let \(E_n(t_1,z_1)\) be the union of the neighborhoods of the five collapsed speeds in (152) and the neighborhoods of the finite-scale profile speeds just chosen. Outside \(E_n\), every retained cold speed at the second point has an axial mismatch bounded below by a fixed positive number for all large \(n\). For \(\varrho\ge\varrho_0\), this follows from (150) and (151), with lower bound \(b_0/2\). For a constant collapsed speed it follows immediately, with lower bound \(\delta/2\). For the remaining branch use the exact identity \[\xi_3-\frac{z_2}{t_2} =\frac{t_1}{t_2} \left(\xi_3-\frac{z_1}{t_1}\right);\] the factor \(t_1/t_2\ge1\) preserves the lower bound. The errors \(\delta/4\) in small-scale approximation and the vanishing thermal errors can be absorbed. We have therefore obtained, on the complement of \(E_n\), \[|\xi_3-p_3^{(2)}| \ge c_*:=\tfrac14\min(\delta,b_0)>0\] for every retained second-core parameter.

The second-kernel axial mismatch estimate in Lemma 36 now tends uniformly to zero as \(|\bar\xi|\to0\). Choose \(b>0\) so small that this second kernel has mass at most \(\epsilon\) whenever \(|\bar\xi|\le b\) and the displayed mismatch holds. For this fixed \(b\), the first-kernel mass on \(|\bar\xi|>b\) tends to zero. Its mass on \(E_n\) is at most \(2\epsilon\), while the remaining first-kernel mass is bounded. Combining the three regions, then removing the initial tail error, gives a bound tending to zero as \(\epsilon\downarrow0\). All choices preceded the final passage \(n\to\infty\), and were uniform in the two times. This proves the lemma.

In a longer integral, all factors following these two gains are bounded by (141); the same proof applies. ◻

Proof. Write \(L=L^\perp+\lambda e\). The map \(L^\perp\mapsto n\times L^\perp\) multiplies two-dimensional area by \(|n|^2\). Integration in \(\lambda\) contributes \(\int\zeta=1\), proving (153) and the support and density bounds. For successive gains, fix all previous parameters before applying the identity. The current normal is then fixed; Fubini’s theorem for nonnegative functions justifies iteration. No derivatives of the variable normals enter this conditional change of variables. ◻

Proof. Boundedness and convergence in measure on the finite-volume set \(B\) imply \(\|F_n\|_{L^1(\mathbb R^d)}\to0\): for every \(\eta>0\), bound the integral by \(\eta|B|+M|\{|F_n|>\eta\}|\).

Fix \(\epsilon>0\). The critical set in \(K\) has measure zero, so choose an open neighborhood \(E\) of it with \(|E\cap K|<\epsilon\). The compact set \(K\setminus E\) consists of regular points. At such a point \(q_0\), choose \(d\) input coordinates giving a nonzero minor of \(D\Phi(q_0)\). Let \(\pi\) be projection onto the complementary \(k-d\) coordinates, and define \[G(q)=(\Phi(q),\pi q),\qquad G_n(q)=(\Phi_n(q),\pi q),\qquad A_0=DG(q_0).\] The matrix \(A_0\) is invertible. Choose a convex ball \(V\Subset\mathcal O\) about \(q_0\) so small that, for all sufficiently large \(n\), \[\sup_{q\in V}\|A_0^{-1}DG_n(q)-I\|<\tfrac12.\] Integration on the line segment between \(q,q'\in V\) gives \[|A_0^{-1}(G_n(q)-G_n(q'))-(q-q')| \le\tfrac12|q-q'|.\] Thus \(G_n\) is injective on \(V\), and its Jacobian has a positive lower bound \(c_V\) independent of \(n\). The inverse function theorem consequently makes it a diffeomorphism onto its image.

Cover \(K\setminus E\) by finitely many such balls \(V_j\). The change-of-variables formula gives \[\begin{split} \int_{V_j\cap K}|F_n(\Phi_n(q))|\,\mathrm dq &\le c_{V_j}^{-1} \int_{G_{n,j}(V_j\cap K)}|F_n(z)|\,\mathrm dz\,\,\mathrm dw\\ &\le c_{V_j}^{-1}|\pi_j(V_j)| \|F_n\|_{L^1(\mathbb R^d)}. \end{split}\] When \(k=d\), the zero-dimensional factor has value one. The integral on \(E\cap K\), including its density, is at most \(AM\epsilon\). Multiplying the regular-chart estimates by \(A\) and summing over the finite cover gives \[\limsup_{n\to\infty} \int_{\mathcal O}|F_n(\Phi_n(q))a_n(q)|\,\mathrm dq \le AM\epsilon.\] Let \(\epsilon\downarrow0\). This proves (154). The argument used the densities only through \(|a_n|\le A\), so arbitrary measurable masks and restrictions may be included in them without any convergence assumption. ◻

Proof. For \(t>0\), the derivative with respect to \(u\) in the defining implicit equation is \[1+\frac{t\alpha\gamma}{\varrho} S\bigl(\gamma(z-tu)/\varrho\bigr)>0.\] The implicit analytic function theorem gives an analytic profile for every \(z\in\mathbb R\). Uniqueness, which also follows from this strict monotonicity, identifies the local profiles on overlaps. The recursive formulas (155) are consequently analytic. They make sense even without ordering the positive times: a negative formal free-flight duration merely changes the point at which the next analytic profile is evaluated. This supplies the connected open domain stated in the lemma.

We exhibit one rank-seven point. Fix \(t_5=\tau\) with \(0<\tau<t_0\), and set all five collision times equal to \(\tau\). After a rotation about the velocity axis, write \(v_0=(R,0,z)\), \(R>0\). All collision positions now coincide at \(x_0-(t_0-\tau)v_0\), independently of the angular parameters. Thus write the five axial velocities simply as \(p_i=a_i e_3\); their values \(a_i\) may be arbitrary.

Choose \[k_1=(0,1,0),\qquad k_2=(1,-R,0).\] They satisfy \(k_1\perp n_1\), \(k_2\perp n_2\), and span the transverse plane. Put \[w=\bar v_2=(R+1,1-R)\ne0,\qquad b_3=v_{2,3}-a_3,\] and choose a sufficiently small nonzero number \(\epsilon\) and \[k_3=\epsilon\left(-\frac{b_3w}{|w|^2},1\right).\] Then \(k_3\perp n_3\), its axial component is nonzero, and \(\bar v_3\ne0\). Hence \(k_1,k_2,k_3\) are linearly independent. Writing \(\bar v_3=(X,Y)\), take \[k_4=(-Y,X,0),\qquad k_5=0.\] The fourth increment is perpendicular to \(n_4\), and \[\det(\bar n_4,\bar n_5)=X^2+Y^2>0.\] Thus \(n_4,n_5\) are nonparallel. All the normals in this construction are nonzero, and every chosen increment can be represented by \(L_i=-n_i\times k_i/|n_i|^2\). Choose \(L_5=0\).

At the equal-time configuration, \[x_5=x_0-v_0(t_0-t_1) -\sum_{i=1}^4v_i(t_i-t_{i+1}).\] All derivatives of \(x_5\) in angular directions vanish. Keeping \(t_5\) fixed, its first three time columns are \[\partial_{t_1}x_5=-k_1,\qquad \partial_{t_2}x_5=-k_2,\qquad \partial_{t_3}x_5=-k_3.\] Derivatives of the \(v_i\), including derivatives of the position-dependent \(p_i\), are multiplied by zero free-flight durations in this formula. In the final velocity block, \[\delta_{L_4}v_5=n_4\times\delta L_4,\qquad \delta_{L_5}v_5=n_5\times\delta L_5,\] because \(L_5=0\) and the \(p_i\) have zero angular position derivatives at equal times. The sum of the planes \(n_4^\perp+n_5^\perp\) is \(\mathbb R^3\). Choose three angular columns spanning this sum. Together with the three time columns above, they give an invertible \(6\times6\) block for \((x_5,v_5)\): its upper-right block is zero and both diagonal blocks are invertible. Appending the \(t_5\) column and target coordinate gives a nonzero \(7\times7\) minor.

This minor is a nontrivial analytic function on the connected domain. Its zero set has Lebesgue measure zero. For completeness, at any zero of a nontrivial analytic function some derivative of finite order is nonzero; otherwise its Taylor series vanishes on a neighborhood, and analytic continuation through overlapping balls in the connected domain makes the function identically zero. Choosing a derivative of minimal nonzero order puts the point on a regular zero set of a derivative of one lower order. There are countably many such derivatives, and each regular zero set is covered by countably many smooth hypersurface charts. Their union is null. Rank failure is contained in the zero set of our particular minor, so it is null as well. Continuity also shows directly that the witness persists at nearby strictly ordered times. It need not obey any spatial mask.

For the prelimit maps replace \(p_i\) by (138) and include \(w_i\in\mathbb R^3\) among the parameters. On compact sets the scales have positive lower bounds, \(d_{r_n\varrho_i}\to0\), and all implicit derivatives have the same positive lower bound one. Differentiating the implicit equations, or applying their implicit function formulas, gives \(C^1\) convergence in the times, positions, and all bounded \(L_i,w_i\). The limit is \(\Phi\), independent of the fifteen spectator \(w_i\)-coordinates. Its rank is still seven almost everywhere on the enlarged parameter domain, by Fubini. Lemma 41 now proves the final assertion. ◻

Proof. Fix an arbitrarily small error tolerance. Remove the tails of Lemma 35. All path velocities and positions are now in fixed bounded sets. At every encountered main core, \(s\le Cr_n\), so its transverse width tends uniformly to zero.

Starting from a positive lower bound for the endpoint transverse speed, apply the nonaxial small-input estimate in Lemma 36 successively at gains \(1,\ldots,5\). Choose positive numbers \(b_1,\ldots,b_5\) so that deleting \(|\bar v_i|<b_i\) costs less than the prescribed error at step \(i\). At each step the previous lower bound has already been fixed; the later gain masses are bounded by (141). Thus this is a finite successive choice, followed by \(n\to\infty\). The sum of the five deleted contributions is arbitrarily small. Write \(b>0\) for a lower bound on all retained transverse speeds, including \(v_0\). For large \(n\), every retained normal \(v_{i-1}-p_i\) has length at least \(b/2\).

Next delete collision points with \(|y_i|<\rho\). Conditional on all earlier parameters, \(y_i\) is an affine function of \(t_i\) with derivative \(\bar v_{i-1}\). Consequently \[\left|\{t_i:|y_i|<\rho\}\right|\le\frac{2\rho}{b}.\] Integrate the current kernel using its mass bound \(C/a\), and bound the later kernels by (141). Summing over the five steps bounds this deletion by \(C_a\rho/b\). Choose \(\rho>0\) after the speed thresholds. The retained scales now belong to a compact subinterval \([\varrho_-,\varrho_+]\subset(0,\infty)\). Mesh convergence gives a fixed finite list of scales on this interval, with positive limits, after an immaterial enlargement of its endpoints. There are consequently only finitely many five-tuples of scales and signs to consider.

The tested time \(t_5\) must remain inside the weak-limit interval. If \(U>1\), it already lies in the compact interval \([a,1]\Subset(0,U)\). If \(U=1\), delete \(t_5>1-\delta\). Ordering forces all five times into \((1-\delta,\tau_n)\), whose length is at most \(\delta\). This costs at most \[ 2(C/a)^5\frac{\delta^5}{5!}. \tag{156}\] Choose \(\delta>0\) to make this as small as desired. No convergence at the event time is being asserted.

We also remove a thin neighborhood of the boundary of the ordered time region. Put \(t_0=\tau_n\), and discard paths on which \(t_5-a<\kappa\) or \(t_{i-1}-t_i<\kappa\) for some \(1\le i\le5\). Integrating velocities successively with the bound \(C/a\), their total contribution is at most \(C_a\kappa\): each of the six time strips has five-dimensional volume at most \(\kappa\) in the unit cube. Choose \(\kappa>0\) after the preceding tolerances. On the retained set all consecutive time gaps are at least \(\kappa\), and the lower time exceeds \(a+\kappa\). Since \(\tau_n\to1\), the retained times lie in a fixed compact subset of \(0<t_5<\cdots<t_1<1\) for all large \(n\).

Fix one retained scale/sign tuple. Drop all spatial masks in constructing its maps, and retain the masks in its nonnegative integrand. Each plane increment is bounded, all normals are bounded above and below, and each cold parameter is bounded. Apply Lemma 40 successively. The integral becomes an integral over five times, fifteen \(L\)-coordinates, and fifteen cold parameter coordinates. Its support lies in a fixed compact subset of \((0,\infty)^5\times\mathbb R^{30}\), and its density is uniformly bounded: the core amplitudes are at most \(C/a\), the factors \(|n_i|\zeta(L_i\cdot n_i/|n_i|)\) are bounded, and all retained ratios \(H(v_i)/H(v_{i-1})\) are bounded. The masks, time ordering, and cut conditions are arbitrary measurable factors between zero and one in this density.

The target \((t_5,x_5,v_5)\) on the actual support lies in a fixed compact set \(B\) inside the open weak-limit region, away from the velocity axis. Set \(F_n=f_n\mathbf 1_B\mathbf 1_{\{a\le t\le\tau_n\}}\) and extend it by zero; it is bounded by two and tends to zero in measure. The maps converge in \(C^1\) to the map in Lemma 42. That lemma and Lemma 41 show that the integral for this tuple tends to zero. Notice that the entire parameter compact need not map into \(B\); only the support of the measurable density must do so.

There are finitely many tuples, so their sum tends to zero. The total limsup of the original integral is bounded by the arbitrarily small errors from the preceding deletions. Letting those tolerances tend to zero proves the lemma. ◻

Completion of the proof of Proposition 33. Assume \(l=0\) and \(g=0\), as in the reduction above. Choose endpoints \((x_n,v_n)\) at time \(\tau_n\) in the common full-measure set fixed above such that \(f_n(\tau_n,x_n,v_n)\ge1/2\). For each fixed \(a\), the residual in (143) tends to zero along any subsequence of these endpoints. Indeed, Lemma 35 first excludes large velocities and escaping positions. Otherwise choose convergent endpoints after a periodic translation. If the limiting velocity is axial, the first two gains vanish by Lemma 39, and the last three have bounded mass by (141). If it is nonaxial, use (142) in Lemma 43. This covers every subsequence; hence the residual tends to zero for the original endpoint sequence as well.

Taking the limsup in (143) at these endpoints gives \[\frac12 \le a\sum_{j=0}^4\frac{(C\log(1/a))^j}{j!}.\] Finally let \(a\downarrow0\). The right side tends to zero, a contradiction. Throughout this argument \(U\) was allowed to be infinite, and the case \(U=1\) was handled by (156). This proves the proposition. ◻

Proof. Set \(G=2P(\widetilde B,f)\). In free coordinates \(f^\sharp(t,x,v)=f(t,x+tv,v)\), the equation becomes \[\partial_t f^\sharp=(G-\nu f)^\sharp.\] On bounded tubes its right side is locally integrable, so \(f^\sharp\) has absolutely continuous time sections for almost every \((x,v)\) and local \(L^1\) time traces. The rate bounds make \(\nu^\sharp\) integrable along every bounded characteristic segment in a compact positive-time tube. The one-dimensional integrating-factor formula therefore consists of the freely transported initial density multiplied by its survival factor, plus the time integral of the nonnegative gain with the same survival factor. Repeated substitution into the gain gives (159), by (157) and (158). Every remainder is nonnegative. Restricting the histories, or the initial density, preserves the lower bound. Tonelli’s theorem justifies the substitutions. On bounded portions with total gain rate at most \(C\), the \(j\)-collision term is bounded by \(C^j(b-a)^j/j!\) times the initial mass.

These formulas apply to point-velocity partners. Indeed Lemma 3 gives a density for the gain against each such partner, with mass (158). Thus free flight and each gain preserve absolute continuity when the starting measure has a density. Null spatial sets in representatives of the coefficients do not affect the integrated formula. ◻

Proof. Set \(s=|v|^2\), \(z=v_3\), and \(R=\sqrt{(s+1)^2-4z^2}\). For either partner sign the squared child speed has the law \[|v'|^2=\frac{s+1+RZ}{2}, \qquad Z\text{ uniform on }[-1,1].\] This follows by expanding the square in (157): a coordinate of a uniform point on \(S^2\) is uniform on \([-1,1]\). The maximum is at most \(s+1\), so there is no exit for \(s\le24\). For \(24<s\le25\), the exit probability, if positive, is \[\frac12+\frac{s-49}{2R} \le\frac12+\frac{s-49}{2(s+1)} =\frac{s-24}{s+1}\le\frac1{26}.\] The inequality uses \(R\le s+1\) and \(s-49<0\). If \(R=0\), the squared speed is deterministic and at most 25. Mixing the signs preserves the bound.

A Markov history of rate \(\lambda\) has the survival factor \(e^{-\int\lambda}\), and one rate times jump-law factor per collision. Multiplication by \(e^{\int(\lambda-\nu_0)}\) changes this exactly to (159), restricted to \(E\). Both history sums converge absolutely on finite intervals because their rates are bounded. Finally \(4\pi\le\nu_0(v)\le24\pi\) on \(E\), and \(2p(v)-1\ge12/13\), giving (161). ◻

Proof. Given \(v\in E\), choose a sign with \(|v-\sigma e_3|\ge1\). Use this partner for the first collision, and select an output within a small fixed distance \(\delta\) of \(\sigma e_3\). Its probability has a uniform positive lower bound: the incident distance lies in \([1,6]\), and a cap of fixed Euclidean radius \(\delta\) at that endpoint has area probability bounded below uniformly on the corresponding diameter spheres. Such first outputs lie strictly in \(E\).

Use partner \(-\sigma e_3\) next. If the first output is exactly \(\sigma e_3\), the second sphere is the unit sphere; restrict its output \(u\) near \(e_1\). For the third collision use partner \(e_3\). The last output is \[T(u,n)=\frac{u+e_3}{2}+\frac{|u-e_3|}{2}n.\] At \(u=e_1\), \(n_0=-(e_1+e_3)/\sqrt2\), this output is zero. The two angular derivatives in \(n\) span \((e_1+e_3)^\perp\). The vector \(e_3\) is tangent to the preceding unit sphere at \(e_1\), and direct differentiation gives \[ \partial_{e_3}T(e_1,n_0)=\frac{e_1+3e_3}{4},\qquad (e_1+e_3)\cdot\partial_{e_3}T(e_1,n_0)=1. \tag{163}\] Thus the map of the four angular coordinates in the last two collisions to the last velocity has rank three.

Take the two coordinates of \(n\) and the coordinate of \(u\) with derivative \(e_3\) as distinguished variables, and hold the fourth coordinate in a short interval. The inverse function theorem, with this fourth coordinate and the first output as parameters, gives neighborhoods on which the distinguished-variable map is one-to-one, its Jacobian and inverse Jacobian are bounded, and its image contains a fixed ball \(B(0,r_0)\). Shrink these neighborhoods and \(\delta\) so this holds for every first output within \(\delta\) of \(\sigma e_3\), for both signs. Sphere area in these coordinates has a positive lower density. Change of variables in the three distinguished coordinates, and integration in the fourth, give a fixed positive multiple of volume on \(B(0,r_0)\).

Force the three times into three fixed, disjoint intervals inside \((0,1)\), and prohibit other jumps. The first selected relative speed is at least 1, the second is close to 2, and the third close to \(\sqrt2\). In the reference Markov process the selected transition intensity is \[4\pi|v-\sigma e_3|\, \mathbf 1_E(\Phi(v,\sigma e_3,n))\,\mathrm d\varsigma(n).\] It has positive uniform lower density on the patches just specified. The no-additional-jump factor is at least \(e^{-C_0}\). Integrating the times proves (162). Keeping only the first two collisions, selecting a small patch near \(e_1\) in the second, and prohibiting later jumps proves the last assertion. ◻

Proof. The normalized measure on the right side of (162) is a common part of positive mass \(\epsilon\) in all rows of \(P_1\). Removing this common part shows that the map \(\mathfrak m\mapsto\mathfrak m P_1\) contracts total variation distance by \(1-\epsilon\). Probability measures form a complete space in that metric, so iteration yields a unique fixed point \(\pi\) and uniform geometric convergence. Since \(P_t\) commutes with \(P_1\), \(\pi P_t\) is another fixed point of \(P_1\), hence equals \(\pi\). Contractivity of Markov kernels gives the estimate between integer times.

Negation of velocity interchanges the two signs and preserves \(E\), so uniqueness of \(\pi\) makes it symmetric and its mean zero. Consequently \(\sup_{v\in E}|\mathbb E_vV_t|\le Ce^{-ct}\). The Markov property and bounded velocities imply \[|\mathbb E(V_s\cdot V_t)|\le Ce^{-c(t-s)}\quad(0\le s\le t)\] uniformly over starting states. Integrating this estimate on \([0,A]^2\) bounds the second moment of \(\int_0^A V_s\,\mathrm ds\) by \(CA\). ◻

Proof. Put \(A_0=(1-\varkappa)A-1\). On the first interval of length \(A_0\), require the reference Markov flight to have integrated velocity of norm at most \(Ad\varkappa/4\). Lemma 48 and Chebyshev’s inequality give probability at least \(1/2\), uniformly in the start when \(A\) is sufficiently large. Over the next unit interval require the last velocity to lie in \(B(e_1,d/4)\). The conditional probability is at least a fixed \(\varepsilon>0\), by Lemma 47. Require no jump during the final interval of length \(\varkappa A\), whose conditional probability is at least \(e^{-C_0\varkappa A}\).

The errors in displacement from \(\varkappa Ae_1\) on the three intervals are at most \(Ad\varkappa/4\), \(D\), and \(Ad\varkappa/4\). For \(A>4D/(d\varkappa)\) their sum is less than \(3Ad\varkappa/4\). The final velocity lies strictly in \(E\). Lemma 46 bounds the branching weight below by \[\frac{\varepsilon}{2}\exp((c_0-C_0\varkappa)A).\] Its exponent is positive by (164). Choose \(L\) large enough that this exceeds 8, and all the preceding lower bounds on \(A\) hold throughout \(I_L\). ◻

Proof. Fix a reference start and duration \((v,A)\in E\times I_L\). The zero-collision term has weight \(e^{-\nu_0(v)A}\le1\). Discarding it from the family in Lemma 49 leaves weight greater than 7. Monotone convergence in the positive history count permits restricting to finitely many counts \(j\ge1\) while retaining weight greater than 7. Every postcollision velocity lies strictly in \(E\) almost surely: the intersection of a nondegenerate diameter sphere with \(\partial E\) has sphere area zero, and a degenerate collision has zero rate. This includes the final velocity. No inward margin is required for the initial velocity. The displacement inequality is strict. Ordered times may be kept apart and away from the endpoints. For each retained collision count and sign sequence, parameterize the ordered times as fractions of \(A\) and partition the angular parameter space into finitely many measurable pieces lying in sphere coordinate charts. Inner regularity of the resulting finite parameter measures allows a further restriction to compact sets with total weight greater than 6. These compact sets have positive distances from the forbidden time coincidences and from every postcollision speed boundary. They also have a positive margin in \(|Z_A-\varkappa Ae_1|<d\varkappa A\), since the reference histories obey the stronger inequality (165). We call these finitely many compact parameter sets a test for \((v,A)\).

The reference paths and weights are continuous in these finite parameters and in \((v,A)\). The fixed compact tests therefore retain weight greater than 5 and at least half their original positive margins in a sufficiently small relative neighborhood of that start and duration in \(E\times I_L\). These neighborhoods cover the compact set \(E\times I_L\). A finite subcover gives finitely many tests, a maximum count \(J_{\mathrm{br}}\), and uniform positive margins.

Use the same initial velocity, signs, directions, and fractional times for each perturbed history and its reference history. Their first flights have speed at most \(D\), including boundary starts. The sphere map satisfies \[|\Phi(v,w,n)-\Phi(\widetilde v,\widetilde w,n)| \le |v-\widetilde v|+|w-\widetilde w|.\] Induction bounds the velocity error after \(j\) collisions by \(j\varepsilon_*\) and the displacement error by \(AJ_{\mathrm{br}}\varepsilon_*\). This comparison does not compare the medium at two nearby positions: its value at the actual perturbed position is close to the same constant reference partner. A sufficiently small \(\varepsilon_*\) preserves all the strict velocity and displacement margins. The actual paths thus have speed at most \(D\), so they stay within the region on which closeness was assumed. Formally this follows by induction up to the first possible failure; the preserved margins exclude such a failure.

The relative-speed errors and full loss-rate errors on these histories are bounded by \(C_{J_{\mathrm{br}}}\varepsilon_*\). The density error gives the same bound for the gain-rate factors. The interval has bounded length, so the error in the integrated loss and its exponential is bounded by \(C_{J_{\mathrm{br}},L}\varepsilon_*\). Finite products of the bounded gain factors then show that (159) changes by at most \(C_{J_{\mathrm{br}},L}\varepsilon_*\) uniformly on each test. The tests have finite parameter measure. Decreasing \(\varepsilon_*\) makes their integrals greater than 3. All coefficient estimates were pointwise; measurability is sufficient. ◻

Proof. Choose \(t_{\mathrm{br}}\) so that \((1-\delta_K)(1-d)\varkappa t_{\mathrm{br}}>1\). Then every speed-\(D\) excursion in (169) stays outside the hole for every \(l\in[0,1]\) and every \(t\ge t_{\mathrm{br}}\). Put \(\Delta=a_{\rm br}t\). At each starting point of \(\mathcal B_t\), apply Lemma 50 using the simultaneous space–time scaling (171). Translating back its displacement restriction gives \[|x_{\mathrm{end}}-x_{\mathrm{start}}-\varkappa\Delta e_1| <d\varkappa\Delta.\] The starting radius \(d\varkappa t\), followed by the triangle inequality, places the endpoint in \(\mathcal B_{t+\Delta}\). The velocity belongs to \(E\): the selected positive-count histories have a strictly interior final velocity. Their first flight need only have speed at most \(D\), exactly as in the medium comparison. History weights are unchanged by this scaling: each gain rate is divided by \(\rho_0\), each time differential is multiplied by \(\rho_0\), and the integrated loss is invariant. The selected histories therefore have total weight greater than 3 uniformly over starts in \(\mathcal B_t\).

Use the kernel obtained by integrating all histories with \(1\le j\le J_{\mathrm{br}}\) that satisfy the block’s velocity and displacement restrictions. The finite compact tests above certify its weight greater than 3; no choice of a test for each starting point is needed. For each integer \(m\ge1\), integrate this positive kernel against \(\min\{g(t),m\}\mathbf 1_{\mathcal B_t}\). This is a bounded, integrable portion of the trace, so Lemma 45 applies. Letting \(m\to\infty\) and using monotone convergence gives (175). ◻

Proof. At some positive time \(a_0\), a bounded, integrable portion of \(g(a_0)\) has positive mass. Such a portion is obtained by restricting to bounded phase-space sets and truncating the density. The velocity axis has Lebesgue measure zero, so a further portion of positive mass satisfies \[|x|\le R,\qquad |v|\le R,\qquad |\bar v|\ge\delta>0.\] For a sufficiently long but finite free flight of length \(s\), \[|y+s\bar v|\ge s\delta-R>l+2\] on this portion. Its survival is bounded below by a positive constant by (160). Lemma 45 gives positive mass in a bounded region strictly outside the hole at time \(a_1=a_0+s\). This freely transported portion remains bounded and integrable.

Choose \(h>0\) small so that paths whose initial speed is at most \(R\) and whose later speeds are at most 3 remain in a bounded region with \(|y|>l+1\) during \([a_1,a_1+h]\). Both sign densities have strictly positive lower bounds there. To see this directly, the time, axial coordinate, and transverse radius are bounded, the transverse radius has a positive lower bound, and \(e^{-\eta}|y|\le\varrho\le|y|\). The argument of \(S\) is therefore bounded, its numerator is positive, and the denominator in (122) is bounded above. The rates on the selected bounded-speed paths also have finite upper bounds.

Confine three collisions to this short interval. Choose a reference sign with \(|v-\sigma e_3|\ge1\) for the first collision. The actual relative speed is at least \(1-\alpha\ge3/4\). Choose its output in a sufficiently small cap about the actual partner. The cap probability has a positive lower bound for inputs bounded by \(R\). Since the actual partner is within \(\alpha\) of \(\sigma e_3\), this first output lies in the reserved first-output neighborhood. Next use the signs \(-\sigma\) and \(+1\), and the compact angular tests defining \(\alpha_{\mathrm{acc}}\). Their final velocity lies in (168). The second output stays near \(e_1\), so the second and third relative speeds have positive lower bounds; all postcollision speeds are at most 3. Put the three times in disjoint subintervals and prohibit other collisions. Positive lower bounds for the selected gain factors, angular probabilities, time-interval lengths, and survival give strictly positive total weight. The sphere-map comparison used here requires only uniform partner closeness, and therefore remains valid for measurable coefficients.

There is consequently a positive portion at time \(a_2=a_1+h\), with bounded spatial support and \(|v-\varkappa e_1|<d\varkappa/4\). Truncating its density gives a bounded, integrable subportion of positive mass. Let \(R_2\) bound \(|x-\varkappa a_2e_1|\) on that portion. After free flight to any sufficiently large finite time \(t\), \[|x+(t-a_2)v-\varkappa te_1| \le R_2+\frac{d\varkappa}{4}(t-a_2)<d\varkappa t.\] Its speed is strictly less than \(D\). For each finite \(t\), survival remains strictly positive by (160). The zero-collision term of Lemma 45 proves the assertion. ◻

Proof of Proposition 44. Suppose that a nonzero density \(g\) solves (124) for all positive times and obeys the cap. Lemma 52 gives a time \(t_0\ge t_{\mathrm{br}}\) with positive mass \(m_0\) in \(\mathcal B_{t_0}\).

Apply Lemma 51 successively. At \(t_n=(1+a_{\rm br})^nt_0\), induction gives \[\int_{\mathcal B_{t_n}}g(t_n,x,v)\,\mathrm dx\,\mathrm dv\ge3^n m_0.\] The cap and \(l\le1\), on the other hand, give \[\int_{\mathcal B_{t_n}}g(t_n,x,v)\,\mathrm dx\,\mathrm dv \le (t_n+1)\frac{4\pi}{3}(d\varkappa t_n)^3\int_E H(v)\,\mathrm dv \le C(1+t_n)^4.\] This contradicts (174) as \(n\to\infty\). The positive mass \(m_0\) and starting time may depend on the particular density; their sizes are immaterial to this comparison. The parameters governing the blocks were fixed before the mesh and \(l\) were chosen. This rules out every nonzero global density in the proposition. Applying this conclusion to the nonzero limits of Proposition 33 gives \(U<\infty\). ◻