This paper proves that the periodic hard-sphere Boltzmann equation in three space dimensions is NOT well-posed among global entropy solutions that obey exact local conservation of mass, momentum, and kinetic energy. Concretely: there is a nonnegative initial density F_0 with bounded velocity support and finite mass, energy, and absolute entropy, and there are two global solutions F^(1) and F^(2) of the same Cauchy problem, both "entropy solutions with local conservation," both in C([0,inf);L^1), and differing on a set of positive spacetime-velocity measure. That is a counterexample to uniqueness inside a class that had been conjectured to be rigid. The mechanism is a self-similar family of "cold jets" whose scale s runs through a geometric progression; a tiny seed at the smallest scale decides which of two limits the solution tracks. The seed is invisible to the initial datum up to (but not including) the smallest scale, and the two global limits genuinely disagree.

The precise statement (Theorem 1.2 of the paper) is:

  There exist R<inf and a nonnegative datum F_0 with
  supp_v F_0  subset  B_R(0),
  int_Omega F_0 (1+|v|^2+|log F_0|) dx dv < inf,
  for which the hard-sphere Boltzmann equation (with periodic space T^3)
  has two global entropy solutions with local conservation (Definition
  1.1 below), differing on a set of positive (t,x,v) measure. Both are
  in C([0,inf); L^1(Omega)). For every finite T and every k>=0, both satisfy
  int_0^T int_Omega <v>^k [Q^+(F,F)+Q^-(F,F)] dx dv dt < inf.

The rest of this file unpacks every lemma, proposition, and theorem of the paper (13 LaTeX sections, ~30 numbered results) in a "learn by questions" format: one short question, one compact answer, then one or two citations. Everything you need to reconstruct the argument is here. If an answer is dense, back up to the previous exercise; the exercises follow the logical order of the proof.

Citations are given as  Author Year, Short title: URL  so you can go read the primary source.

Exercise: What is the Boltzmann equation, physically?
A gas is a large number of tiny particles moving freely, sometimes colliding elastically. Put a distribution function F(t,x,v) on T^3_x x R^3_v: for a measurable set A in phase space, int_A F dx dv is (approximately) the expected number of particles of the gas that are inside A at time t. Between collisions, each particle moves with constant velocity, so F satisfies the free transport equation d_t F + v.grad_x F = 0. Collisions act as a source and sink, so the true equation is d_t F + v.grad_x F = Q(F,F). The collision operator Q(F,F) is the only nonlinearity. This is Boltzmann's equation.
Boltzmann 1872, Weitere Studien uber das Warmegleichgewicht: https://en.wikipedia.org/wiki/Boltzmann_equation

Exercise: What is a hard-sphere collision?
Two incoming velocities v and v_* collide in an "impact parameter" direction omega in S^2 (the unit sphere). Then the outgoing velocities are v' = v - a omega and v_*' = v_* + a omega where a = (v - v_*).omega. This is the elastic collision rule: it conserves total mass (number of particles), total momentum v + v_*, and total kinetic energy (1/2)(|v|^2 + |v_*|^2). The hard-sphere collision kernel is |a| = |(v-v_*).omega|, the relative speed along the impact direction. The full collision operator is Q(F,F) = Q^+(F,F) - Q^-(F,F) where Q^+ = gain = intint |a| F(v') F(v_*') dv_* domega, and Q^- = loss = intint |a| F(v) F(v_*) dv_* domega = F(v) nu_F(v). nu_F is called the collision frequency.
Levermore-Masmoudi 2010, From Boltzmann to Navier-Stokes-Fourier: https://link.springer.com/article/10.1007/s00205-009-0254-5

Exercise: What is a "weak" or "renormalized" solution, and why does it matter here?
The classical PDE has solutions in smooth function spaces, but only for special data. DiPerna and Lions built a global existence theory for arbitrarily large data by using a weaker notion of solution. Renormalized solutions satisfy beta(F) for appropriate test functions, and are characterized by: continuity in L^1, finite mass/energy/entropy, and an entropy-production inequality. These are the physically natural objects because they include limits of smooth solutions, particle systems, and finite-volume schemes. The entropy inequality alone does NOT close the equation algebraically -- many distinct densities could satisfy it -- so uniqueness is a subtle add-on, not automatic. That is precisely the gap the present paper exploits.
DiPerna-Lions 1989, Cauchy problem for Boltzmann equations: https://annals.math.princeton.edu/1989/130-2/p03
DiPerna-Lions 1991, Global solutions and entropy inequality: https://link.springer.com/article/10.1007/BF00375684

Exercise: What exactly is an "entropy solution with local conservation" (Definition 1.1)?
It is the class the paper considers. A nonnegative density F(t,x,v) belongs to it if:
(i)   F in C_w([0,inf); L^1(Omega)) and F(0) = F_0, with a uniform bound sup_{0<=t<=T} int F(t)(1+|v|^2+|log F(t)|) dx dv < inf for each finite T;
(ii)  renormalized equation: for every C^1 beta with beta(0)=0, |beta'|<= C/(1+z), and every smooth test phi periodic in x and compactly supported in velocity,
      int_0^inf int_Omega [ M beta(G) (d_t phi + v.grad_x phi) + beta'(G) Q(F,F) phi ] dx dv dt + int_Omega M beta(F_0/M) phi(0,x,v) dx dv = 0,
      where G = F/M and M(v) = (2pi)^{-3/2} e^{-|v|^2 / 2};
(iii) total momentum is non-increasing, |v|^2 moment and entropy-dissipation inequality hold: |v|^2 mass non-increasing, H_M(F(t)) + int_0^t D(F(s)) ds <= H_M(F_0);
(iv)  int_0^T int_Omega (1+|v|^3) F dx dv dt < inf for every finite T;
(v)   local conservation: for psi in {1, v_1, v_2, v_3, |v|^2/2} and every phi in C_c^inf([0,inf)xT^3),
      int_0^inf int_Omega F psi (d_t phi + v.grad_x phi) dx dv dt + int_Omega F_0 psi phi(0,x) dx dv = 0.
Condition (v) is stronger than a mere global mass/momentum/energy inequality: it says mass, momentum, and kinetic energy flow conservatively at each spatial point, not just globally.
Levermore-Masmoudi 2010, From Boltzmann to Navier-Stokes-Fourier: https://link.springer.com/article/10.1007/s00205-009-0254-5

Exercise: What does Theorem 1.2 claim, exactly?
There is R<inf and a datum F_0 >= 0 with supp_v F_0 subset B_R(0) and finite mass, energy, |log F_0| integral, such that the Boltzmann equation has two global entropy solutions with local conservation F^(1) != F^(2). Both are in C([0,inf);L^1), and both have, for every finite T and every k>=0, int_0^T int_Omega <v>^k (Q^+ + Q^-)(F,F) dx dv dt < inf. This is a stronger conclusion than mere non-uniqueness of renormalized solutions: it delivers two solutions whose collision operators are individually integrable with every polynomial velocity weight, so both local balance identities hold -- no "defect" in the momentum or energy flux.

Exercise: Why rotate space by the specific direction (1,sqrt2,0)/sqrt3?
The jet axis in the paper is (1,sqrt2,0)/sqrt3. To simplify everything (annulus geometry, velocity profile, entropy weights), the paper rotates space and velocity simultaneously by the same orthogonal map taking this axis to the third coordinate axis. After rotation, space is R^3/Lambda with Lambda the rotated integer lattice, still covolume 1, and the local coordinates are x=(y,z) with y in R^2 transverse, z in R axial; similarly v=(bar v, v_3). The rotation preserves the transport operator and the hard-sphere collision operator, so nothing physically changes. The only leftover from the irrational axis lives in one lemma (Lemma 2.1) quantifying lattice separation from the transverse axis: |bar n| >= c/(1+|n|) for all nonzero n in Lambda. This Diophantine-like separation is what prevents distinct lattice copies from intersecting at small scales.
Original paper, Section 2 (framework), Lemma 2.1.

Exercise: Why the specific singular velocity weight H(v) = e^{-|v|^2} <v>^{-100} (1+|bar v|^{-q})?
Because the cold jets concentrate onto the velocity axis bar v = 0, so any velocity weight that is uniformly bounded near the axis would be a fatal obstruction. The factor (1+|bar v|^{-q}) with 1<q<2 provides an integrable singularity: in the transverse plane R^2 (two dimensions), |bar v|^{-q} is in L^1_loc, and in L^{1+eps} for small eps in (0, 2/q-1). The exponent m=100 in <v>^{-m} gives every polynomial moment -- needed for the local energy flux (which needs |v|^3). The Gaussian e^{-|v|^2} controls large velocities. The weighted sup norm ||f||_H = ess sup |f(x,v)| / H(v) is the natural topology in which the cold-estimate supersolution operates, and the sheet-bound shows hot products fit in this norm too. The value assigned on the exact axis bar v=0 is immaterial (sets of (x,v)-measure zero).
Original paper, Sections 2 and 3.

Exercise: What is the Carleman bilinear gain formula (Lemma 3.1)?
The hard-sphere bilinear gain P(c,h) = intint |(v-v_*).omega| c(v') h(v_*') dv_* domega equals
  P(c,h)(v) = 2 int c(dp) int delta((v-p).(xi-v)) h(xi) dxi ,
where the inner delta is surface measure on the hyperplane (v-p).(xi-v)=0, divided by |v-p|. In other words, for each fixed incoming velocity p, the partner xi must satisfy the "energy sphere" condition: |v-p|^2 + |v-xi|^2 depends only on the outgoing velocity v. Expanding: the change-of-variable identity makes the delta collapse a free angular integral into an exact planar measure. The constant 2 is the angular mass of a full collision sphere of relative speed |p-xi|. P is symmetric in c,h, and P(F,F) = Q^+(F,F). The formula is used all through the paper to reduce every gain estimate to a geometric planar-integral statement.
Gamba-Panferov-Villani 2009, Upper Maxwellian bounds: https://link.springer.com/article/10.1007/s00205-009-0250-9

Exercise: What is Lemma 3.2 (geometric slice estimate)?
For a "product" cold velocity measure c = lambda (x) rho, where lambda is a probability mixture of transverse Gaussians N_d (width d>=0) and of delta_0, and rho is an arbitrary axial probability measure (no density or support bound), and for any R>=1, any v with bar v != 0, and any axial interval I of length 0 < l <= 2R:
  (1/W(v)) int c(dp) int_{|xi-v|<=R} delta((v-p).(xi-v)) W(xi) dxi <= C_q R^2,
  (1/W(v)) int c(dp) int_{|xi-v|<=R, xi_3 in I} delta((v-p).(xi-v)) W(xi) dxi <= C_q (Rl + R^{q-1}l^{2-q}).
Both bounds are uniform in the transverse width d and in the axial law rho. In other words, the W-normalized planar gain for a cold input is bounded independently of how thin the transverse Gaussian is -- this is the fundamental reason the cold-jet mechanism works at all. The proof expands the delta in transverse 2D alone (transverse line measure), then integrates in the axial slice using (1+s^2)^{-q/2}.
Original paper, Section 3, Lemma 3.2.

Exercise: What is a "uniform sheet" (Definition 3.3)?
A sheet c in S_L is a probability measure c = N_d (x) rho on velocity space where 0 <= d <= 1/8 is a transverse Gaussian width and rho is any axial probability with exponential moment int_R e^{8a^2} rho(da) <= L. A "sheet mixture" is any positive probability mixture of elementary sheets. A function is "under sheet domination with amplitude A_0" if it is bounded above pointwise by A_0 times such a mixture. The point: sheets capture exactly the "concentrated in velocity, unbounded axial density" regime that the cold jets create. Sheet dominations provide UNIFORM constants even as the transverse Gaussian collapses (d -> 0) to a velocity atom.
Original paper, Section 3, Definition 3.3.

Exercise: What is the uniform sheet gain bound (Lemma 3.4)?
For every L<inf there is C_{q,m,L} <inf such that for almost every v (away from the velocity axis), P(c,H)(v) <= C_{q,m,L} H(v) for every c in S_L. Quantitatively, the "shell tail" for P(c,H) at |p| + |xi-v| in [R, 2R] is bounded by C_{q,m,L} R^2 e^{-cR^2} for an absolute c>0. Same conclusions with amplitude factor A_0 for dominations. Consequence: every cold jet, no matter how concentrated, hits the H-weight at gain order 1 -- the jet never blows up the gain and never has unbounded collision strength in the H-topology.
Original paper, Section 3, Lemma 3.4 (proof of the sheet-gain hinges on the tilt factor e^{2|p|^2} being absorbed into the Gaussian's own slope).

Exercise: What does the hot-hot bound (Lemma 3.5) say?
For every m>5 there is C_{q,m} with P(H,H)(v) <= C_{q,m} H(v) for almost every v. So two full-H densities have finite normalized gain. The proof uses the symmetry of P and the fact that on the Carleman collision plane |p|^2 + |xi|^2 = |v|^2 + |w|^2, so at least one of |p|, |xi| >= |v|/2; then the weight decay e^{-|w|^2} <v>^{-m} picks up polynomial-vs-Gaussian suppression on shells. The value m=100 is far above the threshold m>5.
Original paper, Section 3, Lemma 3.5 (proof uses disk-mixtures and dyadic shell sums).

Exercise: What does the small-mass gain bound (Lemma 3.6) say?
For each pure constant B<inf and every eps>0 there is C_{B,eps}<inf such that for EVERY measurable j with 0<=j<=BH and for almost every v:
  P(H,j)(v) <= H(v) ( eps + C_{B,eps} int_R^3 j(p) dp ) .
Same conclusion with P^b (cutoff) in place of P. In words: small total velocity-mass of the cold input forces small normalized gain of the hot compoment, uniformly in the "shape" of j as long as j stays under BH. This is the correction term the global continuation argument needs: without it, bootstrapping a hot bound w <= BH through P(w,w) <= B^2.(something) would fail to converge.
Original paper, Section 3, Lemma 3.6.

Exercise: What is a "cold jet at scale s"?
Fix a scale 0<s<=1/8. A cold jet is a velocity distribution concentrated in a very thin tube around the velocity axis v=(0,0,v_3) with a small spread d_s = d_0 s^{10}, where d_0 is a fixed tiny number. Its spatial support is a thin transverse annulus |y| ~ s (thickness ~s^3 because the boundary gap is c_eta s^2 with c_eta = (e^eta-1)/16 << s) crossed with a thin axial slab |z|<1/8. Physically, a jet is a puff of almost-axis-aligned particles moving with axial velocity near v_3 = a_s^sigma(z) = sigma + alpha tanh(gamma z/s). As s -> 0, the jet concentrates in both velocity and transverse space; its velocity-integrated mass is of order K/s but total mass is O(s^2), small. The subscript sigma in {+1,-1} selects one of two axial-speed signs (particles moving in +z or -z). The mass is invariant under (j,sigma,n) -> (j,sigma,n+lambda) on the lattice.

Exercise: What are the scales and the stripe pattern?
Pick 0<s_0<=1/8 and 0<eta<=log2, and define s_j = s_0 e^{-j eta} for j>=1 (geometric). Put p_j = s_j^2. The transverse support of a scale-s_j jet is the annulus s_j + c_eta p_j < |y| < e^eta s_j - c_eta p_j, INTERSECTED with a "stripe pattern" y_1/p_j mod 1 in (0,1/4) for sigma=+1 (resp. (1/2,3/4) for sigma=-1). The stripe period is p_j (so the stripes are much finer than the annulus thickness in y_1). This split into two signs prevents one collision path from containing both signs in the same stripe cell. Because p_j = s_j^2 << s_j, each annular ring contains many periods; the two signs' stripes are separated by empty half-periods. Fraction occupation of each sign is 1/4 (hence the "1/4" coefficient later).

Exercise: What is the background bath, and why the "hole"?
The bath is a fixed Maxwellian-like density u^0(x,v) = 1_{D_hole^c}(x) mu(v) 1_{|v|<=R_mu}, where mu(v)=e^{-beta_0|v|^2} with beta_0 large, and R_mu is a velocity cutoff. D_hole is the open 2delta-neighborhood of the axial segment [-1/3,1/3] in the spatial torus. "Hole" means: near the axis (where the jets live), there are NO bath particles. The reason for the hole is technical: at time zero the jets would collide with the bath and immediately wash out; by opening a gap the paper keeps the jets from interacting with the bath until the jets have had enough time to congeal into their cold self-similar shape (and then retire). Once the jets retire and are absorbed into the hot component, the hole is no longer needed. The bath is what supplies a uniform positive collision frequency kappa > 0 after time t_1, which forces jets to decay exponentially.

Exercise: What is the initial datum F_0, precisely?
Let c_i^0(x,v) = 1_{E_j^sigma}(y_c) 1_{(-a,a)}(z_c) (K/s) S(gamma z_c/s) d_s^{-3} J((v - a_s^sigma(z_c) e_3)/d_s), where s = s_j, (y_c, z_c) = x-n, a = 1/8, S(r)=sech^2(r), a_s^sigma(z) = sigma + alpha tanh(gamma z/s), and J is a fixed smooth bump with int J = 1 supported in the unit ball. Let i range over labels i = (j, sigma, n) with j>=1, sigma=+/-1, n in Lambda. Then F_0 = u^0 + sum_i c_i^0. The parameters K>=1, 0<eta<=log2, 0<alpha<=1/4, 0<gamma<=1 are all fixed early and only later optimized. Each c_i^0 has velocity mass of order K/s_j and total mass O(s_j^2) (because the annulus area is O(s_j^2) and the axial length contributes another s_j). The lattice sum over i is absolutely convergent in L^1, so F_0 is well-defined and F_0 >= 0.

Exercise: What do the datum bounds (Lemma 4.2) give?
F_0 >= 0, has bounded velocity support (|v| <= 2, say), and satisfies int F_0 (1+|v|^2+|log F_0|) dx dv < inf. The partial sum approximations (with j <= N, and with optional seed h^0_{N,T,lambda} = lambda m_T s_N sum_n chi(x/s_N - n, v) where 0<=lambda<=L_0 and m_T=T^p) converge to F_0 in weighted L^1 (all polynomial v-weights) and in relative entropy H_mu or H_M. The convergence is uniform in N, T, lambda. Moreover the initial mu-relative entropy can be made arbitrarily small by shrinking delta, increasing R_mu, and then shrinking s_0. The seed contribution to entropy is O(s_N^4 (1+|log s_N|)) which vanishes because s_N -> 0.

Exercise: What are the "colored equations"?
For fixed N, b, T, lambda, the initial datum is decomposed into colored components c_i = c_i^0 + colored evolution, plus bath u and hot h. Up to time t_1 = 1/32 or so:
  D c_i + nu_F^b c_i = P^b(c_i, c_i),  (each cold color)
  D u + nu_F^b u = P^b(u, u),          (bath)
  D h + nu_F^b h = P^b(h, h) + 2P^b(c+u, h) + S,
  S = 2P^b(c, u) + sum_{i!=i'} P^b(c_i, c_{i'}),
where c = sum_i c_i and P^b denotes the cutoff gain with kernel min{1, b/|v-v_*|} (cutoff b_N -> inf in the limit). The colored decomposition is a formal algebraic reorganization of the single Boltzmann equation D F = Q^b(F,F): you compute F = c+u+h and it works out. The color labels track each jet so we can control them individually before retirement. At retirement time L_s = s^{-1/2}, all colors at scale s are absorbed into the hot component w; identities are rearranged so that F stays continuous.

Exercise: What does the finite-evolution proposition (4.3) give?
For each fixed (N,b,T,lambda), the colored system has a unique nonnegative local mild solution with bounded total H-norm. It continues while the H-norm stays bounded, with prescribed group-transfers at retirement times, and depends continuously in H-norm on its initial data on every common existence interval. Each component's norm is continuous between transfers. On every such interval:
  - total mass, momentum, and kinetic energy are conserved;
  - the exact local collision-invariant balances hold, with all polynomial velocity fluxes integrable;
  - for rho = M or rho = mu, entropy-dissipation inequality holds: H_rho(F(t)) + int_0^t D^b(F(s)) ds <= H_rho(F(0)).
The proof is by Picard iteration in free coordinates with Gronwall-type bounds from the H-norm estimates of Section 3, and the entropy is controlled by approximating F with F_eps = F + eps g (g = Gaussian truncation of the v-dependence), differentiating, passing to the symmetric form, and using integrability of the weight.

Exercise: Why are the cold "tubes" (spatial supports of the jets) disjoint until retirement?
Lemma 5.1. Define O_i(t) = n + E_j^sigma x [-a-2t-b_sp, a+2t+b_sp] where E_j^sigma is the (c_eta/4)s_j^2-neighborhood of E_j^sigma. These are the natural enlargements of a jet at time t. The claim is: for distinct active labels, O_i(t) AND O_{i'}(t) = {} whenever t <= min{L_{s_i}, L_{s_{i'}}}. Within a fixed copy, the annular gaps and stripe gaps are wider than the enlargements (both scale as s_j^2 but with different constants; the empty stripe space is s_j^2/4 >> c_eta s_j^2/2). Between distinct lattice copies, an intersection would force |bar n - bar n'| <= 4S_* = 4 max{s_i, s_{i'}}, but Lemma 4.1 gives |bar n - bar n'| >= c/(1+|n-n'|) >= c' S_*^{1/2}, which is much larger than 4S_* for small s_0. Disjointness is what makes the jets noninteracting with each other, so each can be bounded independently.

Exercise: What is the barrier supersolution m_i, and what does it control?
Fix a label i = (j, sigma, n). Let U_{s,sigma}(t,z) be the axial profile satisfying U = a_s^sigma(z - t U) (implicit, smooth, monotone). Put tau = z_c - t U, xi = (bar v, v_3 - U) = velocity minus axial drift, D_y = dist(y_c, E_j^sigma), D_z = dist(tau, [-a,a]). The supersolution is
  m_i(t,x,v) = C_m (K/s) d_s^{-3} e^{t/L_s} exp[ -(2 - t/L_s)|xi|^2 / d_s^2 - (D_y^2 + D_z^2) / (8 d_s^2 L_s t) ].
Lemma 5.2 says: the free-transported datum f_i(t,x,v) = c_i^0(x-tv,v) satisfies f_i <= m_i for 0<=t<=L_s; the barrier m_i satisfies D m_i >= (1/L_s)(1 + (7/8)|xi|^2/d_s^2) m_i along free flights; and the gain P(m_i,m_i) = m_i nu_{m_i} is bounded by C (K/s)(d_s + |xi|) m_i. Also f_i > 0 forces x in O_i(t). The barrier m_i is a Gaussian in velocity minus axial drift, with the extra D-distance factor recording how far the free-transported position is from the initial annular+axial support. It grows by e^{t/L_s} (so that its loss-frequency term overwhelms gain), while the D-distance ensures nonmain labels decay.

Exercise: What does cold control (Proposition 5.3) say?
There is a fixed constant C_* such that, with eps_s = C_* L_s d_s / s (so eps_s ~ s^{15/2}.s^5.s^{-1} = s^{19/2}, tiny) and U_i = f_i + eps_s m_i (free core + small Gaussian remainder), the actual color obeys
  0 <= c_i(t,x,v) <= U_i(t,x,v)  for 0 <= t <= L_{s_i},
as long as the color is still active. Furthermore, if on an interval beginning at t_1 the collision frequency satisfies nu_F^b >= kappa > 0 (which holds once the bath contributes), then
  c_i(t,x,v) <= e^{-kappa(t-t_1)} U_i(t,x,v)  for t_1 <= t <= L_{s_i}.
So each jet tracks its free core plus a tiny Gaussian, with no dependence on the hot component. Once the bath kicks in, every jet decays exponentially at rate kappa. This is the crucial "the cold part of the gas can be summarized as free-transport + Gaussian dust" statement that unlocks the global estimate.

Exercise: What is the "core formula" (Lemma 5.4)?
At a fixed (t,x), the velocity measure f_i(t,x,p) dp is the pushforward of
  [K S(gamma(z_c - t p_3)/s)] / [s + t alpha gamma S(gamma(z_c - t p_3)/s)] . J(w) dw,
  with bar p = d_s bar w, p_3 = a_s^sigma(z_c - t p_3) + d_s w_3,
and with the initial spatial masks evaluated at x - t p. Key point: even though the jet has velocity-integrated mass ~K/s at t=0, the mass at any fixed (t,x) while 0<t<=L_s is bounded by C/(t+s). The core formula shows precisely how the initial strong axial concentration (mass ~1/s) gets diluted by the axial expansion (the amount of t in the denominator). Also: for every k>=0, int <p>^k c_i(t,x,p) dp <= C_k/(t+s), and the tail past |p|>R is <= C_k e^{-c_k R^2}/(t+s). Same bounds hold for the remainder eps_s m_i dp (amplitude <=Cs^{15/2}).

Exercise: What are the nonmain tails (Lemma 5.5)?
There is a function theta_0(s_0) -> 0 as s_0 vv 0 such that, uniformly at all times of existence, the sum of U_i over active labels i whose O_i(t) does NOT contain x is bounded by theta_0(s_0) e^{-4|v|^2}. So the total mass at (t,x) from jets whose spatial supports do not cover x is uniformly small. For 0<t<=1, even sharper: bounded by C_{A,B} t^A s_0^B for any fixed A,B. The proof uses the D-distance exponential in m_i combined with a lattice-sum count on the shell architecture (the number of lattice points in a ball of radius R is ~C(R+1)^3, and the exponential dominates every polynomial).

Exercise: What is the retirement size (Lemma 5.6)?
At retirement time L_s = s^{-1/2}, the H-norm of the sum of jets at scale s is bounded by C s^{-C'} for some fixed C,C' >0 (polynomial in 1/s). Also, the volume of the union O(t) of active enlargements is bounded by C s_0^{3/2}, uniformly in t and label set. So as s_0 -> 0, the total transverse "footprint" of all active jets shrinks to zero. This is the smallness that eventually lets the hot component's H-norm decay below any prescribed size before the next stage of the argument.

Exercise: What is the bath barrier (Lemma 6.1)?
There are t_1 in (0, 1/32], beta' > 4, and constants C,c > 0 (independent of delta, R_mu, s_0), such that every finite evolution satisfies
  0 <= u(t,x,v) <= C e^{-beta'|v|^2} exp(-c D_0(x)^2 / t)  for 0 < t <= t_1,
where D_0(x) = dist(x, D_hole^c). Additionally, for s_0 sufficiently small, D_0(x) >= delta for all 0<=t<=t_1 and x in O(t): the jets' tubes stay strictly away from the bath hole's boundary. The proof constructs an explicit Gaussian-in-velocity, distance-in-space barrier B(t,x,v) = exp(C_1 t - (beta_0 - C_1 t)|v|^2 - c D_0(x)^2/t) and checks that its transport-derivative dominates its own quadratic gain (which reduces to B nu_B), so B is a supersolution; the smallness of t makes the factor e^{C_1 t} <= 2 and beta_0 - C_1 t >= beta' > 4 so the velocity decay is preserved.

Exercise: What are the source estimates (Lemma 6.2)?
For s_0 sufficiently small (depending on delta), every finite evolution satisfies, on its interval of existence inside (0,t_1):
  P(c+u,H) <= C/(t+s_N) . H,
  0 <= S <= C_{A,delta} t^{A-1} H,
for every fixed A>0. Here S is the source in the hot equation: S = 2P(c,u) + sum_{i!=i'} P(c_i, c_{i'}). The first bound says the total "cold + bath" input to the hot gain is bounded by the sheet amplitude of the cold part. The second says the source feeds the hot part with a vanishing rate as tvv0 -- very strong smoothing at very short times. The proof uses the sheet-bound plus the nonmain-tail decay, and the hole-distance from Lemma 6.1 upgrades the bath estimate to e^{-c delta^2 / t} H inside the jet tubes.

Exercise: What does the cap selection (Proposition 6.3) do?
Fix delta>0. Choose exponents in the seed (4.5) so p > C_2 + 1 and A > p + C_2 + 3, with m_T = T^p. Then there are s_0-bar, T_0 > 0 (uniform in N and cutoff) with the following property. For each 0<s_0<=s_0-bar, 0<T<=T_0, and each finite approximation, there is a seed parameter lambda in (0, L_0) such that the associated finite solution exists on [0,T] and satisfies
  Z(t) = ||h(t)||_H <= m_T (t + s_N)  for 0<=t<=T,
  max_{0<=t<=T} Z(t) / (m_T (t+s_N)) = 1.
The dormant solution (lambda=0) obeys a stronger bound Z(t) <= C_{A,delta} t^A for t in an initial interval. The proof is a continuity/goniometric argument: at lambda=0, Z stays below the cap; at lambda=L_0, Z exceeds it at t=0 by construction (L_0 ||chi||_H > 1); the set of "lambda for which Z stays strictly below cap on [0,T]" is relatively open and contains 0 but not L_0, so it has a right endpoint lambda_* at which the cap is exactly touched (max ratio = 1). This is the "cap selection": a solution whose hot component has a prescribed size, normalized by the cap.

Exercise: What is the "path source estimate" (Lemma 6.4)?
Fix 0 < t_b < t_1. There is a function E_{t_b}(s_0) -> 0 as s_0 -> 0 such that, for every finite evolution and every terminal time t in [t_b, t_1] in its interval of existence,
  || int_{t_b}^t S(s, x - (t-s)v, v) ds ||_H <= E_{t_b}(s_0).
So the SOURCE TERM fed to the hot component (integrated along free flights) is small in the H-norm. Uniform in hot seed, cap horizon, N, and cutoff. The proof splits S into nonmain-tail terms (already O(eps(s_0)H)) plus the main-bath term (bounded by the sheet-gain estimate as C/(t_b)H), then uses the Gaussian-parameter majorant for the bath-plus-jet convolution along paths and lattice-shell sums to convert pointwise estimates into weighted H-norm bounds.

Exercise: What does uniform early continuation (Proposition 6.5) give?
Fix delta>0 and zeta in (0,1). There are s_0-bar>0 and T_0 in (0,t_1), depending on (delta, zeta, fixed jet params), such that for every 0<s_0<=s_0-bar, every finite approximation, and every 0<T<=T_0, both the dormant solution and every cap solution continue to time t_1 and satisfy
  sup_{0<=t<=t_1} ||h(t)||_H <= zeta.
All choices are uniform in N, cutoff, R_mu, T. This is the smallness of the hot component at time t_1 that later bootstraps into a global bound. The proof orders the parameter choices: at horizon T, Z(T) <= T^p; on [T,t_b], Z grows like O(t^{C_2}) + O(t^A) with exponents p > C_2 and A > C_2 (so the T-dependence vanishes); on [t_b, t_1], Z grows exponentially but is bounded by path-source estimate E_{t_b}(s_0) -> 0. Ordering s_0 << T_0 << t_b << t_1 gives the strictest bound.

Exercise: What is the c + w decomposition after the fixed time t_1?
After t_1, color labels are still present but only until their retirement times L_{s_i}. Write F(t,x,v) = c(t,x,v) + w(t,x,v) where c = sum_i c_i is the sum of the still-active cold jets and w = u + h + (sum of retired colors) is everything else. w is a single component satisfying
  D w + nu_F^b w = P^b(w, w) + 2P^b(c, w) + S_c,  S_c = sum_{i!=i'} P^b(c_i, c_{i'}),
and F satisfies the same cutoff Boltzmann equation D F = Q^b(F,F) across the whole evolution. The decomposition is bookkeeping: it isolates the cold jets (which decay at rate kappa thanks to the bath) from everything else (which is controlled by a bootstrap norm on ||w||_H). At a retirement time L_s, all colors of scale s are added to w; F is continuous across the transfer.

Exercise: What does the averaging-along-flights lemma (7.1) give?
Let A(s,x) >= 0 be measurable with ||A_s||_inf <= A_* and ||A_s||_1 <= a_*. For a bounded velocity set V and duration >= rho:
  int_V A_s(x - tau v) dv <= C_{V,rho} a_*, uniformly in x, s, tau >= rho.
Moreover if A_* is fixed, then as a_*->0 the following quantities tend uniformly in (x, t) to zero:
  sup_{t>=t_1,x} int_{t_1}^t e^{-kappa(t-s)} int H(v) A_s(x-(t-s)v) dv ds,
  sup_{t>=t_1,x} int_0^{t_1} int H(v) A_s(x-(t-s)v) dv ds.
So the H-weighted space-time average of a small-L^1 function is uniformly small: this is the "dilution along free flights" mechanism. The proof splits H into a bounded support part H_L with tail delta_L, then applies the free-flight change of variables and the local averaging over velocity set V.

Exercise: How is a uniform positive collision frequency produced (Lemma 7.2)?
Fix the bootstrap size B>B_0+2. If the initial entropy smallness eps_E is small enough depending on B, then every finite evolution satisfying the bootstrap w <= B.H obeys, for t >= t_1,
  nu_F^b(t,x,v) >= kappa > 0,    uniformly in b >= 1 and in N.
The value of kappa >= pi min(a_1, a_2) where a_k = |V_k| (Delta/2) m_* e^{-C Delta}, with Delta = t_1/2 and m_* > 0 a lower bound for P^b(mu,mu)/mu on V_1 OR V_2. V_1, V_2 are two closed velocity balls at distance >3 in R^3. The approach: on V_k, P^b(F,F) is close to P(mu,mu); each point x has a recent Duhamel contribution from velocities in V_k (via the time-integrals of Lemma 7.1 and the small initial entropy), so the loss-frequency integral over V_k is bounded below. Since every velocity v is at distance at least 1 from ONE of V_1, V_2, the cutoff min{1,|v-p|} >= 1 there, giving kappa. This is the key structural input that turns "small entropy" into "positive frequency at every point".

Exercise: What does the uniform nonlinear-gain lemma (7.3) give?
Fix B > B_0+2, and suppose the bootstrap w <= B.H holds on an interval beginning at t_1. After taking (eps_E, zeta, s_0) sufficiently small depending on B, one has on that interval
  P^b(w,w) <= C_5 H,    C_5 := ||P(mu,mu)||_H + 1.
C_5 does NOT depend on B, on N, or on the horizon T. The proof writes w = mu + j where j = (w - mu)_+ (positive excess mass over the Maxwellian), uses small relative entropy to bound int j in L^1_x, uses Lemma 6's flight-average to convert that into pointwise small excess mass m_j(x) = int j dv at almost every point, then uses the small-mass gain bound (Lemma 3.6) to conclude P(w,w) is comparable to P(mu,mu) up to the sheet-bound constant C_5. This is the whole reason the bootstrap works: the quadratic self-gain of w behaves as if it were a fixed Maxwellian.

Exercise: What does global continuation (Prop 7.4) give?
Fix the four parameters K, eta, alpha, gamma (all small/large as per (4.1)), the profile J, the Maxwellian mu, t_1, and the seed exponents p, A. Then there is a B < inf and kappa > 0, both independent of N, b_N, and T, such that every dormant finite evolution and every cap solution continue globally and satisfy
  ||w(t)||_H <= B,    nu_F^{b_N}(t,x,v) >= kappa,    for all t >= t_1.
Every active jet satisfies the exponential-decay cold bound; the total retirement increments bounded by J_0 = sum_s C s^{-C} e^{-kappa(s^{-1/2} - t_1)} -> 0; the hot component is bounded by zeta on [0,t_1]. The proof is a Gronwall bootstrap: on the bootstrap interval (||w||_H <= B), Duhamel's formula gives ||w(t)||_H <= B_* < B-3 with B_* = (B_0 + (C_4+C_5)/kappa + 1) exp(4C_3/kappa) fixed BEFORE B, so the bootstrap cannot first be touched; at retirement times the increment is a bounded jump, still under B-3; and each fixed-N system has the local existence from Prop 4.3, so we get existence all the way to t=inf.

Exercise: What is the compactness strategy (Section 8), at tree level?
We now pass to the limit N->inf with b_N->inf (cutoff removed). Along a fixed horizon T (or a global limit), the finite solutions F_N converge weakly in L^1([0,tau]xOmega) to a density F. To identify the limit (a) as an entropy solution with local conservation (Definition 1.1), we need (b) strong compactness of F_N in t (for a time-representative), (c) uniform integrability of the two-velocity product F_N(t,x,v)F_N(t,x,v_*) (for identifying the collision operator limit Q^+(F,F)) and of Q_N^+/- with weights, (d) strong compactness of velocity averages (Golse-Lions-Perthame-Sentis), (e) passing the collision terms and (f) passing renormalization and the local balances. The bulk of Section 8 is (b)-(f). The main new input vs classical DiPerna-Lions is that the R_N (velocity-integrated mass) has a self-similar envelope (comp:moment-envelope) which is what makes the uniform integrability decay physical and explicit.

Exercise: What is the small-scale uniform-integrability bound (8.moment-envelope + 8.small-annuli)?
Let A_s(t) be the union of active-enlargements at scale s (including all lattice copies); |A_s(t)| <= C_tau s^2 (its transverse 2D area times a bounded axial length). The velocity-integrated mass satisfies
  R_{k,N}(t,x) = int <v>^k F_N(t,x,v) dv <= C_{k,tau} (1 + sum_j 1_{A_{s_j}(t)}(x)/(t+s_j)),
so the SUM of the singular 1/(t+s_j) factors weighted by indicator of annular shells bounds the k-moment. When you square this, the different shells don't overlap (disjointness), so the cross terms vanish and each shell contributes O(s_j) in int_0^tau dt (because int 1/(t+s_j)^2 = 1/s_j - 1/(tau+s_j) <= 1/s_j). This gives, for any eps>0, sup_N int_0^tau int_{A_{<eps}(t)} R_{k,N}(t,x)^2 dx dt <= C sum_{s_j<eps} (s_j + s_j^2) -> 0. So the "very small scales" cost arbitrarily little in the L^2 velocity-integrated sense, hence in the two-velocity product, hence in the collision integrands.

Exercise: What does the velocity-averaging lemma (8.1) say?
Let f_n, g_n be locally uniformly integrable in L^1 on an open region of time-space-velocity, satisfying (d_t + v.grad_x) f_n = g_n in distributions. Then for every smooth compactly supported velocity test psi, the velocity averages int psi(v) f_n(t,x,v) dv are relatively compact in local L^1_{t,x} (on the same region), provided uniform integrability holds for f_n and g_n on a neighborhood of supp psi. The proof is the Golse-Lions-Perthame-Sentis averaging lemma: localize, Fourier transform in (t,x), integrate the resolvent formula and use Cauchy-Schwarz to bound high-frequency Fourier tails; then convolution with a mollifier recovers compactness. This is why a solution that's merely distributionally equal to an integrable source becomes a strong local L^1 function after velocity averaging -- which is exactly what we need to identify Q^+(F,F) from Q_N^+(F_N,F_N).

Exercise: How do the collision terms pass to the limit?
Step 1. From the compactness of small scales, we get a further subsequence with F_N -> F weakly in L^1([0,tau]xOmega), and (F_N(t,x,v)F_N(t,x,v_*)) converging weakly to a two-velocity function W. Step 2. The separated-test expansion int psi(v)chi(v_*) F_N(t,x,v)F_N(t,x,v_*) dx dv dv_* = (intpsiF_N dv)(intchiF_N dv_*) converges in L^1 by velocity-averaging (8.1) plus uniform integrability. Step 3. A determining family argument shows W = F(t,x,v) F(t,x,v_*). Step 4. The elastic substitution (v,v_*) -> (v', v_*') is measure-preserving and involutive, so the primed products also converge weakly. Step 5. Fix a bounded output test phi; the collision factor (v-p).omega vanishes at the boundary of the sphere, and the cutoff b_N equals 1 for all large N on a bounded velocity region; standard shell decomposition removes the velocity complement. Step 6. Hence Q_N^+/-(F_N,F_N) ~> Q^+/-(F,F) in L^1([0,tau]xOmega), with every polynomial weight k >= 0. This is where the two-velocity product uniform integrability pays off.

Exercise: How is the single time-representative chosen?
For a smooth periodic-velocity test phi compactly supported in velocity, the finite path equations give
  |<F_N(t) - F_N(s), phi>| <= int_s^t int_Omega ( F_N |v.grad_x phi| + (Q_N^+ + Q_N^-) |phi| ) dx dv du <= C_phi |t-s| + modulus(|t-s|).
The first term is Lipschitz in t; the second has a modulus going to zero uniformly in N, by the short-time integrability of Q_N^+/-. Arzela-Ascoli applies on a countable family of dense tests. Then {F_N(t)} is relatively weakly compact in L^1(Omega) at EVERY t, and every cluster point agrees with the smooth pairings. Consequence: one subsequence converges weakly in L^1(Omega) at every t, giving a representative F(t) whose smooth pairings are continuous and agree with the spacetime limit.

Exercise: What does the global-limit proposition (Prop 8.1) state?
Fix the parameters of global continuation. Let b_N -> inf. Either the dormant family, or a cap family at one fixed horizon T in (0,T_0], has a subsequence of total densities F_N converging weakly in L^1([0,tau]xOmega) for every tau < inf to a nonnegative F. The subsequence can be chosen so F_N(t) ~> F(t) in L^1(Omega) at EVERY t >= 0, with uniformly small polynomially-weighted velocity tails on each bounded time interval. The limit F:
  - has initial datum F_0;
  - is a global entropy solution with local conservation (Definition 1.1);
  - F in C([0,inf); L^1(Omega));
  - for every tau<inf and k>=0: int_0^tau int_Omega <v>^k (F + Q^+(F,F) + Q^-(F,F)) dx dv dt < inf;
  - conserves total mass, momentum, energy at every time;
  - satisfies entropy inequality H_M(F(t)) + int_0^t D(F(s)) ds <= H_M(F_0);
  - for a dormant sequence: is bounded from above by the sum of the cold majorants over all scales, plus the bath and dormant-hot majorants.

Exercise: How do renormalization and the local balances pass?
Renormalization: From the strong compactness in free coordinates (obtained via free transport F(t,x+tv,v) and time-Lipschitz regularity of Q^+/-), the distributional equation d_t F# = Q(F,F)# holds with Q(F,F) in L^1. Hence F# is absolutely continuous in time for a.e. (x,v), with the chain rule D[M beta(F/M)] = beta'(F/M) Q(F,F). Integrating against a smooth test yields exactly the renormalized equation + initial trace. |beta'| <= C/(1+z) makes the source integrable. |Mbeta(f/M) - Mbeta(g/M)| <= C_beta |f-g| makes the strong initial and terminal traces pass through the renormalization. Local balances: for psi in {1, v_1, v_2, v_3, |v|^2/2} the collision integral vanishes by symmetry (int psi Q(F,F) dv = 0), so passing to the limit is just a weighted weak convergence in L^1; integrability of |v|^3 F is what gives energy flux finiteness.

Exercise: What is the dilatation rescaling (eq 9.1)?
Pick horizons T_Nvv0, terminal scales s_N/T_N->0, cutoffs b_N->inf, and cap solutions with cap times t_{*,N}. Define r_N = t_{*,N} + s_N, m_N = m_{T_N} = T_N^p. The dilated hot component is
  h_{r_N}(t,x,v) = h_N(r_N t, r_N x, v) / (m_N r_N),
and the dilated cold component is C_{r_N}(t,x,v) = r_N c_N(r_N t, r_N x, v). Set l_N = s_N/r_N -> l in [0,1], U_N = T_N/r_N -> U in [1,inf], and note r_N -> 0 and U_N >= (1 + s_N/T_N)^{-1} -> U >= 1. The rescaled cap is 0 <= h_{r_N}(t,x,v) <= (t + l_N) H(v) for 0 <= t <= U_N, with ||h_{r_N}(1-l_N)||_H = 1 (this is where the "hit" happens in dilated coordinates). This rescaling is the "blow-up" that isolates what happens near the cap-hitting time t_{*,N} on the shrinking spatial scale r_N -> 0.

Exercise: What does the dilated cold limit converge to? What is B?
Under dilation, the annular structure becomes a geometric mesh M of scales rho_1 > rho_2 > ... -> 0. On each mesh annulus rho < |y| < e^eta rho, the cold velocity measure converges (weak-star in local L^inf_{t,x}) to a two-atom velocity measure
  B(t,x,dp) = sum_{sigma=+/-1} b_{rho,sigma}(t,z) delta_{u_{rho,sigma}(t,z) e_3}(dp),
where u_{rho,sigma}(t,z) is the unique solution of u = sigma + alpha tanh(gamma(z - t u)/rho), and the axial amplitude is
  b_{rho,sigma}(t,z) = theta K S(gamma(z - t u_{rho,sigma}) / rho) / [rho + t alpha gamma S(gamma(z - t u_{rho,sigma})/rho)]
with theta = 1/4 the stripe fraction. In the hole |y| < l (when l > 0), B = 0. On annular interfaces and on the transverse axis (both null measure), values are set to any conforming bound. The bound int <p>^k B(t,x,dp) <= C_k / (t + rho) is what makes the linear equation well-posed (below).

Exercise: What is the hot linear equation for the dilated limit g?
The dilated hot density h_{r_N} converges weakly in local spacetime L^1 on (0,U) x R^3_x x R^3_v to a limit g satisfying
  0 <= g(t,x,v) <= (t + l) H(v),   D g = 2P(B,g) - g nu_B,
where B is the two-atom medium above and nu_B is its collision frequency. Both P(B,g) and g nu_B are locally integrable with position-uniform velocity majorant, and h_{r_N} has strongly-converging compact-velocity averages. The limit has a locally strongly continuous L^1 representative at positive times obeying the integrating-factor formula
  g(t,x,v) = g(a,x-(t-a)v,v) exp[-int_a^t nu_B(tau,x-(t-tau)v,v) d tau]
             + 2int_a^t P(B,g)(s,x-(t-s)v,v) exp[-int_s^t nu_B(tau,x-(t-tau)v,v) d tau] ds,
for 0 < a < t < U. When l > 0, one may set a=0 and the initial trace is g(0,x,v) = lambda_0 l chi(x/l, v). This is Proposition 9.1.

Exercise: How do the "errors" vanish in the dilatation?
Write the actual dilated h_{r_N} equation as D h_{r_N} = 2P^{b_N}(C_{r_N}, h_{r_N}) - h_{r_N} nu^{b_N}_{C_{r_N}} + E_N, where
  E_N = r_N [2P^{b_N}(u_N^r, h_{r_N}) - h_{r_N} nu^{b_N}_{u_N^r}]
        + m_N r_N^2 [P^{b_N}(h_{r_N}, h_{r_N}) - h_{r_N} nu^{b_N}_{h_{r_N}}]
        + m_N^{-1} S_N(r_N t, r_N x).
The first line is the bath interaction, which shrinks by a factor r_N -> 0. The second line is the hot-self-gain, which shrinks because m_N r_N^2 -> 0. The third line is the source in the hot equation, controlled by the arbitrary-power source-bound (S_N <= C_M t^{M-1} H) and by m_N = T_N^p: the dilated source is O(T_N^{M-1-p}) -> 0 for any M > p+1. So E_N -> 0 in local spacetime L^1, and one may pass D h_{r_N} -> D g = 2P(B,g) - g nu_B directly.

Exercise: What does the free-core comparison (Lemma 9.2) say?
Let A_N(t,x,dp) = r_N sum_{i:j<=N} f_i(r_N t, r_N x, p) dp be the "dilated collection of free cores" (each f_i is the free transport of c_i^0). Then:
  - int <p>^k A_N(t,x,dp) <= C_k/(t+l_N),
  - P(A_N,H) <= C/(t+l_N) H,
  - There are eps_N -> 0 such that on 0 <= t <= min(1, U_N), along almost every free characteristic:
      D h_{r_N} <= 2P(A_N, h_{r_N}) + eps_N H.
The comparison holds on full-measure endpoint sets, because the actual c_i differs from free-core f_i only by a small Gaussian dust (from cold control) plus tiny correction terms; those are absorbed into eps_N uniformly. The consequence: for the next section (nonconcentration), one may replace "actual cold + bath" by "free cores only" without losing any essential information.

Exercise: What does the "nonconcentration" proposition (10.1) state?
Consider any sequence of cap solutions with horizons T_n vv 0, approximation indices N_n -> inf, and s_{N_n}/T_n -> 0. Make the dilation rescaling of eq 9.1, and pass to any subsequence supplied by Prop 9.1. Then the dilated limit g is NOT identically zero on (0,U) x R^3_x x R^3_v. This holds whether U is finite, U=1, or U=inf. In words: magnifying at the cap-hit point produces a genuinely nonzero solution of the linear medium equation. The proof occupies the whole rest of Section 10 and is a two-track argument: near the velocity axis two gains suffice, and away from it five gains plus a rank argument suffice. The two tracks together rule out every way for g to disappear.

Exercise: How does the reduction to finite gain integrals work?
The reduction uses the positive majorizing equation D h_n <= 2P(A_n, h_n) + eps_n H along free characteristics, with A_n = dilated free cores. Iterate the gain five times using the Volterra operator V_n and positivity: h_n <= d_n + V_n h_n + e_n, where d_n is the free-transport of h_n at a slightly earlier time a, and e_n <= eps_n -> 0. Five iterations give
  h_n(t) <= (a + l_n + eps_n) sum_{j=0}^4 (C log(1/a))^j / j! + (V_n^5 h_n)(t).
The first term tends to 0 as (n->inf then a->0). What remains is to show V_n^5 h_n vanishes at endpoints carrying a nonzero fraction of the cap event. Then h_n would hit zero at the cap too, contradicting the cap-normalization.

Exercise: What does the endpoint-tightness lemma (10.3) give?
Fix a>0 and a sequence of times t_n in [a,1]. In a five-fold integral (V_n^5 h_n)(t_n, x_n, v_n) with 0 <= h_n <= 2, one may restrict:
  - all |p_i| + |v_i - v_{i-1}| to fixed bounded sets (each at arbitrarily small cost),
  - all sheet parameters to fixed bounded sets,
with the bounds uniform in n, t_n, x_n, v_n. Moreover:
  (i)  if |v_n| -> inf, the whole five-fold integral -> 0.
  (ii) after retaining bounded endpoint velocities, an endpoint escaping every dilated periodic origin also has a vanishing integral (because the amplitude of the exponential tail S(gamma z/rho) is uniformly small).
The consequence: any sequence with a POSITIVE LOWER BOUND on its five-fold integral must have bounded velocities and bounded distance from a periodic origin, hence has a translated limit.

Exercise: What do the axis-kernel estimates (Lemma 10.4) say?
Fix 1<=L<inf. Uniformly over sheets c in S_L:
  (i)   For fixed M,delta>0, as epsvv0, sup over |v|<=M and 0<|bar v|<=eps of K_c(v;|bar xi|>=delta) -> 0, and similarly for the axial mismatch |v_3-p_3|>=delta.
  (ii)  The axial input marginals are uniformly non-atomic: for any R>=1 and axial interval I of length 0<ell<=1, K_c(v; xi_3inI) <= C_{q,m,L,R}(ell + ell^{2-q}). This yields a common axial modulus lim_{ell->0} sup K_c(v; xi_3inI) = 0.
  (iii) Off the axis: for |v|<=M and |bar v|>=r_0>0, as the transverse width of the sheet tends to zero and then the input small transverse threshold goes to zero, K_c(v; |bar xi| < delta) tends to zero.
These are refinement lemmas for the axial argument: (i) says a single gain's kernel mass is small near the axis for transverse-concentrated sheets; (ii) says the axial partner is never concentrated on an interval; (iii) says away from the axis, transverse-concentrated cold input has small normalized gain.

Exercise: What is the characteristic-matching lemma (10.5), and what does it buy?
For the profile u_{rho,sigma}(t,z) defined by u = sigma + alpha tanh(gamma(z - t u)/rho):
  (i)  |w - u_{rho,sigma}(t_1,z_1)| <= |w - u_{rho,sigma}(t_2,z_2)| whenever a <= t_2 <= t_1, Delta = t_1-t_2, and z_2 = z_1 - Delta w. So the profile is CONTRACTIVE along free-flight characteristics.
  (ii) As rhovv0, the profile u_{rho,sigma}(t,z) is within a small distance of the three collapsed speeds {z/t, sigma-alpha, sigma+alpha}. And |p_3 - u_{rho,sigma}(t,z)| <= d_s |w_3| for the perturbed axial velocities.
This is what allows "two gains at an axial endpoint" (Lemma 10.6) to choose a finite list of forbidden speeds at the second gain (the collapsed speeds z_1/t_1, +/-1+/-alpha, plus the finitely many scaled-profile speeds at scales in [rho_0, M]) such that the first-kernel mass on their neighborhood is arbitrarily small. Outside the neighborhood, a fixed-gap axial mismatch survives to the next gain, where estimate (i)/0.5 in Lemma 10.4 does the job.

Exercise: What is "two axial gains" (Lemma 10.6)?
Fix a>0. Let t_n in [a,1] with bounded endpoint positions and velocities converging to a pure axial velocity (bar v_n -> 0). Then (V_n^2 1)(t_n, x_n, v_n) -> 0. In words: two successive gains ALREADY exclude endpoints converging to the velocity axis. The proof: choose delta to make the first kernel's mass on the delta-neighborhoods of the collapsed speeds << eps. Then choose rho_0 to make every profile with rho<rho_0 sit within delta/4 of a collapsed speed. Then choose b_0 small enough that the finite many scaled-profile speeds have small first-kernel neighborhoods. Outside these neighborhoods, a fixed positive mismatch c_* separates the first output from any second-core parameter. The second-kernel axial-mismatch estimate makes the second gain small whenever the transverse part is small. Combining, one shrinks the two-gain integral below any eps.

Exercise: What does the redundant plane parameter lemma (10.7) say?
Given n!=0, e = n/|n|, and a fixed bump zeta supported on [-1,1] with integral one: for every nonnegative measurable F,
  int_{n^perp} F(k) dA(k)/|n| = int_{R^3} F(n x L) |n| zeta(L.e) dL.
So the planar measure divided by |n| is realized as the pushforward of a specific easy-to-integrate Lebesgue measure on R^3 (the (n x L) hitting parameter). If delta<=|n|<=R and the plane integral is restricted to |k|<=R_1, the pushed measure is supported in |L|<=sqrt((R_1/delta)^2+1) and has density at most R.||zeta||_inf. Same parametrization iterates: successive gains with different normals just compose these (measurable-in-previous-parameters) changes of variables. This is the abstract machinery that converts a sequence of collision planes into one big Lebesgue integral over time-parameters and plane-parameters.

Exercise: What does the uniform pullback lemma (10.8) say?
Let O subset R^k open (k>=d), Phi_n -> Phi in C^1_loc, rank DPhi = d a.e. Let K subset O and B subset R^d be compacts. Let a_n be supported in K with |a_n|<=A. Let F_n be supported in B with |F_n|<=M and F_n -> 0 in measure. Then
  int_O |F_n(Phi_n(q)) a_n(q)| dq -> 0.
The proof: Fix eps>0. On the compact "critical neighborhood" (small measure), the integral is trivially <= AM eps. Off it, DPhi has full rank, and for large n so does DPhi_n; a finite-cover inverse-function argument converts the integral into the L^1 norm of F_n against a bounded Jacobian factor. Since F_n -> 0 in measure with bounded support, its L^1 norm -> 0. This is the key comparison that lets us evaluate the five-fold gain integral at a small (measure-zero) set of near-cap events.

Exercise: What is "five-gain rank" (Lemma 10.9)?
Fix a nonaxial endpoint velocity v_0, endpoint (t_0,x_0) with t_0>0, five positive scales rho_i and signs sigma_i. Given positive times t_i and L_i in R^3, define recursively
  x_i = x_{i-1} - (t_{i-1}-t_i) v_{i-1},
  p_i = u_{rho_i,sigma_i}(t_i, z_i) e_3,
  n_i = v_{i-1} - p_i,  k_i = n_i x L_i,  v_i = v_{i-1} + k_i
for i = 1,...,5. The map Phi : (t_1,...,t_5, L_1,...,L_5) -> (t_5, x_5, v_5) is real analytic on (0,inf)^5 x R^15 and has rank 7 almost everywhere. In particular, it is a submersion a.e. on the ordered region 0<t_5<...<t_1<t_0. The key point: five gains on the (parameters)-space have ENCODED the endpoint (t_5,x_5,v_5) with full rank 7, so pushing back a Lebesgue-small family of endpoint states gives a Lebesgue-small family of parameters. The proof exhibits one witness point (equal-time configuration, five specific L_i's giving (bar v_i) that span the transverse plane and produce rank-7 partial derivatives), then notes that any zero set of the analytic 7x7 minor is Lebesgue-null (nontrivial analytic function argument), so rank failure is null in dimension 7.

Exercise: What does "five nonaxial gains" (Lemma 10.10) say?
Suppose ell_n -> 0 (the dilation's l=0 case), the hit endpoints converge after periodic translation, and their limiting velocity is nonaxial (bar v_n NOT -> 0). Suppose also 0<=h_n<=2 on a<=t<=tau_n and h_n -> 0 in measure on every compact subset of (0,U)xR^3_xx{v: bar v!=0}. Then (V_n^5 h_n)(tau_n,x_n,v_n) -> 0. In other words: away from the axis, five gains cannot hide input on a measure-zero-in-parameter set. The proof: successively delete the (uniformly small) input near the axis at each step so that all velocities are bounded below by b>0; delete spatial neighborhoods |y_i|<rho of various radii (their effect is controlled by the derivative of y_i in t_i at rate |bar v_{i-1}|>=b/2); then the parameter map of the five gains is exactly the rank-7 submersion of Lemma 10.9, and the uniform pullback gives a Lebesgue-small pushforward. The limiting "h_n small in measure" hypothesis does the rest.

Exercise: How does Proposition 10.1 (nonvanishing) follow?
Assume l=0 and g=0 (the contradiction case). Choose endpoint positions (tau_n, x_n, v_n) with h_n >= 1/2. Lemma 10.3 excludes unbounded velocity/position; the sorted subsequence has a translated limit. If the limit velocity is axial, use Lemma 10.6 (two gains already vanish), plus Lemma 10.6's extension to further gains, to bound the five-fold integral by a small constant. If it is nonaxial, use Lemma 10.10. In every case, limsup V_n^5 h_n(tau_n,x_n,v_n) = 0. Plugging into the five-fold iteration (10.five-residual) with these endpoints:
  1/2 <= a sum_{j=0}^4 (C log(1/a))^j / j!   for all small a>0,
which is a contradiction as a vv 0. So g != 0, proving the cap-dilation always produces a nonzero linear limit.

Exercise: What is Proposition 11.1 (finite diluted horizon)?
There are fixed parameters K>=1, 0<eta<=log2, 0<alpha<=1/4, 0<gamma<=1 (chosen before the thermal, seed, bath, and final horizon parameters) such that for every allowed geometric mesh with lin[0,1], the ONLY nonnegative g solving D g = 2P(B,g) - g nu_B on 0<t<inf, with locally integrable gain/loss, mild formula, and cap g<=(t+l)H, is g==0. Consequence: every nonzero limit in Propositions 9.1 and 10.1 has U<inf (finite dilated horizon). This is the final structural result needed to compare dilations across different horizons in Section 12.

Exercise: What are "positive histories" (Lemma 11.2)?
For the two-beam medium B = sum_sigma b_sigma(t,x) delta_{w_sigma(t,x)} with rate coefficients q_sigma = 2pi b_sigma |v - w_sigma| and frequency nu=sum_sigma q_sigma: consider a collision "history" where at ordered times a<tau_1<...<tau_j<b we perform elastic collisions with the wave velocity w_{sigma_i}(tau_i, X_tau_i) in direction n_i. The weight of such a history is
  W = e^{-int_a^b nu(s,X_s,V_s) ds} prod_{i=1}^j 2 q_{sigma_i}(tau_i, X_{tau_i}, V_{tau_i-}),
generalizing the usual Feynman-Kac formula. The lemma: if g >= 0 solves D g = 2P(B,g) - nu g with q_sigma bounded on compact positive-time sets, then the integral of W over any bounded measurable collection of bounded finite histories starting from any bounded measurable portion of g(a) is bounded by g(b). This is a "positive kernel domination" that lets us construct a lower bound on g(b) from any positive amount at time a.

Exercise: What is the "retention" lemma (11.3)?
Fix D=5, E = {|v|<=D}, B_0 = delta_{e_3} + delta_{-e_3}, nu_0(v) = 2pi (|v-e_3| + |v+e_3|). Let J(v,dv') be the single-child velocity law (choose sign with probability proportional to |v - sigma e_3|, then use the sphere rule Phi). Let p(v) = J(v,E) = Probability that a single collision starting from v lands a child in E. Lemma: p(v) >= 25/26 for every v in E. Moreover, the reference history sum with velocities in E is represented by a Markov velocity process with jump rate lambda(v) = 2p(v) nu_0(v), jump law J(v,dv')[. AND E]/p(v), and multiplicative weight exp(int_0^A W(V_s) ds) with W(v) = (2p(v) - 1) nu_0(v) >= c_0 := 48pi/13 > 0. Also lambda(v) <= C_0 := 48pi. The proof: expand the collision sphere and compute exit probability; on |v|<=5 the escape probability 1-p(v) <= 1/26 comes from a simple bound on sqrt((s+1)^2-4z^2). The weight is positive because each collision on average produces 2.p(v) children in E while consuming one, and the loss rate in the Markov is 2.p(v) < 1 at nu_0 scale.

Exercise: What is the three-collision minorization (Lemma 11.4)?
There exist r_0 > 0 and eps_0 > 0 such that the one-step transition law P_1(v, dA) of the retention Markov process satisfies
  P_1(v, A) >= eps_0 |A AND B(0, r_0)|  for every v in E and Borel A subset E.
Equivalently, three successive collisions produce an absolutely continuous component on the ball B(0, r_0) that is uniform in the starting v. Similarly, for every zeta>0 there is eps_zeta>0 with P_1(v, B(e_1,zeta) AND {|v'|<D}) >= eps_zeta for every v in E. The proof: force the first collision with a partner at distance >=1 (always possible); require the first output in a fixed small cap around the chosen partner; force the next two collisions with partners -sigma e_3 and +e_3; then the composite sphere map from three angular parameters to the third output has full rank (compute at the special point v=e_1, and note that reversing e_3 gives an extra rank-3 direction). Thus a uniform ball's worth of output velocities is reachable, with a bounded Jacobian lower bound.

Exercise: What does the mixing lemma (11.5) give?
The retention Markov process has a stationary probability law pi on E with mean zero (int v d pi(v) = 0), and there are C,c > 0 such that
  sup_{vinE} ||P_t(v,.) - pi||_TV <= C e^{-c t},
  sup_{vinE} E_v [ |A^{-1} int_0^A V_s ds|^2 ] <= C/A     (A>=1).
The proof: the three-collision minorization (Lemma 11.4) plus contractivity of P_t in total variation gives uniform geometric ergodicity, hence unique stationary pi. Negation of velocity is a symmetry, so pi is symmetric and its mean is zero. The integrated-velocity second moment is O(1/A) by a standard mixing estimate: the autocorrelation decays exponentially, so the double-time integral of <V_s.V_t> is bounded by C.A.

Exercise: What is the reference block (Lemma 11.6)?
There is a finite L such that for every A in I_L := [L/(1+d), L/(1-d)] (with d = 1/10 fixed) and every vinE, the reference branching histories (with velocities confined to E, final velocity strictly inside E, and displacement Z_A satisfying |Z_A - varkappa A e_1| < (3/4) d varkappa A) have total weight greater than 8. Here varkappa is a fixed small constant: 0 < varkappa < min{1/4, c_0/(2C_0), r_0/4}. The proof constructs the history explicitly in three phases:
  - phase 1 (length A_0 = (1-varkappa)A - 1): require integrated velocity <= Ad varkappa/4 (Chebyshev + mixing lemma ==> probability >= 1/2 uniform in start).
  - phase 2 (length 1): require last velocity in B(e_1, d/4) (minorization ==> fixed positive probability).
  - phase 3 (length varkappa A): require no collision (probability >= e^{-C_0 varkappa A}).
The displacement error stays below (3/4) d varkappa A. The potential gain for the branching weight (per Lemma 11.3) is exp((c_0 - C_0 varkappa)A), strictly positive; the multiplicative prefactor eps/2 from Lemmas 11.4 and 11.5 is fixed. For L large enough, this exceeds 8.

Exercise: What is the robust block (Lemma 11.7)?
Fix varkappa, L. There are J_br in N and eps_* in (0,1/4) with the following property. Suppose a measurable medium satisfies
  |b_sigma - 1| <= eps_*,  |w_sigma - sigma e_3| <= eps_*
uniformly along every speed-D excursion from the specified start over the duration AinI_L. From every starting velocity vinE, histories with between 1 and J_br collisions, with speeds at most D on every free flight, with all post-collision and final velocities strictly inside E, and with displacement |Z_A - varkappa A e_1| < d varkappa A, have total weight greater than 3. No continuity of the medium is needed. The proof fixes a compact test for each (v,A) using inner regularity and strict margins from Lemma 11.6, takes a finite subcover of ExI_L, and checks that the perturbed media preserve all the strict margins; the total parameter measure of "bad" histories is uniformly small.

Exercise: How are the parameters (eta, alpha, gamma, K) chosen to support the robust block?
Set delta_K = DL / (theta K (1-d)), a_br = L varkappa / (theta K). Enforce delta_K <= 1/4 and a_br <= 1/4 by increasing K. Pick eta > 0 small and then alpha in (0, min{1/4, eps_*, alpha_acc}) small and then gamma > 0 small so that the total medium-comparison error
  C_2 (eta + delta_K + (gamma/varkappa)^2 + alpha gamma / varkappa)
is below eps_*/2. Finally increase K further to make delta_K < eps_*/2 and (1+a_br)^4 < 3. These four parameters are fixed before the thermal, seed, bath, and horizon
parameters. All the robust block estimates are preserved under the later parameter choices.

Exercise: What does the block growth lemma (11.8) give?
For the parameters just fixed, there is a time t_br > 0 (independent of mesh and l) such that every density g >= 0 with D g = 2P(B,g) - g nu_B, cap 0<=g<=(t+l)H, mild formula, on an interval containing [t, (1+a_br)t] with t>=t_br, obeys
  M((1+a_br)t) >= 3 M(t),     M(s) := int_{B_s} g(s,x,v) dx dv,
where B_s = {(x,v): |x - varkappa s e_1| <= d varkappa s, |v| <= D} is a moving "block" whose transverse distance from the axis grows linearly in s. The proof: at each start in B_t, the robust block (11.7) provides a family of positive histories ending in B_{(1+a_br)t} with weight > 3, uniformly in the start; the transport equation preserves the history-weight bounds; summing over B_t gives M((1+a_br)t) >= 3M(t). The parameter a_br = L varkappa/(theta K) is small enough that (1+a_br)^4 < 3, matching the volume growth of the moving block (V ~ t^3) and the cap (one more factor of t).

Exercise: What does the accessibility lemma (11.9) give?
For any nonzero g >= 0 solving the linear equation on 0<t<inf under the cap 0<=g<=(t+l)H, at arbitrarily large finite times t the moving-block mass M(t) := int_{B_t} g(t,x,v) dx dv is strictly positive. In words: any positive mass in the linear equation eventually finds its way into the moving blocks. The proof:
  - Pick a bounded, integrable, non-axis-portion of g at some time a_0 to get a positive mass outside the hole and off the axis.
  - Free-flight for a long-but-finite time keeps it off the hole (bar v != 0 dominates).
  - In a small time window (bounded by h << t), steer the velocity to near varkappa e_1 via three collisions with partners +/-e_3, using the alpha_acc reserved for this (unlike the reference block, we just need one specific steering path, and we can use any medium bounds in this window because t is huge so the medium is well-behaved on annular meshes);
  - Free-flight for a long time t to land the trajectory inside the moving block B_t.

Exercise: How is Proposition 11.1 (finite dilated horizon) proven?
Suppose g is nonzero on all positive times, obeys the cap. Lemma 11.9 gives a time t_0 >= t_br with M(t_0) > 0. Apply Lemma 11.8 inductively: at t_n = (1+a_br)^n t_0, M(t_n) >= 3^n . M(t_0). On the other hand, the cap bound and the definition of B_t give
  M(t_n) <= (t_n + 1) . (4pi/3)(d varkappa t_n)^3 . int_E H(v) dv <= C (1+t_n)^4.
If we match exponents: 3^n grows exponentially, (1+(1+a_br)^n t_0)^4 grows like ((1+a_br)^n)^4 = (1+a_br)^{4n}. Since (1+a_br)^4 < 3, we get (1+a_br)^{4n} < 3^n, contradiction as n -> inf. So no nonzero global g exists; hence the dilated horizon U must be finite in Prop 9.1/10.1.

Exercise: What is the dormant vanishing lemma (12.1)?
Let F^d be the dormant global limit (from Prop 8.1 with zero hot seed). On the periodic lift centered at the origin,
  F^d(Tt, Tx, v) / (T m_T) -> 0     as Tvv0,
in local L^1 on V := (0,1) x R^3_x x {v : bar v != 0}. The proof: use the common early-time pointwise majorant of the dormant family, sum cold majorants over all scales plus bath and dormant-hot; each majorant is integrable with every polynomial weight. On a compact set with |bar v|>=eps and t in [a,1], each piece independently vanishes: bath because the time-dilated bath is inside the initial hole (D_0 >= delta), hot because Z(t) <= C t^A = C(Tt)^A after division by (Tm_T) = T^{1+p}, cold because the main spatial enlargement has scale s <= C_R T and the Gaussian in m_i with |bar v|>=eps dominates every polynomial in 1/T.

Exercise: How is Theorem 1.2 (main result) finally proven?
Assume there are horizons T_k vv 0 at which a cap limit equals F^d (dormant) on (0,T_k). For each k, choose approximations to that specific limit along the cap subsequence. A diagonal argument picks N_k such that s_{N_k}/T_k -> 0 and the normalized approximations f_k(t,x,v) := F_k(T_k t, T_k x, v) / (T_k m_{T_k}) -> 0 in local L^1 on V, using Lemma 12.1.
Each approximation has its cap time t_{*,k}, so define r_k = t_{*,k} + s_{N_k}. Prop 9.1 gives h_k(r_k t, r_k x, v)/(m_{T_k} r_k) ~> g, nonzero on 0<t<U, with Prop 11.1 forcing U<inf. Hence rho_k := r_k/T_k -> 1/U > 0, so the two dilations (hit-dilation vs horizon-dilation) are comparable up to a positive factor.
Compare F_k = c_k + u_k + h_k >= h_k. Then
  0 <= h_k(r_k t, r_k x, v) / (m_{T_k} r_k) <= rho_k^{-1} f_k(rho_k t, rho_k x, v).
Push forward a compact subset K subset (0,U)xR^3_xx{bar v != 0} under (t,x) -> (rho_k t, rho_k x). The image sits in a fixed compact K' subset V (since rho_k -> 1/U > 0). The spacetime Jacobian is rho_k^4, so
  int_K h_k(r_k t, r_k x, v)/(m_{T_k} r_k) dt dx dv <= rho_k^{-5} int_{K'} f_k(t,x,v) dt dx dv -> 0.
Hence g = 0 off the velocity axis, and g is a density bounded by cap on a null-measure axis; so g == 0. Contradiction with Prop 10.1. Therefore for all sufficiently small T, no cap limit can equal F^d on the corresponding early interval. Choose one such T and its cap limit F^(1). Then F^(1) and F^d are both global entropy solutions with the same datum and local conservation, and they differ on a set of positive spacetime-velocity measure. Rotating back from the lattice Lambda to Z^3 preserves all these properties and gives Theorem 1.2.

Exercise: Why does this NOT contradict known uniqueness results?
Because those results assume strictly MORE than Definition 1.1's class. Kaniel-Shinbrot 1978 and Ukai 1974 assume amplitudes bounded by a fixed multiple of a Maxwellian, or smallness of a weighted norm; Mischler-Wennberg 1999 for the homogeneous case assumes boundedness and integrability of specific velocity moments; Duan-Huang-Wang-Yang 2017 assume small relative entropy AND small spatial integral of velocity supremum. The datum F_0 in the present paper has unbounded amplitudes at shrinking spatial scales: sup_v over a single point can blow up like 1/s_N while the mass and energy integrals stay bounded. No known uniqueness theorem covers such a datum, and DiPerna-Lions-style compactness does not select a unique limit. In fact recent well/ill-posedness bifurcation work (Chen-Holmer 2024, Chen-Shen-Zhang 2024, Chen-Guo-Shen-Zhang 2026) has already shown that nearby-datum uniqueness fails in various weighted norms; the present paper upgrades "nearby data" to "exactly the same data" for solutions in the natural physical class. So the result is a sharp counterexample at the exact boundary between where existence is known (large class) and where uniqueness is known (strictly smaller class).

Exercise: What is the physical interpretation of the nonuniqueness?
The mechanism manufactures a minutely fine structure (jets at scales s_j -> 0) whose initial energy is bounded, but whose pointwise amplitudes blow up. Two different tiny seeds -- differing only at the smallest scale s_N with total mass O(s_N^2), which goes to zero -- can drive the solution into two different macroscopic long-time states. Physically this is a statement that a small, highly concentrated perturbation can remain hidden in the velocity axis and only later reveal itself through a genuinely different macroscopic evolution. It does not mean the Boltzmann equation is "wrong"; the real gas is finite-N and the N->inf limit has more structure than any single renormalized solution class as encoded in Definition 1.1. What it does mean: the local-conservation + entropy class is not, by itself, a predictive theory, and any uniqueness claim needs additional structural control (e.g., bounded amplitudes, smallness, positivity at every point). The result also confirms the intuition behind known ill-posedness bifurcations, and suggests that any numerical scheme resolving only the macroscopic fields cannot distinguish the two global limits without additional fine-structure data at time zero.

What you now have: a complete path through the paper. Every lemma, every proposition, every theorem has a plain-English summary answer, and every proof idea is at least sketched. If you want to see the raw LaTeX, the whole original is in hard_sphere_local_conservation_original.txt alongside this file. If you want to reconstruct ANY specific number (e.g., the choice of m=100, or the role of theta=1/4), search for the lemma name in the original. Thematically, the paper reduces to four ideas:
  1. Concentration: build a self-similar family of jets at shrinking scale s_j, with a barrier that controls each jet uniformly.
  2. Selection: use a tiny seed to pick a finite solution with a prescribed hot size at a prescribed time.
  3. Compactness: pass to the limit to get two global entropy solutions with local conservation.
  4. Separation: show the two limits genuinely differ by dilating at the cap-hit point and showing that a positive linear solution appears (via branching growth), while the dormant solution vanishes under the same dilation.
Everything else is technical scaffolding to make each of those four steps rigorous.