Proof. The comparisons follow by multiplying each plank denominator by \((b/a)^{A_2-A_1}\) and using \(1\le b/a\le N\). Thus a fixed plank’s contribution to \(N^A\Delta_A\) has logarithmic slope \(\log_N(Na/b)\in[0,1]\). Taking the supremum preserves these comparisons. Monotonicity of the bound and closedness of the set of valid candidates follow, the latter by letting \(A_2-A_1\) tend to zero after applying the bound on an arbitrary fixed sequence.

It remains to prove validity at \(A=d-1\). We bound multiplicity by the weighted count of pairs meeting in fine cells, then sum by slope separation. Let \(\mu(Q)\) be the total weight of events in a fine space–time cell \(Q\). Cauchy–Schwarz gives \[m=\mathcal I/|E|\le \frac{\sum_Q\mu(Q)^2}{\mathcal I}.\] Fix one indexed line. Other lines with relative slope magnitude comparable to \(\theta\ge N^{-1}\) that pass within bounded fine distance of it at some time are contained in a full-time plank of both widths \(O(\theta N)\) around it. Their total mass is at most \(O(\Delta_{d-1}(\theta N)^d)\). Two such lines can be within bounded fine distance for at most \(O(1/\theta)\) time bins: their spatial difference is affine with velocity of magnitude comparable to \(\theta\). For relative slopes at most \(N^{-1}\), use mass \(O(\Delta_{d-1})\) and at most \(N\) bins. Fixed large slope ranges or patch boundaries require only fixed finite coverings. Summing over dyadic \(\theta\) and then the reference line gives \[\sum_Q\mu(Q)^2 \lesssim n\Delta_{d-1}N^d \left(1+\sum_{\substack{\theta\text{ dyadic}\\N^{-1}\le\theta\lesssim1}} \theta^{d-1}\right) \le n\Delta_{d-1}N^{d+o(1)}.\] This includes \(d=1\), where the sum costs a logarithm, and includes repeated indices. Consequently \[m\mathrel{\leq_{\mathrm{exp}}}\Delta_{d-1}N^d/k \mathrel{=_{\mathrm{exp}}}\Delta_{d-1}N^{d-1+F(1)}.\] Since \(\Pi_F(1)\ge(p-1/2)F(1)\ge F(1)\), (9) holds at \(A=d-1\). When \(z=-1\), one has \(d=1\) and this is the only possible candidate, proving \(h(p,-1)=0\). ◻

Proof. Start with \(A\). Whenever a tested cell has positive remaining mass less than \(\vartheta_0\mu(C)\), delete its remaining points. A cell used in such a deletion becomes empty and is never charged again. Each deletion costs less than \(\vartheta_0\mu(C)\), measured against the original measure. For each partition the sum of the original cell masses is \(\mu(\Omega)\). Thus all deletions together cost at most \(L\vartheta_0\mu(\Omega)\). The process terminates because \(\Omega\) is finite. Cascades between the partitions cause no additional charge. ◻

Proof. Fix first a finite mesh \(0=\kappa_0<\cdots<\kappa_J=1\) and an accuracy \(\nu>0\). At resolution \(N\), round each interior mesh point to a multiple of \(1/\log_2N\), keeping the endpoints. For sufficiently large \(N\) these rounded points remain ordered; they specify actual dyadic block sizes and differ from the prescribed points by at most \(1/\log_2N\). In this finite construction \(\kappa_r\) denotes the rounded value. The rounding error tends to zero before the mesh is refined. At every event label the original counts in its index’s occupied ancestral blocks at these scales, rounding \(\log_N\) of each count to intervals of length \(\nu\). There are at most \((2+1/\nu)^{J+1}\) labels. Include the other prescribed labels, retain a label class of maximal event mass, and clean all the partitions into pairs consisting of an index and a block. Those partitions may have many cells, which is permitted in Lemma 6. Let \(c_j\) be the lower endpoint, a multiple of \(\nu\), of the common rounded count interval at scale \(\kappa_j\). Every surviving occupied block there has count between \[\vartheta_0N^{c_j} \quad\hbox{and}\quad N^{c_j+\nu}.\] The same assertion at the root gives uniform counts per retained index. The weights cancel in these within-index comparisons.

Put \(\lambda_N=\log_N(1/\vartheta_0)\) and \(\epsilon_N=\nu+\lambda_N\). An occupied coarse block contains an occupied finer block. Conversely it has at most \(N^{\kappa_k-\kappa_j}\) subblocks across the actual dyadic scale gap \([\kappa_j,\kappa_k]\). The preceding count bounds therefore give, for every \(j<k\), \[-\epsilon_N\le c_k-c_j \le\kappa_k-\kappa_j+\epsilon_N.\] The error does not accumulate with the number of intervening scales. To replace these approximate count exponents by an admissible profile, first define a branching-count exponent and then its deficit: \[b_N(s)=\min_{0\le j\le J}\{c_j+(s-\kappa_j)_+\}, \qquad F_N(s)=s-b_N(s).\] Each function in the minimum is nondecreasing and \(1\)-Lipschitz, and so is their minimum. Since \(c_0=0\) and every \(c_j\ge0\), one has \(0\le b_N(s)\le s\) and \(b_N(0)=0\). Thus \(F_N\) is an admissible profile. At a tested scale the preceding increment bounds imply \[c_j-\epsilon_N\le b_N(\kappa_j)\le c_j.\] Taking an occupied tested descendant and a tested ancestor for an arbitrary occupied intermediate block \(I\) of \(K\) fine bins proves the finite, all-scale estimate \[ \left|\log_N|S_i'\cap I|- \bigl(s-F_N(s)\bigr)\right| \le \max_j(\kappa_{j+1}-\kappa_j)+2\nu+\lambda_N, \qquad s=\log_N K. \tag{13}\]

For clarity, the passage to arbitrarily many tests is sequential. At stage \(r\) take a fixed finite mesh of gap at most \(1/r\), the first \(r\) prescribed tests and their required joint or domain partitions, and a fixed label accuracy at most \(1/r\). Let \(C_r\) be the reciprocal of the retained mass fraction guaranteed by the finite construction, including cleaning, and let \(L_r\) be the number of cleaned partitions. Choose the resolutions for that stage so large that \(\log(C_rL_r)/\log N\le1/r\) and all other fixed-stage errors are at most \(1/r\). Let the stage grow sufficiently slowly along the original resolution sequence. The resulting refinements are extensive. Compactness of the finite profiles \(F_N\) gives a uniformly convergent subsequence with admissible limit \(F\). Equation (13) gives uniformity on all scales. This proves (i).

For (ii), let \(s<t\) be fixed. Uniformity shows that the number of occupied \(s\)-blocks in any occupied \(t\)-block on an original index has exponent \[(t-s)-\bigl(F(t)-F(s)\bigr).\] For the restricted and regularized index the corresponding exponent is \((t-s)-(G(t)-G(s))\). Its occupied subblocks are a subset of the original ones. Comparison proves (11); rational scales suffice, followed by continuity. Let \(n'\) be the remaining total index weight. Since \(n'\le n\), the assumed event-mass retention gives \[n'N^{1-G(1)+o(1)} \ge nN^{1-F(1)-e+o(1)}.\] This proves the upper bound in (12); the lower bound follows from (11) with \(s=0,t=1\). If \(e=0\), all increments of \(G-F\) are nonnegative and its total increment is zero, so \(G=F\).

For (iii), apply the same child-block upper bounds within each derived fiber. The sum of its possible starting weights is at most \(N^{o(1)}\) times the sum for the original index–starting-block pairs. The retained event mass supplies the same lower bound used in the preceding displayed formula. Thus the total branching loss is at most the event-loss exponent, and every subinterval has the inherited upper branching bound. After normalizing a logarithmic interval of length \(L>0\) to length one, both the logarithmic loss and the profile deficits are divided by \(L\). No estimate uniform in a vanishing \(L\) is asserted before fixing that interval. ◻

Proof. The original event mass assigned to \(C\) is at least \(N^{a-\varepsilon}M_C\). Each pair contributes at most \(N^{a+\varepsilon}\omega_i\) to its retained event mass, so division by \(N^{a+\varepsilon}\) gives the first bound. The discarded pieces in a cell have total mass at most \(\sigma_N\) times that cell’s mass. An occupied retained piece has mass at least \(\vartheta_0\sigma_N/L_N\) times the original cell’s mass; the same division proves its active-weight bound. ◻

Proof. For a partition \(0=t_0<\cdots<t_J=t\), Jensen’s inequality gives \[\sum_{j=1}^J(t_j-t_{j-1}) P_{p,q}\!\left(\frac{F(t_j)-F(t_{j-1})}{t_j-t_{j-1}}\right) \le\Pi_F(t).\] The supremum over refining dyadic partitions equals \(\Pi_F(t)\): the interval averages of the bounded function \(F'\) converge to it in \(L^1\), and \(P\) is Lipschitz. Each finite sum is continuous in \(F\) for uniform convergence, proving (i). By (11), \(G-F\) is nondecreasing and Lipschitz, so \(G'\ge F'\) almost everywhere. The slope bound gives \[0\le\int_0^1[P(G')-P(F')]\le p\int_0^1(G'-F')\le pe,\] proving (ii). A change of variables proves (iii).

The \(p\) derivative is immediate. For \(q>0\) the displayed average for \(\mathcal D_q\) exists also when \(e=0\); for \(e>0\) its possible equality set in the \(r\) integral has measure zero. It is continuous in \((q,e)\) with \(q>0\), and bounded between zero and one. Dominated convergence gives (14), with the positive sign because \(z=-q\). ◻

Proof. Increasing \(p\) increases the cost and therefore cannot increase \(h\). For \(z_2=z_1+s\), \(s>0\), the dimension parameter increases by \(s\). Starting with a valid candidate \(A\) at \(z_1\), use \(A+s\) at \(z_2\). Its \(a\) exponent is unchanged and its \(b\) exponent increases by \(s\). Writing the dimension in a subscript temporarily, we have \[\Delta_{A+s,d+s}\ge N^{-s}\Delta_{A,d}.\] Also \(q\) decreases, so \(P_{p,q}\) increases pointwise. The new right side of (9) is therefore at least the old one. The candidate is still in range since \(A+s\le d+s-1\). Thus \[ h(p,z+s)\le h(p,z)+s, \tag{15}\] including admissible endpoints. This proves the separate monotonicity of \(g(p,z)=h(p,z)-z\).

Here is the passage from this monotonicity to the stated form of differentiability. On each compact rectangle in the interior, \(g\) is bounded, and its variations on coordinate lines are bounded by its range. Integration along those lines shows that both distributional first derivatives are finite measures. Thus \(g\) is locally of bounded variation. We use the approximate differentiability theorem for BV functions (Ambrosio et al. 2000, Theorem 3.83): away from a null set there are a value and a linear map giving a first-order approximation on sets of density one. Coordinatewise monotonicity upgrades this approximation to ordinary total differentiability, as follows.

At such a point \(x\), fix an error tolerance and a density-one set on which the approximate linear estimate holds. For a displacement \(u\) of size \(r\) and any fixed \(\varepsilon>0\), consider small squares of side comparable to \(\varepsilon r\) located coordinatewise just above and just below \(x+u\). They lie within \(O(r)\) of \(x\). For sufficiently small \(r\) their area is larger than the bad part of the density-one set in that neighborhood, so each square contains a comparison point where the linear estimate holds. Monotonicity bounds \(g(x+u)\) between the two comparison values. The comparison displacements differ from \(u\) by \(O(\varepsilon r)\). The same sandwich first identifies the actual value \(g(x)\) with its approximate value. Divide the resulting error by \(r\), let \(r\to0\), and then let the approximation tolerance and \(\varepsilon\) tend to zero. This is total differentiability at \(x\).

For fixed \(p\in I\), the decreasing function \(z\mapsto g(p,z)\) has only nonpositive singular variation. Its ordinary derivative therefore obeys \[g(p,b)-g(p,a)\le\int_a^b g_z(p,z)\,dz.\] Adding \(b-a\) gives the same inequality for the increment of \(h\) and the integral of \(h_z\). There is no lost upward jump at \(z=0\): (15) gives \(h(p,-t)\ge h(p,0)-t\). At the other endpoint, \(0\le h(p,z)\le1+z\) and \(h(p,-1)=0\). Letting \(a\downarrow-1\) and \(b\uparrow0\) consequently gives \[0<h(p,0)\le\int_{-1}^0h_z(p,z)\,dz.\] Fubini and almost-everywhere total differentiability yield an interior point of total differentiability with \(h_z>0\). At a differentiability point where \(h=0\), nonnegativity of \(h\) would force its gradient to vanish. Hence the selected point has \(h>0\) as well. The derivative signs follow from the two monotonicity comparisons. ◻

Proof. For every uniform-profile sequence form the score \[\limsup_{N\to\infty} \left\{\log_N(m/\Delta_{h_0})-\Pi_F(1)\right\},\] and let \(H\) be the supremum of these scores. The critical bound at \(h\) and \(\Delta_h\le\Delta_{h_0}\) give \(H\le h\). If \(H\le h_0\), then the bound defining \(h\) would hold at \(h_0\) for every sequence, contrary to \(h_0<h\). Thus \(H>h_0\). The supremum definition also proves (18) at \(H\) itself.

Choose for each integer \(j\) a sequence with profile \(F_j\) and score at least \(H-1/j\). Profiles form a compact set for uniform convergence; pass to a subsequence with \(F_j\to F\). From the \(j\)th sequence choose a realization at a resolution so large that its uniform-profile count error relative to \(F_j\) is at most \(1/j\) over all occupied indices and all dyadic time scales, that all chosen resolutions increase, and that \[\log_N(m/\Delta_{h_0})\ge H+\Pi_{F_j}(1)-2/j.\] Any other finite list of prescribed tests is accommodated before this choice. The selected diagonal sequence has uniform profile \(F\). Lower semicontinuity of the cost gives \[\liminf\log_N(m/\Delta_{h_0})\ge H+\Pi_F(1).\] The universal upper bound (18) supplies the reverse limsup. Equality follows. This argument also explains why a downward jump of the cost at the profile limit cannot cause loss of an extremizer: it would instead improve the score, which is forbidden by the same sharp upper bound. Multiplying all weights by \(\Delta_{h_0}^{-1}\) is permitted if a normalized realization is desired.

For an extensive graph refinement, its incidence mass loses only a subpower and its support is a subset of the old support. Hence \(m\mathrel{\leq_{\mathrm{exp}}}m'\). Its plank masses cannot increase, so \(\Delta'_{h_0}\le\Delta_{h_0}\). Its profile remains \(F\) by Lemma 7. Applying (18) to the refinement sandwiches both quantities: \[m\mathrel{\leq_{\mathrm{exp}}}m'\mathrel{\leq_{\mathrm{exp}}} N^{H+\Pi_F(1)}\Delta'_{h_0} \le N^{H+\Pi_F(1)}\Delta_{h_0}\mathrel{=_{\mathrm{exp}}}m.\] This proves (19).

For the final assertion, let \(H_j\) denote the corresponding sharp hybrid power. By (10), \[0\le\log_N(\Delta_{h_{0,j}}/\Delta_h)\le h-h_{0,j}, \qquad h_{0,j}<H_j\le h.\] Thus a sufficiently accurate realization of the \(j\)th hybrid equality has \[\log_N(m/\Delta_h)-h-\Pi_{F_j}(1) =o_j(1),\] where the error can be made to tend to zero by first taking that realization far enough along its sequence. Take a uniformly convergent subsequence of \(F_j\) and a finite-test diagonal as above. Lower semicontinuity gives the lower bound in (20); the critical bound at \(h\) gives its upper bound. Uniform convergence also preserves the lower endpoint-deficit bound \(F_j(1)\ge\alpha_*\). ◻

Proof. We sample the indices according to their normalized weights. A finite family of box tests will control all convex clusters in the sample, with only a logarithmic loss.

Write \(v_\delta\) for the common tube volume, and normalize the weights by \(v_i=\omega_i/A\). Testing the convex body \(T_i\) gives \(v_i\le1\). Testing a fixed ball containing the family gives \[S:=\sum_i v_i\le C_R\delta^{-2}.\] Choose each index independently with probability \(v_i\). The expected number of chosen indices is \(S\).

We first reduce all convex tests to finitely many box tests. If a convex body \(K\) contains a tube from the family, put \(K_R=K\cap\overline{B(0,R)}\); this intersection contains every tube counted in \(K\). Enlarge an inscribed John ellipsoid of \(K_R\) by a dimensional factor to obtain an ellipsoid containing \(K_R\), with volume at most a dimensional constant times \(|K|\), largest axis \(O_R(1)\), and every axis at least a constant times \(\delta\): the body contains a ball of radius \(\delta\). An enclosing rectangular box has the same properties. Quantize its center, axis lengths, and orthonormal frame with accuracy \(c_R\delta^2\), and enlarge each side by \(C_R\delta^2\). There is thus a collection \(\mathcal B_\delta\) of at most \(C_R\delta^{-C_R}\) convex boxes such that every relevant \(K_R\) is contained in some \(Q\in\mathcal B_\delta\) satisfying \[|Q|\le C_R|K|,\qquad |Q|/v_\delta\ge c_R.\] The quantization error is smaller than a fixed fraction of the shortest side; this proves both containment and the volume comparison.

For \(Q\in\mathcal B_\delta\), the number \(Z_Q\) of chosen indices whose tubes lie in \(Q\) is a sum of independent Bernoulli variables, and (23) gives \[\mathbb EZ_Q\le |Q|/v_\delta.\] The elementary exponential-moment bound \[\mathbb P(Z_Q\ge t) \le \inf_{a>0}\exp\bigl((e^a-1)\mathbb EZ_Q-at\bigr)\] shows that, for a sufficiently large fixed \(L_R\), simultaneously \[ Z_Q\le L_R(1+\log(1/\delta))\,|Q|/v_\delta \qquad(Q\in\mathcal B_\delta) \tag{25}\] with probability at least \(3/4\). Indeed each failure probability can be made smaller than \(\delta^{C_R+2}\), and a union bound applies. If \(S\ge L_R(1+\log(1/\delta))\), the Bernoulli lower-tail bound \[\mathbb P(N_{\rm sample}<S/2)\le e^{-S/8}\] gives at least \(S/2\) chosen indices with probability at least \(3/4\), after increasing \(L_R\). This lower-tail estimate follows by applying the exponential moment method to \(e^{-aN_{\rm sample}}\). Fix a sample with both properties.

By the box comparison, its convex clustering is \(O_R(1+\log(1/\delta))\). The multiplicity of any repeated tube is bounded by the same quantity, since a box of volume \(O(v_\delta)\) contains that tube. Keep one chosen index for each distinct tube. The resulting set has at least \(c_RS/(1+\log(1/\delta))\) members, and each retained shading still has the required minimum density. It is not necessary to identify different shadings on repeated tubes.

Apply Theorem 13 with target loss \(\varepsilon/2\), and choose \(\eta_{\mathrm{in}}\) no larger than its density tolerance. For all sufficiently small \(\delta\), the logarithmic clustering bound satisfies the theorem’s hypothesis. Consequently \[\left|\bigcup_iY_i\right| \ge \frac{c_RS v_\delta\delta^{\varepsilon/2}} {1+\log(1/\delta)}.\] Since \(\sum_i\omega_i|Y_i|\le ASv_\delta\), this proves (24). If \(S<L_R(1+\log(1/\delta))\), use instead the pointwise bound \(\sum_i\omega_i\mathbf 1_{Y_i}\le AS\) and absorb the logarithm. The bounded range of remaining \(\delta\) is covered by \(\sum_i\omega_i\le C_RA\delta^{-2}\), with a larger constant. ◻

Proof. There are two geometric steps: a plank bound controls weighted convex clustering of thickened traces, and disjoint balls at marked bin centers convert incidence counts to shading volume.

It is harmless to allow \(a,b\) up to a fixed multiple of \(D\) in (27). A larger spatial coordinate can be handled by partitioning its two endpoint values, at \(t=0\) and \(t=1\), into intervals of fixed length in physical units. Linear interpolation puts every part inside a permitted full-time plank. There are only a bounded number of parts, depending on the chart. Apply the given bound to each part; the sum of their area factors is at most a fixed multiple of \(ab\). A fixed decrease in the lower width cutoff is handled by enlargement, at a fixed cost in the area factor.

Enclose each trace in a tube \(T_i\) of radius \(C_0\delta\) and a common fixed Euclidean length \(L\). Choose \(L\) large enough that the trace, and balls of radius \(C_0\delta\) about its points, lie away from the end caps. Both constants depend only on the bounded chart.

We check weighted convex clustering for these tubes. Let \(K\) contain some of them, intersect \(K\) with a fixed ambient ball containing all \(T_i\), and take an enclosing ellipsoid \(\mathcal E\) of volume \(O(|K|)\) by dilating an inscribed John ellipsoid. Its diameter is bounded. Its horizontal slices have affine centers and a fixed pair of principal axes; their semiaxes are a common scalar multiple of those of the maximal slice. The time diameter of \(\mathcal E\) is at least one, because every counted tube contains its full chart trace. The ellipsoid volume is a dimensional constant times this time diameter times the maximal slice area. Thus that area is \(O(|K|)\).

Circumscribe a rectangle about the maximal horizontal slice and translate it to the affine center of each slice. The resulting full-time plank contains all the counted traces and has cross-sectional area at most \(C|K|\). Its spatial widths are at least a constant times \(\delta\), since a counted tube contains a spatial disk of comparable radius at every chart time. Applying (27) therefore gives \[\sum_{i:T_i\subset K}\omega_i\le C A\delta^{-2}|K|.\] Since \(|T_i|\asymp\delta^2\), the weighted convex clustering of the family is at most \(CA\).

At each marked bin center put a ball of radius \(c_0\delta\) on the trace, with \(c_0>0\) fixed and small. Balls belonging to different bins of one index are disjoint. Their union \(Y_i\subset T_i\) has volume comparable to \(\#J_i\delta^3\); its density in \(T_i\) is at least \(c\delta^{\eta_{\mathrm{in}}}\). Moreover \[\left|\bigcup_iY_i\right|\le C\delta^3\#E.\] Rescale the common length \(L\) to one and apply Lemma 14. Choose the present \(\eta_{\mathrm{in}}\) smaller than half the density tolerance of that lemma, so fixed length and density constants are absorbed for small \(\delta\). Canceling \(\delta^3\) proves (28) for large \(D\). For bounded \(D\), a fixed collection of widest planks bounds the total weight by \(CA D^2\). The pointwise multiplicity bound by this total weight gives the conclusion after increasing \(C\) on that bounded range. ◻

Proof. It suffices to select separated line parameters whose uniform distribution inherits the weighted Frostman bound, and to keep one original shading at each selected parameter.

Push the normalized weights to the \(O(\delta^{-2})\) parameter cells. Discard cells of mass below \(\delta^4\); for small \(\delta\) the lost mass is less than \(1/2\). There are \(O(1+\log(1/\delta))\) remaining dyadic mass levels, so one level has mass at least \(c/(1+\log(1/\delta))\). On this level the uniform distribution of occupied cells has Frostman constant at most \(C\delta^{-\eta_{\mathrm{in}}}(1+\log(1/\delta))\): compare the comparable masses of the cells in an enlarged ball with the original probability. Color the parameter mesh by \(J\) congruence classes, where \(J\) is a fixed constant, and retain a class containing at least \(1/J\) of the retained cells. Choose one actual line and its original shading from each retained cell. The resulting line parameters are separated, the shadings retain their Frostman bounds, and their union is contained in the original union. Apply Theorem 16, having chosen the current \(\eta_{\mathrm{in}}\) smaller than half its allowed tolerance. ◻

Proof. For unit weights this is Guth (2010, Theorem 1.3, p. 264), the endpoint form of multilinear Kakeya introduced in Bennett et al. (2006, Theorem 1.15). Integer weights are represented by repetition. There is no distinctness requirement; alternatively one can first separate repeated cylinders by distinct parallel translations tending to zero and use Fatou’s lemma. Off their boundaries, the indicators then converge to the repeated indicators, and directions are unchanged.

For rational weights clear denominators separately in the three families; the factors cancel on the two sides of (30). Approximate real weights monotonically from below by rational weights. Monotone convergence proves the displayed estimate. Truncating the cylinders can only decrease the integrand. ◻

Proof. Increase each cylinder’s radius to \(1+\sqrt3\). Every cube meeting the original cylinder lies entirely in the enlarged cylinder, so the enlarged multiplicity dominates \(m_j(Q)\) everywhere on \(Q\). Integrate over the disjoint cubes and use Theorem 19, followed by the fixed radius rescaling. Finally the determinant of three unit vectors changes by at most a constant times the sum of their perturbations. Caps of radius \(c\theta\) therefore give cross-family determinant at least \(\theta/2\) when \(c\) is small enough. ◻

Proof. The rescaling and its cell count. We first describe the operation that determines the threshold \(T\). Suppose active indices on selected \(K\)-blocks have been assigned to packets of widths \(a,b\). Suppose also that the indices of each packet lie in one cell of an isotropic grid in matrix space, of side \(b/K\); call this cell its full-time parent. Subtract the parent center matrix and divide space by \(b/K\), leaving time unchanged. Use resolution \[ N'=\frac{Nb}{Ka}. \tag{36}\] The new matrices lie in a bounded patch. A new spatial fine cell corresponds to \(a\) old spatial fine cells, and an original \(K\)-block contains \(b/a\) new time bins.

Fill each selected block with those new bins, keeping all selected blocks of one original index together in its full-time parent. A packet now has transverse widths \(1,b/a\). It occupies at most \(C(b/a)^2\) space–time cells: at most \(Cb/a\) spatial cells per bin, over at most \(Cb/a\) bins. If its inherited index mass is \(M\), its incidence mass is at least \(cMb/a\). Thus a collection of disjoint indexed-block assignments, each of mass at least \(M\), has aggregate raw multiplicity at least a constant times \(Ma/b\). We count the supports in different parents separately and bound each union by the sum of its packet supports. Overlaps within a parent only improve this lower bound.

This calculation explains the factor \(a/b\) in the comparison below. To use the sharp upper bound we must also preserve the original time tree above scale \(K\). We now construct the extensive parent ensemble for which that profile comparison is valid.

Proof. The full bounded patch is covered by boundedly many planks with both widths comparable to \(N\), so \(n\mathrel{\leq_{\mathrm{exp}}}\Delta_{h_0}N^d\). Since \(k\mathrel{=_{\mathrm{exp}}}N^{1-\alpha}\) and \(m\mathrel{=_{\mathrm{exp}}}\Delta_{h_0}N^{H+\Pi(1)}\), the ratio in (40) is at least \(N^{q+H+\Pi(1)-2\alpha}\) in exponent. Inequality (32) gives the claim.

Lines meeting a fixed fine cell with slope diameter \(O(1/L)\) stay within a bounded-width plank about one of them throughout the dyadic \(L\)-block containing that cell. They are active on that block. Apply (33) with \(a,b=O(1)\), using bounded coverings to handle fixed enlargements. Bounded slope charts identify spatial slope diameter and spherical direction diameter up to constants. This proves (41). ◻

Proof. Choose a lattice cover by balls of radius a sufficiently large fixed multiple of \(K\) in fine space-time units. Its overlap is bounded. The radius is large enough that one of the balls assigned to a reference index-block contains its whole segment and the whole \(K\)-block segment of every bounded-slope line meeting it at a fine cell. Only boundedly many choices are needed for any index-block.

By (33), the active mass near such a ball is at most \(mK^{C_0}\), for an absolute constant \(C_0\): cover its time extent by boundedly many \(K\)-blocks and its transverse extent by boundedly many planks with \(a,b\asymp K\). Multiplying by the at most \(CK\) events of each active line gives an incidence upper bound \(mK^{C_0+1}\) for the ball.

Divide directions into caps of radius \(K^{-20}\) and call a cap rich at the ball if its active mass there is at least \(mK^{-100}\). There are \(O(K^{40})\) caps. A ball is bad if three rich caps have direction determinant at least \(K^{-10}\). Apply Theorem 19 to the unshaded tubes at spatial resolution \(K/N\), summing over the possible cap triples. The transversality loss, the cap count, and the thresholds are fixed powers of \(K\). Therefore, for a fixed constant \(C_1\) and any fixed \(\varepsilon>0\), \[ \#\{\text{bad balls}\} \lesssim_\varepsilon N^\varepsilon K^{C_1}(n/m)^{3/2}. \tag{42}\] Indeed, the multilinear summand at a tested ball is at least \((mK^{-100})^{3/2}\), and the product of the three full tube masses is at most \(n^{3/2}\). Bounded ball-to-cube comparisons do not change the estimate.

The fraction of all weighted incidence in bad balls is consequently at most \[N^\varepsilon K^{C_2} \left(\frac{n}{k^2m}\right)^{1/2}.\] Choose first \(\kappa_{\mathrm{ml}}\) small enough that the \(K^{C_2}\) factor uses less than one quarter of the positive slack exponent, and then choose \(\varepsilon\) smaller still. By (40), bad balls carry a power-small incidence fraction. The same estimate applies to the bounded enlargements and choices of balls just used.

At a remaining ball, any three rich directions have determinant \(O(K^{-10})\). Choose a maximally separated pair. If its separation is at most \(K^{-1}\), all rich directions are within \(O(K^{-1})\) of any plane containing one of them. Otherwise the norm of the pair’s cross product is bounded below by \(c/K\), and dividing the determinant bound by this norm places every rich direction within \(O(K^{-9})\) of their spanning plane. In either case there is a single plane with the claimed \(O(K^{-1})\) tolerance. Its normal has spatial component bounded below: it is approximately perpendicular to a direction of the form \((1,u)\) with bounded \(u\).

The total mass in light caps at the ball is at most \(CmK^{-60}\). Cells whose original multiplicity is less than \(mN^{-\varepsilon_1}\) carry at most \(N^{-\varepsilon_1}\) of all incidence, because their number is at most \(|E|\). Let \(\varepsilon_1\) decrease slowly to zero. Remove central incidences requiring bad balls and central incidences in the light caps of their assigned ball. At each relevant fine cell the light-cap bound is negligible compared with \(mN^{-o(1)}\); the bounded number of possible adjacent balls does not alter that conclusion. Choose the ball once for each retained reference index-block, and keep this choice fixed at all its times. Regularize this central graph, preserving \(F\).

For neighbors at a retained time use the original rich incidences at that fine cell. Their total mass is at least \(mN^{-o(1)}\). The cap bound (41) with a radius \(N^{-\gamma}\) shows that neighbors within that angle of the reference have mass at most \(mN^{-H\gamma+o(1)}\). First take any fixed small \(\gamma>0\), then diagonalize \(\gamma\downarrow0\) after the previous errors are smaller. This leaves separation at least \(N^{-o(1)}\) and does not use an estimate at an uncontrolled shrinking scale.

Finally, all these directions are \(O(K^{-1})\) from the one chosen plane, and each neighbor meets the reference within bounded fine distance. Over a block of length \(K\), their displacement normal to the affine plane through the reference remains bounded. Their tangential and temporal displacements are \(O(K)\). A plate of the stated dimensions contains them all, including the filled reference segment. Neighbor shadings have not been restricted in this step, so their original profile and cap upper bounds remain available. ◻

Proof. Take any measurable linearized choice \(P_x\) of an averaging plate containing \(x\), and put \[Af(x)=\frac1{|P_x|}\int_{P_x}f(z)\,dz.\] The symmetric kernel of \(AA^*\) is \[\mathcal K(x,y)=\frac{|P_x\cap P_y|}{|P_x||P_y|}.\] Let \(\vartheta\in[0,\pi/2]\) be the angle between the unoriented central-plane normals of the two plates. Intersecting two unit slabs gives transverse cross-sectional area \(O(1/\sin\vartheta)\), while the available extent along their intersection line is \(O(K)\). The trivial overlap bound is \(O(K^2)\). Thus \[ |P_x\cap P_y|\lesssim \min(K^2,K/\vartheta) \lesssim\frac{K^2}{1+K\vartheta}. \tag{45}\]

Fix \(P_x\) and let \(\mathcal P_x\) be its central plane. If the plates intersect, choose a point in their intersection. Since \(y\in P_y\), the normal displacement from that point to \(y\), measured relative to \(\mathcal P_x\), is at most \(C(1+K\vartheta)\). It follows that \[\mathop{\mathrm{dist}}(y,\mathcal P_x)\leq C(1+K\vartheta).\] Also \(|x-y|\leq CK\), since both plates contain their respective query points and have a common point. Because \(|P_x|,|P_y|\asymp K^2\), (45) gives \[ 0\leq\mathcal K(x,y) \lesssim \frac{\mathbf 1_{\{|x-y|\leq CK\}}} {K^2(1+\mathop{\mathrm{dist}}(y,\mathcal P_x))}. \tag{46}\] In coordinates tangent and normal to \(\mathcal P_x\), the tangential area in this ball is \(O(K^2)\). Integrating the normal coordinate therefore proves \[\sup_x\int\mathcal K(x,y)\,dy\lesssim1+\log K.\] Symmetry gives the same column bound. Schur’s test yields \(\|AA^*\|_{2\to2}\lesssim1+\log K\), proving the claimed squared norm bound uniformly in the linearization.

For a finite family of plate parameters, choose a maximizing plate measurably at each point. The uniform bound passes to the supremum over that family. A countable dense family suffices, by continuity of plate integrals under translation and rotation for locally integrable functions, and monotone convergence then gives (43). Apply it to \(f=\mathbf 1_E\) and use \(\mathcal N_K\mathbf 1_E\geq\rho\) on \(E^+\) to obtain (44). Replacing counted cells by their unit cubes and allowing fixed plate enlargements proves the discrete version. ◻

Proof. The planar neighbor measure. Use the central graph and the fixed block planes of Lemma 23. Choose a retained reference index-block. Normalize its block length to one and project orthogonally onto the corresponding affine plane. The projection changes its relevant unit directions by an additive \(O(K^{-1})\). The reference–neighbor separation is \(N^{-o(1)}\), which is much larger than \(K^{-1}\) for this fixed \(\kappa>0\); that separation is therefore preserved up to fixed factors. Also, projected direction diameter \(O(r)\) implies original direction diameter \(O(r+K^{-1})=O(r)\) whenever \(r\geq K^{-1}\).

We use physical time, rather than distance along the reference, as the graph coordinate on this plane. After a translation let its unit normal be \(\nu=(\nu_0,\nu_{\mathrm{sp}})\), where \(|\nu_{\mathrm{sp}}|\geq c>0\). Put \(n_{\mathrm{sp}}=\nu_{\mathrm{sp}}/|\nu_{\mathrm{sp}}|\) and choose a unit spatial vector \(e_{\mathrm{sp}}\) perpendicular to it. The coordinates \((\sigma,y)=(t,e_{\mathrm{sp}}\cdot x)\) on the plane have inverse \[(t,x)=\left(\sigma, y e_{\mathrm{sp}}- \frac{\nu_0}{|\nu_{\mathrm{sp}}|}\sigma n_{\mathrm{sp}}\right),\] so they are uniformly bi-Lipschitz. For a relevant original direction \((1,u)\), the projected time component is \(1-\nu_0\nu\cdot(1,u)=1+O(K^{-1})\), while its projected \(y\) component is \(e_{\mathrm{sp}}\cdot u\). All projected lines are consequently bounded-slope graphs in these coordinates. This also preserves lengths, time resolution, and shading counts up to fixed factors.

First, the marked-time probability distribution on the reference block is \(s\)-Frostman at resolution \(K^{-1}\). To check this, an interval of relative length \(r\in[K^{-1},1]\) is covered by boundedly many dyadic blocks of \(L\asymp rK\) bins. Put \(u=\log_N L\). The relative number of reference events it contains is at most \[ N^{o(1)}\frac{LN^{-F(u)}}{KN^{-F(\kappa)}} \leq N^{o(1)}r^s, \tag{50}\] by (47). The same estimate holds for the original marked times of every neighboring line on this block. The errors are subpowers of \(K\) because \(\kappa>0\) is fixed.

Construct a probability measure on planar neighbor lines as follows. Choose a marked reference time with the preceding distribution, then choose one of its coplanar separated neighbors with probability proportional to its original raw weight at that fine cell. The normalizing neighboring mass is at least \(mN^{-o(1)}\). For any cap of radius \(r\in[K^{-1},1]\), use (41) with \(L\asymp1/r\) and (32) to obtain the conditional probability bound \[ \mathbb P(\text{neighbor direction in that cap}\mid\text{time}) \leq N^{o(1)}r^H N^{-p_*F(\log_N(1/r))} \leq N^{o(1)}r^{a_{\mathrm{ang}}}. \tag{51}\] The coplanar directions fit in \(O(K)\) caps of radius \(K^{-1}\). Their normalized mass is one, so summing (51) at this radius gives \(1\leq N^{o(1)}K^{1-a_{\mathrm{ang}}}\). Hence \(a_{\mathrm{ang}}\leq1\) in the limit.

We claim that the resulting planar line-parameter probability measure is \(t\)-Frostman. A line-parameter ball of radius \(r\) restricts the projected slope to an interval of size \(O(r)\), and therefore the original three-dimensional direction to a cap of size \(O(r)\): the lost normal component is only \(O(K^{-1})\leq O(r)\). It also restricts the encounter time with the reference to an interval of length \(N^{o(1)}r\). For the latter assertion, subtract the projected reference’s affine graph by a bounded shear in the physical-time coordinates just defined, and write a projected neighbor as \(y=c+v\sigma\). The neighbor-reference angle is at least \(N^{-o(1)}\), so \(|v|\geq N^{-o(1)}\). The encounter condition is \(|c+v\sigma|\lesssim K^{-1}\). Varying \((c,v)\) in an \(r\)-ball changes \(-c/v\) by at most \(N^{o(1)}r\), provided the ball is smaller than a fixed fraction of \(|v|\). For larger balls the same assertion follows from the trivial unit bound on the time interval. The fine encounter error contributes at most \(N^{o(1)}K^{-1}\leq N^{o(1)}r\).

Use (50) for that encounter interval and (51) at each admissible time. Their product is at most \(N^{o(1)}r^{s+a_{\mathrm{ang}}}\), as required. This argument is conditional probability followed by integration; it does not assume that the encounter time and direction are independent.

Proof. Suppose that there is no such interval, and put \(e=\liminf_{u\downarrow0}F(u)/u\). Then \(F(u)>0\) for every \(u>0\). For any sufficiently small \(\varepsilon>0\), choose a small \(r_0>0\) so that \(F(u)\geq(e-\varepsilon)u\) for \(0\leq u\leq r_0\). There are arbitrarily small \(r\leq r_0\) where \(F(r)<(e+\varepsilon/2)r\). The continuous function \(F(u)-(e+\varepsilon)u\) has a negative minimum on \([0,r]\). Choose a minimizing point \(\kappa>0\). It is a record minimum on \([0,\kappa]\), and hence \[ F(\kappa)-F(u)\leq(e+\varepsilon)(\kappa-u),\qquad F(u)\geq\max(e-\varepsilon,0)u \quad(0\leq u\leq\kappa). \tag{55}\] We may take \(\kappa\) as small as needed for Lemma 25.

At a typical fine incidence the neighboring mass is at least \(mN^{-o(1)}\) and lies within \(O(K^{-1})\) of one plane. Cover these directions by \(O(K)\) caps of radius \(K^{-1}\) and use (41) together with (55) and (32). This gives \(1\leq N^{o(1)}K^{1-H-p_*\max(e-\varepsilon,0)}\), hence \(H+p_*\max(e-\varepsilon,0)\leq1\), without first requiring a positive time exponent. Letting \(\varepsilon\downarrow0\) yields \(H+p_*e\leq1\). In particular \(e<1\), and we may arrange \(s=1-e-\varepsilon>0\). Use (55) in Lemma 25, with \[\gamma=\max(e-\varepsilon,0),\qquad t=1-e-\varepsilon+H+p_*\gamma.\]

If \(e=0\), choose \(\varepsilon\) small compared with \(H\). Then \(s=1-\varepsilon\), \(t=1+H-\varepsilon\), and \(\mu=1\). Since \(F(\kappa)>0\), \[\Pi(\kappa)-F(\kappa) \geq(p_*-1)F(\kappa)>0,\] contradicting (49). This argument does not need a lower bound for \(F(\kappa)/\kappa\).

Suppose \(e>0\). At \(\varepsilon=0\) one has \[s=1-e,\qquad t=1+H+(p_*-1)e>1,\] and therefore \[2(1-\mu)=\max\{0,(2-p_*)e-H\}.\] The gap between \((p_*-1)e\) and the right-hand side is strictly positive. If the maximum is zero this is immediate; otherwise the gap equals \((2p_*-3)e+H>0\). By continuity, for sufficiently small fixed \(\varepsilon\), \[(p_*-1)(e-\varepsilon)>2(1-\mu).\] But (55) and (32) now give \[\Pi(\kappa)-F(\kappa) \geq(p_*-1)F(\kappa) \geq(p_*-1)(e-\varepsilon)\kappa >2\kappa(1-\mu),\] again contradicting (49). Both cases are impossible, proving the flat delay. ◻

Proof. Split into temporal \(K\)-blocks and bounded-overlap position charts of diameter \(CK\) fine cells containing complete block segments. Each original index-block uses boundedly many charts. The normalized resolution is \(K\), and the local shading density is \(N^{-o(1)}\) because \(F(\kappa)=0\). Count charts separately. Their total incidence is extensive and their aggregate support is at most a fixed multiple of the old support, so every extensive remainder has aggregate raw multiplicity at least \(m\) in exponent.

Fix a small \(\varepsilon>0\). Greedily remove and assign all remaining local lines in a plank whenever its density \(n_W/(ab)\) is at least \(mN^{-\varepsilon}\). If an extensive remainder survives, its clustering parameter is at most \(mN^{-\varepsilon}\) in every local chart. The weighted set estimate, Lemma 15, applies with local resolution \(K\) and arbitrarily small fixed theorem loss: choose that loss below \(\varepsilon/(2\kappa)\), then take \(N\) large enough for the subpower local shading density to meet its hypothesis. It bounds the remainder’s multiplicity by at most \(mN^{-\varepsilon/2}\) in exponent. Uniformity over charts follows by representative selection. This contradicts its extensive incidence and inherited support. Thus the captured graph is extensive, with disjoint index ownership on each block.

Combine its lower bound \(mN^{-\varepsilon}ab\) with (33), where \(\Pi(\kappa)=0\). It gives \[N^{-\varepsilon}\leq N^{o(1)}K^{-H}a^{-q-H}b^H,\] or, in logarithmic coordinates, \[(q+H)\log_N a+H\log_N(K/b) \leq\varepsilon+o(1).\] Both terms are nonnegative and \(H>0\). Let the capture tolerance decrease to zero after each finite set of tests. The captured planks have \(a=N^{o(1)}\), \(b=KN^{o(1)}\), and their mass is at least \(mK N^{-o(1)}\). Cleaning the assignment and chart partitions preserves these lower bounds on occupied packets, as in Proposition 21. ◻

Proof. The coordinate map and multiplicity gain. Replacing \(\theta\) by \(\min(\theta,1)\) changes it only by a subpower factor, so we may take \(0<\theta\le1\) in the geometric construction. First consider a chart centered on one affine line, with normal–tangent coordinates \(z_n,z_t\) and time \(r\in[0,K_1]\), all in old fine-cell units. The chart bounds we will need are \(|z_n|\lesssim\theta K_1\), \(|z_t|\lesssim K_1\), together with the corresponding bounds on displacement over the larger block. For such a chart use normalized coordinates \[ s=\frac r{K_1},\qquad X=\frac{z_n}{\theta K_1},\qquad Y=\frac{z_t}{K_1}, \tag{59}\] at resolution \(N'=K_1\theta\). Equivalently, in the new fine-cell units the coordinate map is \[ (r,z_n,z_t)\longmapsto(\theta r,z_n,\theta z_t). \tag{60}\] The new slopes and intercepts are bounded. An old normal fine cell remains a fine cell; old time and tangent cells are grouped in intervals of length \(1/\theta\).

Suppose a fine plate’s axis differs from the chart normal by \(|\delta|\le\theta/10\), as the angular partition below will ensure. Its thin condition is transformed from \(|\cos\delta\,z_n+\sin\delta\,z_t-c(r)|\lesssim N^{o(1)}\) to \[|\cos\delta\,z_n'+(\sin\delta/\theta)z_t'-c'(r')| \lesssim N^{o(1)}.\] The coefficients have bounded or subpower size and the normal coefficient is bounded below. Hence it has subpower fine thickness and tangential extent at most \(N^{o(1)}K\theta\) in the new chart. Fill its selected \(K\)-intervals. Each now spans \(K\theta\) new bins, up to rounding, and occupies at most \(N^{o(1)}(K\theta)^2\) cells. If its retained index mass is at least \(mKN^{-o(1)}\), its incidence mass is at least \(mKN^{-o(1)}\cdot K\theta\). Thus disjoint indexed-block assignments with these retained masses give aggregate raw multiplicity at least \(m/\theta\) in exponent. Packet support overlaps can only improve this lower bound.

Proof. We show that a sufficiently small fixed upper bound on \(\alpha\) is incompatible with any such hybrid equality. All fixed tolerances in the next paragraphs will be chosen independently of \(H-h_0\). Restrict to \(h_0\geq h/2\), so that \(H\geq h/2\).

We will capture an extensive incidence graph in nearly maximally elongated global slabs, then align the saturated local plates with the fixed slab axis assigned to each index. Anisotropic normalization will make the sharp hybrid upper bound smaller than the retained raw multiplicity.

Proof. Decrease \(p\) by a positive amount \(\varepsilon\). Differentiability gives \(h(p-\varepsilon,z)=h+v\varepsilon+o(\varepsilon)\), with \(v\ge0\). Increasing the clustering exponent cannot increase \(\Delta_h\): for each plank, its dependence on that exponent is through \((a/b)^h\). The cost changes by \(-\alpha\varepsilon\). Applying the perturbed upper bound to an equality therefore gives \(\alpha\le v\). Since a positive-deficit equality exists, \(v>0\).

For the local assertion fix the perturbation size first. Denote the nearby parameters by primes, and let \(\Pi'\) denote the cost with the changed integrand and the same profile. On each prefix block greedily assign and remove the remaining active indices in planks whose mass is at least \[ mN^{-\mathrm{tol}-\Pi'(\kappa)}K^{-h'} a^{d'-1-h'}b^{1+h'}. \tag{69}\] Here \(\mathrm{tol}>0\) is fixed during this operation. If the captured events were not extensive, the remainder would be extensive. Regularize that remainder to its inherited profile and apply the perturbed critical bound separately on the prefix blocks and bounded spatial charts. Its plank parameter is bounded by the threshold used in the greedy removal, so this bound gives aggregate raw multiplicity at most \(mN^{-\mathrm{tol}+o(1)}\). The extensive remainder has aggregate raw multiplicity at least \(mN^{-o(1)}\), a contradiction. Spatial charts may be assigned by position at the block center; bounded slopes give bounded overlap in their support counts.

The captured graph is therefore extensive. Pigeonhole a common dyadic width type and use Lemma 7 to preserve the packet lower counts. Comparing (69) with (67) yields \[ \Pi'(\kappa)-\Pi(\kappa) +(h'-h)\chi-(d'-d)\mathfrak a\ge-\mathrm{tol}. \tag{70}\] The first term records the change in temporal cost. The remaining terms record the resolution lost in the cut and the change of its thin-width exponent. The shapes lie in the compact triangle \(\mathfrak a,\mathfrak u\ge0\), \(\mathfrak a+\mathfrak u\le\kappa\). Take the perturbation to zero along the specified direction, with \(\mathrm{tol}\) of smaller order, and pass to a convergent sequence of shapes. First choosing the resolution sufficiently large at each fixed perturbation makes the preceding exponent errors smaller than \(\mathrm{tol}\). The captured masses are squeezed to saturation in the limit.

Since \(z=-q\) and \(d=2+z\), \[\partial_p\Pi(\kappa)=F(\kappa),\qquad \partial_z\Pi(\kappa) =\int_0^\kappa\mathcal D_q(q,F'(s))\,ds, \qquad d_z=1.\] The averaging in \(\mathcal D\) makes its \(q\) derivative available at the interior parameter in use. Differentiating (70) proves (68). Taking \((\dot p,\dot z)=(1,0)\) and \((-1,0)\) gives the two final selections. The same argument applies to a preceding extensive regular refinement because its profile, equality exponents, and inherited local upper bounds are unchanged. ◻

Proof. At the initial flat prefix, Lemma 30 gives \(\chi_0=0\). Thus \(a=1\), \(b=N^\tau\), and the packets are plates of mass \(\mathrel{=_{\mathrm{exp}}}mN^\tau\). Cut to the outer model, keeping its full ensemble of matrix parents with inherited weights, rather than selecting a single parent. Apply derivative capture at the next physical prefix in this ensemble, and continue.

The parameter calculation for one step explains the choice of cuts. Suppose the preceding cumulative parameters are \((\mathfrak a_{\rm old},\mathfrak u_{\rm old})\), with \(\chi_{\rm old}=\mathfrak a_{\rm old}+\mathfrak u_{\rm old}\), and put \(L=1-\chi_{\rm old}\). If the next cut in the normalized outer model has shape \((\mathfrak a',\mathfrak u')\), its pullback has \[(\mathfrak a,\mathfrak u) =(\mathfrak a_{\rm old},\mathfrak u_{\rm old}) +L(\mathfrak a',\mathfrak u').\] Indeed spatial cell widths multiply. Physical time is unchanged, but in the new cell units \[K_{\rm new}=K N^{-\chi_{\rm old}},\qquad b_{\rm new}=b N^{-\mathfrak a_{\rm old}},\qquad \frac{K_{\rm new}}{b_{\rm new}} =\frac Kb N^{-\mathfrak u_{\rm old}}.\] This also gives the asserted increment in \(\mathfrak u\); both increments are nonnegative. If the preceding physical cut was at \(\kappa_{\rm old}\), the outer deficit up to the next physical scale \(\kappa\) is \([F(\kappa)-F(\kappa_{\rm old})]/L\), by (72). The first selection in Lemma 30 therefore gives \[\chi-\chi_{\rm old}\le [F(\kappa)-F(\kappa_{\rm old})]/v.\] Iteration gives (73). Since \(F\) is \(1\)-Lipschitz and \(v>0\) is fixed, adjacent changes in both shape parameters are bounded by a constant times the mesh size.

We give the retention details for one step and then for its pullback. Each current parent has bounded normalized matrix coordinates, the common profile, and the inherited upper bounds. Summing support counts with the different parents counted separately preserves the aggregate lower multiplicity in (71). The greedy argument of Lemma 30 applies in each parent and physical block with this common raw threshold. If it could not capture an extensive piece of the ensemble, the uniform perturbed upper bound on the remainder, summed over the parents, would contradict that aggregate lower estimate. Retain a common width type among the captured packets, and reunite the selected blocks of each index in its current full-time parent. There is one such parent per index, so this reunion does not duplicate its weight. Regularization preserves the profile and the packet masses, after which the next isotropic parent transformation is defined.

All the derived blocks used at the next stage are coarser physical blocks than those already filled. A filled time is present only because its index occupied the earlier, finer block. Hence activity in a derived block implies original activity in the same physical block. The pulled back upper bounds therefore count subsets of the original active indices, as required by Proposition 21.

For the pullback, consider an earlier registered pair consisting of an index and its occupied cut block. Before filling, this pair supplied a common number in exponent of selected events; after filling and rebinning, it supplies another common number. The index weight is unchanged. An extensive selection of the later events therefore hits an extensive weight of these pairs. Keep all the preceding selected events of each hit pair. A later assignment is constant on that earlier pair: its temporal block is a coarser member of the same nested dyadic grid, and its ownership decision concerns the index on that block. Thus this pullback preserves all later assignment memberships. Repeating the operation restores an extensive graph of original events carrying all assignments in the finite chain.

One must also preserve the masses of the used parts of earlier packets. At any fixed earlier level, its assignments are disjoint on each physical block, and every assigned index had the same event count in exponent there. A packet’s event mass is consequently a common factor times its index mass. Include the packet partitions at every preceding level in the deficiency deletion of Lemma 7. For a finite list, choose the deletion cutoff smaller than the total retained proportion, while still subpower. The charge from each partition is at most its cutoff times the original total event mass. Iterating deficient cell deletion over the finite list leaves an extensive graph and retains each packet’s lower mass exponent. Any other fixed finite list of tests is included in this same deletion procedure. When only an upper packing count is needed, one may also keep the whole earlier assigned index set, which remains a valid active set for the original local bound.

Selections obtained by a derivative limit are made before the next cut: for each fixed perturbation and tolerance perform the finite selection, then take a subsequence attaining its limiting shape. Only finitely many such choices occur in the present lemma. At any stage at which a pure outer equality is needed, rather than its full ensemble, one may choose a representative parent subsequence: the inherited upper bounds are uniform at a fixed positive remaining length, and at least one parent attains the aggregate lower estimate in exponent. The construction itself continues with the ensemble. Finally, each \(\kappa_i<1\) and \(\chi_i\le\kappa_i\) leave positive outer length, so no zero-length iteration is used. ◻

Proof. Packing adjacent packets. Use the finite chains of Lemma 31. Write \(\delta\) for their mesh size. Consider one smaller-block packet of widths \((a,b)\) and the larger-block packets meeting it along assigned indices at dyadic angle \(\theta\gg a/b\). Adjacent widths differ by \(N^{O(\delta)}\). Each meeting is witnessed by an affine index belonging to both packets, so their centers are tied to that same segment throughout the smaller block. At a fixed time, a strip of length comparable to \(b\) tilted by \(\theta\) has normal spread \(O(a+\theta b)=O(\theta b)\) relative to the smaller packet. Extrapolating the shared segment to the larger block costs another factor \(N^{O(\delta)}\). Thus all these larger packets lie in a plank of widths \(N^{O(\delta)}(\theta b,b)\), aligned with the smaller packet. Their assigned index sets on the larger block are disjoint. Dividing the local upper mass of this enclosing plank by their saturated mass bounds their number by \[N^{O(\delta)+o(1)}\theta^x(b/a)^x.\] On the smaller block, the two thin conditions bound transverse motion along their intersection by \(O(a/\theta)\): solving the two strip inequalities divides by the sine of their angle. Their intersection is therefore contained in a plank of widths \(N^{O(\delta)}(a,a/\theta)\). Figure 1 shows the two width calculations. Apply the local upper bound again and divide by the smaller packet’s saturated mass. The fraction of its assigned indices accounted for at this angle is at most \[ N^{O(\delta)+o(1)} \left[\theta^x(b/a)^x\right] \frac{a^x(a/\theta)^y}{a^xb^y} =N^{O(\delta)+o(1)} \left(\frac{a}{\theta b}\right)^{y-x}. \tag{76}\]

The factors omitted in the displayed quotient are changes of time length, profile cost, and width type across one mesh step; their logarithms are \(O(\delta)\). If an extrapolated width passes the full spatial size, cover it by the corresponding \(N^{O(\delta)}\) bounded planks. The same bound holds for event mass because the active indices in a packet have uniform event counts on its block.

Since \(y-x>0\), angles larger by a fixed positive exponent than \(a/b\) can be discarded with a power-small loss, after making \(\delta\) sufficiently small. Include the earlier packet partitions in the subsequent cleaning. At cut \(i\) the aspect exponent is \(\kappa_i-\chi_i\), which is at least \(\kappa_i-F(\kappa_i)/v\). Summing the angular errors along the finite chain, rather than adding their exponents, bounds the total angle by \(N^{-D_*(\kappa)+O(\delta)+o(1)}\). The final packet has one owner per index on its physical block, so all retained initial plates of that index on the block compare to the same final normal. Sending the mesh to zero in the specified order proves the orientation assertion.

Proof. Attaining the maximal deficit. First, equalities with deficit \(v\) exist. Start with any positive-deficit equality and put \(g(s)=s-F(s)/v\). If the running minimum of \(g\) from \(\tau\) remained equal to \(\tau\) throughout its nontrivial part, Lemma 32, applied at a prefix with positive deficit, would produce a nontrivial equality with zero initial delay. This contradicts Lemma 26. There is therefore a new minimum \(D<\tau\), attained at a prefix \(\kappa\) with \(g(\kappa)=D\). At this prefix the inner deficit is \[\frac{F(\kappa)}{\kappa-D}=v.\] Let \(\mathcal E_v\) denote the nonempty class of equality profiles with deficit \(v\), and set \[\tau_* =\inf_{F\in\mathcal E_v} \sup\{r:F=0\text{ on }[0,r]\}.\]

Proof. An outer cut of a deficit-\(v\) equality again has deficit at most \(v\). Applied to the canonical profile, this forces \(\chi\le b'\). The reverse derivative selection attains \(\chi=b'\). Formula (72) then shows that the outer profile is the same canonical profile: its flat delay is \[\frac{\tau+\ell b'-b'}{1-b'}=\tau,\] and its nonflat slope is still \(\beta\). The same statement applies incrementally in every current outer model, in the original logarithmic units.

For one mesh increment \(\Delta b'\), the two coefficients in (68), written in those units, are \[v(\Delta b'-\Delta\chi),\qquad h_z\Delta\chi+\ell\mathcal D_q(q,\beta)\Delta b' -\Delta\mathfrak a.\] Every available saturation has \(\Delta\chi\le\Delta b'\) by the outer deficit bound. Take either sign of \(\dot z\) and take \(\dot p<0\) with an arbitrarily large ratio \(|\dot p|/|\dot z|\). First fix that ratio and perform derivative capture. Along a subsequence as the ratio increases, the nonnegative quantity \(\Delta b'-\Delta\chi\) must tend to zero; otherwise its negative \(\dot p\) contribution cannot be offset by the bounded second coefficient. The two signs of \(\dot z\) therefore provide choices with \(\Delta\chi=\Delta b'\) and, respectively, \[\Delta\mathfrak a\le w\Delta b', \qquad\text{or}\qquad \Delta\mathfrak a\ge w\Delta b'.\] Since \(0\le\Delta\mathfrak a\le\Delta\chi\), this implies \(0\le w\le1\). Its defining expression and \(h_z>0\) give \(w>0\).

It is unnecessary to realize an exact intermediate shape on a single step. If the running value of \(\mathfrak a\) is above \(wb'\), choose the next increment from the lower side; if below, choose from the upper side. The error can then overshoot zero by at most the mesh size, because both \(\Delta\mathfrak a\) and \(w\Delta b'\) lie in \([0,\Delta b']\). This tracks the desired total shape within one mesh increment. The current outer profile stays canonical after each choice, so both sides are available at every step. All perturbation limits are resolved on a fixed finite chain before taking its mesh diagonal, and Lemma 31 preserves the required earlier assignments. ◻

Proof. Choose the finite chains from Lemma 34 and use the scales defined above. On the canonical profile the running aspect exponent is decreasing above \(\tau\), and at \(\kappa=1-\ell c\) its value is \[\kappa-F(\kappa)/v=\tau c.\] The proof of Lemma 32, which applies unchanged to the tracked finite chains, shows that every initial plate used by this packet has its normal within \(\Theta_cN^{o(1)}\) of the packet normal. Since a packet is assigned only once to each index on its block, this also gives the range assertion for all event normals on that index and block. The canonical profile gives the number of occupied subblocks: subtracting its deficits between the prefixes \(1-\ell c\) and \(1-\ell b\) yields (86).

We now verify the width bounds in the event frames. The widths in (85) are first computed in the packet’s own normal frame. Changing to one of its used event normals adds at most \(\Theta_cN^{o(1)}\) times the transverse coordinate; the ratios of transverse to normal widths are exactly \(\Theta_c^{-1}\). Changing the time center within \(I_c\) adds at most \(R_c\) times the velocity coordinate. Both changes preserve the four widths up to subpower factors.

For this profile, \[\Pi(\kappa)=\ell(1-c)P_{p,q}(\beta),\qquad m\mathrel{=_{\mathrm{exp}}}\Delta_hN^{B_*}.\] Substituting (83) in (67), the exponent of the packet mass relative to \(\Delta_h\) is \[\begin{align*} &B_*-\ell(1-c)P_{p,q}(\beta)-h(1-\ell c) +xw(1-c)+y\bigl(w+(\tau-w)c\bigr)\\ &\hspace{12mm}=dw(1-c) +c\bigl(\ell P_{p,q}(\beta)+h\ell+y\tau\bigr)\\ &\hspace{12mm}=dw(1-c)+(B_*+\tau)c, \end{align*}\] using \(\ell+\tau=1\) and \(y=1+h\). This proves (87).

We next verify the passage to the collapsed law. Before collapse, the number of finer events in every occupied index–\(I_1\) pair has a common exponent. Thus an extensive selection of atoms of weight \(\omega_i\) pulls back to an extensive selection of the uncollapsed graph. Restricting the normal to one of finitely many direction charts also retains an extensive piece, and normalized covectors and unit normals are comparable on that chart. The orientation and time ranges proved above imply that one index and \(I_c\) use only subpower many \(e_c\) bins. Whenever the chart restriction or these labels split a packet, retain nondeficient occupied pieces. The packet lower bounds therefore survive for the law specified before the proposition.

For completeness, scale interpolation does not require new unrelated assignments. At a finite mesh, use the packets already constructed. A nearby intermediate interval is assigned the packet of its nearest coarser mesh interval, restricted to the indices active in the smaller interval. Extrapolation, width changes, and subdivision have \(N^{O(\text{mesh})}\) factors. Regular time counts and deficiency deletion preserve the packet lower masses on typical occupied pieces. Along the slow diagonal these factors are subpower. Alternatively, include any specified intermediate scales in the next finite mesh. Thus a new finite or countable list of scale tests can be accommodated by further refinement, with the same limiting data.

Finally include, in the finite cleaning list, the assignment partitions from every level in use. Lemma 31 then gives all the simultaneous memberships, the original active-index interpretation, and stability under extensive regular refinement in (iv). On a fixed physical \(I_b\), ownership was a decision for an index on that block; after registration in a chart it is therefore constant on the tuple specified in (iv). A cell of that chart’s fixed matrix partition is likewise a function only of the index and chart. The construction needs outer steps only for \(b'<1\), or \(c>0\); it makes no cut at zero remaining length. The endpoint \(c=1\) is the initial assignment and has already been included. ◻

Proof. We distinguish two reference graphs in the cleaning. For the new homogenization tests use the finite label construction of Lemma 7: these are its within-index integer counts and conditional-probability tests with polynomially many children. Perform its negligible conditional-probability tail deletions before computing the labels. Arbitrary-cardinality original packet partitions are retained only as cleaning tests; no absolute weight label is imposed on their cells. Let \(W\) be the original event weight and let \(R\) be the restricted, routed graph after those tail deletions and before the finite labels are selected. Write \(M_0=|R|\); thus \(M_0\ge N^{-E-o(1)}W\). Compute the prescribed rounded labels on \(R\), including their joint and domain tests for conditional counts, and retain one common finite list of labels. All graph masses denoted by \(|\cdot|\) here and in the registration rule below are weighted event masses. Denote this label class by \(A\), and write \(M=|A|\). The number of label lists is subpower, so \(M\ge N^{-o(1)}M_0\ge N^{-E-o(1)}W\). Counts within index fibers are event counts, with the inherited index weight cancelling in their ratios. No bound on the number of distinct weighted indices is required.

There are two collections of reference masses. Original packet, index–time, and chart tests use their masses in the original graph; the sum of these references is at most \(L_{\rm old}W\). New regularization tests use their masses in \(R\), the graph on which their labels were computed; their reference sum is at most \(L_{\rm new}M_0\). Here \(L_{\rm old},L_{\rm new}=N^{o(1)}\) include the finite number of partitions and any subpower chart copies. In particular, one does not substitute the possibly much smaller post-label cell mass for an \(R\)-reference without recomputing its label.

Choose \(\epsilon_N\downarrow0\) slowly enough to dominate the label loss \(\log_N(M_0/M)\), the error in \(M\ge N^{-E-o(1)}W\), and the \(\log_N L_{\rm old}\) and \(\log_N L_{\rm new}\), with margins tending to infinity after multiplication by \(\log N\). Starting from \(A\), delete all events in a deficient original-reference cell when its current mass is less than \(N^{-E-\epsilon_N}\) times that reference. For a new regularization cell or domain, use instead \(N^{-\epsilon_N}\) times its \(R\)-reference. Continue either type of deletion until no deficient cell remains. A cell is charged only once, because it is empty after deletion. Even with cascades, the two total charges are at most \[L_{\rm old}N^{-E-\epsilon_N}W=o(M),\qquad L_{\rm new}N^{-\epsilon_N}M_0=o(M).\] The final graph therefore has mass \((1-o(1))M\). Every surviving old cell retains at least \(N^{-E-o(1)}\) of its original mass. Every surviving new cell retains \(N^{-o(1)}\) of its \(R\)-mass and is a subset of that cell, so the rounded exponents selected from \(R\) remain valid, including conditional exponents obtained by ratios of joint and domain masses. This proves the mass assertion: the new labels incur only a subpower loss in addition to the original loss \(E\). The finite-test diagonal allows further prescribed tests.

For the profile assertion fix a finite mesh of time scales and uniformize the branching counts on its gaps. A subtree cannot have more children on any gap than the original tree. Since the branching exponent on \([s,t]\) is \((t-s)-(F(t)-F(s))\), this proves the first inequality in (89). The total available index weight after routing is at most \(N^{o(1)}\) times its original value. Comparing this with the retained weighted event count shows that the number of events per surviving uniform index is at least \(N^{-E-o(1)}\) times the original number. This proves the endpoint inequality. Refine the fixed mesh only after these comparisons hold. The Lipschitz profile limits give (89) on every interval, hence \(G'\ge F'\) almost everywhere and \(\int(G'-F')\le E\). The function \(P\) is increasing and has Lipschitz constant at most \(p\), which proves (90). ◻

Proof. Angular concentration would produce a configuration whose normalized temporal deficit is at least \(\beta\). Since \(\beta>v=-h_p\), decreasing \(p\) makes the critical upper bound for that configuration strictly smaller than its packet lower bound. We quantify how close the two bounds remain after selecting a narrow angular bin: a selection loss \(E\) decreases the packet mass by at most \(E\) in exponent and increases its temporal cost by at most \(pE\).

Write \(g=b-c>0\) and \[S=n_bN^{-\tau b},\qquad K'=N^{b-\ell c}, \qquad\theta=\Theta_b=N^{-\tau b}.\] First pass to the isotropic outer parents associated to \(B_b\), as in Lemma 31. Their effective resolution count is \(N^b\). The assigned \(B_b\) plates have time and long spatial length \(N^{\tau b}\) in these cell units, unit thin width, and mass at least \(SN^{\tau b-o(1)}\). On the longer interval \(I_c\), which has \(K'\) bins in this model, the local bound of Proposition 21 pulls back to \[ n(\text{plank of widths }a_1,b_1) \le N^{o(1)} S N^{-\ell gP(\beta)}(K')^{-h}a_1^xb_1^y. \tag{93}\] For example, the isotropic physical cell width is \(N^{w(1-b)}\) in original cell units. Substitution in the original local bound gives the coefficient \[mN^{-\ell(1-c)P(\beta)}N^{-h(1-\ell c)} N^{dw(1-b)} = S N^{-\ell gP(\beta)}(K')^{-h}\] in exponent, using \(m\mathrel{=_{\mathrm{exp}}}\Delta_hN^{B_*}\) and (87). Thus (93) counts original active indices; it is not a bound only for indices occupying a selected angular bin.

Suppose that fibers violating (91) account for an extensive event piece. On each such index and \(I_c\) choose one violating bin and keep its events. This costs at most \[E=\gamma\tau g\] in log probability, apart from a vanishing error. First register and complete every occupied \(I_b\) token hit by this selection, before any chart-portion cutoff. The preceding registration rule gives token weight at least \(N^{-E-o(1)}\) of the original token graph. The normal of its assigned \(B_b\) packet is within \(\Theta_bN^{o(1)}\) of the selected bin, since the packet contains the event that registered the token and its normal label is constant on the token. Completing the block does not change its packet owner. Route these whole tokens through the outer parents and bin choices. Each packet has only subpower many relevant choices, and each index on \(I_c\) has only subpower many charts. Any deficient chart portions are pruned relative to this restricted token graph. Now apply the two-budget cleaning of Lemma 36: original packet and time-tree references use the loss \(E\), while the new homogenization references, computed after registration and routing, use only a subpower loss. The final token graph still has weight \(N^{-E-o(1)}\) of the original one, and each surviving packet has raw assigned mass at least \[ SN^{\tau b-E-o(1)}. \tag{94}\] The uniform token trees are subtrees of the original occupied \(I_b\) trees below \(I_c\), with subpower many routed copies. Thus their endpoint deficit loss is at most \(E\) and their cost increase is at most \(pE\). Packet loss \(E\) and cost increase \(pE\) are the two consequences of this same retained graph; the selection is not performed a second time.

Apply the geometric construction in the proof of Lemma 28 on \(I_c\), with angular width \(\theta\). Here we use only its bounded-chart routing, rectangle pullback, and cell-count calculation. That calculation is linear in the packet mass, which here is the reduced value \(SN^{\tau b-E-o(1)}\) from (94). We do not use that lemma’s profile-preservation conclusion for an extensive graph: the endpoint loss \(E\) and cost increase \(pE\) were proved by Lemma 36 and are used explicitly in (97) below. The resulting normalization has resolution \[ N_{\mathrm{new}}=K'\theta=N^{\ell g}. \tag{95}\] The selected plates become bounded-cell packets: their old time and long spatial length multiplied by \(\theta\) is one, and their orientation error is \(N^{o(1)}\theta\). Filling their occupied \(I_b\) blocks before this normalization therefore gives aggregate raw multiplicity at least the quantity in (94). The full-plank parameter of the new models is at most \[ \Delta_{\mathrm{new},h} \le N^{o(1)}S N^{-\ell gP(\beta)}(K')^{-h}\theta^{-y}. \tag{96}\] Here the factor \(\theta^{-y}\) follows by pulling back a rectangle: its area expands by \(\theta^{-1}\) and its largest width by at most \(\theta^{-1}\), and \(a^xb^y=(ab)^xb^{y-x}\) with \(0\le x\le y\).

We next quantify the time cost. Prior to selection the occupied \(I_b\) blocks below \(I_c\) have constant deficit slope \(\beta\) over a log interval of length \(\ell g\). Filling and then normalizing removes only the flat range below them. The restricted uniform time tree cannot gain branching on any fixed gap. Its normalized deficit \(\alpha_{\mathrm{new}}\) is therefore at least \(\beta\). By (90), its cost, expressed in \(\log N\) units, is at most \[ \ell gP(\beta)+pE+o(1). \tag{97}\] The comparison uses the whole routed ensemble: the number of chart copies per original index and \(I_c\) is subpower, so the total branching loss is at most \(E\), as in the proof of Lemma 36.

Without the controlled losses, the sharp upper bound from Definition 4 and (96) matches \(SN^{\tau b}\) exactly. Indeed its exponent relative to \(S\) is \[\begin{align*} -\ell gP(\beta)-h(b-\ell c)+y\tau b +h\ell g+\ell gP(\beta) &=\tau b. \end{align*}\] Thus (94) and (97) place the new models within \((p+1)E+o(1)\) of their sharp upper bound in \(\log N\) units.

Choose a small fixed \(t_*>0\) with \(p-t_*>2\) and \[ 0\le h(p-t_*,z)-h(p,z) \le \frac{v+\beta}{2}\,t_*. \tag{98}\] This is possible by differentiability and \(\beta>v\). Increasing \(h\) at fixed \(d\) cannot increase the plank parameter, because its value on a plank is multiplied by \((a_1/b_1)^{h'-h}\le1\). Also \(P_{p-t_*,q}(e)=P_{p,q}(e)-t_*e\). Applying the critical bound at \((p-t_*,z)\) to these same models therefore saves at least \[N_{\mathrm{new}}^{(\beta-v)t_*/2}\] relative to the old sharp upper estimate. Choose once and for all \[ 0<\gamma<\min\left\{ \frac12,\frac{\ell(\beta-v)t_*}{8(p+1)\tau} \right\}. \tag{99}\] The saving then strictly exceeds the error \((p+1)E\). This contradicts the aggregate lower multiplicity. Uniformity over bounded parent charts follows from the uniform all-configuration critical bound; otherwise a parent subsequence would violate that bound.

It follows that the violating fibers are not an extensive piece. In particular their probability tends to zero. This proves (91) for each fixed gap. The choice (99) is independent of \(b-c\): both the selection loss and the strict saving were proportional to that gap before normalization. Fix finitely many gaps first and absorb their errors; the diagonal convention then gives all required scale tests.

Within an index and \(I_c\) the normals occupy only subpower many \(e_c\) bins. Delete bins retaining less than a subpower fraction of the fiber, by the same deficiency argument. Dividing by their remaining mass changes (91) by only a subpower factor. An extensive subsequent regular refinement has the same property by Lemma 36.

Finally suppose \(\beta=1\). For any fixed \(0<c<1\), an occupied index–\(I_c\) fiber has only \(N^{o(1)}\) occupied \(I_1\) blocks by (86), and there is one collapsed event per such block. One angular bin of width \(\Theta_1\) therefore has at least \(N^{-o(1)}\) of its event mass. This contradicts (91) at \(b=1\), since \(\gamma\tau(1-c)>0\). Hence \(\beta<1\); positivity follows from \(v>0\). ◻

Proof. The geometric argument uses coarse matrix cells within registered charts. We will replace each cell by its center, weighted by the full original active-index mass in that cell, and regard the selected original indices as its children. After selecting a common ratio of child mass to representative mass, their normalized event laws are comparable. The critical inequality bounds the representatives’ multiplicity, whereas packet saturation gives a lower bound for the children’s mass in a typical spatial cell. These bounds force a typical packet to contain a sufficiently large fraction of its queried cell’s reference mass. We first construct the charts in which this comparison is made.

Put \[ g=b-c=\frac{a+\Delta}{w},\qquad D_0=N^g,\qquad D=N^\Delta,\qquad Z'=N^{\tau g},\qquad S'=\frac{n_b}{Z'}. \tag{102}\] The conditions (101) imply \(c<b<1\) and \(\Delta<\tau g\), so a positive flat range remains in the time filling used below.

Proof. For a finite partition \(\mathcal P\) of \([s,t]\), sum the values of \(K\) on its intervals, and set \[m(s,t)=\inf_{\mathcal P}\sum_{[u,v]\in\mathcal P}K(u,v).\] Concatenation of partitions gives \(m(s,u)\le m(s,t)+m(t,u)\). Inserting \(t\) into any partition of \([s,u]\) can only decrease its sum, by superadditivity of \(K\). Taking infima gives the reverse inequality. Thus \(m\) is additive and \(0\le m(s,t)\le C(t-s)\). It is the interval function of an absolutely continuous measure, so \(m(s,t)=\int_s^t\phi(r)\,dr\) for some \(0\le\phi\le C\).

The excess \(E=K-m\) is nonnegative and superadditive. Its infimum of partition sums is zero on every interval. Fix a compact interval \(J\) and \(\epsilon>0\). Consider points in its interior that admit arbitrarily small centered intervals \(I\) with \(E(I)>\epsilon|I|\). Choose a finite partition of \(J\) with sum of excesses less than an arbitrary \(\eta>0\). Except at its finitely many endpoints, the small test intervals can be confined to individual partition cells. The Vitali covering theorem supplies a disjoint countable selection of these intervals covering the exceptional set up to a null set. For every finite selection, superadditivity and nonnegativity give \[\epsilon\sum_I|I|<\sum_I E(I) \le\sum_{J'\in\mathcal P}E(J')<\eta.\] Pass to the countable union. Its length, and hence the measure of the exceptional set, is at most \(\eta/\epsilon\). Since \(\eta\) was arbitrary, this set is null. Exhaust the domain by compact intervals and take countably many positive rational \(\epsilon\). It follows that the excess on centered intervals has upper density zero almost everywhere.

At a point with this property that is also a Lebesgue point of \(\phi\), the excess on any subinterval of \([a-C_0b,a+C_0b]\) is bounded by the excess on the entire surrounding interval. The additive part has error at most \[\int_{a-C_0b}^{a+C_0b}|\phi(r)-\phi(a)|\,dr=o(b).\] This proves (122), first for positive integral \(C_0\) and therefore for every fixed \(C_0\). ◻

Proof. Use first the fixed \(e_c\) frame and define \[F_\sigma(a)=\mathrm I_\infty (Q(c,a);e_{c+\sigma}\mid e_c).\] Nested grids and the chain rule show that \(F_\sigma\) is nondecreasing, and that its increment on \([a_1,a_2]\) is exactly \[ F_\sigma(a_2)-F_\sigma(a_1) =\mathrm I_\infty (Q(c,a_2);e_{c+\sigma}\mid Q(c,a_1),e_c). \tag{126}\] It is bounded by \(4(a_2-a_1)\), and its total variation on any fixed interior interval is at most \(\mathrm H_\infty(e_{c+\sigma}\mid e_c)=O(\sigma)\). If \(\sigma_1\le\sigma_2\), the increment measure for \(\sigma_1\) is at most that for \(\sigma_2\), again by the chain rule and label nesting. Their bounded densities therefore have decreasing versions along \(\sigma\downarrow0\). Their integrals tend to zero, so their density limit is zero almost everywhere. This monotonicity is the reason that the conclusion is almost everywhere, rather than just in \(L^1\).

We spell out how to use this measure for partial observations, since mutual information need not decrease under arbitrary conditioning. Write \(L_0=e_c\) and take nested extra labels \(L_1=e_{c+\sigma_1}\) and \(L_2=e_{c+\sigma_2}\), where \(0<\sigma_1\le\sigma_2\). First use the \(L_1\) frame for every partial observation. Let \(A\) consist of the full starting cell and any preceding nuisance observations, and let \(Z\) be the partial increment being revealed. Choose containing full \(L_0\)-frame observations \(R=Q(c,a-Cb)\) and \(T=Q(c,a+Cb)\). Given \(L_1\), the observation \(A\) determines \(R\), and \((T,L_1)\) determines \((A,Z)\), up to bounded overlap. Adjoin the bounded-overlap observations at ordinary entropy cost \(O(1)\). The chain rule gives \[\begin{align*} \mathrm I(Z;L_2\mid A,L_1) &\le \mathrm I(T;L_2\mid R,L_1)+O(1)\\ &\le \mathrm I(T;L_2\mid R,L_0)+O(1). \tag{127}\end{align*}\] For the second inequality use \(L_1\subset L_2\): its difference is the nonnegative term \(\mathrm I(T;L_1\mid R,L_0)\). After dividing by \(\log N\) and taking the limit, the last line is \(F_{\sigma_2}(a+Cb)-F_{\sigma_2}(a-Cb)\). The first line bounds exactly the change in \(\mathrm H(Z\mid A,L_1)\) on adding \(L_2\). Thus nuisance data are permitted because they are determined by the containing full ending grid and the older label; no arbitrary-conditioning assertion is used. A difference of rectangular entropies is treated by applying this bound to its finitely many terms.

Only after conditioning on \(L_2\) do we change the partial observations to the \(L_2\) frame. Full starting cells in the two frames have bounded mutual overlap. For partial refinements retain this full base cell and take \(b\ll\min\{\ell,\tau\}\sigma_1\). The unresolved base coordinates in (118) acquire the extra factors \(N^{-\ell\sigma_1}\) or \(N^{-\tau\sigma_1}\), which put all their errors inside the ending widths. Known coordinate centers cause only known translations. The joint partial tests therefore have bounded mutual overlap given \(L_2\) and the base cell. This comparison costs \(O(1)\) ordinary entropy, which vanishes in the \(N\) limit before any division by \(b\).

Apply Lemma 39 and differentiate at a common Lebesgue point of the information densities. The difference of two \(\sigma\)-rates is bounded by a constant times the information density for their total label refinement. That density tends to zero almost everywhere. The rates are thus Cauchy there, proving convergence and the limiting layering rule. For the full subset, conditional increments differ from increments of \(\mathscr H(c,a)\) by exactly the information in (126). This proves (125). Taking a countable intersection for subsets, extra depths, and stable tests, and then using Fubini in \(c\), gives the assertions almost everywhere in \((c,a)\). The stable comparison estimates give measurable versions of all the functions used here. ◻

Proof. Fix an extra label depth \(\sigma>0\) with \(c+\sigma<1\). Lemma 38 bounds the normalized conditional \(X\) entropy at every sufficiently small fixed gap \(r\) by \(xr\) in the limit \(N\to\infty\). Its base cell is \(Q(c,a)\) in the \(e_c\) frame. Given \(e_{c+\sigma}\), this full cell and the full cell \(Q_a\) in the finer frame determine one another up to bounded overlap, by (118). Replacing the base cell therefore changes the ordinary conditional entropy by \(O(1)\), which vanishes in the \(N\) limit before division by \(r\). The resulting entropy is exactly \(K_{\{X\}}^\sigma(a,a+r)\). Dividing by \(r\) and taking \(r\downarrow0\) gives \(\phi_{\{X\}}^\sigma\le x\) at almost every point. Finally Lemma 40, with \(\sigma\downarrow0\), gives \(a_1=\phi_{\{X\}}\le x\). The positive-offset range is the range required by Lemma 38. ◻

Proof. For \(b>0\), label nesting gives the exact normalized limiting identity \[\begin{align*} \mathscr H(c+b,a)-\mathscr H(c,a) ={}&\mathrm H_\infty(e_{c+b}\mid e_c,Q(c,a))\\ &+\mathrm H_\infty(Q(c+b,a)\mid e_{c+b}) -\mathrm H_\infty(Q(c,a)\mid e_{c+b}). \tag{133}\end{align*}\] Condition the first entropy additionally on the index. The old matrix cell is then deterministic given that index and \(e_c\). The time branching in (86) contributes at least \(s_0\ell b\). The normal range on an index in the old time bin has only subpower many coarse angular choices, so conditioning on the old joint label does not cost a positive exponent of that time branching. The finite regularization includes these conditional label tests. This proves the stated lower bound for the first term.

Fix now \(\sigma>b\) and condition both remaining terms further on \(e_{c+\sigma}\). Given \(e_{c+b}\), the two full observations are sandwiched between full observations at offsets \(a-Cb\) and \(a+Cb\) in the \(c\) frame, up to bounded overlap; all four speeds are bounded. The change in their entropy difference is bounded by the extra-label information on this containing gap, as proved in Lemma 40. At a common differentiation point it is at most \(b\) times a quantity tending to zero as \(\sigma\downarrow0\), plus \(o(b)\).

In the common finer frame, changing \(c\) to \(c+b\) changes the four thresholds by \((w-k_i)b\). Use a common full starting offset below all the thresholds, and subtract the two rectangular layering formulas. The axes with smaller \(k_i\) extend farther. The contribution is therefore \[b\sum_G(w-k_G)j_G^\sigma+o(b).\] Let \(b\downarrow0\) with \(\sigma\) fixed, and then let \(\sigma\downarrow0\). Combining this expression with the label contribution gives (132). ◻

Proof. At fixed \(\lambda\), the last conditional rate, before removing an extra label, is the derivative at \(r=0+\) of \[ \mathrm H_\infty\bigl(V_{\lambda+r}\mid V_\lambda,X_{\lambda+c+r},Y_{\lambda+\ell c+r}, U_{\lambda+\tau c+r},e_{c+\sigma}\bigr). \tag{134}\] This follows by subtracting the rectangular entropy for the first three axes from that for all four, using (124). Compare two values of \(c\) with the same \(\lambda\), using extra precision \(\sigma\) at both locations. The later label refines the earlier one. The later other-coordinate thresholds are also finer. In converting from the later frame to the earlier frame, the unknown part of \(V\) beyond \(V_\lambda\) enters \(Y,U,X\) with factors bounded respectively by \[O(N^{-\ell(c+\sigma)}),\qquad O(N^{-\tau(c+\sigma)}),\qquad O(N^{-(c+\sigma)}),\] where \(c\) is the earlier location. These errors fit in the earlier end cells when \(r\ll\min\{\ell,\tau\}\sigma\). For clarity, write the two locations as \(c_1<c_2\) and put \(\Delta c=c_2-c_1\). In the conversion \[X_1=X_2-\Delta t\,U_2-\Delta\vartheta\,Y_2 +\Delta t\Delta\vartheta V,\] the four uncertainty terms, divided by the earlier \(X\)-cell width \(N^{-\lambda-c_1-r}\), are bounded respectively by constants times \[N^{-\Delta c},\quad N^{-\tau\Delta c-\ell\sigma},\quad N^{-\ell\Delta c-\tau\sigma},\quad N^{r-\sigma}.\] The unresolved \(V\) terms in \(Y_1\) and \(U_1\), divided by their respective earlier widths, are bounded by \(O(N^{r-\ell\sigma})\) and \(O(N^{r-\tau\sigma})\). The remaining observed-coordinate errors are smaller. Hence the later conditioning data determine the earlier conditioning data up to bounded overlap. The variable \(V\) itself is identical in both frames. Conditional entropy decreases when information is added, proving the required inequality for (134). First differentiate in \(r\) and then remove the extra label by Lemma 40. Fubini in the common threshold and the two locations gives the essential monotonicity assertion. More explicitly, the change of variables \((c,a)\mapsto(c,\lambda)\) is an invertible shear, and its interior domain is \(c>0\), \(\lambda>0\), \(c+\lambda<1\). Rate-existence and label-removal null sets remain null in these variables. After intersecting over countably many extra precisions, Fubini gives, for almost every \(\lambda\), the comparison for almost every ordered pair \(c_1<c_2\). Integrating over two ordered small neighborhoods of Lebesgue points and shrinking them gives the comparison at every pair of Lebesgue points on that line. Extend by one-sided limits to obtain a monotone version. Taking limits of one-sided local averages supplies a jointly measurable choice of these versions.

For the trace statement suppose \(\alpha_0=1-w>0\), and put \(f(c,a)=v_1(c,a)\). On a compact interior strip define \[h(r,a)=f(r+a/\alpha_0,a).\] Here the common threshold is \(\rho-\alpha_0r\). For almost every \(r\), the function \(a\mapsto h(r,a)\) has a bounded nonincreasing version, and hence has one-sided limits \(h_+(r)\) and \(h_-(r)\) at zero. Dominated convergence gives strong \(L^1\) convergence on compact \(r\) intervals on either side. For \(J\Subset J'\Subset(0,1)\), undoing the shear yields, for small \(|a|\) on the appropriate side, \[\begin{align*} \|f(\cdot,a)-h_\pm\|_{L^1(J)} \le{}&\|h(\cdot,a)-h_\pm\|_{L^1(J')}\\ &+\|h_\pm(\cdot-a/\alpha_0)-h_\pm\|_{L^1(J)}\longrightarrow0. \end{align*}\] The last term tends to zero by continuity of translation in \(L^1\). Fubini gives one full-measure set of offsets whose original slices agree almost everywhere with these representatives. Thus convergence holds through every sequence in that set. Averaging the \(L^1\) estimate over a one-sided stripe gives the same limit for stripe averages.

If \(w=1\), the fixed-threshold lines are constant-offset lines. Every generic slice is a bounded monotone function of \(c\). Helly’s selection theorem, followed by dominated convergence, gives strong \(L^1\) compactness on each compact \(c\) interval. A compact exhaustion and subsequence diagonal give strong \(L^1_{\mathrm{loc}}\) compactness. No choice of values on exceptional offset slices is needed in either case. ◻

Proof. The fundamental theorem of calculus in the offset variable and (121) give \[\mathscr H(c,\epsilon)-\mathscr H(c,0) =\int_0^\epsilon g(c,a)\,da\ge d\epsilon.\] On the left, that same inequality at \(a=-\epsilon\) gives \(\mathscr H(c,0)-\mathscr H(c,-\epsilon)\le d\epsilon\). This proves the two stripe bounds; they are not assertions about individual offset slices of \(g\).

Local Lipschitz continuity gives uniform convergence \(\mathscr H(\cdot,a)\to\mathscr H(\cdot,0)\) on \(J\) and a common bound for the \(c\) derivatives. Against a smooth compactly supported test function \(\psi\), integration by parts therefore gives \[\int_J\psi(c)\partial_c\mathscr H(c,a)\,dc =-\int_J\psi'(c)\mathscr H(c,a)\,dc \longrightarrow (dw-B_*-\tau+s_0\ell)\int_J\psi(c)\,dc.\] The common \(L^\infty\) bound and \(L^1\) approximation extend this to all \(L^1\) tests, proving (137). It also proves the corresponding statement for stripe averages. Finally Proposition 42 rearranges as \[\mathcal M-w(g-d)\ge s_0\ell+wd-\partial_c\mathscr H.\] The right side converges weak-star to \(\tau+B_*\). Nonnegative \(L^1\) tests preserve the inequality under every weak-star subsequential limit, proving the claim. ◻

Proof. The widths in (141) satisfy \[ \frac{\mathfrak s_X}{\mathfrak s_U}=R_{c+b},\quad \frac{\mathfrak s_X}{\mathfrak s_Y}=\Theta_{c+b},\quad \frac{\mathfrak s_Y}{\mathfrak s_V}=R_{c+b},\quad \frac{\mathfrak s_U}{\mathfrak s_V}=\Theta_{c+b},\quad \frac{\mathfrak s_X}{\mathfrak s_V}=R_{c+b}\Theta_{c+b}. \tag{144}\] The last equality uses \(\tau+\ell=1\). Substituting these ratios in the exact change of the four linear forms gives (143). Using uncentered forms adds only known translations. The time and angular errors of \(L\) relative to \(E_0\) are respectively \(D^{-\ell(C_0-1)}\) and \(D^{-\tau(C_0-1)}\). The formula, on a bounded set of coordinates and coefficients, gives the asserted error bound. Actual dyadic widths may be used as the units, or bounded rounding factors may be retained throughout. ◻

Proof. We prove the rectangular entropy formula first. We then realize its values as uniform finite cell masses and verify the time and normal laws needed to choose a second event.

Proof. Given \(E_0\), each label \(L_i\) has at most \(C N^{C_1b}\) possible values, where \(C_1\) depends on \(C_0\) and the fixed speeds. This is an alphabet bound from the time and normal grids, independent of the probabilities of these labels. It remains true under either marginal bias. Consequently \[ \mathrm H(L_0,L_1\mid E_0)\le C_2(1+b\log N). \tag{153}\]

Temporarily write \(J_0(a)\) for the rebase grid at offset \(a\) and set \[f(a)=\mathrm I(J_0(a);L_0,L_1\mid E_0).\] This is nondecreasing, because the matrix grids are nested as \(a\) increases. Its range is contained in \([0,C_2(1+b\log N)]\). The chain rule gives the exact identity \[ f(a+Kb)-f(a) =\mathrm I(J_0(a+Kb);L_0,L_1\mid E_0,J_0(a)). \tag{154}\] For a nondecreasing function with range of length \(H_*\), \[\int_A^B[f(a+q)-f(a)]\,da\le qH_* \qquad(q>0),\] as follows either by changing variables in the two integrals or by integrating its nonnegative increment measure. For a grid replacement \(a_n\) with \(|a_n-a|=o(b_n)\), monotonicity also bounds the left side of (154) by \[f(a+(K+1)b)-f(a-b)\] for large \(n\). Integrate on a compact interval whose slightly larger neighborhood stays admissible. Equations (153) and (154) give \[ \int \frac{\mathrm I(J_f;L_0,L_1\mid E_0,J_0)}{\log D}\,da \le C_3\left(b+\frac1{\log N}\right). \tag{155}\] The left side uses the stepwise chosen grid offsets. The containing-gap estimate proves the bound regardless of the sizes of the grid’s inverse images.

Choose \(b_n\) summable, and increase \(N_n\) so that \(1/\log N_n\) is summable as well as satisfying all the finite tests of Lemma 47. Tonelli’s theorem applied to the nonnegative functions in (155) shows that their sum is finite almost everywhere. In particular the summands tend to zero, which proves (152). A further subsequence, such as the one used for pinning, preserves this conclusion. For countably many pair experiments use the same integral argument for each and intersect the full-measure sets. Finally exhaust the admissible offset domain by compact intervals. ◻

Proof. Suppose the inequality fails strictly. Choose fixed \(0<t_*<t\), \(0<s_*<s\), and \(u>a_*\), still satisfying \[ u<\min\{t_*,(t_*+s_*)/2,1\}. \tag{162}\] The choices are possible by continuity, and are made before any pinning loss is chosen. Let \(\kappa>0\) be an allowance from Lemma 50 for these three fixed exponents. In the pinned case choose the loss tolerance \(\epsilon\) so small that \[ \epsilon<s-s_*,\qquad 2\epsilon<\kappa/2. \tag{163}\] Choose the pinned coupling and its fixed positive depth \(\zeta\) with this tolerance. For the ordinary pair law any fixed rational \(0<\zeta\le\zeta_{\max}\) works. The depth does not subsequently shrink with \(D\). Subpower errors in the original single-event estimates therefore tend to zero also after division by \(\zeta\).

Proof of Proposition 45. Fix a good interior \(c\). Work at offsets where the entropy densities, the local counts, and the pair information conclusions all hold. Strict violations of any inequality can be witnessed by rational choices of the smaller exponents and loss tolerances in (162)–(163). There are countably many such choices, so the preceding constructions apply on a common full-measure set of offsets. Fubini then gives the asserted almost-everywhere statement in \((c,a)\).

For distinct speeds the four tests needed below are displayed in Table 1. Each uses the ordinary pair law. Source exponents are conditional on the listed nuisance bins in the original matrix law. They follow from the all-depth probability ratios in Lemma 47, in either of the two possible speed orders.

For example the first formula of (143), with \(U,V\) frozen, is \[X'=X+\Delta\vartheta Y +\Delta t U+\Delta t\Delta\vartheta V.\] The last two terms are a full-label-dependent translation up to \(O(D^{-\zeta})\). With \(Y,V\) frozen, the same expression is \(X+\Delta t U\) plus such a translation. The second and third formulas give the other two rows. Thus every row has exactly the form required in Lemma 51.

It is useful to spell out the numerical implication. For \(0\le a,b\le1\) and \(s>0\), suppose the test with source exponent \(a+b>0\) gives \[ a\ge\min\{a+b,(a+b+s)/2,1\}. \tag{166}\] If \(a<1\) and \(b>0\), both the first and the third entries of this minimum exceed \(a\). The second must therefore be at most \(a\), so \(b\le a-s\). If \(b=0\) there is no further restriction. In either case \[ a<1\quad\Longrightarrow\quad b\le(a-s)_+. \tag{167}\] When \(a+b=0\), that conclusion is immediate without applying the projection lemma.

The first row gives \(y_1\le(a_1-\gamma)_+\) when \(a_1<1\). Together with \(y_1\le1\), this implies \(a_1\ge y_1\) also when \(a_1=1\). The second row gives \(u_1\le(a_1-s_0)_+\) whenever \(a_1<1\). The third row analogously gives \(u_1\ge v_1\). For the last row, if \(v_1>0\) and \(y_1<1\), (167) gives \(y_1\ge s_0+v_1\). If \(y_1=1\), the required \(y_1\ge\min\{1,s_0+v_1\}\) holds automatically. These are all assertions of (139).

Suppose now that \(\tau=\ell\). Put \[r_Y=\phi_{\{X,Y\}}-\phi_{\{X\}}.\] Conditional entropy monotonicity and the one-dimensional alphabet bound give \(0\le r_Y\le1\) and \(0\le m_1-r_Y\le1\). Use the second row’s time projection with \(Y,V\) frozen. For every \(0\le u\le\zeta\), the tied rectangle formula gives the conditional source exponent \[t=a_1+(m_1-r_Y).\] The target exponent is \(a_1\) and the slope exponent is \(s_0\). Equation (167), with \(b=m_1-r_Y\), yields \[a_1<1\quad\Longrightarrow\quad m_1-r_Y\le(a_1-s_0)_+.\] Since \(r_Y\le1\), this proves the bound for \(m_1\) in (140).

Finally freeze \(Y,U\) and use the pinned pair distribution. The first formula in (143) becomes \[X'=X+(\Delta t\Delta\vartheta)V +\Delta t U+\Delta\vartheta Y.\] The source pair is \((X,V)\), of conditional exponent \(a_1+v_1\), by Lemma 47. Its target exponent is \(a_1\). The product slope has available exponent \(s_0\) by Lemma 48 and the preceding coupling construction. Lemma 51 includes both its marginal bias and its small direction loss. Thus (167) gives \(v_1\le(a_1-s_0)_+\) when \(a_1<1\), as required.

Every construction above takes place on a compact subset of \(\mathcal U\) from (117). Indeed the smallest and largest of the four unshifted thresholds are respectively \[(1-w)(1-c)+a,\qquad 1-w(1-c)+a.\] If \(w<1\), a compact interior \(c\)-interval admits sufficiently small offsets of either sign, and the \(O(b)\) rebase and refinement shifts stay in range. If \(w=1\), only positive small offsets are interior. This proves the stated conclusion for every permitted sign and completes the proposition. ◻

Proof. Put \(\Delta=h_{\mathrm{pr}}-u>0\) and \(\varepsilon=\Delta/4\). Let \(\eta_{\mathrm{pr}}=\eta_{\mathrm{pr}}(\varepsilon,s,t)>0\) be the Frostman tolerance in Ren and Wang (2025, Theorem 4.1). We may take \[ \kappa_0=\min\{1/20,\Delta/16,\eta_{\mathrm{pr}}/12\}. \tag{168}\] All constants below may depend on \(R,C_0,s,t\), but not on the number of labels or on their weights. We decrease \(\delta_0\) finitely many times.

First put the incidences in the bounded coordinates used for dyadic tubes. Set \(B_0=R+1\) and \[v=\frac{y+B_0}{2B_0},\qquad w=\frac{x+B_0}{2B_0^2},\qquad a_\lambda=-\frac{\vartheta(\lambda)}{B_0}.\] Then \((v,w)\in(0,1)^2\), \(a_\lambda\in(-1,1)\), and \[w-a_\lambda v =\frac{x+\vartheta(\lambda)y+B_0(1+\vartheta(\lambda))} {2B_0^2}.\] Apply this affine transformation to \(A_\lambda\), writing the result as \(J_\lambda\). Each \(J_\lambda\) has at most \(C\delta^{-u}\) constituent intervals of length at most \(C\delta\). The transformed point and slope measures satisfy (158) with an additional fixed factor \(C\). We keep the notation \(\mu\) for the transformed point measure.

Choose a dyadic number \(\varrho\) with \(\delta\le\varrho<2\delta\), and put \[L=8\lceil\log_2(1/\delta)\rceil+8, \qquad q_*=\delta^{2\kappa}.\] Suppose that the left side of (159) is at least \(q_*\). Write \(f(\lambda)=\mu\{(v,w):w-a_\lambda v\in J_\lambda\}\). Partition the slopes into dyadic \(\varrho\)-intervals \(I\). Give \(I\) its total label weight \(W_I=\nu\{\lambda:a_\lambda\in I\}\), and choose a label \(\lambda_I\) maximizing \(f\) in that interval. There are finitely many labels, so \[ \sum_I W_If(\lambda_I)\ge q_*. \tag{169}\] There are at most \(C\delta^{-1}\) occupied slope intervals. Discarding those with \(W_I<\delta^4\) loses at most \(C\delta^3\le q_*/2\) from (169). The remaining weights have at most \(L\) dyadic levels. One level, consisting of \(n\) slope intervals, satisfies \[H_*:=\sum_{I\text{ in the level}}W_If(\lambda_I) \ge\frac{q_*}{2L}.\] Let \(a_* =\sum_I W_I\) on this level. Its weights are within a factor of two, and \(H_*\le a_*\le1\). Consequently the uniform distribution of its slope intervals satisfies \[ \frac{\#\{I:I\cap B(b,r)\ne\varnothing\}}n \le CL\delta^{-3\kappa}r^s \quad(\varrho\le r\le1). \tag{170}\] Indeed, relative counting costs at most \(2/a_*\) times the original slope mass of a fixed enlargement of the ball, and \(a_*\ge q_*/(2L)\). For enlarged radii exceeding one, the same bound follows from the total mass bound.

Let \(d(p)\) be the number of selected representative tests satisfied by \(p=(v,w)\). Comparability of the slope weights gives \[\int\frac{d(p)}n\,d\mu(p) \ge\frac{H_*}{2a_*}\ge\frac{q_*}{4L}.\] Since \(0\le d(p)/n\le1\) and \(\mu(\mathbb R^2)\le1\), the set \(G_0=\{p:d(p)\ge q_*n/(8L)\}\) has mass at least \(q_*/(8L)\). Pigeonhole its positive integer degrees into at most \(L\) dyadic levels. This gives a Borel set \(G\subset G_0\) and an integer \(M\ge1\) such that \[ \mu(G)\ge\frac{q_*}{8L^2},\qquad M\le d(p)<2M\ (p\in G),\qquad M\ge\frac{q_*n}{16L}. \tag{171}\]

Partition the point square into dyadic \(\varrho\)-squares and give \(Q\) the weight \(\mu(G\cap Q)\). There are at most \(C\delta^{-2}\) squares. Discard weights below \(\delta^4\); their total is at most \(C\delta^2\le q_*/(16L^2)\). A dyadic weight pigeonhole gives a family \(\mathcal P\) of squares with comparable weights and total retained mass \[ b_*\ge\frac{q_*}{16L^3}. \tag{172}\] For each \(Q\in\mathcal P\) choose \(p_Q\in G\cap Q\). Comparison between counting and these comparable cell weights shows that \(\mathcal P\) has relative \(t\)-Frostman constant \[ CL^3\delta^{-3\kappa} \tag{173}\] down to \(\varrho\): passage to relative counts costs at most \(2/b_*\), and the original measure bounds the mass of a fixed enlargement of each testing ball.

Here a dyadic \(\varrho\)-tube is the union of lines \(w=av+b\) with \((a,b)\) in a fixed dyadic parameter square of side \(\varrho\). For each selected slope representative \(a_I\) and each \(b'\in J_{\lambda_I}\), include the tube whose parameter square contains \((a_I,b')\). Call the resulting family \(\mathcal T\). Every \(J_{\lambda_I}\) meets at most \(C\delta^{-u}\) intercept intervals, so \[ |\mathcal T|\le Cn\delta^{-u}. \tag{174}\] For \(Q\in\mathcal P\) and each representative test satisfied by \(p_Q=(v_Q,w_Q)\), select the tube with parameter cell containing \((a_I,w_Q-a_Iv_Q)\). This tube intersects \(Q\). Different slope cells give different tubes. Denote this fiber by \(\mathcal T(Q)\); then \(M\le|\mathcal T(Q)|<2M\).

A ball of radius \(r\) in tube-parameter space restricts its slope coordinate to an interval of comparable radius. By (170) and (171), \[ \frac{\#\{\text{parameter cells of }\mathcal T(Q) \text{ meeting }B(z,r)\}} {|\mathcal T(Q)|} \le CL^2\delta^{-5\kappa}r^s \quad(\varrho\le r\le1). \tag{175}\] Thus the base cells and selected incident fibers form exactly the quantitative configuration of Ren and Wang (2025, Definition 3.1 and Theorem 4.1). The selected fiber cardinalities lie between \(M\) and \(2M\), as permitted by the cited theorem. For sufficiently small \(\delta\), the constants in (173) and (175) are at most \(\varrho^{-\eta_{\mathrm{pr}}}\). In detail, arrange \(CL^3\le\delta^{-\kappa}\) and use \(6\kappa\le\eta_{\mathrm{pr}}/2\) together with \(\varrho<2\delta\).

The cited theorem now gives \[|\mathcal T|\ge c_\varepsilon M\varrho^{-h_{\mathrm{pr}}+\varepsilon} \ge\frac{c'_\varepsilon}{16L} n\delta^{-h_{\mathrm{pr}}+\varepsilon+2\kappa}.\] Combining this with (174) yields \[1\ge\frac{c''_\varepsilon}{L} \delta^{-(\Delta-\varepsilon-2\kappa)}.\] But (168) implies \(\Delta-\varepsilon-2\kappa\ge5\Delta/8>0\), so the right side tends to infinity. This proves (159). Finally, (160) would make the product hit probability at least \(\nu(B)\delta^\kappa\ge\delta^{2\kappa}\), a contradiction. ◻

Proof. We localize the second point to a small square and linearize the output map there. Symmetry supplies the reverse rays from the second point to the anchors; those rays provide the directions for the planar projection estimate. A joint pruning then makes the local estimates iterable over a finite ladder of depths.

Choose \[ \sigma_*<\sigma_m<\min\{s,(s+\sigma)/2\}. \tag{177}\] We first prove the bound at a fixed depth \(T\in(0,1]\). Choose a small starting depth \(d_0>0\) and a finite ladder \[ d_0<d_1<\cdots<d_n=T, \qquad d_{j+1}<2d_j. \tag{178}\] The choices of \(d_0\) and of a further small parameter \(\lambda>0\) will be specified below. Delete pairs of distance less than \(\delta^\lambda\). Their probability is at most \(\delta^{s\lambda-o(1)}\), by the square-mass upper bounds.

Proof of Lemma 48. It is enough to prove the assertion for small \(\epsilon<s/10\); we reserve a fixed fraction of \(\epsilon\) for each normalization. Choose a small constant \(\eta_0>0\), depending on \(s,\gamma',\epsilon\), to be fixed in the first case below.

Proof. Suppose otherwise. Choose an open interval \(I\) of \(p\)-values, with lower endpoint \(p_{\min}>2\), on which \(h(p,0)>0\). Fix the smoothing parameter before making any of the subsequent selections, with \[ 0<\eta\le \min\left\{\frac1{32},\frac{p_{\min}-2}{16}\right\}. \tag{186}\] This choice is legitimate because the critical inequality at \(q=0\) is independent of \(\eta\). Lemma 11 supplies an interior differentiability point with \(h>0\) and \(h_z>0\). The construction in Proposition 33, Proposition 35, and Lemma 37 then supplies fixed parameters satisfying \[\begin{gather*} p>2,\qquad 0\le q<1,\qquad d=2-q,\qquad 0<h\le1-q,\qquad x=1-q-h<1, \tag{187}\\ 0<\tau<1,\qquad \ell=1-\tau,\qquad 0<\beta<1,\qquad s_0=1-\beta, \tag{188}\\ B_*=h+\ell\bigl(p\beta-\mathcal D(q,\beta)\bigr),\qquad w=h_z+\ell\mathcal D_q(q,\beta),\qquad 0<w\le1. \tag{189}\end{gather*}\] In particular, Lemma 37 supplies the strict inequality \(s_0>0\).

Recall that \(g\) is the total matrix entropy rate and \(\mathcal M\) its speed-weighted rate from (128). For brevity in this proof put \[ b_+=(\beta-q)_+,\qquad T_*=\tau+B_*,\qquad \mathcal Q=\mathcal M-w(g-d), \tag{190}\] where \(r_+=\max\{r,0\}\). The smoothing definition in (5) gives \[ 0\le\mathcal D(q,\beta)\le\min\{q,\beta/2\}. \tag{191}\] If \(q<(1/2-\eta)\beta\), then the integrand defining \(\mathcal D\) equals \(q\) throughout its averaging interval. Consequently \[ \mathcal D(q,\beta)=q,\qquad \mathcal D_q(q,\beta)=1,\qquad w>\ell. \tag{192}\]

We record precisely the limiting statements to be used. Fix a compact interval \(J\subset(0,1)\). Proposition 42 and Lemma 44 imply that every weak-star limit of \(\mathcal Q(\cdot,a)\), along generic offsets \(a\to0\), is at least \(T_*\) almost everywhere on \(J\). The same lower bound holds for every weak-star limit of stripe averages. Moreover, the right stripe averages of \(g\) are at least \(d\), and the left stripe averages are at most \(d\), whenever the latter offsets are available. In formulas, \[ \frac1\epsilon\int_0^\epsilon g(c,a)\,da\ge d, \qquad \frac1\epsilon\int_{-\epsilon}^0 g(c,a)\,da\le d \tag{193}\] for almost every \(c\in J\) and sufficiently small \(\epsilon>0\), on the sides being used. These inequalities may be integrated against every nonnegative \(L^1\) test function supported on \(J\). The rates are uniformly bounded, so weak-limit inequalities initially established with smooth tests extend to such tests by \(L^1\) approximation. We will use this extension to localize on measurable subsets of \(J\).

For \(w<1\), Lemma 43 gives strong one-sided \(L^1(J)\) traces for \(v_1\) through a fixed full-measure set of generic offsets. The corresponding stripe averages converge to the same trace. For \(w=1\), that lemma gives strong subsequential \(L^1(J)\) compactness along every sequence of generic offsets. We use this strong convergence to localize the constraints that hold where \(v_1>0\); weak-star convergence suffices for the other rates.

By Corollary 41, on the right we also have \[ a_1\le x=1-q-h<1. \tag{194}\] It follows in particular that \[ (a_1-s_0)_+\le(\beta-q-h)_+\le b_+. \tag{195}\]

The domain \(\mathcal U\) in (117) contains small offsets of both signs uniformly on \(J\) when \(w<1\), and small positive offsets when \(w=1\). Thus Proposition 45 applies on every stripe used below. The bound (194) is available on the right.

The choice of stripe follows from the identity \[\mathcal Q=(\mathcal M-rg)+rd+(r-w)(g-d), \qquad r\in\mathbb R.\] After averaging, an upper bound for \(\mathcal M-rg\) gives an upper bound for \(\mathcal Q\) on the right when \(r\le w\), and on the left when \(r\ge w\). For distinct speeds the projection inequalities pair \(a_1\) with \(y_1\) and \(u_1\) with \(v_1\), making \(r=\tau/2\) useful. Direct bounds for \(\mathcal M\), corresponding to \(r=0\), handle two further regimes. Table 2 records the five cases. In the last case, compactness first gives a gap common to all weak limits of generic slices; averaging then gives the stripe contradiction.

Proof. We first establish a uniform discrete estimate, then an indicator estimate, and finally (203). All geometric constants in this proof are absolute unless their dependence is displayed.

Proof of Theorem 1. Fix \(\varepsilon>0\). Proposition 54 allows a choice of \(p>2\) with \(h(p,0)=0\) and \((p-2)/3<\varepsilon/2\). Apply Proposition 55 with \(\kappa=\varepsilon/2\). Since \(0<\delta<1\), \[\|K_\delta f\|_{L^3(S^2,\sigma)} \le C_{p,\varepsilon/2} \delta^{-\varepsilon}\|f\|_{L^3(\mathbb R^3)}.\] Choose this \(p\) once as a function of \(\varepsilon\) and rename the constant \(C_\varepsilon\). It is independent of \(\delta\), of the complex-valued function \(f\), and of all tube centers and directions, as required. ◻

Proof. Write \(\widetilde K_\delta\) for the Kakeya operator of Gao et al. (2025, sec. 2.2), normalized by \(\delta^{-2}\) over \(\delta\)-neighborhoods of unit segments. For \(0<\delta<1\), each such neighborhood is contained in the union of three of the cylinders \(T_\delta(a,\omega)\) used here. Consequently \(\widetilde K_\delta f\le3\pi K_\delta f\), so Theorem 1 gives its \(L^3\)-to-\(L^3\) bound with every positive power loss. Also \[\|\widetilde K_\delta f\|_{L^\infty(S^2)} \le\delta^{-2}\|f\|_{L^1(\mathbb R^3)}.\] Interpolation for this sublinear operator, with \(1/p=1-2\theta/3\) and \(1/q=\theta/3\), gives \(q=2p'\) and the power \(-2(1-\theta)-\eta\theta=1-3/p-\eta\theta\) from an \(L^3\) loss \(\delta^{-\eta}\). Since \(\eta>0\) is arbitrary, including the two endpoint bounds gives \(K(3)\) in the notation of Gao et al. (2025, equation (6)).

Their Proposition 2.3 identifies \(K(3)\) with the Euclidean Nikodym estimate \(N(3)\), and with the geodesic Nikodym estimate on a constant-curvature manifold, in equations (7) and (9) respectively. The negligible exponent loss in that transfer is absorbed by starting with a smaller positive loss above. Its manifold comparison uses a geodesic-straightening diffeomorphism whose bi-Lipschitz and volume comparison constants are bounded on the fixed compact chart. Applying that comparison to \(U\Subset V\) gives the stated local estimate, with constants allowed to depend on this geometry. ◻

Proof. The Euclidean comparison and interpolation in the preceding proof give \(K(3)\) in the notation of Gao et al. (2025, equation (6)). Their Proposition 4.1 gives the equivalent assertion \(\mathcal K(3)\) for the phase class above, which is exactly the displayed \(p,q\) range. Their straightening theorem is local (Gao et al. 2025, Theorem 1.12); on the fixed chart its spatial and direction-parameter comparison constants may depend on the phase and the chart. This gives the stated estimates. ◻