The Kakeya maximal conjecture in three dimensions

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

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

This includes d=1, where the sum costs a logarithm, and includes repeated indices. Consequently

Since \Pi_F(1)\ge(p-1/2)F(1)\ge F(1), (2.8) 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 2.5. 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

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,

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:

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

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

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 (2.12) 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

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 (2.10); rational scales suffice, followed by continuity. Let n' be the remaining total index weight. Since n'\le n, the assumed event-mass retention gives

This proves the upper bound in (2.11); the lower bound follows from (2.10) 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

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 (2.10), G-F is nondecreasing and Lipschitz, so G'\ge F' almost everywhere. The slope bound gives

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 (2.13), 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

Also q decreases, so P_{p,q} increases pointwise. The new right side of (2.8) is therefore at least the old one. The candidate is still in range since A+s\le d+s-1. Thus

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 [1, 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

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: (2.14) 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

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

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 (2.17) 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 jth 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

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

The universal upper bound (2.17) 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\leexp m'. Its plank masses cannot increase, so \Delta'_{h_0}\le\Delta_{h_0}. Its profile remains F by Lemma 2.6. Applying (2.17) to the refinement sandwiches both quantities:

This proves (2.18).

For the final assertion, let H_j denote the corresponding sharp hybrid power. By (2.9),

Thus a sufficiently accurate realization of the jth hybrid equality has

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 (2.19); 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

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

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 (3.3) gives

The elementary exponential-moment bound

shows that, for a sufficiently large fixed L_R, simultaneously

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

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 3.1 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

Since \sum_i\omega_i|Y_i|\le ASv_\delta, this proves (3.4). If S<L_R(1+\log(1/\delta)), use instead the pointwise bound \sum_i\omega_i\one_{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 (3.7). 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 (3.7) therefore gives

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

Rescale the common length L to one and apply Lemma 3.2. 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 (3.8) 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 3.4, having chosen the current \eta_{\mathrm{in}} smaller than half its allowed tolerance.□

Proof. For unit weights this is Guth [10, Theorem 1.3, p. 264], the endpoint form of multilinear Kakeya introduced in Bennett et al. [2, 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 (3.10). 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 3.7, 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

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.

Capturing and routing an excessive family. Fix a proposed excess N^\varepsilon and a finite scale and shape test. On each temporal block, greedily assign all remaining active indices in a plank with mass greater than N^\varepsilon T(\kappa,a,b), and remove their events on that block. Continue until no such plank remains. The plank tests may be meshed and thickened by a fixed factor, so the procedure is finite. Its assignments are disjoint in original indices on each block. Suppose the removed events form an extensive graph. Pigeonholing dyadic widths costs only a subpower, so retain a common pair a,b.

Two matrices whose segments stay in the same local plank differ in spatial slope by O(b/K): evaluate the difference at the endpoints of the block and divide its O(b/N) transverse diameter by the duration K/N. Their intercepts likewise differ by O(b/K), since all original times belong to a bounded interval. Each packet therefore meets only boundedly many full-time isotropic matrix-grid parents of side b/K. Partition its indices among those parents. For a fixed width type each original matrix has one parent, so an index is not duplicated once for each selected block.

Before this subdivision, every assigned index-block contributes KN^{-F(\kappa)+o(1)} events. Include the assignment cells and their parent subdivisions among the regularization tests. The inherited-weight cleaning of Lemma 2.6 and Corollary 2.7 preserves the threshold N^{\varepsilon-o(1)}T on surviving packets: delete active packets retaining less than an inverse subpower fraction of their original incidence, and retain nondeficient parent subdivisions. Upper branching is inherited on each original index and starting block. Total incidence retention and bounded routing then preserve the whole profile F on the parent ensemble.

Apply the rescaling and filling just described. Its cell count gives aggregate raw multiplicity at least

The sharp upper bound. A new full-time plank of widths A,B pulls back to one of widths aA,aB in original fine units. These are legitimate original tests: 1\leq aA\leq aB\leq aN'=Nb/K\leq N. Since the shape exponents for \Delta_{h_0} sum to d, each parent’s raw parameter obeys

The old occupied time tree above scale K is unchanged; below it we have grafted full time intervals of b/a bins. The profile is exactly (4.4) in normalized logarithmic coordinates. In particular its temporal cost, expressed in original \log N units, is \Pi(1)-\Pi(\kappa).

The sharp hybrid upper bound in each parent is consequently at most

The same upper bound holds uniformly for the ensemble. Otherwise a sequence of failing parent representatives would contradict the definition of the sharp exponent after regularizing to their common limiting profile. This contradicts (4.6).

If N'=N^{o(1)}, then \chi\to1. Because \chi\leq\kappa\leq1, one has \kappa\to1 and \Pi(1)-\Pi(\kappa)=o(1). The full-box mass bound in a parent, at most a^d\Delta_{h_0}(N')^d, supplies the same exponent upper bound as (4.8). Thus this endpoint also cannot carry the proposed fixed excess.

It follows that removing excessive assignments costs no extensive piece. Perform these removals successively for a finite mesh of scales and shapes, regularizing after each removal. The observation at the start of the section preserves equality for the next test. Nearby exponents are controlled by enlarging widths and extending to nearby nested temporal blocks; endpoint extrapolation costs only the prescribed mesh error. Let the mesh and error tolerances tend to zero in the stated order. This proves (4.2).

For the converse assertion, start with the extensive saturated assignments in the statement and perform the same routing, rescaling, and filling, with packet mass at least T in exponent and no fixed excess. The lower bound is now Ta/b and the upper bound is exactly the same expression. When \chi<1, the derived resolution grows with a positive exponent and the profile in (4.4) is well-defined. Equality of the ensemble bounds provides either the ensemble formulation or realizing representatives. At H=h_0=h these are equalities for the original critical problem.□

Proof. The full bounded patch is covered by boundedly many planks with both widths comparable to N, so n\leexp\Delta_{h_0}N^d. Since k\eqexp N^{1-\alpha} and m\eqexp\Delta_{h_0}N^{H+\Pi(1)}, the ratio in (4.9) is at least N^{q+H+\Pi(1)-2\alpha} in exponent. Inequality (4.1) 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 (4.2) 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 (4.10).□

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 (4.2), 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 3.7 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,

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

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 (4.9), 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 (4.10) 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

The symmetric kernel of AA^* is

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

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

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, (4.14) gives

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

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 (4.12). Apply it to f=\one_E and use \mathcal N_K\one_E\geq\rho on E^+ to obtain (4.13). 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 4.3. 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

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

by (4.16). 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 (4.10) with L\asymp1/r and (4.1) to obtain the conditional probability bound

The coplanar directions fit in O(K) caps of radius K^{-1}. Their normalized mass is one, so summing (4.20) 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 (4.19) for that encounter interval and (4.20) 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.

Plate density and the filled support. The line-parameter measure is now t-Frostman. We apply the planar theorem to the neighbors’ shadings, and use the resulting plate density to fill the reference blocks. Each neighbor index carries its original shading on the entire current block. It consists of KN^{-F(\kappa)+o(1)} cells, with the s-Frostman bound (4.19). Repetitions of a neighbor through different sampled times merely add its line weight; they do not shorten this shading. The bounded projection preserves its cell count and Frostman estimates. Thus the weighted and repeated-line formulation in Corollary 3.5 applies to this probability law and these shadings. It gives at least

planar cells in their union. More explicitly, one first fixes any small theorem loss, includes all required cell and conditional mass tests at that tolerance, and then lets the loss decrease. The strict inequality tested below is fixed before these choices. This is sufficient for the subpower notation in (4.21).

All the original neighboring segments lie in the plate from Lemma 4.3. Its thickness is bounded, so a projected cell receives only boundedly many relevant normal layers. In particular (4.21) lower-bounds the number of original union cells in this plate. The plate has volume comparable to K^2. It therefore has original-union density at least

It contains the full reference segment, including times that were not originally marked. The same construction applies to every retained reference index-block.

Fill all such reference blocks with every fine time bin, keeping each original index and its inherited weight. Write E^+ for the new support. Every point of E^+ lies in a plate with density (4.22) in a fixed enlargement of the original support. Lemma 4.4 gives

The squared density loss in this formula is retained as a power of N.

The number of incidences increases by N^{F(\kappa)+o(1)}. Indeed each retained occupied block had KN^{-F(\kappa)+o(1)} times and now has K. The filled profile is zero below \kappa and equals F(u)-F(\kappa) above it. Its cost is \Pi(1)-\Pi(\kappa), while its global raw clustering parameter is at most \Delta_{h_0}. The support estimate gives

The sharp hybrid upper bound gives m^+\leq mN^{-\Pi(\kappa)+o(1)}. These are incompatible with a strict reverse inequality to (4.18). This proves the lemma.□

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

We may take \kappa as small as needed for Lemma 4.5.

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 (4.10) together with (4.24) and (4.1). 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 (4.24) in Lemma 4.5, with

If e=0, choose \varepsilon small compared with H. Then s=1-\varepsilon, t=1+H-\varepsilon, and \mu=1. Since F(\kappa)>0,

contradicting (4.18). This argument does not need a lower bound for F(\kappa)/\kappa.

Suppose e>0. At \varepsilon=0 one has

and therefore

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,

But (4.24) and (4.1) now give

again contradicting (4.18). 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 3.3, 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 (4.2), where \Pi(\kappa)=0. It gives

or, in logarithmic coordinates,

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 4.1.□

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

at resolution N'=K_1\theta. Equivalently, in the new fine-cell units the coordinate map is

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

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.

Charts and preservation of the time tree. We now construct charts with the stated bounds, keeping each original index and its selected time blocks together and retaining the packet masses required by the cell count. Use angular bins of width \theta/10 for the unoriented thin spatial axis. Since the axes used along any one original index and larger block lie within subpower times \theta of its fixed axis, that index-block meets only subpower many bins. In each angular bin choose a normal–tangent frame. Subdivide matrix space, expressed over the larger interval, into parent cells of widths \theta K_1,K_1 in normal and tangent position and also in the normal and tangent displacement over that interval. A fixed matrix has only boundedly many adjacent grid parents in each bin.

An entire fine plate packet meets only subpower many such parents. Indeed its thin axis belongs to the angular bin and differs from the bin normal by at most \theta/10. Its cross-sectional spread measured in that normal is at most N^{o(1)}(1+\theta K), and its tangential spread is at most N^{o(1)}K. Divide the endpoint differences by the fine-block duration, then extrapolate over the larger interval. The resulting spreads are at most

and N^{o(1)}K_1 in the tangent coordinate. This uses \theta\geq K^{-1} in exponent. The same bounds hold for intercepts after subtracting a parent center. Thus the asserted parent routing follows from endpoint extrapolation, rather than from the number of fine subblocks used by the index.

Within a chosen larger block, bin, and matrix parent, keep all selected fine intervals of one original index together as a single derived index. Do not create a separate derived index for each fine interval. This preserves inherited weights and ensures that a full-plank test in a chart counts a subset of original active lines on the larger block. Include packet subdivisions and parent assignments in the finite cleaning. Since every original index-block has subpower routing, Corollary 2.7 shows that surviving packets retain active mass mKN^{-o(1)}.

The routing also preserves the time tree. On every fixed logarithmic subrange of the larger block, each derived time tree is contained in its original tree. Its branching exponents therefore cannot increase. The chart ensemble retains an extensive part of the weighted incidence, while its total available original index-block mass has increased by only a subpower. After uniform regularization there can be no loss in the total branching exponent. The nonnegative losses on each fixed subrange must consequently all be zero. This is the ensemble profile-preservation argument of Lemma 2.6, with its routing hypothesis now verified. Thus the inherited prefix is F|_{[0,\kappa_1]}. The charts and retained packets now meet the hypotheses of the first calculation. Applying (4.29) and filling the selected blocks gives the asserted lower multiplicity m/\theta.

The plank parameter and temporal cost. For the clustering calculation, pull back a new full-time plank of spatial widths a,b by the inverse of (z_n,z_t)\mapsto(z_n,\theta z_t). Its cross-section is a parallelogram contained in a rotated rectangle of side lengths A\leq B with

For example, singular-value axes of the linear image of its inscribed ellipse give this rectangle with fixed comparison constants. Its smallest width is bounded below by a constant times a, and B\lesssim b/\theta\leq K_1; thus these are permitted old tests after bounded coverings or clipping. Since 0\leq x'\leq y', the pulled-back mass is at most

Each chart index is an original index counted only once there, so no extra weight factor enters this inequality. This proves (4.26).

Finally, the fine scale 1/\theta=N^\lambda lies in the flat prefix, since \lambda\leq\kappa. Filling the selected K-blocks changes neither the limiting profile nor its cost. Rebinning at 1/\theta deletes the initial flat prefix of length \lambda and translates the remaining logarithmic scale axis. After dividing by the new total logarithmic length \kappa_1-\lambda, this is precisely (4.27). Its cost in original units is \Pi(\kappa_1)-\Pi(\lambda)=\Pi(\kappa_1). When \kappa_1-\lambda=0 in the limit, the packet and clustering bounds still have their asserted exponents; an upper estimate can use total mass instead of a positive-resolution profile.□

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.

Fine-scale planarity and global slab capture. We first establish a fine-scale version of the multilinear discard. Choose a small constant c_*>0, depending only on the positive lower bound h/2 and the multilinear input. At a fine cell use direction caps of radius N^{-2c_*}, declaring a cap rich if its active mass is at least mN^{-10c_*}. Mark a cell bad if three rich caps have determinant at least N^{-c_*}. The same multilinear count as in (4.11), now at fine resolution, gives

for a fixed C_3. Local bound (4.2) at K=1 bounds active mass at a fine cell by mN^{o(1)}. By (4.9), the fraction of incidence in bad cells is at most

Choose c_* and then \varepsilon so this is a negative fixed power. At every remaining cell all triples of directions in rich caps have determinant O(N^{-c_*}). The total mass in light caps is at most CmN^{-6c_*}, since there are O(N^{4c_*}) caps. These choices are independent of any later flat scale.

Choose a fixed e_1>0 sufficiently small compared with c_*h and h, and require \alpha to be smaller still, as well as small enough for Lemma 3.3 with target loss e_1/2. Greedily assign and remove all original full-time indices in any global plank S of widths a,b with density

If an extensive remainder survives, its profile remains F, so its shading density is N^{-\alpha+o(1)}. All its plank densities are at most mN^{-e_1}. The weighted set estimate then gives multiplicity at most mN^{-e_1/2}, contrary to extensivity and the inherited support bound. Thus the assigned graph is extensive. Each original index has exactly one global assigned slab, and all of its retained times keep that label.

Compare (4.30) with the original \Delta_{h_0} bound and the hybrid equality. Put A=\log_N a, B=\log_N b just for this calculation. Then

In particular

If (4.31) has no feasible shape, the capture already gives a contradiction; otherwise these width bounds hold on every retained slab.

The assigned slab S has at least n_Sk weighted incidences and support in O(abN) cells. Its average assigned multiplicity is therefore at least mN^{-e_1-\alpha-o(1)}. In each slab delete cells with assigned multiplicity smaller by an inverse subpower factor. Their incidence is bounded by that factor times the slab incidence, by the support count and (4.30). Summing over slabs and cleaning the corresponding partitions gives the following lower bound at every retained query incidence:

This is a lower bound for the incidence from its own assigned slab. No independence from any later assignment is asserted.

Alignment with saturated local plates. The global slab assigned to an index has a fixed axis. We next compare it with the axes of the local saturated plates used by that index. By Lemma 4.6, this particular hybrid equality has a positive flat delay. Now choose a fixed

smaller than that delay, and set K=N^\xi. Lemma 4.7, applied within the extensive S-assigned graph, supplies saturated local plates W of widths 1,K up to subpowers and mass at least mK in exponent. Each such packet has mK^2 incidence in exponent and at most K^2N^{o(1)} support cells. The analogous low-cell deletion therefore gives, at its retained query incidences,

Retain (4.32) as another partition test. Both lower bounds thus hold on the same extensive central graph. Fine multilinear bad cells can be discarded from it by their power-small original incidence bound, followed by the same subpower deficiency cleaning.

At such a query cell, remove directions in the global light caps when choosing comparison directions. Their mass CmN^{-6c_*} is negligible compared with both (4.32) and (4.33), by the choices of e_1 and \alpha. Among the remaining directions in S there are two separated by at least N^{-c_*/10}. Otherwise one cap of this radius would contain all their mass, whereas (4.10) with L=N^{c_*/10} bounds that cap mass by mN^{-Hc_*/10+o(1)}. Choose e_1+\alpha<Hc_*/20 to contradict (4.32). Likewise there are two remaining directions in W separated by at least K^{-1/2}: the cap bound with L=K^{1/2} is mK^{-H/2}N^{o(1)}, which is smaller than (4.33) for this fixed \xi>0.

We justify how these two pairs compare the actual plate axes. A plank with spatial thin axis n_0, thickness a_0, and duration K_0 confines n_0\cdot u to an interval of length O(a_0/K_0). Its direction plane has normal proportional to (-v_0,n_0), where v_0 is the central normal slope. Since the original slopes are bounded, this normal has spatial component bounded below after normalization. Thus all directions in S are within N^{-1+O(e_1/h_0)} of its actual direction plane; all directions in W are within K^{-1+o(1)} of its direction plane.

Let v_1,v_2 be the separated rich pair from S. Every rich direction is within

of their span, by the absence of a bad triple. The span of v_1,v_2 is within N^{-1+O(e_1/h_0)+c_*/10} of the actual S plane. If w_1,w_2 is the separated rich pair from W, their span is within N^{-9c_*/10+\xi/2+o(1)} of the span of v_1,v_2, and within K^{-1/2+o(1)} of the actual W plane. Consequently the two actual direction planes differ by at most

With the fixed choices above and \xi\ll c_*, this is smaller than \theta=N^{-\xi/4}. Taking the normalized spatial components of the plane normals is a uniformly Lipschitz map here. Hence the thin spatial axes of W and of the index’s assigned global slab S differ, up to sign, by at most C\theta.

The hybrid contradiction. For each original index its S axis is fixed over all time. Thus the hypotheses of Lemma 4.8 hold globally, with K_1=N and the finer flat scale K=N^\xi. All previous discards were extensive or removed a power-small incidence fraction. The fixed losses e_1 and \alpha were used only to obtain the query lower bounds and identify the axes; they are not losses in the retained incidence or packet mass. After subpower cleaning and routing, the lower raw multiplicity supplied by that lemma is therefore m/\theta, without an extra factor N^{-e_1-\alpha}.

We now use the strict gap between the hybrid exponents. Take (x',y')=(d-1-h_0,1+h_0) in the anisotropic clustering bound; these satisfy 0\leq x'\leq y'. The new resolution is N\theta, and the temporal cost in original units is still \Pi(1). The sharp hybrid upper estimate becomes

Its ratio to the lower bound m/\theta is \theta^{H-h_0}, a fixed negative-power saving in N because H-h_0>0 and \xi>0 are fixed for this hybrid equality. This is a contradiction. The possibly small value of \xi(H-h_0) causes no problem: that hybrid and its flat scale are fixed before its realizing resolution tends to infinity.

Passage to an exact equality. The choices of c_*,e_1 and the upper bound on \alpha depended only on h,p,q and the external theorem tolerances, not on h-h_0. This proves a uniform positive lower bound for \alpha as h_0\uparrow h. Finally, for every configuration,

because 1\leq b/a\leq N in each plank test. Thus the hybrid equality exponents tend to the exact critical one. Apply the profile compactness and equality diagonal from Lemma 2.11. A downward jump of the convex cost would improve the critical score and contradict the sharp upper bound, so it cannot spoil equality. Uniform convergence of the profiles preserves their positive endpoint lower bound. The resulting exact equality has F(1)>0, as claimed.□

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

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 2.6 to preserve the packet lower counts. Comparing (5.3) with (5.1) yields

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,

The averaging in \mathcal D makes its q derivative available at the interior parameter in use. Differentiating (5.4) proves (5.2). 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 5.1 gives \chi_0=0. Thus a=1, b=N^\tau, and the packets are plates of mass \eqexp 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

Indeed spatial cell widths multiply. Physical time is unchanged, but in the new cell units

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 (5.6). The first selection in Lemma 5.1 therefore gives

Iteration gives (5.7). 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 (5.5). The greedy argument of Lemma 5.1 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 4.1.

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 2.6. 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 5.2. 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

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

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.

The inner equality. The reverse selection in Lemma 5.1 has \chi\ge F(s)/v, while every admissible shape has \chi\le s. Therefore s-F(s)/v\ge0. Its value at s=\tau is \tau, proving the bounds on D_*. Apply Lemma 4.8 over the N^\kappa-block with angular scale \theta=N^{-D_*}. Its fine assigned plates are the flat-prefix plates of length N^\tau, and D_*\le\tau ensures \theta\ge N^{-\tau}. The effective resolution is N^{\kappa-D_*} and the raw lower multiplicity is mN^{D_*-o(1)}. By the local upper bound its full-time plank parameter is at most

Only an initial flat log interval of length D_* is dropped. Multiplying this parameter by the critical factor N^{h(\kappa-D_*)+\Pi(\kappa)} gives mN^{(y-h)D_*}=mN^{D_*}, exactly the lower exponent. The normalized profile thus gives (5.9). As for outer parents, one can extract a representative equality from the aggregate chart ensemble whenever this statement is used as a fact about equality profiles.□

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 5.3, applied at a prefix with positive deficit, would produce a nontrivial equality with zero initial delay. This contradicts Lemma 4.6. There is therefore a new minimum D<\tau, attained at a prefix \kappa with g(\kappa)=D. At this prefix the inner deficit is

Let \mathcal E_v denote the nonempty class of equality profiles with deficit v, and set

A positive lower bound for the delay. We claim that \tau_*>0. Suppose otherwise, and choose F_j\in\mathcal E_v with initial delays \tau_j\to0. Equality compactness, as in Lemma 2.11, gives a subsequence converging uniformly to an equality F_\infty with deficit v>0. The cost is lower semicontinuous; a strict downward jump would give a violation of the critical inequality, so this limiting configuration is still an equality. By Lemma 4.6, F_\infty has a flat interval of positive length.

The small-prefix multilinear argument underlying Lemma 4.5 has slack depending only on the fixed critical parameters. In particular, its allowed prefix length can be chosen independently of the equality profile: the estimate used there is

Indeed n\leexp\Delta_hN^d, m\eqexp\Delta_hN^{h+\Pi(1)}, and \Pi(1)\ge p\alpha-q. Choose fixed 0<a_0<b_0 inside both that allowed prefix range and the flat interval of F_\infty.

For a sufficiently late j,

Choose a minimizer \kappa_j of F_j(u)-(h/4)u on [0,b_0]. At b_0 this function is strictly less than -ha_0/4, whereas for u\le a_0 it is at least -ha_0/4. Hence \kappa_j>a_0>\tau_j, so \delta_j=F_j(\kappa_j)>0. The minimizing property gives

Fix this particular j, its prefix \kappa_j, and its positive \delta_j before passing to the realizing resolution sequence.

Apply Lemma 4.5 to this fixed equality and prefix, with s=1-h/4 and \gamma=0. Its uniform small-prefix range includes \kappa_j, by the choice of b_0. The record inequality (5.13) is precisely its temporal hypothesis; the other hypothesis is simply F_j\ge0. Its angular exponent is h, so

The lemma gives \Pi(\kappa_j)\le F_j(\kappa_j)=\delta_j. But \Pi(\kappa_j)\ge(p-1/2)\delta_j>\delta_j, since \delta_j>0 and p>2. This is a contradiction. The positive gap is fixed after choosing j and before taking its realizing resolution to infinity; no uniform lower bound for the numbers \delta_j is needed. We have proved \tau_*>0.

Rigidity from the inner and outer delays. We now determine a profile attaining this infimum. For any F\in\mathcal E_v with initial delay \tau, g(1)=0. Given D'\in(0,\tau), let s be the first passage below D' after \tau:

Continuity gives \tau<s<1, g(s)=D', and \min_{\tau\le r\le s}g(r)=D'. There are values below D' arbitrarily soon to the right of s. The inner equality from Lemma 5.3 has deficit v, so its delay is at least \tau_*:

Any saturated outer cut at s has \chi\le s<1 and deficit (v-F(s))/(1-\chi). By Lemma 5.1 this is at most v, whence \chi\le F(s)/v. The reverse selection of that lemma therefore attains \chi=F(s)/v=s-D', and its outer deficit is exactly v. Moreover F increases arbitrarily soon to the right of s: g(r)<g(s) implies F(r)-F(s)>v(r-s). Formula (5.6) consequently gives the exact outer delay D'/(1-s+D'), not merely a lower bound for it. Its membership in \mathcal E_v yields

The two inequalities compare complementary rescalings of the same profile, both with deficit v. For a sequence approaching the least delay, they will force the same first-passage location.

Choose F_j\in\mathcal E_v with \tau_j\to\tau_*. The Lipschitz bound gives v\le1-\tau_j, so \tau_*\le1-v<1. For each fixed D'\in(0,\tau_*), the two passage inequalities give

Thus

Pass to a compact equality limit F. The rational values of D' in (0,\tau_*) suffice; continuity then determines the profile at every point above \tau_*. Below \tau_* all these profiles vanish. We obtain

Its delay is exactly \tau_* and its slope is at most one by the Lipschitz bound. This proves (5.11).□

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 (5.6) then shows that the outer profile is the same canonical profile: its flat delay is

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 (5.2), written in those units, are

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,

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 5.2 preserves the required earlier assignments.□

Proof. Choose the finite chains from Lemma 5.5 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

The proof of Lemma 5.3, 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 (5.20).

We now verify the width bounds in the event frames. The widths in (5.19) 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,

Substituting (5.17) in (5.1), the exponent of the packet mass relative to \Delta_h is

using \ell+\tau=1 and y=1+h. This proves (5.21).

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 5.2 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 2.6: 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

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 (6.1). 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 (6.1) 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 (6.2).□

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

First pass to the isotropic outer parents associated to B_b, as in Lemma 5.2. 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 4.1 pulls back to

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

in exponent, using m\eqexp\Delta_hN^{B_*} and (5.21). Thus (6.5) counts original active indices; it is not a bound only for indices occupying a selected angular bin.

Suppose that fibers violating (6.3) 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

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 6.1: 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

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 4.8 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 (6.6). 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 6.1 and are used explicitly in (6.9) below. The resulting normalization has resolution

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 (6.6). The full-plank parameter of the new models is at most

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 (6.2), its cost, expressed in \log N units, is at most

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 6.1.

Without the controlled losses, the sharp upper bound from Definition 2.3 and (6.8) matches SN^{\tau b} exactly. Indeed its exponent relative to S is

Thus (6.6) and (6.9) 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

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

relative to the old sharp upper estimate. Choose once and for all

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 (6.3) for each fixed gap. The choice (6.11) 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 (6.3) by only a subpower factor. An extensive subsequent regular refinement has the same property by Lemma 6.1.

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 (5.20), 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 (6.3) 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

The conditions (6.13) imply c<b<1 and \Delta<\tau g, so a positive flat range remains in the time filling used below.

Chart geometry. Fix e_c and its interval I_c. In its frozen frame normalize time by R_c, spatial normal and transverse position by R_c\Theta_c,R_c, and velocity by the corresponding spatial units divided by R_c. In these coordinates a Q(c,a) cell has isotropic matrix diameter, up to dimension constants,

Take isotropic matrix parents of diameter N^{-\rho(1-b)}, subtract their centers, and divide the spatial and velocity coordinates by that diameter. The resulting charts are bounded and have resolution D_0^{-1}. In each such parent, the diameter of a cell of Q(c,a) is

We used wg=a+\Delta in the equality. One may choose compatible nested grids; if a prescribed cell meets a parent boundary, splitting it among boundedly many parents adds only O(1) entropy at the end. Hereafter C denotes the resulting cell within its registered chart, including this subdivision when needed.

A B_b packet fits in subpower many of these charts. Its used normals lie in \Theta_bN^{o(1)}, hence in subpower many e_c bins, and its four width bounds extrapolated from I_b to I_c have matrix diameter at most N^{-\rho(1-b)+o(1)} in the I_c normalization. Use the packet’s own normal and its own time center for the following four widths after parent normalization. Their negative log exponents are

In the normalized I_c chart the change of normal is (\vartheta_P-\vartheta_c)/\Theta_c=O(1), and the change of time center is (t_P-t_c)/R_c=O(1). These are bounded shears. They need not preserve the four coordinate widths in the frozen e_c frame; they preserve the following geometric bounds, up to fixed factors. At resolution D_0 the packet is a plate of thin spatial width one cell, long spatial width Z', and time length Z' bins, with raw assigned mass at least S'Z'N^{-o(1)}. Each index and I_c is in only subpower many registered charts. After chart splitting, Lemma 6.1 preserves these packet masses on typical occupied pieces.

The reference-mass claim. For each cell C in a registered chart define

For a packet P=B_b on an I_b block, write n(C\cap P) for the inherited mass of its assigned active indices that lie in C. We will prove that, outside a set of query events of probability tending to zero,

The claim also holds if the query events were first restricted to an extensive regular piece, with the same full original active mass in (6.17). This reference mass belongs to the chart cell C just defined.

The critical comparison uses the following inherited bound. Let \Delta_0 denote the full-time plank parameter in one of these bounded normalized charts, with exponents x,y. Its physical normal cell width in original cell units is N^{w(1-b)}; the transverse cell width is \Theta_c^{-1}N^{w(1-b)}. Pulling back an arbitrarily oriented rectangle multiplies its area by N^{2w(1-b)}\Theta_c^{-1} and its largest width by at most N^{w(1-b)}\Theta_c^{-1}. Since 0\le x\le y, the original local estimate gives

For the last equality, its exponent relative to \Delta_h is cB_*+\tau c+dw(1-b); the same exponent results from S'D_0^{-B_*} using (5.21). All these bounds count original indices active on I_c, whether or not those indices carry the particular normal label used to name the chart. An upper bound for a subcollection is therefore inherited without changing the reference weights.

Registering the failed queries. Suppose the reference-mass claim fails. After a subsequence, for some fixed \epsilon>0 the events satisfying

form an extensive piece. Register tuples (i,\mathfrak c,J), where \mathfrak c specifies the chart, including e_c,I_c and the parent, and J=I_b is occupied. Give each tuple weight \omega_i. The cell C is fixed by (i,\mathfrak c), and its packet owner is fixed by (i,\mathfrak c,J), by Proposition 5.6(iv). Consequently (6.20) is constant on the tuple. This constancy is the reason that replacing its finer events by a token and later filling J does not change the failed test.

Before selection each occupied J has a common number of original events in exponent. Register every tuple (i,\mathfrak c,J) hit by a bad query and complete its J block in that fixed chart, before chart-portion pruning. The chart component is retained throughout: the failed predicate depends on it. Because the predicate is constant on this tuple, completion preserves it. The bad tuples consequently have extensive inherited weight in the original token graph. Any chart cutoff is taken relative to the restricted completed token graph; there are only subpower many chart portions per index and I_c. Clean the packet partitions, using the original token masses for packet references and the graph where labels were computed for new regularization references. The registered assigned mass of each surviving packet in a chart is then at least S'Z'N^{-o(1)}. The assigned index sets are disjoint for a fixed chart and J.

Frequency selection. We next make the selected child mass comparable to the reference mass by a common factor. This will allow a multiplicity estimate sampled under the representative law to be used for the child events. On the retained tuple graph define

This is at most n_C. Select one common exponent of \mathsf w(C,J)/n_C across the ensemble. Here is the finite justification even when the original weights are arbitrarily unequal. There are polynomially many charts, cells, and time blocks, and each n_C\le n. If their total number is at most N^A, ratios below N^{-A-3} contribute at most N^{-3}n to the tuple weight and may be discarded. The extensive original token graph has weight at least nN^{-o(1)}: every original index had a common total event count, and the restriction cannot create more events per index. Partition the remaining logarithmic ratios in [0,A+3] on a mesh tending slowly to zero. There are subpower many levels, so one retains an extensive level. After further cleaning the (C,J) cells, using their masses at this level as references, there is an exponent r\ge0 such that

uniformly on all surviving pairs. It is the selected \mathsf w(C,J), not n_C, that is used as the reference for this deficiency deletion. Therefore the sum of reference masses in this cleaning partition is the tuple weight, not a possibly much larger sum of n_C’s. Include the packet cells and all prescribed time-tree tests in the same deletion process. The packet lower masses, the ratio (6.21), and the inherited uniform child branching then hold simultaneously.

Representatives and their event law. Fill every registered tuple over its whole interval J and rebin time at the coarser resolution

Each J contains Z'/D such bins, up to the harmless dyadic rounding, and this number has positive exponent. The operation drops only part of the full-branching range grafted below J. For each occupied C put one representative matrix at its center, of weight n_C. Occupy that representative at every coarse time bin belonging to a registered pair (C,J). Charts are counted separately. A child tuple at a coarse bin is assigned to the spatial cell hit by its representative at that bin center; this is the hashing used in the argument.

On each representative event the total weight of its children is \mathsf w(C,J). By (6.21), the ratio of that weight to the representative weight is N^{-r+o(1)}, uniformly. The same common factor compares the total masses. Thus the two normalized event laws are mutually dominated up to subpower factors. In particular, an extensive set of representative events pulls back to an extensive set of interpolated child events. This statement does not identify the individual index weights n_C with the child weights.

The representative multiplicity bound. We apply the critical inequality to this representative configuration. Its plank parameter is inherited from the reference indices; its temporal cost will be bounded using the selected children. The full-plank parameter for this representative model is at most

Indeed, by (6.15), every original line counted in n_C stays within a bounded coarse cell of the representative throughout the whole normalized time interval. A coarse plank containing the representative therefore contains all these lines after bounded enlargement. In the old D_0 cell units its widths are at most constant multiples of Da_1,Db_1. The cells C partition the original active indices within a chart, so summing their weights does not duplicate any index in that chart. Apply (6.19) and use x+y=d to obtain (6.22).

We claim that the sum of the occupied representative weights in a typical hashed space–time cell is at most

If cells with a strict excess carried extensive representative incidence weight, restrict to those cells. Their aggregate support is at most their weighted incidence count divided by the proposed excess threshold. Make the representative time profiles uniform over the ensemble, retaining an extensive piece.

It remains to bound the cost of this profile. Up to the creation of representatives, all selections (the bad predicate, chart routing, frequency-level selection, and their regularizations) are extensive restrictions of inherited-weight child tokens. The reference mass n_C used to define the frequency ratio never becomes a child index weight. Filling each registered J adds the same Z'/D coarse events per token. It follows from Lemma 6.1, on every tested gap, that the interpolated children have profile

The bottom interval is filled and has zero cost, and the remaining interval has length \ell g.

Keep the extensively restricted and regularized representative graph fixed from now on. Pull its event restriction back to its children. Equation (6.21) compares normalized representative and child event weights by subpower factors on every event, so this pullback is an extensive restriction of the child graph. Uniformize these children, again extensively, and retain their profile by the same lemma. This last pruning removes only child events; it does not alter the already uniform representative graph. Every surviving child block is therefore still occupied in that fixed representative graph. Descendant inclusion can be compared within that very block on each tested gap, regardless of whether different children of the same representative occupied disjoint branches before pruning.

Each surviving child index stays in one cell C in its chart. Its surviving times form a subtree of the surviving times of that representative. On any fixed log gap, choose an occupied block of this child. Uniformity says that the representative’s number of occupied descendants of that block has its common representative branching exponent; it is at least the child’s number. Therefore, if G_{\mathrm{rep}} is the representative deficit profile in these same log units,

First make this comparison on the finite tested gaps and then use the Lipschitz profile limits. It implies G_{\mathrm{rep}}'\le F_{\mathrm{ch}}' almost everywhere. Since P is increasing, the representative cost is at most

This comparison uses the children on every gap, not just their total number of events.

Apply the sharp critical bound at resolution D_0/D using (6.22) and (6.25). The resulting upper bound for raw average multiplicity is

because B_*=h+\ell P(\beta). This contradicts the strict excess on the restricted support. Hence (6.23) holds typically. At each fixed excess margin its exceptional fraction is power-small, by Convention 2.8. The subpower comparison of the two event laws therefore makes it negligible for the child events as well.

The packet comparison. Fix a surviving packet and one of its filled coarse time bins. Its mass is at least S'Z'N^{-o(1)}. Its child positions lie in a rectangle of thin width one and long width Z' in D_0 cell units. Replacing a child by its representative moves it by O(D) such units, by (6.15). Hence at this time the packet hashes into at most

spatial cells. In its typical hashed cell its assigned child mass is therefore at least S'DN^{-o(1)}. To make this last typicality explicit, cells of packet mass below S'DN^{-\epsilon/4} have total weight at most N^{o(1)}(Z'/D)S'DN^{-\epsilon/4} at this time, which is a vanishing fraction of its packet mass. Sum this estimate over packets and filled bins; assignments are disjoint on each block.

Every child remaining in the test satisfies (6.20). Partition the mass of one packet in a hashed cell according to C. Its contribution from C is at most the original n(C\cap P), hence is at most n_CD^{-x}N^{-\epsilon}. All eligible C representatives are occupied in that hashed cell. Outside the representative exceptional set their total n_C is at most S'D^{d-h}N^{\epsilon/4}. The packet mass in that cell is therefore at most

using d-h-x=1. This contradicts the preceding lower bound on a set of child events of probability tending to one. It proves (6.18). Choosing the strict margin through a slow diagonal gives its o(1) form and a vanishing exceptional event fraction. All selections used only extensive restrictions and the original upper bounds, so the same proof applies after any extensive regular refinement of the query law.

From reference mass to normal position. The reference-mass claim is now proved. We use full-time packet containment to place those reference indices near the query in the finer label’s normal-position bins, and then apply cross-entropy. Return to original coordinates. Let M be the queried matrix and \widetilde M any reference index in C\cap P counted in (6.18). Write C_j=L_j(c)N^{-a} for the four cell widths. Full-time containment in the packet over I_b gives, at the actual query time t and in the packet normal \vartheta_P,

This uses containment of each assigned line throughout I_b, not an assumption that each of its shadings meets the query time.

Inside C, replacing its frozen frame by an actual e\in e_c changes difference bounds only by dimension constants: the identity

has |\delta t|\lesssim R_c, |\delta\vartheta|\lesssim\Theta_c, and C_U=C_X/R_c, C_Y=C_X/\Theta_c, C_V=C_X/(R_c\Theta_c). Changing \vartheta_P to the query normal adds at most \Theta_bN^{o(1)}C_Y, whose ratio to C_X is N^{-\tau g+o(1)}. Changing the query to the center of L=e_{c+\sigma} adds, relative to C_X, at most

The three terms are the normal, time, and mixed terms of (6.26). Our choices make \tau g,\tau\sigma,\ell\sigma strictly greater than \Delta. Consequently all the reference indices counted in (6.18) lie within a subpower number of additional-precision X_L bins of the queried bin.

Reference cross-entropy. For each fixed base (C,L) let p_{C,L} be the histogram obtained by evaluating X_L on all indices in (6.17), with their weights divided by n_C. Let q_{C,L} be the actual conditional distribution of the query bin under the event law. These distributions need not agree. The preceding geometry and (6.18) say that, outside a vanishing event fraction, the p_{C,L} mass in a subpower enlargement of the queried bin is at least D^{-x-o(1)}.

Choose an integer r_N=N^{o(1)} large enough for that enlargement on the good events. Replace p_{C,L} by its convolution with the uniform distribution on \{-r_N,\ldots,r_N\}. At a good query bin the new probability is at least D^{-x}N^{-o(1)}. The image of a full C under X_L has length O(C_X) by (6.26); hence there are only N^{\Delta+o(1)} possible bins, including the enlargement. Take the equal-weight mixture of the convolved histogram and the uniform law on this finite support, and call it \widetilde p_{C,L}. It is a positive reference probability on every possible query bin. On good events

and on every event it is O(\log N). The vanishing exceptional fraction thus has vanishing normalized contribution.

Nonnegativity of relative entropy, on each conditional base, gives

The auxiliary conditioning can now be removed with a conditional count, rather than the global number of charts. Given Q(c,a) and L, the coarser label e_c and its interval I_c are determined. In this fixed normalization, the Q(c,a) cell has diameter N^{-g+\Delta} times its isotropic parent diameter, so it meets only boundedly many parent cells. The possible bounded grid and boundary choices have the same property; any further subpower routing multiplicity contributes only o(\log N). Hence their conditional entropy given Q(c,a),L is o(\log N), even if the global collection of parents is polynomial. No I_b or packet-owner label is conditioned upon in the reference cross-entropy: those labels were only used to prove the reference-mass inequality, and the histogram itself uses only C,L. The chain rule therefore gives (6.12) with exactly its stated conditioning. No independence of the matrix and the finer event label has been used.□

Proof. For a finite partition \mathcal P of [s,t], sum the values of K on its intervals, and set

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

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

This proves (7.7), first for positive integral C_0 and therefore for every fixed C_0.□

Proof. Use first the fixed e_c frame and define

Nested grids and the chain rule show that F_\sigma is nondecreasing, and that its increment on [a_1,a_2] is exactly

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

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 (7.3) 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 7.1 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 (7.11). This proves (7.10). 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 6.3 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 (7.3). 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 7.2, with \sigma\downarrow0, gives a_1=\phi_{\{X\}}\le x. The positive-offset range is the range required by Lemma 6.3.□

Proof. For b>0, label nesting gives the exact normalized limiting identity

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 (5.20) 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 7.2. 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

Let b\downarrow0 with \sigma fixed, and then let \sigma\downarrow0. Combining this expression with the label contribution gives (7.17).□

Proof. At fixed \lambda, the last conditional rate, before removing an extra label, is the derivative at r=0+ of

This follows by subtracting the rectangular entropy for the first three axes from that for all four, using (7.9). 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

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

the four uncertainty terms, divided by the earlier X-cell width N^{-\lambda-c_1-r}, are bounded respectively by constants times

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 (7.19). First differentiate in r and then remove the extra label by Lemma 7.2. 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

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,

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 (7.6) give

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

The common L^\infty bound and L^1 approximation extend this to all L^1 tests, proving (7.22). It also proves the corresponding statement for stripe averages. Finally Proposition 7.4 rearranges as

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 (8.3) satisfy

The last equality uses \tau+\ell=1. Substituting these ratios in the exact change of the four linear forms gives (8.5). 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.

Rectangular entropy increments. Fix \sigma>0 and, temporarily, take b so small that C_0b<\sigma and c+\sigma<1. Condition additionally on e_{c+\sigma} and read all forms in that finer frame. Given L, Lemma 8.2 shows that the required observations in the two frames determine one another up to bounded overlaps. The full starting cell is included in this comparison; errors from unobserved components are below the finest tested width.

Apply the rectangular layering conclusion of Lemma 7.1. The starting offsets of the four increment intervals are

Intervals belonging to different speed groups are disjoint, because \zeta_{\max}<|k_i-k_j|. During the interval for a group of speed k, every group of smaller speed has already been required to a depth beyond that interval, even in its base observation. Every group of larger speed ends below the interval. If S is the union of the groups of smaller speed and G is the current group, subtraction of the base rectangle from the refined rectangle leaves the density \phi_{S\cup G}^{\sigma}-\phi_S^{\sigma}=:j_G^{\sigma}. Thus the group’s contribution is its ordered conditional rate times its refinement depth. This reasoning applies both to the order X,Y,U,V, when \tau<\ell, and to X,U,Y,V, when \ell<\tau. It gives (8.7) with the \sigma-conditioned rates, to error o(b) in entropy divided by \log N.

For a tied group, first reveal Y and U together to the smaller of their two depths; then refine only the deeper coordinate. The common part costs m_1 times that depth. The remaining increment uses r_Y or r_U, according to the deeper coordinate. This gives (8.8), again first with \sigma-conditioned rates. Equivalently, one may subtract the layering formulas for a joint rectangle and its conditioning rectangle. All their thresholds lie within a\pm Cb for a fixed C.

We next remove the extra label. Given L, the joint and conditioning observations in question are sandwiched between full e_c-frame observations at offsets a-Cb and a+Cb, up to bounded overlaps. By Lemma 7.2, the change of a conditional increment upon adding e_{c+\sigma} is bounded by the incremental label information on this full-cell interval. For almost every a, its upper first-order density tends to zero as \sigma\downarrow0. Consequently the limsup of the error divided by b, as b\to0 with \sigma fixed, is bounded by a constant times that density. Now let \sigma\downarrow0 through a countable sequence. The rates converge to \phi_S and j_G, and the error is o(b). This proves the asserted formulas for the limiting normalized entropies without requiring a convergence speed uniform in a.

Uniform finite cell masses. Here is a simultaneous implementation on finite configurations. Fix a summable sequence of positive rational b_n\downarrow0. At stage n, prescribe a finite offset grid of spacing o(b_n) on the compact interval under consideration, and a finite grid of relative depths whose union is dense in [0,\zeta_{\max}]. Include the following in the countable family of tests: E_0,J_0,L; every subset of the axis observations at equal and unequal depths; their joint and conditioning cells; the full-cell sandwiches just used; and the same observations with the fixed additional labels e_{c+\sigma}. Include also the within-index parameter-cell tests described below.

Use Lemma 2.6 on finitely many tests at a time, including the deficient-cell deletions relative to their original mass. This makes the joint and conditional cell masses uniform in exponent. Previously uniform tests keep their limits: surviving cells retain their original masses within the prescribed subpower factor, and the total restriction has the same order of loss. Thus the already defined stable entropy function and its rates do not change. Take subsequential limits on this countable family first. Their values satisfy the estimates established above with fixed \sigma and then \sigma\downarrow0.

Finally choose N_n sufficiently large along the resulting configurations that the first n finite collections of tests have errors o(b_n) in exponents normalized by \log N_n. The finite pigeonholings and cleaning can have total loss N_n^{-o(b_n)}: for a fixed finite number of tests their number of mass levels grows at most polynomially in \log N_n and in the inverse requested accuracy, so increasing N_n makes their normalized logarithmic cost smaller than the chosen multiple of b_n. Choose the deletion cutoffs below that retention while still with exponent o(b_n). Also require b_n\log N_n\to\infty. For each generic a, use its nearest offset-grid point. The stable full-cell and finer-label comparisons change the exponents by o(b_n) at this replacement. The first-order density estimates hold at the actual generic point for every sequence b_n\to0; no uniform differentiability rate over all such points was used.

Uniform joint cell masses now prove (8.9), rather than only its entropy average. For example, in the tied case and u\le\zeta,

The probability of an occupied U_u cell within an occupied Y_\zeta cell is the ratio of their joint and marginal probabilities, hence D^{-(m_1-r_Y)u+o(1)}. For distinct speeds the same subtraction removes all nuisance terms and leaves u times the sum of the selected rates. For (X,V) with Y,U fixed, the tied-group contribution cancels and leaves (a_1+v_1)u. These statements hold at all the tested intermediate depths. Bounded square coverings and increasingly fine depth meshes give the asserted Frostman upper bounds at every intermediate radius.

Time and normal parameters. It remains to verify the parameter hypotheses for these finite experiments. Lemma 6.2 and the uniform time branching give the two upper bounds conditional on the index and E_0. Nondeficient E_0 cells have the same exponent as their time blocks, since each index uses only subpower many angular bins there. Add these conditional cells and their refinements to the tests before choosing the triangular sequence. When \tau=\ell, refining the label from c+b to c+b+ub/\ell multiplies both relative widths by D^{-u}. The time mass is D^{-s_0u+o(1)}. Within each of its time blocks, the normal range occupies only subpower many corresponding angular bins. Its nondeficient occupied joint cells therefore have that same mass exponent. Cleaning them gives the lower as well as the upper bound, uniformly through the required depths. All of this cleaning precedes the choice of a pair law; a later bias is never applied to an uncontrolled exceptional set of single-event cells.□

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

Temporarily write J_0(a) for the rebase grid at offset a and set

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

For a nondecreasing function with range of length H_*,

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 (8.16) by

for large n. Integrate on a compact interval whose slightly larger neighborhood stays admissible. Equations (8.15) and (8.16) give

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 8.3. Tonelli’s theorem applied to the nonnegative functions in (8.17) shows that their sum is finite almost everywhere. In particular the summands tend to zero, which proves (8.14). 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

The choices are possible by continuity, and are made before any pinning loss is chosen. Let \kappa>0 be an allowance from Lemma 8.6 for these three fixed exponents. In the pinned case choose the loss tolerance \epsilon so small that

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.

The product probability bound. Fix a good base B, and initially use the original conditional matrix law \mu(\,\cdot\mid B), independently of the pair experiment’s conditional law of L_1. Within this original matrix law, freeze the nuisance bins at depth \zeta. Lemma 8.3 gives the point Frostman bounds, with exponent t_*, in every occupied fiber. For sufficiently large D their constants are smaller than \delta^{-\kappa}. The slope bounds, with exponent s_*, have the same allowed constant: in the pinned case they cost C\delta^{-\epsilon}, and for the ordinary pair law a subpower.

The frame formula (8.5) writes the target coordinate as a two-coordinate scalar projection plus a translation depending on the frozen nuisance bins and the full label L_1. The error from a nuisance bin has size O(\delta), since all coefficients are bounded. For each L_1, enlarge all its allowable target bins by this error and subtract the known translation. Their number is at most C\delta^{-u}, because u>a_*. Their lengths are at most C\delta. The label may contain more information than the projection slope; Lemma 8.6 permits precisely this dependence of the target intervals on the entire label.

Apply that lemma in each original nuisance fiber, with fixed bounded coordinate charts. Its independent product hit probability is less than \delta^{2\kappa}, uniformly in the fiber. Integrating over the original nuisance distribution gives the same bound for the entire original conditional matrix law. Only now replace that whole matrix law by its biased marginal. Equation (8.19) bounds the new product hit probability by

for large D. For the ordinary pair law the domination factor is one. This argument does not assume that biased nuisance fibers have the original Frostman estimates, or that a slope remains independent after conditioning inside the actual coupling.

The support event for the same-index law. Under the true coupling, the target matrix coordinate always belongs to the allowed support for (E_0,J_0,L_1). Both pair marginals are supported on the original single-event graph and both events have exactly the same index matrix. Thus this support test has probability one in the true coupling. It requires no probability lower bound for target cells after the change of measure.

To make it a finite information test, replace the matrix by a representative of its J_f cell. The relative coordinate error is O(D^{-(1+\zeta_{\max})}), which is smaller than D^{-\zeta} by a fixed positive power. The bounded frame coefficients and the precision of L_0,L_1 leave an O(\delta) error. Enlarge the target bins once more. The resulting event A_B depends only on (J_f,L_1) at fixed B, has conditional probability one in the true coupling, and still has probability at most \delta^\kappa under the product of the conditional marginals of J_f and L_1. For the last assertion, sample the exact matrix from its conditional marginal and then round it. A rounded hit implies an exact hit in one further bounded enlargement, already allowed in the preceding application of Lemma 8.6.

The information contradiction. Let P_B be the conditional joint law of (J_f,L_1) and Q_B the product of its marginals. Data processing of relative entropy by the indicator of A_B gives

The good bases have pair probability tending to one by (8.19). Averaging the last inequality therefore gives

On the other hand the chain rule and nonnegativity imply

by Lemma 8.5. Since \kappa\zeta>0 is fixed, this contradicts (8.27) and proves the lemma.□

Proof of Proposition 8.1. 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 (8.24)–(8.25). 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 8.3, in either of the two possible speed orders.

For example the first formula of (8.5), with U,V frozen, is

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 8.7.

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

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

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, (8.29) 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 (8.1).

Suppose now that \tau=\ell. Put

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

The target exponent is a_1 and the slope exponent is s_0. Equation (8.29), with b=m_1-r_Y, yields

Since r_Y\le1, this proves the bound for m_1 in (8.2).

Finally freeze Y,U and use the pinned pair distribution. The first formula in (8.5) becomes

The source pair is (X,V), of conditional exponent a_1+v_1, by Lemma 8.3. Its target exponent is a_1. The product slope has available exponent s_0 by Lemma 8.4 and the preceding coupling construction. Lemma 8.7 includes both its marginal bias and its small direction loss. Thus (8.29) 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 (7.2). Indeed the smallest and largest of the four unshifted thresholds are respectively

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 [17, Theorem 4.1]. We may take

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

Then (v,w)\in(0,1)^2, a_\lambda\in(-1,1), and

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 (8.20) 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

Suppose that the left side of (8.21) 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

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 (9.2). The remaining weights have at most L dyadic levels. One level, consisting of n slope intervals, satisfies

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

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

Since 0\le d(p)/n\le1 and \mu(\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

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

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

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

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 (9.3) and (9.4),

Thus the base cells and selected incident fibers form exactly the quantitative configuration of Ren and Wang [17, 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 (9.6) and (9.8) 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

Combining this with (9.7) yields

But (9.1) implies \Delta-\varepsilon-2\kappa\ge5\Delta/8>0, so the right side tends to infinity. This proves (8.21). Finally, (8.22) 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

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

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.

A local projection estimate. Consider one step d_1\to d_2 of the ladder, and put h_1=d_2-d_1. Choose \lambda small enough that

Let J be an occupied square of side \delta^{d_1} and let \mu_J=\mu|_J/\mu(J), rescaled to a bounded unit square. For 0\le u\le h_1, the ratio of child and parent masses is \delta^{su+o(1)}. A bounded square covering therefore makes \mu_J an s-Frostman probability through resolution

This uses the lower bound on \mu(J) as well as the upper bound on its children.

For a fixed \upsilon\in J, consider the anchors \xi paired with \upsilon by the old relation and satisfying the separation condition. Symmetry and (9.9) give an absolute \sigma-Frostman bound on their reverse rays. Replacing \upsilon by the center of J changes the direction by O(\delta^{d_1-\lambda}). By (9.12), this is smaller than \delta_J. The same absolute bound thus holds for the directions perpendicular to the rays from these anchors to the center of J.

Apply Lemma 8.6 at resolution \delta_J, with point exponent s, direction exponent \sigma, and target exponent \sigma_m. Fix its positive power allowance \kappa. A reverse section of mass less than \delta_J^{\kappa/4} may be discarded at total pair cost at most \delta_J^{\kappa/4}. For larger sections, normalization increases the Frostman constant by at most \delta_J^{-\kappa/4}. All inherited subpower constants are smaller than \delta_J^{-\kappa/4} for sufficiently small \delta, since h_1 is fixed. Decreasing \kappa if necessary leaves the loss required by the projection lemma.

For each direction, call a projection bin of width \delta_J heavy if its \mu_J-mass exceeds \delta_J^{\sigma_m}. The heavy bins number at most O(\delta_J^{-\sigma_m}). The robust projection lemma implies that the directions for which their union has mass at least \delta_J^\kappa occupy at most a fixed positive power of \delta_J in every normalized reverse section. One can use the product bound of that lemma and Markov’s inequality; shrinking \kappa accommodates constants. Call these directions exceptional for J.

There are two different removals here. First remove pairs whose anchor direction is exceptional for the cell containing the other point. For each fixed \upsilon, its reverse-anchor mass is power-small; integration against \mu(d\upsilon) gives a power-small total loss. Second, for each nonexceptional (\xi,J), remove the points in J lying in heavy projection bins. Their unconditional \mu-mass is at most \delta_J^\kappa\mu(J). Integrating over \xi and summing over the disjoint cells J again gives a power-small loss. Exceptionality is a test of J and the anchor direction, independent of the point within J. Neither removal assumes independence on the old relation G.

For the ray map, its derivative on J is a perpendicular scalar projection multiplied by the inverse distance to the anchor. That multiplier is bounded below by a positive constant, since the ambient square is bounded. Its quadratic error is O(\delta^{2d_1-2\lambda}). Consequently (9.12) implies the following local bound on the surviving section:

Finite angular charts and overlapping projection grids only change C.

For the product map, the derivative is

Its direction is the swapped ray, whose direction law has the same Frostman exponent. Thus the product test still uses the old ray bound as its direction input; it does not require a previous product bound. Its size is at least \delta^\lambda. An output interval of width \delta^{d_2} consequently uses at most \delta^{-O(\lambda)} projection bins at resolution \delta_J. The quadratic error is O(\delta^{2d_1}). Thus the product version of (9.13) has right side C\delta^{\sigma_mh_1-C\lambda}\mu(J). In what follows this slightly weaker bound works for both maps.

Pruning and iteration. Perform these removals for every step in the finite union of tested ladders, also with the legs interchanged. After all these removals, perform one joint pruning process over every tested level and both legs: if, for an anchor and a tested cell J, the remaining nonempty section has mass less than \delta^\lambda\mu(J), delete it. Continue with either leg until no deficient section remains at any tested level. Each anchor–cell combination is charged at most once, because it is empty after its deletion. Integrating over anchors and summing over J bounds the charge by \delta^\lambda per tested level and per leg, even if the deletions cause a cascade. There are finitely many levels, so the total loss tends to zero. All operations are finite on the finitely supported measures. Any subrelation satisfying every section threshold survives each deletion, so the terminal relation is the unique largest subrelation with those thresholds. Interchanging the legs preserves both the initial relation and the tests; hence the terminal relation is symmetric.

For the final relation, let Q(d) be the largest absolute section mass in an output ball of radius \delta^d. The retained mass in a nonempty anchor section of J is now at least \delta^\lambda\mu(J). The local estimate divided by this lower bound is a factor

times that section mass. If the section has an output in a chosen fine ball, all its outputs lie in an enlarged parent ball of radius O(\delta^{d_1-\lambda}). That ball is coverable by \delta^{-O(\lambda)} balls of radius \delta^{d_1}. The cells are disjoint, so summing their section masses gives the recurrence

Restrictions made at other levels preserve every upper bound.

Starting with Q(d_0)\le1, the ladder gives

First choose d_0 so that \sigma_m d_0<(\sigma_m-\sigma_*)T/4. After fixing the ladder, choose \lambda so that C_2n\lambda<(\sigma_m-\sigma_*)T/4 and all the strict inequalities (9.12) hold. The positive remaining margin absorbs C^n and the subpower errors. This proves Q(T)\le\delta^{\sigma_*T} for small \delta.

The same construction works for finitely many target depths at once: take the union of their ladders and choose all loss parameters below the minimum of their finitely many positive allowances. To obtain all scales, take successively finer finite rational meshes of target depths. For each fixed mesh first make \delta small enough that its finite collection of projection and deletion bounds holds and its total deleted mass is as small as prescribed. Then let the mesh refine sufficiently slowly. Between tested depths monotonicity costs at most \delta^{-\sigma_*e}, where e is the mesh spacing; at depths below the first mesh point the trivial mass bound has the same cost. Since e\to0, this gives the claimed subpower constant at all radii. This order never compares an uncontrolled error with a shrinking ladder step. All bounds used above are uniform for the stipulated ensembles.□

Proof of Lemma 8.4. 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.

A line cluster. Suppose first that, for some fixed rational \zeta\in(0,\zeta_{\max}], a set of initial indices of probability bounded below has a line neighborhood of width \delta=D^{-\zeta} carrying mass at least \delta^{\eta_0}. Pass to that subsequence, restrict the initial index there, and choose one such cluster C_i on each index. The normalization of the initial-index law costs a fixed factor.

Let (v_t,v_\vartheta) be a unit vector parallel to its line. The time and normal-coordinate ranges of the cluster have lengths at most C(|v_t|+\delta) and C(|v_\vartheta|+\delta). The coordinate bounds and its mass lower bound imply

for small \delta, with C depending only on the fixed exponents. Thus its slope k_i=v_\vartheta/v_t satisfies

Here the subpower constants have been absorbed by slightly enlarging C. Choose K with Ks>3, and require K\eta_0<1 in the standing choice of \eta_0. Sample two points from the normalized cluster, restricting to

For every anchor the rejected conditional mass is at most \delta^{(Ks-1)\eta_0-o(1)}=o(1). The pair restriction can be normalized symmetrically. Both marginals are bounded by (1-o(1))^{-1}\mu_i/\mu_i(C_i).

On these pairs,

The quadratic is monotone on each separated branch and its derivative there has magnitude at least \delta^{C'\eta_0}. For every r\ge\delta, the preimage of an output interval of radius r is contained in at most two time intervals of total length

Using the time Frostman bound before normalizing the cluster proves

The estimate includes r=\delta: the larger approximation error costs the displayed small power, which has not been suppressed. Fix \eta_0 sufficiently small compared with \epsilon and all the constants above, and then take D large. Equation (9.17) implies (8.12); the two marginal losses, including the initial-index restriction, are at most D^{\epsilon\zeta}.

No line cluster. In the other case, the proportion of indices with such a cluster tends to zero at every fixed rational depth. Testing finite depth meshes and discarding their exceptional indices, and only then refining the meshes, gives uniformly on the retained indices

for all lines L. To verify the interpolation, replace a radius by the next larger tested radius; if the depth mesh has spacing e, the loss is at most D^{\eta_0e}. The discarded index probability can tend to zero while e\to0, since every fixed finite test has exceptional probability tending to zero.

Fix any rational 0<\zeta\le\zeta_{\max} and put \delta=D^{-\zeta}. A ray-direction ball through an anchor is contained in a line neighborhood of comparable width. Equation (9.18) therefore supplies (9.9) with a positive exponent, decreasing \eta_0 if necessary to make it smaller than s. Start with all distinct pairs. Their probability tends to one, because the mass of any point is at most that of its square at resolution \delta, which is \delta^{s-o(1)}.

Repeatedly apply Lemma 9.2. For example, choosing the next exponent halfway between the old one and \min\{s,(s+\sigma)/2\} decreases its distance to s by a fixed factor. Finitely many iterations give a ray exponent as close to s as desired. Apply the product version once more to obtain an absolute product-section exponent larger than s-\epsilon/2, with subpower constant, on a symmetric relation of probability 1-o(1).

Restrict first to anchors whose surviving section has mass at least 1/2, and then normalize on their sections. The discarded anchor mass tends to zero. The first marginal is bounded by a constant times \mu_i. The second marginal is also so bounded: its density is an integral of section densities bounded by 2, against a first-anchor law bounded by a constant times \mu_i. The product-section bound survives these normalizations. Uniformity makes the same construction possible on every retained initial index; the initial-index normalization is 1+o(1). All fixed and subpower constants fit inside the reserved D^{\epsilon\zeta} allowance for sufficiently large D. This proves the lemma and its independence from subsequent offset tests.□

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

This choice is legitimate because the critical inequality at q=0 is independent of \eta. Lemma 2.10 supplies an interior differentiability point with h>0 and h_z>0. The construction in Proposition 5.4, Proposition 5.6, and Lemma 6.2 then supplies fixed parameters satisfying

In particular, Lemma 6.2 supplies the strict inequality s_0>0.

Recall that g is the total matrix entropy rate and \mathcal M its speed-weighted rate from (7.13). For brevity in this proof put

where r_+=\max\{r,0\}. The smoothing definition in (2.4) gives

If q<(1/2-\eta)\beta, then the integrand defining \mathcal D equals q throughout its averaging interval. Consequently

We record precisely the limiting statements to be used. Fix a compact interval J\subset(0,1). Proposition 7.4 and Lemma 7.6 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,

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 7.5 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 7.3, on the right we also have

It follows in particular that

The domain \mathcal U in (7.2) contains small offsets of both signs uniformly on J when w<1, and small positive offsets when w=1. Thus Proposition 8.1 applies on every stripe used below. The bound (10.9) is available on the right.

The choice of stripe follows from the identity

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.

Distinct speeds, w\ge\tau/2. Suppose first that \tau\ne\ell. In coordinate notation,

On the right, Proposition 8.1 and (10.10) give a_1\ge y_1, u_1\ge v_1, and u_1,v_1\le b_+. Thus

If \ell-\tau/2\ge0, bound both last rates by b_+. If \ell-\tau/2<0, first use u_1\ge v_1; the resulting coefficient of v_1 is \ell+1-\tau=2\ell>0. This verifies (10.11) for either ordering of the middle speeds. Since \tau/2-w\le0, the right stripe inequality in (10.8) implies that every upper stripe limit of \mathcal Q is at most

Its gap below the required lower limit is

For q<\beta, this is at least h+\tau q/2+\ell((p-2)\beta+q)>0. For q\ge\beta, it is at least h+\tau q/2+\ell(p-1/2)\beta>0. Either possibility contradicts the lower stripe limit T_*.

Distinct speeds, w<\tau/2 and \tau<2\ell. Here \ell>1/3. If q<(1/2-\eta)\beta, (10.7) would give w>\ell>\tau/2. We must therefore have q\ge(1/2-\eta)\beta. On the right, y_1\le a_1\le1-q and u_1,v_1\le b_+, so

The right stripe bound and w>0 make U_2 an upper stripe limit for \mathcal Q as well. If q<\beta, subtraction gives

The last inequality follows from \ell>1/3, p>2, and \eta\le1/32; its coefficient is greater than 1/6-1/16>0. If q\ge\beta, the gap is at least h+\tau q+\ell(p-1/2)\beta>0. This again contradicts the demand.

Distinct speeds, w<\tau/2 and \tau\ge2\ell. Use the left stripe. Since w<\tau/2<1/2, its offsets are available and v_1 has a strong left trace, denoted by v_*. Take a weak-star limit \bar g of left stripe averages. The projection constraints imply, wherever v_1>0,

We give the localization estimate, since convergence of the averaged rate alone would not suffice here. The full generic-slice trace implies

Indeed the integrand tends to zero through a fixed full-measure set of offsets, so its essential supremum over (-\epsilon,0) tends to zero. For E_\delta=\{c\in J:v_*(c)\ge\delta\}, where \delta>0, let B_a=E_\delta\cap\{c:v_1(c,a)=0\}. Then

The function f_0(t)=2\min\{1,s_0+t\}+2t, for 0\le t\le1, is bounded by 4 and has Lipschitz constant at most 4. For every bounded nonnegative test \varphi supported on E_\delta, (10.12) and g\ge0 consequently give

Take the stripe weak-star limit, then let \delta=1/m\downarrow0. The resulting localized inequality is

On the other hand, \bar g\le d\le2. If s_0+v_*\ge1, the displayed lower bound is greater than 2. Otherwise it is 2-2\beta+4v_*. It follows that

including the set where the trace vanishes.

Because \ell-\tau/2\le0, the inequalities a_1\ge y_1 and u_1\ge v_1 also imply

Now \tau/2-w>0. Combining the left stripe bound for g with (10.15), every upper stripe limit of \mathcal Q is bounded by

But

This excludes the last distinct-speed case, including \tau=2\ell.

Tied speeds with q\ge(1/2-\eta)\beta. It remains to consider \tau=\ell=1/2. Write m_1 for the middle-group rate. Then

On the right, Proposition 8.1 yields m_1\le1+b_+ and v_1\le b_+. Therefore

This is again an upper stripe limit for \mathcal Q. For q<\beta, its gap below demand satisfies

Indeed (10.1) bounds the coefficient below by 13(p-2)/32. For q\ge\beta, the gap is at least h+(p-1/2)\beta/2>0. This case is impossible as well. In particular, the same \eta fixed before the canonical construction works independently of the subsequently selected h,\tau,\ell,\beta.

Tied speeds with q<(1/2-\eta)\beta. Here \mathcal D(q,\beta)=q and w>1/2. Put

Take any sequence of generic positive offsets tending to zero. On a fixed compact interior interval J, extract a subsequence for which v_1\to v_* strongly in L^1(J) and the other bounded rates converge weak-star. This is allowed also when w=1. Denote their limits by bars. The demand gives \overline{\mathcal Q}\ge T_*.

On E_\delta=\{v_*\ge\delta\}, use (10.14) along the selected sequence of slices: the zero-set measure is at most \delta^{-1}\|v_{1,n}-v_*\|_1\to0. Since a_1<1 on the right, positive v_1 and the projection constraint v_1\le(a_1-s_0)_+ force a_1\ge s_0+v_1. Thus

on the part with positive v_1. Explicitly, for bounded \varphi\ge0 supported on E_\delta,

Weak-star convergence of a_{1,n}, strong convergence of v_{1,n}, and then \delta=1/m\downarrow0 justify the inequality on \{v_*>0\}. Passing to this localized limit gives

Consequently \bar g\le d-K_0/(w-1/2) on \{v_*>0\}. On \{v_*=0\}, the bounds m_1\le1+\beta-q and strong convergence of v_1 give \overline{\mathcal M}\le U_0. The same demand then gives T_*\le U_0+w(d-\bar g), and hence \bar g\le d-K_0/w. Since w-1/2<w, the two parts imply the common bound

Its positive gap depends only on the fixed parameters, not on the chosen sequence or the subsequential trace.

We now convert the common gap in (10.17), obtained from arbitrary offset sequences, to a stripe contradiction. Fix a nonnegative test \varphi\in L^1(J) with positive integral. If arbitrarily small generic positive offsets satisfied

choose a sequence of them tending to zero. The preceding compactness argument would give a weak-star limit violating (10.17). Thus all sufficiently small generic positive offsets satisfy the reverse inequality. Averaging it over 0<a<\epsilon contradicts the first inequality of (10.8); the exceptional offsets form a null set.

Every possible choice of the stationary parameters has now been excluded. This contradicts the differentiability point obtained under the assumed failure of critical vanishing, and proves the proposition.□

Proof. We first establish a uniform discrete estimate, then an indicator estimate, and finally (10.18). All geometric constants in this proof are absolute unless their dependence is displayed.

A discrete estimate in one bounded chart. Let N\ge2 be dyadic. Consider unit-weight lines in a bounded matrix chart, with pairwise c/N-separated spatial slopes, where c>0 is a fixed constant. Let A be a finite collection of N^{-1}-cubes. Suppose each of n such lines has at least c_1\max\{1,\lambda N\} marked time bins whose centerline points lie within C_1/N of a cube in A. Here 0<\lambda\le1 and c_1,C_1 are fixed constants. Then, for each \xi>0,

where |A| is a cell count. Fixed bounded changes of the chart, of the resolution, and of the neighboring-cell allowance only change the constant.

Here is the uniformity argument. For a full-time plank of widths a,b in finest cell units, the slopes of lines contained in it lie in a rotated rectangle with side lengths O(a/N) and O(b/N). This follows by comparing the two endpoint cross-sections of the plank. Since 1\le a\le b\le N, slope separation bounds their number by O(ab). Consequently

for the original family and all its subfamilies.

If (10.19) failed uniformly for a fixed \xi>0, choose a violating sequence with N\to\infty. Bounded N cannot cause this failure, because n\le CN^2 and any nonempty A has at least one cell. On each line retain exactly k\asymp N\theta of the allowed bins, where \theta=\max\{N^{-1},\lambda\}; integer rounding is harmless. The union of their event cells lies in a fixed enlargement of A, of cell count at most C|A|. Along a subsequence write \theta=N^{-\alpha+o(1)}, with 0\le\alpha\le1. Apply Lemma 2.6 to this indexed event graph. It retains nkN^{-o(1)} events in weighted count and gives a limiting profile F. The number of events per active index is at most k, whereas retained index mass is at most n. Total incidence retention therefore forces this number to be kN^{-o(1)} in exponent. It follows that F(1)=\alpha. The retained graph has raw multiplicity at least

along a sequence violating (10.19). At q=0, the temporal cost of every profile is exactly \Pi_F(1)=pF(1)=p\alpha. The critical inequality with h(p,0)=0 and (10.20) bounds this same multiplicity by N^{p\alpha+o(1)}, a contradiction. This proves (10.19) with a constant uniform over all the finite configurations under discussion.

Cylinder geometry and translations. Use an auxiliary family of cylinders of length 4 and radius 4/N, and let \mathcal K_N denote the supremum of their normalized averages over all centers, as a function of direction. Thus 0\le\mathcal K_N\one_E\le1 for every measurable set E. Let E be a finite union of cubes of the fixed N^{-1} grid, and suppose one of these cylinders has average of \one_E greater than \lambda>0.

Cover S^2 by finitely many fixed small direction charts. After an axis permutation and, if needed, reversal, the time component in each chart is bounded away from zero. A cylinder in such a chart meets only O(1) fine spatial cubes in each time bin of length comparable to N^{-1}. Its intersection with any one bin has volume O(N^{-3}), while its full volume is comparable to N^{-2}. An average greater than \lambda therefore supplies at least c\lambda N occupied time bins, and at least one if this quantity is smaller than a constant. The centerline point in each such bin is within C/N of an occupied cube of E. These are the events needed in (10.19).

To allow arbitrary translations, group cylinder centers into unit grid cubes Q. Every cylinder in a group lies in a fixed enlargement Q^*, and the regions Q^* have bounded overlap as Q varies. For one group and one direction chart, all centerlines have bounded intercepts and slopes. Extend their lines across one fixed common time interval containing their cylinder traces, then translate and rescale that interval and the spatial chart by fixed factors to fit the indexed model. Only the marked events on the original cylinder are used. A comparable dyadic resolution and bounded neighboring-cell enlargements account for these fixed changes. Thus (10.19) applies to the cubes of E meeting Q^*, with constants independent of the location of Q. Pairwise angular separation in a fixed chart is equivalent, up to fixed constants, to separation of the spatial slopes.

A directional distribution bound. For 0<\lambda<1, let

In each fixed chart, choose a maximal c/N-separated subset of S_\lambda, of cardinality n, and choose for each selected direction a cylinder witnessing the strict inequality. Such a finite maximal subset exists by the packing bound on the sphere. The C/N-caps around its elements cover the high-value directions in that chart, so their measure is at most CnN^{-2}.

Apply (10.19) to the spatial groups just described and sum over the groups. Their ambient regions overlap a bounded number of times, whence

where |E| now denotes Lebesgue volume. Sum over the finitely many direction charts. The cap measure bound gives

In particular, for N^{-1}\le\lambda<1,

The factor N^{p-2} comes from \lambda^{2-p}\le N^{p-2}. Thus the density cost has been retained throughout this reduction.

For a nonempty union of fine grid cubes, |E|\ge N^{-3}. The distribution formula, \sigma(S^2)=4\pi, and (10.22) imply

The empty set is immediate. This separately accounts for the entire range \lambda<N^{-1}.

Cell averages and general functions. Let f\in L^3(\R^3) be complex-valued. The case f=0 is immediate, and homogeneity reduces the proof to \|f\|_3=1. Choose a dyadic N\ge2 with N^{-1}\le\delta<2N^{-1}, and let F_N be the function equal to the average of |f| on each cube of the N^{-1} grid. Hölder’s inequality on a cube and Jensen’s inequality give

Every grid cube meeting T_\delta(a,\omega) lies in the cylinder with the same center and direction, length 4, and radius 4/N. Indeed its points are within \sqrt3/N of a point of the original cylinder, and N\ge2. Summing the integrals of |f| on all such cubes therefore gives

The enlarged volume is 64\pi N^{-2}, while \pi\delta^2\ge\pi N^{-2}. Consequently

The part of F_N below N^{-1} contributes at most N^{-1} pointwise to \mathcal K_N F_N. For its remaining values use the disjoint level sets

There are J_N=2\log_2N+1 such levels. Each is a finite union of fine grid cubes: its volume is finite by (10.24), and every occupied cube has volume N^{-3}. Sublinearity, the triangle inequality in L^3(S^2), and (10.23) yield

Hölder’s inequality for this finite sum gives

Together with (10.25), this proves

Given \kappa>0, take \xi=\kappa and absorb the logarithm into C_\kappa N^{2\kappa/3}. Since N<2/\delta, this is (10.18) for normalized f. Homogeneity gives the general case. Fixed bounded resolutions are included by enlarging the constant; directly, Hölder gives K_\delta f\le(\pi\delta^2)^{-1/3}\|f\|_3, uniformly in direction, when \delta is bounded below.

Measurability. For clarity, all suprema used above define measurable functions of direction. If (a_n,\omega_n)\to(a,\omega), the indicators of the corresponding cylinders converge in L^{3/2}(\R^3): their boundaries have measure zero and their supports lie in one bounded set for a convergent sequence of parameters. Hölder’s inequality therefore shows that the integral of |f| over the cylinder is continuous in its center and direction. For each fixed direction, its supremum over all centers equals its supremum over the countable dense set \mathbb Q^3. This is a countable supremum of continuous functions of direction, hence is measurable. The same reasoning applies to \mathcal K_N and to the finite-volume indicator functions used above. It also shows that changing f on a null set changes none of the tube integrals. Thus all distribution and L^3 estimates are valid for the stated L^3 equivalence classes.□

Proof of Theorem 1.1. Fix \varepsilon>0. Proposition 10.1 allows a choice of p>2 with h(p,0)=0 and (p-2)/3<\varepsilon/2. Apply Proposition 10.2 with \kappa=\varepsilon/2. Since 0<\delta<1,

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. [9, Section 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 1.1 gives its L^3-to-L^3 bound with every positive power loss. Also

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. [9, 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. [9, 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 [9, 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.□