Endpoint pointwise convergence for the Schrodinger equation in higher dimensions

Proof. The unweighted unit-radius assertion with arbitrary transverse directions is Guth [16, Theorem 2]; the near-coordinate-axis multilinear formulation originates in Bennett et al. [2]. Integral weights are obtained by repeating cylinders. Dividing by a common denominator gives rational weights, and monotone approximation gives arbitrary nonnegative weights. Dilation gives h^d. Every side-a cube meeting a cylinder lies in its enlargement of radius h+C_da\lesssim h. Integration over each cube, of volume a^d, proves the lattice version, also with separate suprema for the factors. For paraboloid normals in a bounded patch, normalization to unit vectors changes the determinant by fixed factors, so (2.2) gives an inverse power of \delta.□

Proof. This is He [21, Theorem 1], with m=d-1, after reducing its error parameter and, if necessary, replacing \rho by a smaller positive number. Here is the match to its Grassmannian hypothesis. The competing subspaces have dimension d-m=1; if one is spanned by a unit vector w, its Schubert angle to the projection plane v^\perp is |v\cdot w|. Thus (2.7) is exactly the required nonconcentration near Schubert cycles. The theorem excludes a direction set of small \nu-mass and is simultaneous for all sufficiently large subsets of A. Its size and nonconcentration errors can be made larger than those displayed above, leaving a fixed positive projection gain \gamma. Shrinking this gain and the error tolerance absorbs the fixed comparison constants. Its theorem tolerance is chosen larger than \eta: our lower size bound then implies its required lower bound, and our subset threshold is stronger than its subset threshold. No choice of a new size exponent is necessary.□

Proof. Choose a smooth partition \sum_j\psi_j=1 on native space, where \psi_j is supported within fixed distance of j. Choose a compactly supported smooth Fourier multiplier equal to one on the original Fourier box, and let K be its Schwartz inverse Fourier transform. Then

These inequalities follow by integrating the Schwartz bounds of K over the fixed support of \psi_j; the decay at j comes from the original pointwise bound. They prove absolute convergence and every asserted output derivative bound. Convolution without the partition also proves fixed derivative bounds for the original packet.

For completeness, the maximal inequality is valid also for r<1. Let M=\sup_y |F(y)|(1+|y-z|)^{-a}, choosing a so large that ar>L+k+1, and choose a point y_0 realizing at least M/2. Reproduction and its derivative version give |\nabla F(y)|\leq C M(1+|y-z|)^a. On a fixed sufficiently small ball about y_0, this and the choice of y_0 imply |F(y)|\geq cM(1+|y_0-z|)^a. Integrating on that ball proves the weighted maximal estimate with weight (1+|y-z|)^{-ar}, and hence (2.9) after choosing ar large enough. Truncation of the weighted supremum justifies the argument when its finiteness has not yet been established.

Translating a native core by j multiplies its weight comparisons and phase-space multiplicity by a fixed polynomial in 1+|j|. For example, w_{T+j}(z)\leq C(1+|j|)^Pw_T(z). The sum over j therefore converges once P' exceeds those fixed powers and k+1. Crucially, A occurs in the constant C_A, not in that required original power P'. For packet estimates use (2.9) and the polynomial comparison of envelopes at translated points. For products apply it separately to each factor and use Tonelli. The exponent in each convolution is fixed after the product exponents have been fixed.□

Proof. We track the dimension-dependent bookkeeping in the induction of Singh and Singh Parmar [26, Appendix A]; no refined estimate is used as an input. Weighted local Bourgain–Demeter decoupling says that, for any fixed large L, s\geq1, 2\leq p\leq q_k, and a cube Q_s of side s,

for s^{-1/2} tangential caps with O(s^{-1}) normal thickness. The weighted version follows by summing the usual local decoupling estimate over translates of Q_s with rapidly decreasing coefficients. Its constant is uniform under the fixed positive-definite affine normalizations used below.

Let \mathfrak A_d(R,\Lambda,a,b) be the least constant in (2.10) without R^\varepsilon, uniformly for the stated configurations but with the full domain 1\leq a<b\leq20. The theorem starts with b-a\geq1/2; smaller positive margins occur only inside the induction. Define it for every \Lambda\geq1 using the trivial estimate when \Lambda\gtrsim R^{1/4}; it is not defined only on the curve \Lambda=R^\vartheta, since \Lambda stays fixed when R is square-rooted. For the one-step estimate assume R^\vartheta\leq\Lambda\ll R^{1/4}; this lower bound persists at each scale reached from the initial \Lambda=R^\vartheta. Group fine labels into coarse R^{-1/4}-cubes J, with fixed finite coloring at cap boundaries. Translate the midpoint of S to t=0, adjusting tube intercepts accordingly. If v_0 is the center of J, assign each fine tube to one parent box

The entire supporting aT lies in its assigned parent box. This uses |v-v_0|=O_d(R^{-1/4}), the common O(R) time slab, and \Lambda R^{1/2}\ll R^{3/4}. The boxes have bounded global overlap for each J, and packet families assigned to distinct boxes are disjoint. On a parent box, remove the carrier and set

Then x-2vt-u_T=R^{1/4}(x'-2v't'-u_T') and |\det\Phi|=R^{-(k+2)/4}. A fine cap and its \Lambda/R normal thickness become a cap of width R'^{-1/2} and thickness \Lambda/R'; an aT or bT becomes the corresponding dilate at radius R' with the same \Lambda. The transformed fine labels form an allowed translated lattice. The Jacobian occurs on both sides of the desired estimate and cancels. A lower-scale R'^{1/2}-cube pulls back to a cell C of transverse width O_d(R^{1/2}) and longitudinal length O(R^{3/4}).

Apply (2.11) with s=R^{1/2} on an original R^{1/2}-cube Q. Choose its neighborhood radius h=\Lambda^{1/4}R^{1/2} and take L\geq C(B+d)/\vartheta. Outside that neighborhood the decoupling weight is O(R^{-B-d-10}). The relative packet tails make boxes missing the neighborhood negligible too. There are at most C_d\Lambda^{d/4} neighboring R^{1/2}-cubes Q'', only O_d(1) parent boxes of a fixed J meet the neighborhood, and each Q'' meets only O_d(1) rescaling cells C in such a box. Thus, writing F_\Box=\sum_{T\mapsto\Box}F_T,

where \mathcal R denotes the right side of the defining refined inequality without its constant. Finite truncation and the polynomial number of packets and cubes in the local box justify the displayed negligible error after increasing B.

Put b'=b-C_d\Lambda^{-3/4}. If b'T meets a cell C that meets a neighbor Q'' of Q, compare the point of intersection with the center of Q in the adapted coordinates y=x-2v_0t-u. The change in y is O_d(\Lambda^{1/4}R^{1/2}); the change in t is O_d(R^{3/4}+\Lambda^{1/4}R^{1/2}). Since |v-v_0|=O_d(R^{-1/4}), the transverse residual for the fine tube changes by at most C_d\Lambda^{1/4}R^{1/2}. Dividing by its width \Lambda R^{1/2} gives C_d\Lambda^{-3/4}. All strips use the same longitudinal slab, so this implies bT meets Q; no individual tube endpoint is crossed.

For completeness, here is the counting step. Pigeonhole the cube norms, parent packet counts W_\Box, lower-scale cell counts m_{\Box,b'}(C)\asymp M', and numbers n(Q)\asymp n of retained triples. The polynomial configuration size makes the cost (\log R)^{C_d}. By (2.12) and (\sum_{i=1}^n a_i^2)^{p/2}\leq n^{p/2-1}\sum_i a_i^p, a retained class satisfies

A fixed triple is associated with at most C_d\Lambda^{d/4} original test cubes. Rescaling and applying \mathfrak A_d(R',\Lambda,a,b') gives, since the original packet norms are comparable and the assigned families are disjoint,

For a fixed Q, one parent box contributes at most C_d\Lambda^{d/4} triples. Choose one retained triple from each of at least c_dn\Lambda^{-d/4} distinct boxes. Each contributes \asymp M' different fine packets whose bT meets Q by the preceding geometric transfer. Disjoint assignment to boxes therefore gives nM'\lesssim\Lambda^{d/4}M. Substitution into (2.13), followed by a pth root, proves

The two neighborhood counts give \Lambda^{d/(4p)}\Lambda^{(d/4)(1/2-1/p)}=\Lambda^{d/8}; there is no dimension-dependent power of R.

Stop at the first R_j=R^{2^{-j}} with \Lambda\gtrsim R_j^{1/4}. Bounded unclipped overlap per label and O_d(R_j^{k/2}) labels give the direct estimate \mathfrak A_d(R_j,\Lambda,a,b_j)\lesssim R_j^{k/4} \lesssim\Lambda^k. There are j=O(\log(1/\vartheta)) steps. For R sufficiently large, their total count-dilation drift C_dj\Lambda^{-3/4} is less than 1/4, so the initial margin b-a\geq1/2 remains positive. Iterating (2.14) yields

Choose \vartheta first so that \vartheta O_d(\log(1/\vartheta))<\varepsilon/3, then \delta so that the decoupling factors cost less than R^{\varepsilon/3}, and absorb the logarithms into the remaining loss. The weight order L and its fixed constant are chosen after \vartheta; no original packet decay order is being chosen as a function of \varepsilon. This proves (2.10).□

Proof. We derive the weighted conclusion from Lemma 2.5, applying that lemma to relative-tail atoms, not to the unsplit polynomial-tail f_T with a single original-tube count. Fix \eps>0, put R=N^2, and choose \vartheta>0 small enough, after the fixed packet exponents, to pay every power of \Lambda=R^\vartheta below. The original profile has fixed native Fourier support. Choose a reciprocal sampling lattice with fixed spacing \gamma in the k native spatial variables and spacing \gamma/\Lambda in native time. Smooth Fourier cutoffs equal to one on the original Fourier box and supported inside the reciprocal fundamental domains give, by Poisson summation, fixed Schwartz kernels K_x,K_t and the exact expansion

where u_T is the native cell center and, with a fixed normalization,

The temporal Fourier support of an atom is O(\Lambda) in native coordinates. In physical coordinates the Fourier map is

Thus an atom has tangential cap width O(R^{-1/2}) and normal thickness O(\Lambda/R), exactly the range of Lemma 2.5. Its L^p norm is comparable to |c_{T,j,m}|N^{(k+2)/p}\Lambda^{-1/p}, with no lower bound on |c_{T,j,m}| required. A fixed-kernel sampling in time would not give the required relative R^{-B} tail outside an R-long tube; the \Lambda^{-1} native-time compression in (2.16) does.

Fix one sample index (j,m) and write g_T for its atom. Let d_{T,j,m} be native distance to its shifted center in the original coordinates X_{N,v} of (2.3), and set

Partition spacetime into test boxes with time cores of length R and bounded spatial overlap. For a core centered at t_0, retain atoms whose sample time centers lie in [t_0-2R,t_0+2R], and use the common support slab S=[t_0-5R,t_0+5R] for all retained atoms. Test cubes lie in [t_0-R/2,t_0+R/2]. Thus retained centers have at least a 3R margin to \partial S, while omitted centers have at least a fixed R separation from the test cubes. The retained atom has R^{-B} relative time tail outside S; the omitted atoms gain C_A\Lambda^{-A} on the core, with a summable further factor in their time-slab distance. Retain spatially only trajectories that can meet a fixed padded box; the other atoms have an arbitrarily small Schwartz spatial tail. The phase-space count (2.4) sums both discarded classes: with the full shifted envelope kept in the denominator, three-factor Hölder as below and annular counting give an arbitrary \Lambda^{-A_1} or N^{-A_1} gain in their weighted L^p norms, respectively. This charge is made before applying the local lemma. A retained atom is concentrated, relative to its own norm, on the common-slab strip of transverse radius \Lambda N, with C_A\Lambda^{-A} tail outside a fixed support dilate aT, say a=2. Count a larger dilate, say bT with b=3. Since A may be chosen after \vartheta, this is R^{-B} for the B of Lemma 2.5. Each atom belongs to only \Lambda^{O_d(1)} padded test boxes.

The actual v may move within its N^{-1} frequency slot. Relabel it by the slot center and use one of a fixed number of cap-grid colors. Over an O(R) slab, the two trajectories differ by O(N), which is O(\Lambda^{-1}) of the inflated transverse radius; the fixed a=2, b=3 collar absorbs this displacement. For this fixed sample index, the shifted phase-space multiplicity is bounded by a polynomial in J=1+|j|+|m|/\Lambda. Color intercepts on the unclipped strips into at most C\Lambda^{C_d}J^{C_d} families of bounded same-label overlap. There are only O(1) relevant time slabs per test cube for each such family. In every local box there are polynomially many packets. Set a_T=|c_{T,j,m}|b_T^{2/p-1} for b_T>0 and a_* = \max_T a_T in that box. Discard only the classes with a_T<N^{-C_0}a_*, where C_0 is chosen after the polynomial local packet count. This is a cutoff in the weighted amplitude, not in |c_{T,j,m}|. Indeed, writing h_T=|g_T|/|c_{T,j,m}| and taking a Schwartz majorant h_T\lesssim w_{T,j,m}, three-factor Hölder gives

The last sum and the number of local packets are polynomially bounded; \int h_T\lesssim N^{k+2}/\Lambda. Thus this discarded contribution is N^{-C_0+O_d(1)}\Lambda^{O_d(1)}J^{O_d(1)} times (N^{k+2}/\Lambda)^{1/p}(\sum_Ta_T^p)^{1/p}. The retained a_T occupy O(\log N) dyadic classes. Within each b_T\asymp\alpha class, comparability of a_T also makes the atom L^p norms comparable, so Lemma 2.5 applies. No comparison between the multiplicity of shifted atom tubes and the multiplicity of the original tubes is used.

To see the weighted reduction quantitatively, sort N-cubes by B_{b,j,m}\asymp\beta and atoms by b_T\asymp\alpha, then by retained a_T. The slot-center relabeling above changes only the local counted strips; the shifted envelope here keeps v and therefore compares globally with B_b. No absolute cutoff on \alpha or \beta is needed: the cube classes are disjoint, and for each T the moderate relative range below contains only O(\log N) \beta-classes. On a cube meeting a counted bT, its atom center is within O(\Lambda) in native space; hence the atom contributes at least c\alpha^2\Lambda^{-P} to B_{b,j,m}^2. The count M_{j,m} for that class obeys

Use Lemma 2.5 on the moderate ratios N^{-C}\leq\alpha/\beta\leq C\Lambda^{P/2} and sum their O(\log N) relative ranges. For smaller ratios the identity

gives N^{-C(1-2/p)}; choose C to dominate the polynomial local packet count and use pointwise Hölder. For the complement of the counted activity strips, retain the anisotropic atom envelope

It gains \Lambda^{-A_1} outside the strip in either transverse or temporal direction, while \int h_{A,\Lambda}^p\lesssim N^{k+2}/\Lambda. By (2.4), annular counting, and pointwise Hölder, the off-activity pth power is bounded by C_A N^{C_p}J^{C_p}\Lambda^{-A_1}(N^{k+2}/\Lambda) \sum_T|c_{T,j,m}|^pb_T^{2-p}. Choose A after \vartheta to absorb the polynomial factor. Keeping the temporal factor in h_{A,\Lambda} is essential to retain \Lambda^{-1} in this bound. The fixed-index conclusion, including local-box and color summation, is

Here and below the exponents C(d,p,P) are fixed before \vartheta.

Native-distance comparison gives B_{b,j,m}\leq CJ^{P/2}B_b. Since 2-p<0, B_b^{2-p}\leq CJ^{P(p-2)/2}B_{b,j,m}^{2-p}. Apply Minkowski in the original weighted L^p norm and use (2.18) and the sampled coefficient bound. The m-sum of \Lambda^{-1/p}(1+|m|/\Lambda)^{-P'+C} is O(\Lambda^{1-1/p}); the j-sum converges if the original fixed P' is chosen greater than C(d,p,P)+k+1. Consequently

Choose \vartheta sufficiently small that the displayed \Lambda power, all preceding color and logarithmic losses, and the deliberately smaller buffered loss fit below N^{\eps/p}. Taking the pth power proves (2.15). The original P' is chosen once, independently of \eps; only the output Schwartz order and weighted decoupling order depend on the chosen \vartheta. Unit suprema follow from Lemma 2.4 and fixed-scale envelope comparability.

Finally rotate an affine \kappa-plane to the first \kappa coordinates and remove its constant and linear normal terms by modulation and shear. The normal frequencies then have size O(N^{-1}), and their quadratic contribution is O(N^{-2}). Put m=k-\kappa and write x=(u,y) in tangential and normal coordinates. For a packet T, let (y_T,t_T) be the normal position and time of its physical center and, on a fixed normal slice y, put j_T(y)=\lfloor(y-y_T)/N\rfloor\in\Z^m coordinatewise. Write d_{T,\kappa} and w_{T,\kappa}=(1+d_{T,\kappa})^{-P} for the projected tangential-time native cell. Since the normal frequency is O(N^{-1}), its native displacement at time t is (y-y_T)/N-2(Nv_T^\perp)(t-t_T)/N^2. Consequently, uniformly in t,

Choose fixed A,P'' with A+P''\le P' and P'' sufficiently large for the \kappa-dimensional estimate. On the slice y, the packet has projected amplitude at most Cs_T(1+|j_T(y)|)^{-A} and decay order P''. Its projected native Fourier box is fixed: substitution in the normal native variable shifts time frequency by O(Nv_T^\perp)=O(1), while the normal quadratic carrier shifts it by N^2|v_T^\perp|^2=O(1).

Group the sliced packets by j_T(y)=j, and write V_j for their sum. For fixed y,j and each N^2 time slab, this condition restricts the normal intercept to O(1) cells for every projected frequency/intercept cell; there are also only O(1) normal frequency slots. Hence each j-family has bounded projected phase-space multiplicity. Its tangential frequency slots are split into O(1) colors and snapped to a \kappa-dimensional N^{-1} grid. Use the dimension-\kappa estimate already obtained from the preceding full-dimensional argument (or induct on k). If B_j^2=\sum_{j_T(y)=j}b_T^2w_{T,\kappa}, then B_b^2\gtrsim(1+|j|)^{-P}B_j^2. As 2-p<0, applying the \kappa-dimensional estimate to that family gives

For each T,j, the set of y with j_T(y)=j has volume N^m. Integrating in y supplies exactly N^m, and Minkowski sums the j\in\Z^m contributions provided A>m+P(p-2)/(2p). This is possible with the fixed sufficiently large original P', and yields (2.15) in the range 2<p\le q_\kappa. If \kappa=0, put \phi_T=(1+d_T)^{-P'}. The bounded number of slots and phase-space counting give \sum_T\phi_T\leq C. Three-factor Hölder gives

Since \phi_T\leq w_T, multiplication by B_b^{2-p} and integration, using \int\phi_T\leq CN^{k+2}, prove the assertion for every finite p>2.□

Proof of Propositions 2.7 and 2.8. Write Eg(x,t)=\int e^{i(x\cdot\xi-t|\xi|^2)}g(\xi)\dd\xi. We reduce the packets on D to exact extensions, keeping track of the norm. Translate the center of D to (x_0,t_0). For a frequency label v, take an integral-normalized smooth bump g_v of width c/N, centered at v, and multiply it by e^{-i(x_0\cdot\xi-t_0|\xi|^2)}. On a fixed enlargement of D its extension is its carrier times

For fixed sufficiently small c, this differs from 1 by less than 1/2. Both H_v and H_v^{-1} have bounded derivatives in the variables (x-x_0,t-t_0)/N. Also \|g_v\|_2\leq C N^{k/2}. In the multilinear case c can be shrunk by a fixed power of \delta so that enlarged supports retain determinant at least \delta/2; all resulting constants are polynomial in \delta^{-1}.

For a packet T with label v, divide its demodulated profile by H_v on this enlarged ball, multiply by a fixed cutoff, and expand in a Fourier series in (x-x_0,t-t_0)/N. The resulting identity on D has the form

where P''>P/2+k+1 can be fixed below the available original decay margin. Only finitely many derivatives are needed for a chosen L. For each mode, combine the coefficients with the same frequency slot; Cauchy–Schwarz in the spatial and time labels and their summable native tails gives

Here one first uses bounded overlap of the bump supports in distinct frequency slots; within a slot the native-center lattice has a summable weight. This explains why the square bound has no factor equal to the number of spatial labels.

The exact paraboloid Strichartz inequality \|Eg\|_{q_k}\leq C\|g\|_2 follows here directly from the dispersive kernel and the TT^* argument. Indeed the Gaussian oscillatory integral gives \|e^{it\Delta}\|_{1\to\infty}\leq C|t|^{-k/2}, while Plancherel gives its L^2 norm equal to one. Interpolation at spatial exponent q_k gives decay |t|^{-k(1/2-1/q_k)}=|t|^{-2/q_k}. The one-dimensional Hardy–Littlewood–Sobolev inequality in time consequently bounds TT^*:L^{q_k'}(\R^{k+1})\to L^{q_k}(\R^{k+1}), and duality gives the stated extension estimate. Applying it to each mode gives (2.19), after summing the modes. For the multilinear assertion use Tao [30, Theorem 1.7]. In ambient dimension d=k+1, its product exponent 2a, a>1/(d-1), becomes geometric-mean exponent p=2da>p_k. Its quantitative constant is polynomial in a parameter controlling domain size, finitely many graph derivatives, inverse determinant and inverse interior support distance. On bounded paraboloid patches these quantities are bounded by a fixed power of \delta^{-1}: enlarge each relevant domain by c\delta, and, if necessary, subdivide into patches of polynomially small width in \delta. The Hessian is constant, all required higher derivatives are bounded, and the bump supports lie a polynomial distance inside the domains. Thus the theorem applies globally with the displayed C_p\delta^{-C_p} and no scale loss. The external modulation belonging to a fixed Fourier mode has modulus one. Expand the product over modes using subadditivity of the 1/(k+1) power, and then Minkowski when p\geq1. Taking L larger than the dimension times k+1 makes this sum converge.

Finally apply (2.9) separately to the exact extension factors. A spacetime translation of Eg is another extension with input multiplied by a unimodular phase, so it has the same input norm and frequency support. Tonelli and the global estimates bound each translated product. The convolution exponents can be chosen to sum all mode and translation weights. This proves the separate-supremum assertions and permits the independent choices of z_j.□

Proof. Put r=K/K'. In parent coordinates the linear map to child coordinates has spatial coefficients O(r) and time coefficient r^2, determinant r^{k+2}; the possible velocity shear has bounded normalized coefficients because the child frequency lies in the parent cap. Let a=1+d_{T'}(z). Parent centers whose child distance is at least a/2 contribute at most Ca^{-P}\sum_Tw_T(z)\leq Ca^{-P}. With d_T(z)>L, this sum has the additional lattice-tail factor L^{-(P-k-1)}. For the other centers, necessarily a\gg1 and d_T(z)\geq ca/r. Their contribution is at most

The last lattice sum is bounded by its integral at the child scale plus a fixed boundary term, with volume ratio r^{-(k+2)}. This proves both conclusions and shows explicitly why no envelope power is lost in repeated convolutions.□

Proof. A concrete band-limited partition is obtained by taking \chi=|\phi|^2, where \widehat\phi is smooth with sufficiently small support. Poisson summation makes \sum_{j\in\Z^{k+1}}\chi(X-j) a positive constant after normalization; alternatively a bounded lattice average gives the same identity. These nonnegative localizers have arbitrarily rapid tails, compact Fourier support, and bounded sums. If U is the pre-cutoff field in a parent cap, choose

where C_0 exceeds the required output decay order by a fixed margin. Multiplying by the localizer makes the resulting normalized packet satisfy any prescribed fixed decay order. Its native Fourier support stays in a fixed enlarged box.

In parent coordinates the child packets have scale K'/K. Apply Proposition 2.6 there, and then its unit-supremum version to (2.25). The child square sum varies by at most a fixed polynomial over parent distance, since velocity differences are O(K^{-1}). The sufficiently large C_0 absorbs this variation, while (2.23), whose exponent 2-p is negative, gives the required upper bound for the parent weighted norm. Changing coordinates supplies K^{k+2} and yields (2.24). Deleting parent outputs decreases its left side. For modified inputs the same proof uses the same localizers and recomputes (2.25); it never assumes that cancellation or an old amplitude survives deletion. The affine part of Proposition 2.6 gives the larger exponent on strip subtrees: a strip thin enough at the final precision is thin enough at each intervening child precision.

We explain the two uniformity assertions needed in iteration. The new square background formed by sampling child squares satisfies

by Lemma 2.9; the outer exponent has not decreased. Additional decay on an edge only improves this bound. A large displacement along a finite path forces a large displacement on an edge after normalization to the largest relevant cell. Keeping a fixed reserve of decay on every edge therefore bounds paths with a large displacement by the full-background convolution times a negative power of that displacement. The choice of which edge is large costs at most the number of edges, rather than a new decay order at each edge.

For a positive macro weight \Psi, suppose both ratios of its values at an edge’s endpoints are bounded by C(1+d)^J, where d=d_{T'}(z_T) is the child distance. The constants and J must be independent of the main scale; this holds for polynomial weights on macro boxes whose size dominates the affected cells. Conjugate the given denominators by

Writing w_{T'}^{(Q)}=(1+d_{T'})^{-Q}, one has

Apply the one-step proof with this stronger fixed child envelope exponent and its fixed supremum-decay reserve. Since \widetilde b_T^{2-p}=b_T^{2-p}\Psi(z_T), it proves the explicit estimate

The original denominators have been retained. The same value of Q works at each edge; the reserve does not accumulate with depth. For a partition of macro space with summable polynomial weights, summing over macros costs their bounded total overlap. This gives the version for restricted subtrees and frozen masks, without using discontinuous physical cutoffs.

Finally, for separated nested scales, inverse widths added by localization form a geometric series, including the bounded velocity shears. Thus the fixed Fourier enlargement can be chosen once. A block of comparable scales is treated as one transition; finitely many extra masks change constants. Repeated constants can grow with the fixed grid, but not with the main scale. In a later slow diagonal those constants may be absorbed only after the grid is fixed. Arbitrarily high Schwartz output decay is supplied by Lemma 2.4; it imposes no increasing regularity condition on the original packets.□

Proof. Write S=\sum_Ts_T^2. By Proposition 2.6 with b_T=s_T and p=q_k,

At a witness, B\leq1 and 2-q_k<0; the unit-supremum version of the weighted estimate supplies the inequality for arbitrary witness positions. Consequently any range N\leq(2+R)^D, with D fixed, already satisfies (2.27). We may assume N exceeds a sufficiently large fixed power of 2+R.

Discard the packets with s_T>C R N^{-k/2}. Their total field at a witness has modulus at most

Choose C large, so the remaining field retains half the threshold. Its coefficients have maximum s_0=C R N^{-k/2}. Put P_1=(C(2+R))^{10}, partition the frequency patch into bins of side P_1/N, and write the remaining field as \sum_bV_b. There are O((N/P_1)^k) bins. Cauchy–Schwarz therefore gives

In this square sum remove all ordered pairs of frequency labels whose separation is at most C'/N, where C' is a sufficiently large fixed constant compared with the native Fourier box. The removed expression is pointwise bounded by CB^2: each frequency slot has only boundedly many such neighbors, and the sum over the spatial and temporal labels is bounded by Cauchy–Schwarz with the summable packet envelopes. It is therefore O(1) at the witnesses. Let G=\sum_bG_b be the sum of the remaining products. Choosing the constant in P_1 large makes (2.29) imply |G(z_Q)|\geq1.

We verify the required spacetime almost orthogonality. If v_b is the center of a bin, the Fourier support of G_b is contained in

To see this, subtract the two cap Fourier coordinates used in the proof of Proposition 2.6. Removing the C'/N pairs guarantees the lower bound on |\zeta|. The deviation from the tangent plane at v_b is the product of two O(P_1/N) quantities, together with the O(N^{-2}) normal errors. For a fixed (\zeta,\sigma), the bin centers lie in a slab of thickness CP_1^2/(N^2|\zeta|). Counting their grid of mesh P_1/N gives overlap at most CN^{k-1}P_1^{C_k}. Plancherel yields

Within a bin, the effective pointwise packet multiplicity is O(P_1^k). If A_b=\sum_{T\in b}s_T(1+d_T)^{-P'}, then |G_b|\leq A_b^2, A_b\leq CP_1^ks_0, and A_b^2\leq CP_1^k\sum_{T\in b}s_T^2(1+d_T)^{-P'}. Integrating these inequalities proves

All Fourier frequencies of G have size at most CP_1/N in the k+1 spacetime variables. Cover spacetime by cubes of side L=N/P_1^{C_2}\geq1, choosing C_2 fixed and sufficiently large. The rescaled version of (2.9), summed over this lattice, shows that the number of these cubes with \sup|G|\geq1 is at most

Each such cube is contained in an N-ball. If it contains a witness, use that witness as the envelope reference in Proposition 2.7; its unit-supremum estimate bounds the number of high unit cubes in the ball by C R^{q_k}. Multiplying the last two bounds proves the lemma.□

Proof of Theorem 2.11. Fix the frequency box, Fourier enlargement, phase-space bound and original regularity class once and for all. Call an exponent p admissible if |Y|\leq C_pR^pS for all configurations in this class. Fixed changes of multiplicity, amplitude normalization, or cell enlargement change constants and do not change the infimum of admissible exponents. By Lemma 2.12 this infimum p_* is finite. We prove p_*\leq q, where q=q_k.

Suppose instead that p_*>q. There are configurations and parameters R\to\infty with

Indeed use a failing exponent p_*-\epsilon_j and let \epsilon_j\downarrow0, with R large enough to absorb all constants for the finitely many estimates already chosen. Bounded R cannot give such failure by Lemma 2.12. Equation (2.28), used with each fixed positive \eps, now implies

All subsequent R^{o(1)} statements have this finite-parameter meaning: first fix positive exponent tolerances and finitely many tests; then take R sufficiently large, and finally let the tolerances tend to zero. In particular an admissible exponent just above p_*, rather than the possibly inadmissible exponent p_* itself, is used in every application below.

Removing excessively occupied fine packets. Fix C_1 large. Delete a fine packet if its CR native enlargement contains more than R^{C_1} witness points. Double counting with B(z_Q)\leq1 gives

Apply any fixed admissible estimate of exponent p>p_* to the deleted subfield, with threshold a small fixed multiple of N^{k/2}/R; its square envelope is bounded by the original one. The number of points where this subfield reaches that threshold is at most C R^{p+P-C_1}|Y|. Choose C_1>p+P+10. Except on a vanishing fraction of Y, the remaining field still reaches a fixed multiple of the old threshold. Constants of this sort are absorbed by changing R by a fixed factor. Every surviving fine packet now has occupancy at most R^{C_1} in its CR enlargement.

Coarse assembly and saturation of a ratio class. Choose a fixed A much larger than C_1 and all preceding fixed exponents, and put

Round to nested grids within bounded factors. Equation (2.33) ensures K\to\infty faster than every fixed power of R. Assemble the surviving fine fields into K-packets by Proposition 2.10. In the sampling definition of their backgrounds use one slightly larger, fixed fine weight exponent P+\sigma, \sigma>0:

The original decay margin permits this choice. Apply Propositions 2.6 and 2.10 with this exponent for the fine estimate and with the original exponent P for the outer parent envelope. By Lemma 2.9,

The ratio satisfies a_T/b_T\leq CL^{k/2} by pointwise Cauchy–Schwarz in the L^k fine frequency slots and summability over their native centers. The global contribution of ratios less than L^{k/2}R^{-2} is at most C K^{k/2}L^{k/2}R^{-2}B, again by Cauchy–Schwarz, so is negligible at the witnesses. Only O(\log R) dyadic ratio classes remain. By the triangle inequality one class

retains threshold N^{k/2}/R^{1+o(1)} on at least R^{-o(1)}|Y| points. Use this class henceforth. Its amplitude-square envelope is O(r^2), and (2.35) bounds its energy by R^{o(1)}r^{2-q}E. After dividing its field by Cr, its effective threshold parameter at scale K is R^{1+o(1)}rL^{-k/2}. This parameter is bounded below by a fixed constant whenever the threshold is attained, by the pointwise Cauchy bound. Apply an admissible exponent tending to p_*. Since r and L are fixed powers of R, its small exponent error is absorbed in R^{o(1)}, giving

Here kq/2=k+2. Comparison with (2.32), and the upper bound r\leq CL^{k/2}, force

The same fixed packet class remains available at this coarser scale: the inherited Fourier enlargement shrinks by L^{-1} in parent coordinates, whereas the parent localizer adds a fixed amount. A sufficiently large initial enlargement is consequently preserved as L\to\infty. Its Schwartz profiles have the originally fixed required bounds after constant renormalization. This verifies the uniformity needed to use the same p_*.

Localizing the selected square mass. We detail the restrictions used to draw a packet through a point. First discard globally all selected labels with b_T>R^3K^{-k/2}. Since the packet decay exceeds P,

at the witnesses. By (2.37) this is negligible relative to N^{k/2}/R. This global estimate, made before restricting distances, avoids any unsupported estimate of field tails from a square bound alone.

The remaining selected square envelope is at least R^{-o(1)} on R^{-o(1)}|Y| retained points. To verify this, fix \eta>0 and consider points where that envelope squared is at most R^{-\eta}. Normalize the field there by rR^{-\eta/2} and apply an admissible near-Cauchy exponent. Compared with (2.36), the threshold parameter gains R^{-\eta/2} and the coefficient-square sum costs R^\eta. The count of these points is therefore at most

Since p_*>q>2, this is negligible compared with (2.32). Letting \eta tend to zero along the finite-test diagonal proves the asserted lower bound.

For any fixed small \eta>0, Lemma 2.9 gives, term by term in the original fine square sum,

For fine distance a\gg1, the second term comes only from parent centers moving to within a/2 of the fine core; they require parent distance \gtrsim aN/K and cost R^{-A(P-k-2)}a^{-P}. For bounded a only the first term occurs. Thus, choosing the square-mass tolerance smaller than these fixed gains, we may restrict to d_T(z)\leq R^{o(1)} and retain square mass R^{-o(1)}.

We may also require each such selected T to have a surviving fine packet of its cap within R^{o(1)} fine native distance of z. Indeed, if all its fine packets have distance greater than R^{c\eta}, then parent centers at distance at most R^\eta from z still have those fine distances greater than \tfrac12R^{c\eta}, since the displacement in fine coordinates is O(L^{-1}R^\eta). The extra fixed \sigma in (2.34) supplies R^{-c\eta\sigma} when it is dropped back to exponent P. Sum against the parent weights and use Lemma 2.9. These unattached labels have negligible square mass. This is why one larger exponent was chosen in the sampling definition; no stronger square-envelope hypothesis at the original witnesses is needed.

Let I_0(z,T) denote the resulting incidence relation. With

we have, on the retained set still called Y,

The upper bound follows by removing w_T at distance R^{o(1)} in (2.35); the lower bound follows from the localized square mass. Draw T at z with probability b_T^2I_0(z,T)/m_0(z) and attach one of its nearby surviving fine packets. At a fixed z there are only R^{o(1)} local parent cells in each angular 1/K slot. Consequently its angular law \mu_z satisfies

Typical directions and resampling. Restrict the draw to directions v satisfying

This removes O(R^{-1+o(1)}) probability, without a logarithm of the number of scales. Here is a finite covering proof. In a fixed finite collection of shifted dyadic grids, restrict attention to cells meeting the bounded frequency box, with side between c/K and a fixed constant. Select the maximal cells in this finite collection for which \mu_z(Q)<cR^{-1}|Q|. Within each grid these cells are disjoint, so the sum of their masses is at most CR^{-1}. A ball failing (2.40) contains a cell of comparable side through its center in one of these grids; that cell belongs to a maximal bad cell. The bounded number of grids and fixed comparison factors are absorbed by the indicated subpower tolerance. Renormalizing the restricted law costs a bounded factor. Let I(z,T) be the resulting incidence relation and m(z)=\sum_Tb_T^2I(z,T); (2.38) remains true.

Fix z\in Y. For another point y\in Y write \tau=|t_y-t_z|. If \tau\geq N, the probability, under the restricted draw, that y is in the same R^{o(1)}-enlarged K-cell as z is at most

Indeed its directions lie in a ball of radius KR^{o(1)}/\tau; (2.39) counts the slots in that ball. A nonempty incidence also requires \tau\leq K^2R^{o(1)}, so the lowest angular precision 1/K costs only a further R^{o(1)} at the endpoint of this range.

If this probability is nonzero, choose a typical direction v that sees y. In the original draw, the ball of radius cN/\tau about v has mass at least

by (2.40). The radius lies between 1/K and a fixed constant: \tau\geq N gives the upper bound, and \tau\leq K^2R^{o(1)} together with N/K=R^A gives the lower one. Every draw in that ball has an attached fine packet which sees both z and y in its CR enlargement. To check this last statement, its intercept at z differs by at most NR^{o(1)}; the change of angular direction over time \tau costs at most cN; and the discrepancy between a parent label and a fine label is O(1/K), costing at most \tau/K\leq KR^{o(1)}\ll N. The time separation is also much less than N^2, whereas z was already at fine native distance R^{o(1)}. All these errors are smaller than the available CR enlargement.

Let P_f(y) be the probability that the attached fine packet in the original draw sees y in that enlargement. Comparing (2.41) and (2.42) bounds the expected number of far points in the sampled coarse cell by

The last inequality uses the fine occupancy cutoff: every attached fine packet sees at most R^{C_1} points, so \sum_yP_f(y)\leq R^{C_1}. This is a comparison with the original draw, not an independence assertion about a direction and a point.

For \tau<N, a point in the same enlarged coarse cell lies in a ball of radius CN about z, since the velocities are bounded and KR^{o(1)}\ll N. The local linear estimate at z, or a bounded cover by N-balls and envelope comparability, bounds the number of such high points by CR^q. Choose A so large that Ak>C_1+q+10. Combining the near and far parts, the expected occupancy of the sampled enlarged coarse cell, for every retained z, is at most

The contradictory lower occupancy. Put n_T=\#\{z\in Y:I(z,T)=1\}. This is no larger than the geometric occupancy of the enlarged coarse cell. Average that occupancy first over the draw at z, then uniformly over z\in Y. By (2.38) and Cauchy–Schwarz, the result is at least

The middle inequality uses \sum_Tb_T^2n_T=\sum_{z\in Y}m(z)\geq R^{-o(1)}|Y|. All selections retained a subpower fraction, so the last inequality still follows from (2.32). This contradicts (2.44) because p_*>q. Thus p_*\leq q, and every q+\eps is admissible, proving (2.26). The unit-supremum and fixed-enlargement versions used in the proof follow from Lemma 2.4 and bounded changes of the lattice.□

Proof. Use a fixed enlargement of the test interval and a cutoff equal to one there. Fourier transform a compact native profile in its spatial variables and insert a smooth partition into unit frequency boxes centered at a\in\Z^k. In the ath box remove the quadratic evolution factor, and expand the remaining smooth function of native time in a Fourier series on the enlarged interval. This writes the profile as a sum, indexed by (a,n)\in\Z^{k+1}, of exact extensions times common time modulations. Integration by parts in the Fourier coefficient formulas shows that, for prescribed finite J,M,

provided finitely many original derivatives are bounded. The physical frequency center changes by a/h. All intercepts are specified at one common reference time. Recentring about the shifted trajectory on the test interval changes a polynomial envelope by at most a fixed polynomial in 1+|a|.

For completeness, normalize each layer by c_{a,n}=(1+|a|+|n|)^{-M/2} and allocate threshold fractions proportional to (1+|a|+|n|)^{-M/4}. Their sum is finite when M is large. If a sum has magnitude at least A_i, one of its normalized layers exceeds the corresponding allocated threshold divided by c_{a,n}. Apply the exact packet estimate to the tuples of these layers. In its ith energy term the threshold dependence is

Every threshold has a strictly negative power; hence, after taking M large, the resulting mode sums converge, including the polynomial envelope losses. Modes |a|>h^{1/2} contribute less than any specified negative power of h by weighted Cauchy and the same decay. For the remaining modes transversality is preserved when \delta\ge h^{-\eps_0} and \eps_0>0 is sufficiently small. If \delta<h^{-\eps_0}, the polynomial bound in h from the elementary packet estimate is absorbed by \delta^{-C}. This also disposes of the discarded tail.

An exact packet outside its central time bin has only polynomial growth in native time in each of the finitely many spatial localization seminorms: differentiate the compactly supported frequency integral and integrate by parts. Multiplication by a common positive-on-the-slab Schwartz cutoff makes all relevant spacetime integrals finite. Choose its decay after those seminorms. The whole argument therefore uses a fixed finite derivative order. The initial native-time cutoff used for the Fourier expansion is distinct from the final compact-Fourier multiplier \chi_h(t)=\chi((t-t_0)/h^2). The latter is bounded above and below by positive fixed constants on the test slab. Multiplying the exact-layer sum by it therefore changes the witness thresholds and packet bounds by fixed factors, which suffices for the reduction.□

Proof. The positive smoothing functional. We give separately the positive smoothing estimate and its comparison with packet sizes. The original selections and deletions are left unchanged and applied externally, and \chi_h remains in every actual field. Only the auxiliary positive density below uses the uncut original cap field U_b. Fix a sufficiently large integer J\ge1, independent of all later grids, and put

Spatial unitarity and the original fine-packet Gram bound give

Here c_J=\int_{\R}(1+s^2)^{-J}\,\dd s<\infty. Indeed the cap projections have bounded overlap, and integration by parts in the original normalized frequency data gives summable Gram rows for the fine packets, using finitely many prescribed localization seminorms. This charges the original data before every transition mask. No division by \chi_h is used. Choose an even Q sufficiently large compared with the original envelope order P and k, and put j_K(x)=c_QK^{-k}(1+|x/K|^2)^{-Q/2}.

Convolve in space with heat variance K^2 and with independent symmetric jumps having density j_r and intensity \dd r/r at scale r\le K. The limiting small-jump distribution exists because \int_0^K r^2\dd r/r<\infty. Denote the resulting kernel by J_K. It has

Here is a useful direct verification of the polynomial tails. For the lower bound request one jump in [K/2,K] and all other displacement at most CK, an event of fixed positive probability by the second moment bound. The density of that jump is comparable to j_K after such a displacement. For the upper bound group jumps into [2^{-j-1}K,2^{-j}K], with Poisson counts m_j. If their total plus the Gaussian has magnitude comparable to |x|\gg K, either the Gaussian has such magnitude or one of the jumps in group j has magnitude at least c2^{-j/2}|x|/(1+m_j). Conditioning on the other variables and summing candidates bounds its density by

Poisson moments are bounded uniformly in j and the series converges for Q>2k. The same argument after differentiation of a slightly broader Gaussian proves the gradient bound. The Fourier bound follows from the heat factor and the characteristic function of a probability measure.

In the coordinates moving with 2b, write \partial_{t,b}=\partial_t+2b\cdot\nabla_x, and also convolve in time by a Gaussian of width

for H\le K\le h, using any increasing continuation below H. Write \rho_{K,b},F_{K,b} for these convolutions. The spatial generator in logarithmic scale is, up to fixed heat conventions, L_K=K^2\Delta_x+j_K*-I.

Generator bounds. We claim that for some c'>0

Indeed (3.7) and the Gaussian identity for Fourier ball mass give

To see the normalization, the local mass is at most S^Cr^k(H^2+r) and the smoothing volume is r^{k+1}. For r<H the same bound loses only a fixed polynomial in H/r. On |\xi|\asymp1/r the squared spatial multiplier is bounded by C_N(K/r)^2(1+K/r)^{-N}. For r\le T, the time Gaussian restricts |\tau+2b\cdot\xi|\lesssim T^{-1}, with rapidly decreasing dyadic tails. Of the O(H^k) cap centers, a fraction at most C(r/T+H^{-1}) obeys this restriction; this is the elementary count of an H^{-1} grid in a slab. The corresponding spacetime Fourier ball has radius O(r^{-1}). For r\ge T use radius O(T^{-1}) and the trivial cap fraction one. At r=K the resulting factor is

The dyadic sums for r<K, K<r<T, and r\ge T converge, after choosing N large; enlargements in the Gaussian tails cost only fixed powers. This proves (3.11), also with the heat and jump multipliers treated separately.

The exact equation supplies a second estimate:

After carrier removal U_b has frequency support O(H^{-1}), and its exact equation gives

Move this divergence to J_K. The required weighted Bernstein bound is

To prove it, remove the carrier and reproduce with a smooth projection kernel R_H=H^{-k}R(\cdot/H). Polynomial comparison of translates of j_K and weighted Young’s inequality give operator norm at most

Together with (3.10) and Cauchy–Schwarz this bounds the current by C(HK)^{-1}(J_K*_xF_b). The sole time-weight term satisfies |w_h'|\le C_Jh^{-2}w_h, and h^{-2}\le(HK)^{-1}. Positive comoving time convolution preserves the inequality. The argument requires only compact-frequency L^2 data for the summed U_b; smooth approximation in that fixed cap justifies the current identity and the limiting convolutions.

Propagation of the clipped bound. Put \phi(u)=M_c(1-e^{-u/M_c}) and I(K)=\sum_b\int\rho_{K,b}\phi(F_{K,b}). For any symmetric diffusion or jump generator L, concavity gives \phi'(F)LF\ge L\phi(F). Thus differentiating I gives a lower bound consisting of twice the generator pairing with \rho. For the spatial part Cauchy–Schwarz, (3.11), and \phi(F)^2\le M_cF yield

where we used D\le S^C\|\rho\|_1 from (3.6). Notice the cancellation of the factors H^{k/2} and M_c^{1/2}. For the time part integrate by parts once. The Gaussian gives \|T\partial_{t,b}\rho_{K,b}\|_1\lesssim\|\rho\|_1, while (3.12) and u\phi'(u)\le CM_c give cost CM_cT/(HK) per cap. There are O(H^k) caps, so the time error is at most S^C\|\rho\|_1/P_K. Both errors integrate from K_0 to h to at most

when L is fixed sufficiently large. We used H\ge h^{2c/k} and S=h^{o(1)}. Smooth regularization of \rho and piecewise smooth approximation at K=H^2 justify the differentiation; the positive convolution formulas allow passage to the limit.

At K=h, drop clipping. Convolving twice averages w_h|U_b|^2 on spatial width h and time width at most Hh along 2b. The original transition masks remain outside this estimate. For a witness z=(x_z,t_z), the Gram bound for the uncut original fine packets is

The relative displacement in width-h units is O(|\tau|/(Hh)), and elapsed native time is at most C+|\tau|/h^2\le C+|\tau|/(Hh). The prescribed finite localization seminorms of the individual original packets grow polynomially in this variable. Spatial almost orthogonality and summable Gram rows therefore prove (3.13), retaining the original exponent on w_T by choosing the auxiliary kernel order Q larger. Since w_h\le1 and T(h)\le Hh, the Gaussian moments give

At K_0, kernel comparability in space and (3.12) in comoving time show that F_{K_0,b} varies by bounded factors in its central smoothing box. Therefore the extra convolution against \rho dominates a constant multiple of \sum_{z\in Y}H^{k+2}\sum_b\min(F_{K_0,b}(z),M_c). We have proved that this last clipping sum has bounded mean on Y.

Recovery of packet envelopes. We now bound the actual square envelopes by the smoothed auxiliary density on the test interval. This also fixes the required finite orders. Partition the exact data of U_b into smooth caps of width K_0^{-1}. Form the actual packets by localizing \chi_h P_vU_b in their scale-K_0 moving coordinates, and retain the original H masks externally. Since K_0\le h, fixed native derivatives of \chi_h are bounded and its temporal Fourier thickness is at most O(K_0^{-2}). Use weighted suprema as packet sizes. Local constancy, |\chi_h|^2\le C_Jw_h, and bounded overlap of the spatial projections give

Thus (3.9) supplies the total integrated budget CD for this representation of the same actual cutoff field.

Here is an explicit fixed-order comparison. Work in native scale K_0=1 for a cap, write its uncut carrier-removed solution as q_v(u,s), and take

A smooth larger frequency projection has propagation kernel |L_\sigma(r)|\le C_N\braket{\sigma}^{N}\braket r^{-N}. Consequently, for every fixed s_*,

The bounded multiplier \chi_h and its fixed native derivatives only change the constant when estimating the actual packet sizes from this uncut field. Choose the size-supremum exponent at least \max(P'_{\rm work},2N) for some fixed P'_{\rm work}>P_{\rm work}. For the packet centered at (a,m)\in\Z^{k+1} its size then obeys

Summing with the work packet weight of order P_{\rm work} first in m leaves time decay P_{\rm work}-2N-1=N+2, and then summing in a retains spatial decay at least N. Thus the packet envelope at z is bounded by a spatial j_{K_0}-weighted energy at any time |t_*-z_t|\le T(K_0). Here T(K_0)\le K_0^2 and the relative velocity displacement is O(T/H)\le CK_0.

For these cap energies use weighted Bessel:

One proof avoids a nonsummable bound for each projection. With an even polynomial weight, differentiate the cap symbols and use their bounded overlap and Plancherel to bound synthesis (g_v)\mapsto\sum_vP_v^*g_v in positive polynomially weighted L^2(\ell^2)\to L^2. Duality gives the displayed negative-weight analysis estimate. A partition into boxes and summable convolution extends it to translates of j_{K_0}. Apply it to U_b at the fixed time t_*. Average the preceding packet comparison over the central part of the time Gaussian. For z on the test slab and |t_*-z_t|\le T(K_0)\le h^2, the polynomial weight w_h(t_*) is bounded below by a fixed constant. No lower bound for \chi_h outside the test slab is needed. Hence e_b^2(z)\lesssim F_{K_0,b}(z) on the test interval. Markov’s inequality, with any slowly diverging subpower threshold, now proves (3.8).

When K_0\le h, these reconstructed packets are the leaves of the subpower tree: no original scale-h fine squares are inserted below them. Their rapid output decay can be fixed above P'_{\rm work}, and the same work exponent is used in each later application of Propositions 2.6 and 2.10 and Theorem 2.11. The original order P enters only the input square hypothesis and the original-packet Gram bound used at K=h; no comparison of a reconstructed fine array with the original square envelope is asserted. If K_0>h, use the unchanged original scale-h representation as the leaf, with order P. Its square envelope is bounded at the witnesses, so clipping is automatic. All orders are chosen before the finite scale grids and the finite original derivative order in Lemma 3.2; they depend on k,c and the original envelope order, not on hierarchy depth.□

Proof. Fix a small number \beta>0 which bounds the exponents of all norm losses, fixed-grid constants, logarithmic sums, and initial balance errors. Work far enough along the sequence that these losses are at most \Lambda^{C\beta}. Choose the distance cutoff exponent next, the eligibility and enlargement exponents after that, and the occupancy cutoff exponent last. Each is larger than the fixed comparability multiples of its predecessors. They will all tend to zero, in this order, along a slow diagonal.

First prove the discard statement, without assuming selection or saturation. For an error U with actual amplitudes and budget Q_U, Propositions 2.6 and 2.10 through its masks give, in the power case,

One can jump directly from a current layer to the root: no not-yet assembled intermediate masks are needed, and any transition restriction already fixed remains imposed. In the subpower case propagate to the gap edge. Sort the actual output amplitudes there by their ratios to the propagated weights. Cauchy bounds the nonnegligible ratios by fixed powers of S, since only finitely many scales are involved and the clipped square is bounded at the transition. Theorem 2.11 across the gap gives (3.20) with a further S^{C\eta}, where \eta>0 is its arbitrarily small exponent slack. Choose \eta after the desired discard saving. Very small ratios have negligible contribution by absolute Cauchy, and larger ratios have summably small weighted budget. Thus Q_U\le\Lambda^{-\eps}D_i allows both a vanishing threshold fraction and a power-small exceptional set, by choosing \beta,\eta and the exponent in t sufficiently small relative to \eps. This proof uses actual error amplitudes throughout.

Assemble from leaves toward the root, with the original transition masks fixed. At each stage Cauchy gives a polynomial upper bound for the amplitude-to-weight ratios. In the subpower case this includes the quotient \kappa_i(s)^{k/2}/A_i, which is a fixed power of S at each fixed scale. Ratios below a sufficiently small inverse power are harmless by Cauchy and the discard estimate. There remain O(\log\Lambda) dyadic classes. Select a class for the sum actually reaching the root before packetizing a smaller inverse width. A finite number of selections costs \Lambda^{o(1)} in thresholds and high set size.

Here is the occupancy deletion at this same stage. If n_T counts eligible witnesses in the prescribed enlarged cell, local weight comparison and (3.8) give

where \beta' accounts for the already chosen enlargement and eligibility. This also holds with full squares in the power case. For a candidate ratio class r delete cells with n_T>\Lambda^\gamma M_s, choosing \gamma>\beta'+C\beta. Their weighted charge is

Use (3.2) or (3.15) in the last inequality. The discard estimate therefore permits this pruning before the final selection at that layer. Similarly, a size supremum realized only at distance \Lambda^\gamma from the packet center carries a negative power budget if its local amplitude is power-smaller: retain an extra fixed decay margin in the supremum and apply the same estimate.

The upper bound for Q_{i,s} follows from the weighted norm estimate through the already assembled masks. The reverse follows by applying (3.20) to the entire selected field, which still has its root high set. Since a_T\asymp r_{i,s}b_T, the formulas for E_{i,s} and \sum b_T^2 follow algebraically. Taking the diagonal of the specified slacks proves the displayed subpower assertions.□

Proof. For s<s', weighted Cauchy between the two layers gives

Conversely, integrated spatial L^2 orthogonality gives E_{i,s}\le C_{\rm grid}E_{i,s'}. To justify it with masks, first remove the outermost mask by its bounded sup norm; then separate its immediate children by their bounded Fourier overlap. Repeat in this order. One never commutes all outer masks past a final fine frequency decomposition. Polynomial spatial weights can be retained using translated band-limited cutoffs. Formula (3.18), and q>2, now give r_{i,s}\ge\Lambda^{-o(1)}r_{i,s'}. At the fine endpoint the ratio is one by the choice of the input sizes.

In the power case root witness sampling and local constancy give |Y|A_i^2\lesssim\Lambda^{o(1)}E_{i,s}. Together with saturation and |Y|=\Lambda^{o(1)}D_iA_i^{-q} this gives r_{i,s}\le\Lambda^{o(1)}A_i. The reverse endpoint comparison comes from root Cauchy, A_i\le\Lambda^{o(1)}\kappa_i(s)^{k/2}r_{i,s}, as s\downarrow0. The same root Cauchy in the subpower case gives f_i(s)+ks/2\ge0. The fine endpoint and the Lipschitz bound give the other inequality in (3.21).

On each finite grid the logarithms are uniformly bounded and have these two one-sided Lipschitz bounds up to vanishing errors. A diagonal subsequence on rational parameters gives the asserted continuous profiles. Every later test involves only a fixed finite grid: include its parameters first and take the subsequence far enough that its logarithms approximate the limiting values to the desired accuracy. Grid complexity can equivalently increase sufficiently slowly. No rate of convergence to a particular slope is assumed.□

Proof. We first show that the available high points have enough surviving path mass to normalize the law. The weighted norm estimate applied afresh to this restricted tree gives a budget at most \Lambda^{o(1)}D_i, with its actual modified amplitudes. At the root its integrand is

The number of high points where W\le\Lambda^{-\alpha} is at most \Lambda^{o(1)-\alpha(q-2)/2}D_iA_i^{-q}. Across the subpower gap the ratio-class argument in the proof of Lemma 3.4 adds only S^{C\eta}; choose \eta smaller than \alpha(q-2)/(4C). Thus low path mass is exceptional. Full-background convolution and the transition eligibility test give W\le\Lambda^{o(1)}.

Use a fixed extra polynomial decay power on each path edge. A path with an intermediate center farther than \Lambda^\alpha from z has a comparably large normalized edge displacement, since the inverse widths increase toward the leaves and the nested-cap shears are bounded in the appropriate coordinates. Dropping the extra power on this edge still leaves a convolution bounded by the full background; hence such paths have mass o(W(z)) after the fixed extra margin is chosen larger than the path-mass lower-bound exponent requires. This uses the same fixed margin on each edge, not an increasing endpoint weight exponent. Redefine J_z and W to count only the retained near paths, and normalize by this new W. Relabel the final retained root set as Y; its cardinality differs by only a subpower factor from the preceding set. Every retained path is eligible at its transition bin: its surviving transition-mask center has e_b^2\le M_c, and proximity to that center and polynomial weight comparison give e_b^2(z)\le\Lambda^{o(1)}M_c. On the near paths, telescoping local weight comparisons and the subpower neighboring-cell multiplicities give

where a_\omega is the corresponding available leaf square weight. If a final direction label sums many final input cells, only its last localization layer and intermediate active cells need be near z; the input coefficients are summed in a_\omega.

Summing this upper bound over descendants proves the upper half of (3.22). For its lower half, at any fixed root point the total probability of sampled labels satisfying p_z(\tau)<\Lambda^{-\eps}b_\tau^2(z) is at most \Lambda^{-\eps}\sum_\tau b_\tau^2(z)=\Lambda^{-\eps+o(1)}. This is a statement at sampled labels, not a uniform lower bound for all labels.

Here are both sides of (3.23). For a fixed leaf direction \omega and a tilted s cell C, put

At the realized pair (z,\omega) the ratio in question is exactly

For the upper bound, all contributing active s cells lie in a subpower enlargement of C. The occupancy deletion counts at most \Lambda^{o(1)}M_{i,s} eligible points there. The leaf background varies by a subpower factor within the tilted cell, and path domination and the typical leaf lower bound compare each summand to p_z(\omega). This proves the upper bound.

For the lower bound, let J_T^{\rm dom}(\omega) be the unnormalized sum of positive descendant-path weights from an active scale-s mask T, evaluated at its center z_T on the selected tree before angular deletions. Retain the transition-center restriction and proportional clipping, and sum all terminal physical cells with direction label \omega. Later macro-dependent omissions give subcollections of this one majorant, without creating extra copies. Positive convolution gives \sum_\omega J_T^{\rm dom}(\omega)\lesssim C_{\rm grid}b_T^2. At the transition this uses both the full-square convolution bound and the bound by C_{\rm grid}M_c from retained centers; take their minimum. At finer cells use the full descendant weights.

Let \mathcal N_s(\omega,C) consist of active masks in the assigned s-cap whose \Lambda^\eta native enlargements meet the tilted s-cell C, and put

This reference charge is used only for domination; the actual law p_z(\omega)=J_z(\omega)/W(z) is normalized using the fresh surviving mass of the restricted tree. The near-ancestor kernel sum and W(z)\ge\Lambda^{-o(1)} give the following pointwise bound. For fixed (\omega,T), only \Lambda^{o(1)} cells C occur, because the two scale-s frames differ by a bounded normalized shear. Consequently

At a strict transition ancestor, eligible contributions are comparable at the charging center up to the prescribed subpower factor; unique label ancestry charges each clipped bin only once. The total probability of fibers satisfying

is at most \Lambda^{-\eps+o(1)}. For every other sampled fiber, (3.24) proves the lower bound. Replacing boxes by balls or bounded-overlap coverings changes these sampled logarithms by o(1): the total probability of a cell power-smaller than one of its boundedly many neighbors is power-small. The same argument handles subpower distortions.□

Proof. Use as reference direction law the endpoint square weights at the center of C, normalized within \tau_l. Clip whole transition bins proportionally if \tau_l is a strict ancestor. The drift assumption makes these reference weights comparable throughout C. By (3.22) and path domination, the conditional density of the direction relative to this law is at most \Lambda^{o(1)} at typical data. Its average conditional density is at least \Lambda^{-o(1)} at a typical sampled direction: the exceptional mass is bounded by summing reference weights exactly as in the lower half of (3.22). Bayes’ identity now bounds the likelihood ratio for position by \Lambda^{o(1)}.

For clarity, the remaining one-sided information comparison requires no independence assumption. If p and p' are the probabilities of a realized finite label before and after additional conditioning, then \mathbb E[p/p']\le1. Markov’s inequality implies p/p'\le\Lambda^\eps outside probability \Lambda^{-\eps}. Apply this to each member of a finite chain of partitions. Combining it with the reference-law comparison proves equality of the relevant information increments to arbitrary logarithmic tolerance.□

Proof. Suppress the field index. In the native coordinates of \tau_l, use macro boxes of side \Lambda^{3u}. The endpoint descendant weights, divided by the weight of \tau_l, vary within bounded factors on these boxes: all affected cell widths and durations are at most \Lambda^{2u+o(1)}, and the endpoint is separated by Cu. If a transition ancestor occurs, clip each entire bin proportionally.

For each mesh value v, greedily remove all (l+v) caps intersecting an enlarged hyperplane neighborhood whose reference measure is at least \Lambda^{-2\eta v}. There are at most O(\Lambda^{2\eta v}) choices, and the residue has the desired upper bound before normalization. Carry out the omissions from coarser to finer mesh values, so earlier omissions only restrict subsequent masks.

We verify that their contribution at level l is cheap. Let q'=2(k+1)/(k-1)>q, the lower-dimensional exponent. The original selected budget at l+v is bounded by

This follows from the recorded ratio class, before any modification. The plate-supported version of Proposition 2.6 at exponent q' propagates each omitted hyperplane contribution through all intervening masks. Plate support is inherited by each descendant in its own normalized coordinates. Summing the omitted planes costs at most \Lambda^{2\eta q'v}. On the active l masks, Hölder with measure b_T^2|T| and \sum b_T^2|T|\le\Lambda^{o(1)}Dr_l^{-q} gives

Choose 2\eta<x(q'-q)/(2q'). The profile assumption then makes this a negative fixed power budget, and Lemma 3.4 discards it at the root. Earlier restrictions are harmless: use actual modified amplitudes when propagating, and bill the initial restricted subfields to their original fine budgets.

We include the localization justification for the fact that the direction subsets can vary by macro box. Assign each l packet by its center to such a box and alter the field before its Schwartz packet cutoff. No rough physical cutoff is inserted. At a single hierarchy step, packets centered away from a target macro box have an arbitrarily large fixed inverse polynomial box-distance gain, by their cutoff and supremum tails. The denominator weights may be left in the child coordinates, using Proposition 2.10; one need not compare them across an entire macro box. These gains form a summable convolution in macro offsets. An equivalent algebraic formulation is useful. For a summable polynomial macro weight \Psi and exponent p>2, replace

Then a_T^pb_T^{2-p} acquires exactly the factor \Psi(z_T). Polynomial comparability of \Psi requires only a fixed additional edge-decay margin, since all affected widths are below the macro scale. Use this for p=q,q' at each step; the same margin suffices at every depth. Summing the target-box estimates bills each initial packet with bounded total weight. This proves the stated norm charge for varying direction restrictions, and also proves the fresh restricted-weight norm bound used in Lemma 3.6.

For broad witnesses, use separately macro boxes of side \Lambda^{3v_0} before modifying directions. In each box mark a group at level l+v_0 if its original contribution through the frozen intervening masks has supremum at least \Lambda^{-C_kv_0}r_lb_{\tau_l} in a fixed enlargement. There are at most \Lambda^{kv_0+o(1)} groups, so for C_k sufficiently large the unmarked sum cannot supply a high witness defining an active l packet. The far-supremum deletion in Lemma 3.4 ensures such a witness occurs within the stated enlargement.

The elementary maximal-simplex test gives a dichotomy for the marked centers: either k+1 of them have successive affine widths at least a constant times \Lambda^{-v_0}, or all marked centers lie in a constant enlargement of a hyperplane strip of that width. In the first case the determinant is at least a fixed power of \Lambda^{-v_0}; enlarge the separation constants so it holds uniformly on the chosen supports. In the second case the same q' calculation shows that the marked contribution is cheap at level l. Thus the nonbroad masks themselves have negligible weighted q budget and may be deleted first.

Finally reconstruct the positive path law for the modified fields, as in Lemma 3.6. Low total path mass is exceptional by its fresh norm bound. Its conditional law is dominated by the surviving reference descendant weights divided by b_{\tau_l}^2(Z); the typical reverse cap comparison loses only a subpower. The heavy strip cutoff \Lambda^{-2\eta v} therefore implies the claimed \Lambda^{-\eta v+o(1)} angular bound. Occupancy upper bounds still refer to the same frozen masks, and the charging proof of (3.23) is unchanged. The broad witnesses used later are the recorded original subfields, whose selected amplitudes remain available for norm and square estimates.□

Proof. Every point under the cap conditioning is near an active l cell. Such a cell has at most \Lambda^{o(1)}M_l eligible points. Path domination, the typical reverse cap comparison, and background variation bound their relative conditional weights by subpowers. It therefore suffices to count active l cells in an isotropic ball. They have the original broad witnesses from Lemma 3.8. Pigeonholing the separated tuple costs \Lambda^{O_k(v_0)}, since there are only a fixed power of \Lambda^{v_0} possible labels.

Write \kappa_s=\Lambda^s in the l coordinates and, for the jth broad group, let \mathcal E_{j,s} be the local square sum of the original selected packet amplitudes with their native weights at relative scale s. The witness inequalities, including their allowed offsets, imply at every counted unit cell

Masks between l and l+v_0 have bounded absolute sums, and the native weights absorb the witness offsets. The background b_{\tau_l} is comparable over the larger test ball.

For s<t\le2s, average the resulting product of squares with exponents 1/k on a \kappa_t ball. Proposition 2.2 gives

Indeed these are nonnegative sums essentially constant on tubes of width \kappa_s and duration \kappa_s^2\ge\kappa_t. On the ball replace them by columns crossing the ball, where each envelope is comparable. Such a column costs a cross section of width \kappa_s times length at least \kappa_t in the enlarged-ball average. The quantitative determinant factor is \Lambda^{O_k(v_0)}. Weighted offsets are summed with the fixed integrable envelope powers.

We verify that each last average is bounded by the corresponding \mathcal E_{j,t}, even when intermediate mask levels are skipped. This verification prevents an aspect-ratio loss. Put S_0=\kappa_s and T_0=\kappa_t. Reproduction in the native s coordinates bounds the square of a weighted local supremum by a normalized weighted L^2 integral of the pre-cutoff field. After summing s cells, the displacements from an averaged point have the form

For each fixed displacement and time, integrate in space over width T_0, replacing the ball by a smooth polynomial weight. Remove the next mask by its bounded sup norm and separate its immediate child caps by spatial almost orthogonality; localization at width T_0 preserves bounded Fourier overlap. Iterate in the actual order of the masks. Translated band-limited cutoffs positive on boxes prove the same assertion for the polynomial weight. At the final layer, pointwise Cauchy bounds the fields by the t packet envelopes.

In the frame of a final t cap, the normalized shift just displayed has sizes

because its velocity differs from v_s by O(S_0^{-1}). Convolution with this contracted shift density preserves the order P of the final weight by the distance and volume comparison in Proposition 2.10. The auxiliary spatial ball weight has all the finitely many moments needed for this comparison. Thus no factor depending on T_0/S_0 is lost. One can use the same fixed P at each inflation step, with a fixed excess supremum margin.

At the terminal scale, the recorded ratio class and weight convolution give

Iterate (3.26) along a finite chain from v_0 to v, each step satisfying s<t\le2s. There are O(1+\log(v/v_0)) steps; covering larger balls by smaller balls at each step gives the cell-count bound

Since p_kx=(k+1)(1-d_0), this proves (3.25). The finite mesh can include the whole inflation chain from the outset, and v_0/u can be as small as needed before taking logarithmic limits.□

Proof. Suppose a differentiability point, away from the transition seam, has slope -x with 0<x<k/2. Choose a short rational block based at l very near this point and a finite relative mesh so that f(l+v)-f(l)=-xv+O(\eps u) throughout [0,2u]. Differentiability allows errors arbitrarily small relative to every mesh spacing. Put d_0=1-2x/k\in(0,1) and apply the angular test with its fixed positive exponent, together with the broad-witness test at a much smaller v_0.

Condition on \tau_l and Q_{2u}. Rescale this parent isotropically to bounded size and write \Delta=\Lambda^{-u}. The information of the fine isotropic cell is

Projecting parallel to a drawn direction gives information

Its inverse-image cells, clipped to the parent, are comparable to T_{l+u}: transverse width is \Lambda^u and length is \Lambda^{2u}. Thus Lemma 3.6 gives the second formula. Adding independent directions costs o(1) information by Lemma 3.7, because all directions in \tau_l drift by at most the endpoint width on this box.

Draw k+1 directions conditionally independently. The angular test shows, with probability tending to one, that their lifted vectors form a basis with inverse norm at most \Lambda^{O(\eps u)}. Indeed expose them successively; proximity to the span of the previous ones is an affine hyperplane-strip event in the frequency coordinates, and use one mesh precision much smaller than u. In this basis each projection records all but one coordinate, up to the same precision loss. Order the coordinates once and use the chain rule for their information. The information of a coordinate conditioned on all its predecessors is no larger, up to the sampled O(\eps u) slack of Lemma 3.7, than its information when one predecessor has been omitted. Each coordinate is counted k times in the k+1 projections. This is the finite-partition Loomis–Whitney information inequality, and gives

The sample form follows from Markov’s inequality for the finitely many conditional probability ratios, as in the proof of Lemma 3.7; no uniform pointwise entropy inequality is being asserted.

Let g(v)=\log_\Lambda(N_v/M_l). The ratio definition gives

The preceding inequality becomes

Lemma 3.9 bounds g(u) and g(2u) above by (k+1)d_0u and 2(k+1)d_0u, respectively. Their weighted sum is exactly the last right side. Hence both upper bounds are equalities up to O(\eps u).

We now apply the robust discretized projection theorem in Proposition 2.3, with ambient dimension k+1, projection dimension k, and \alpha=(k+1)d_0\in(0,k+1). We give the set extraction because exceptional points can depend on direction. Choose a cap and parent box with conditional success probability close to one for all the finite comparisons. Within this fixed cap and parent box, the probabilities of typical \Delta cells, without conditioning on the drawn direction, lie between \Delta^{\alpha+O(\eps)} and \Delta^{\alpha-O(\eps)}. Take their centers to form one set \mathcal A. It is \Delta separated and

For coarser balls, the mesh instances of (3.25), divided by the saturated parent occupancy, give the probability upper bounds with exponent \alpha. Passing from probability to number by the preceding individual-cell bounds gives precisely the nonconcentration inequalities for \mathcal A, with O(\eps) exponent tolerance. Intermediate radii follow by nesting between consecutive mesh values, whose ratio is chosen sufficiently fine on the logarithmic scale.

Restrict the joint position-direction law first to successful positions, then to directions for which conditional spatial and projected success have probability bounded below. Starting with success close to one makes both restrictions retain fixed positive mass. Mixing the conditional angular laws preserves their hyperplane nonconcentration, and these fixed-mass restrictions change it by only a constant. For every retained direction, its successful positions occupy at least \Delta^{O(\eps)}|\mathcal A| cells, because its conditional individual-cell probabilities have the same upper bound. The conditional projected information bound shows that these cells project to at most

projection cells. The direction law obeys the required Schubert nonconcentration: for lifted vectors (2\xi,1), the relevant Schubert test is exactly an affine hyperplane-strip test on \xi, up to fixed constants. The theorem supplies a fixed positive projection gain for every sufficiently large subset for some retained directions, contradicting this upper bound once \eps and the finite mesh tolerances are sufficiently small. All nonconcentration exponents were fixed using x>0 before \eps was sent to zero. This proves (3.27).□

Proof. Use pairs for which all relevant parameters are differentiability points, avoiding the one possible transition seam. Fubini’s theorem permits this for almost every pair. Set b=(l+a)/2 and choose 0<\eta\ll\sigma\ll a-l. By the angular test in a short block above l, two independently drawn paths inside \tau_l have velocity difference \Delta v satisfying

with probability tending to one. Work inside the second path’s parent tube at b+\sigma/2. Adding the first direction costs no logarithmic information by Lemma 3.7; the maximal relative drift there is

which is below the endpoint width when the block and \sigma are chosen in the interior. We used the exact scale identity \kappa_i(b)^2=\kappa_i(l)\kappa_i(a).

Suppose b were a zero-slope point. The occupancy identity between its scales b-\sigma/2 and b+\sigma/2 gives information increment (k+2)\sigma+o(\sigma). Spatial labels have at most k\sigma+o(\sigma) information, so the time label has at least 2\sigma+o(\sigma).

On the other hand, a spatial intercept label for the first path at scale a-\sigma almost determines this time label. The spatial uncertainty in the second parent is \kappa_i(b+\sigma/2)\ll\kappa_i(a-\sigma). Once the first intercept is specified to width \kappa_i(a-\sigma), the relative motion determines time to accuracy at most

There are therefore at most \Lambda^{\eta+o(1)} possible smaller time labels. The whole second parent meets only boundedly many first parents at a+\sigma, by the preceding drift identity and the smaller time extent. Since a is full slope, the first-path information increment from a+\sigma to a-\sigma is o(\sigma). It remains an upper bound after conditioning on the second parent, and the extra first-parent label has negligible cost. The chain rule would bound the second time-label information by \eta+o(\sigma), contradicting 2\sigma+o(\sigma). First let the sequence tend to its limit at fixed rational approximations, then let \eta/\sigma\to0 and \sigma\downarrow0.

In the power case repeat the same geometry with the second path in another field and with l=0. Root transversality gives |\Delta v|\ge\Lambda^{-o(1)}, so the same time-recovery bound holds and the drift is O(h^{a+\sigma}). The conditioning rule applies to the other field’s full weights. Full slope at a in the first field therefore forces full slope at a/2 in the second field.

We finish with the measure-theoretic assertion. Let E be a measurable set satisfying the midpoint rule for almost every pair of its points. For two density points x,y choose equally short intervals centered there on which E has relative measure greater than 9/10. The convolution of their restricted indicators is strictly positive on a neighborhood of x+y: after a small translation the two high-density subsets still intersect in positive measure. Fubini and the midpoint rule imply that a neighborhood of (x+y)/2 belongs to E modulo a null set. Taking x=y shows that the density points form an open set modulo null sets; taking arbitrary x,y shows its midpoint convexity. Repeated midpoints, followed by openness, fill the interval between any two such points. Thus E is an interval modulo null sets.□

Proof. Cover B_R(z) by boundedly overlapping K-balls D. On one meeting Y, the separate-unit-supremum form of Proposition 2.8, with p=p_k+\eps, and comparability of packet envelopes on D give

Since kp_k/2=k+1 and p_k/(2m)=1/k, this is

The bracket is at most U, up to a fixed factor. Summing over D and replacing the sum by an integral uses only the fixed variation of each envelope on a K-ball.

For these globally transverse groups, the local weighted cylinder consequence of Proposition 2.2 is

To verify its normalization, expand the polynomial envelopes on B_R into summably weighted translated columns of radius K. Their time length is at least R because R\le K^2. A column meeting the ball contributes a fraction comparable to (K/R)^k of its coefficient to the average on a fixed enlargement. The radius-K multilinear Kakeya estimate contributes K^{k+1}; the m=k+1 averages contribute (R/K)^{k+1}, giving exactly R^{k+1}. The fixed reserve in the packet envelope power sums the translated-column tails. This is the profile-free geometric calculation in (3.26), with the determinant factor now \delta^{-C}.

For each field separately, the weighted spatial L^2 calculation following (3.26) gives

Indeed reproduce the coarse weighted suprema, remove each intervening bounded mask before separating its immediate child caps by spatial almost orthogonality, and then use the final R-packet envelope. In the final cap frame the normalized shift has sizes ((K/R)(U_0+O(1)w),(K/R)^2w), so convolution preserves the same fixed polynomial weight, as in the calculation preceding (3.26) and following it. No power of R/K is lost. Combining these estimates proves (3.29).□

Proof. Use the path law without an extra angular test, retaining each path’s attachment to an original surviving H_i mask. Fix a field, write H=H_i, K=HS^s, and let \mu_z be the original local direction law at a typical root point z. The clipped cap bound and path domination give the slot estimate

Indeed a slot at inverse width K contains at most its frequency volume times H^k, up to the appropriate bounded grid enlargement, transition bins, each with clipped weight at most H^{-k}S^{C_3}; path normalization loses only a subpower. The same bound summed over slots proves the displayed formula.

There is also a lower bound at a typical sampled direction, simultaneously at all relevant radii:

To prove this without a number-of-radii loss, remove maximal dyadic cubes on which the measure is less than cS^{-1} times volume. The maximal cubes are disjoint and lie in a bounded frequency region, so their total measure is O(S^{-1}). Use finitely many shifted dyadic grids to put a comparable cube containing \omega inside any ball in question. Every violating ball is thereby charged to one of the removed cubes.

Let y\in Y and \tau=|t_y-t_z|. The probability, under the typical-direction restriction, that y belongs to the sampled enlarged K tube is at most

Only \tau\lesssim K^2S^{o(1)} matters. Hence the angular window K/\tau is no smaller than the slot precision up to a subpower, and (3.31) proves the bound.

If this event is nonempty, choose a typical direction \omega_0 realizing it. Every original path with direction within c\min(1,H/\tau) of \omega_0 has its attached enlarged H mask containing y. Those masks are already near z, the extra transverse displacement is O(H), and their enlarged duration contains the relevant time interval. Thus (3.32) gives

For \tau\ge KS^{C_4}, divide (3.33) by this last bound. For \tau\le H use K/\tau\le S^{-C_4}; for \tau\ge H use K/H=S^s. This gives the ratio bound

Sum this comparison once over y. Every attached original mask has occupancy at most S^{C_1} by the deletion preceding (3.7). Therefore

for sufficiently large fixed C_4 and s<-2C_4. There is no iteration over a number of intermediate scales.

The occupancy identity (3.23), together with the typical leaf probability comparisons, turns its effective occupancy into an ordinary count up to S^{o(1)}. With probability tending to one there are no far points, so a ball of radius O(KS^{C_4}) contains at least

root points. The identity is exact on the logarithmic scale:

because kq/2=k+2.

For the upper bound use a common ball scale at s'=s+C_4+1<0. The inverse widths in the fields differ by only subpowers, by (3.15). Choose a common first width below all field-specific first widths and a common terminal width above all field-specific terminal widths, each within an S^{o(1)} factor. Inserting these levels into the fixed grid preserves the tested field; the local polynomial-weight convolution and bounded-multiplicity comparison in Proposition 2.10 change its square-envelope bounds by at most S^{o(1)}. Begin at the first gap-edge parameter s_{\min}\le s of a fixed finite grid, where no earlier smaller-inverse-width masks intervene. Write \mathcal E_{j,s}(z)=\sum_Ta_{j,T}^2w_T(z) for the local square envelope of the original selected outputs at that layer; this is distinct from the integrated energy E_{j,s} in (3.18). On every first-edge ball carrying witnesses, the selected ratio class and clipped square give \kappa(s_{\min})^{k/2}\mathcal E_{j,s_{\min}}^{1/2}/A_j\le S^C: the remaining exponent is d_j(s_{\min})\le ke_j/2. The transversality loss is S^{o(1)}. We may apply Lemma 3.12 once from this first edge to s', because for every fixed finite range

Here H_i grows faster than every fixed power of S. The single step keeps the determinant power and the \eps cost independent of grid length. At the root-centered terminal ball, \mathcal E_{j,s'}\le S^{o(1)}r_{j,s'}^2 by the clipped square bound. Substitution in (3.29) gives the count upper bound

The H_j factors cancel since their ratios are subpowers. Take limits for this finite comparison and then send \eps to zero. Equations (3.34) and (3.35) yield

By (3.21), every d_i is bounded, nonnegative, and nondecreasing. Let L_i=\lim_{s\to-\infty}d_i(s). Sending s to -\infty and summing the preceding inequalities gives

Since q>p_k and L_i\ge0, all the limits vanish.

The full-slope interval must extend to the unbounded left endpoint: otherwise f_i is constant sufficiently far to the left and d_i(s)=f_i(s)+ks/2 becomes negative. On this interval d_i'=0, so the just proved limit gives d_i=0. After its right endpoint f_i is constant, and its fine endpoint value makes that constant zero. Continuity at the right endpoint therefore places it at s=0. If the full-slope interval fills the domain, the endpoint value instead gives e_i=0. In either case f_i(0)=0.□

Proof. The exact external input is Du and Zhang [15, Theorem 1.6] with counting exponent n. On a union X of unit cubes for which every occupied \sqrt S lattice parent has comparable occupancy \lambda, it states

The original unit-ball frequency hypothesis is changed to our fixed box by a fixed parabolic dilation and bounded coverings. The same holds for balls with arbitrary centers by translating the extension, which only modulates its input. Squaring and renaming \eps gives the exponents in (4.7) for an integral over X. No comparable-field-norm assumption occurs in this theorem; that additional hypothesis belongs to Proposition 3.1 of the cited paper, which is not needed here.

Choose a comparable ambient scale with a compatible square-root grid. Group whole square-root parents according to their dyadic occupancy. A parent is covered by boundedly many radius-\sqrt S balls, so its occupancy is at most C\lambda_0. Each group inherits the counting bound. Sum the squared theorem over these groups; the resulting series \sum_{2^a\le C\lambda_0}2^{a\gamma_n} is bounded by C\lambda_0^{\gamma_n}. This removes the equal-occupancy hypothesis.

Finally, \mathcal E g has spacetime Fourier support in a fixed compact paraboloid patch. Choose a Schwartz reproducing kernel equal to one on that patch in Fourier space. Cauchy–Schwarz and local boundedness of translates of this kernel imply, for every sufficiently large A,

For each fixed h, the translated field is an exact extension of an input of the same norm and support. Apply the integral estimate on the unchanged cube collection, then integrate in h. This proves (4.7) without enlarging the ambient radius for distant translations.□

Proof. Normalize m=\norm{g}_2=1; zero cases are immediate. Sort unit cubes by a<\mu(q)\le2a, where a runs over a fixed multiple of the nonpositive dyadic powers. Cubes meeting a radius-r ball lie in a fixed enlargement, and their disjoint masses are at least a times their number. Thus this class has counting parameter at most Ca^{-1} and square-root occupancy at most CdS^{n/2}a^{-1}. Applying Theorem 4.2 and multiplying by (2a)^\beta gives

The two cancellations are

This exact cancellation leaves no positive power of a to sum, so a small-mass truncation is necessary. Bounded frequency support gives \norm{\mathcal E g}_\infty\le C, and there are O(S^{n+1}) relevant unit cubes. Those with \mu(q)\le S^{-A} contribute at most CS^{n+1-A\beta}. Choose A so large, in terms of A_0 and a prescribed error power, that this is bounded by S^{-B}d^{\gamma_n}. There remain O_A(1+\log S) mass classes. Use half the requested loss in the preceding class estimate and absorb this count into the remaining S^\eps. Restoring homogeneity proves (4.9).□

Proof. Fourier transform \phi_\alpha in its native variables, denoting the dual variables by (\eta,\zeta). Insert a smooth bounded-overlap partition \sum_k\vartheta_k(\eta)=1 into bounded boxes about k\in\Z^n, and change variables u=\zeta+|\eta|^2. For fixed (k,u) define the input, in \xi=\omega_\alpha+\eta/L coordinates, by

Fourier inversion gives

Compact support and sufficiently many derivatives of \phi_\alpha give any prescribed polynomial decay in (\eta,\zeta), also after a fixed number of frequency derivatives. The change \zeta=u-|\eta|^2 costs only polynomial factors. Extract \rho(k,u)=C_A(1+|k|+|u|)^{-A} with C_A chosen to bound the needed seminorms, and put the remaining normalized sum into g_{k,u}. Increasing the finite initial derivative order supplies these bounds simultaneously.

The input norm estimate follows by a summable Gram-matrix bound. Two normalized input bumps can interact only if |L(\omega_\alpha-\omega_{\alpha'})|\le C, since the shift k is common. For these pairs, integration by parts in the native frequency variable gives any prescribed decay in |y_\alpha-y_{\alpha'}|/L. The coefficient factors are c_\alpha\overline{c_{\alpha'}} and all remaining seminorms are uniform. Phase-space counting therefore bounds the absolute row and column sums of the Gram matrix. Schur’s inequality proves (4.11), without requiring the subarray to be a Fourier projection.

For spatial decay, the phase in native variables is z\cdot\eta-\tau|\eta|^2. On bounded \tau and \eta near k, integration by parts gives rapid decay in z away from the original trajectory, at the cost of a fixed polynomial in k. This was included in the orders used to extract \rho; derivatives of the extension are handled at the same time. The common shift preserves all relative frequency labels and phase-space counts. Finally, Cauchy–Schwarz over labels and the prescribed rapid spatial decay bound every normalized layer by a fixed polynomial in (k,u) times E(U)^{1/2}. Increase A, sum over |k|>L^\eta, and integrate in u to obtain C_B L^{-B}E(U)^{1/2} for any required B. All orders here are fixed before the varying scale L.□

Proof. Set R=\sqrt T and absorb the initial time into the input by a unit-modulus multiplier. Partition frequency into R^{-1} caps and expand each rescaled cap density in a Fourier series on a fixed larger box. With smooth bumps equal to one on the cap supports, this gives exact packets

Here H_\theta is the extension of a fixed smooth native frequency bump. It and any fixed derivatives decay arbitrarily rapidly in the first variable for bounded second variable. Parseval proves the coefficient bound; differentiating the input density is unnecessary. The same cap-overlap and integration-by-parts Gram bound used in Lemma 4.4 bounds the squared norm of any retained exact input by its coefficient energy.

For a spatial box B of side T, retain packets whose trajectories on the time slab come within CT of B. A bounded-velocity trajectory of duration O(T) can be retained for only O(1) such boxes. Therefore these retained energies sum to C\norm{g}_2^2. For the omitted packets, the native distance from the box is at least c\sqrt T. Extract an arbitrary negative power of T from the rapid packet decay and keep a further weight in the distance from the trajectory to B, measured in units of T. Cauchy–Schwarz and phase-space counting bound the squared supremum of the omitted field on B by

The right-hand budgets sum boundedly in B. This proves the asserted error control and also permits arbitrarily long spatial unions of such boxes. Translations modulate the exact inputs and preserve their norms, so the construction is uniform in the slab and box centers.□

Proof. Apply Lemma 4.4, discarding |k|>L^\eta with a sufficiently high negative power. For 0<\eta<1, every retained layer has support in a fixed bounded frequency box. Apply Proposition 4.3 with S=L^2 and d=1 to each exact layer. Minkowski’s inequality for the weighted \ell^2 norm of unit-cube suprema, followed by (4.11), gives the result because \sum_k\int\rho(k,u)\dd u<\infty. The common time modulation has absolute value one. The discarded error is negligible: each cube has mass at most Cm, there are O(L^{2(n+1)}) relevant cubes, and its supremum has any required negative power of L. Rename the arbitrarily small scale loss after replacing S by L^2.□

Proof. Rotate frequency into tangent and normal coordinates for the affine plane, remove its central normal carrier, and shear away the normal linear time phase. The remaining normal frequencies have size O(K^{-1}), so their quadratic contribution is O(K^{-2}). On each fixed normal physical slice, the tangential spacetime Fourier support is therefore within O(K^{-2}) of the positive j-dimensional paraboloid. Multiply first by a Schwartz cutoff bounded below on Q with Fourier width O(K^{-2}). It preserves this support statement with fixed enlargement.

Apply paraboloid decoupling [7] in the range P\le q_j. After this localization, use smooth tangential cap projections; their spatial convolution kernels have bounded L^1 norm. Only boundedly many original enlarged grid bins in the strip can meet a projected bin, since both the normal thickness and tangential cap size are O(K^{-1}). Bounded overlap and Minkowski in the normal physical variables give the squared-norm estimate with a Schwartz weight outside Q. Decomposing that weight into translates of Q gives (4.14), with any fixed A. This proof uses only supports and not the way a subfield was selected. Without a strip, there are O(K^n) bins, and Cauchy–Schwarz gives the stated alternative directly.□

Proof. The Jacobian of F is K^{-(n+2)}. Thus, with A_Q=F(Q+K^2a),

No density factor accompanies the mass: the measure is the direct pushforward. Let \nu_a=(\tau_{F(K^2a)})_\#F_\#\nu. Then \nu_a(A_Q)=\nu(Q), and the A_Q are disjoint. Group them according to the child K^2 cube C containing their centers. Each A_Q has diameter O(K), so all sets in a group are contained in C^*=\bigcup_{r\in\mathcal R}(C+K^2r) for a fixed finite \mathcal R. Hölder with exponents 1/b and 1/(1-b) gives

Subadditivity of the powers b and 1-b bounds this by

The two indices allow the mass and the field norm to occupy different neighbors. Reindex by D=C+K^2s to obtain precisely (4.16) and (4.15). None of the translations depends on which parent cube was selected.

The inverse image of a radius-r ball under F has n transverse widths O(Kr) and longitudinal length O(K^2r). It is covered by O(K) balls of radius CKr. Hence its mass is at most CmK^{n+1}r^n for r\ge1, and translations preserve this bound. For the tube moment, writing L'=L/K, \omega'=K(\omega-v), and y'=y/K gives the identity

This proves the direct-pushforward assertion. When the tube centers are kept fixed, translated measures preserve the moment up to a fixed factor only if their native displacements are bounded. Accordingly, applications requiring this moment truncate the Schwartz shifts while the child packet widths are much larger than the retained shifts. The freely chosen exponent A in (4.14) makes the omitted tail smaller than any prescribed power once the global suprema and occupied cube counts are polynomially bounded. For exact-field applications using only ball growth, arbitrary translations are harmless.□

Proof. Choose a sufficiently large integer J, fixed independently of S and \alpha, and dyadic K so that H=K^J\asymp S^\alpha, with fixed rounding factors. Set P=2/(1-b). Since b<2/n, one has P<q_{n-2}=2n/(n-2). Use the counting measure of the selected unit cubes, assigning unit mass to their centers. Local constancy and Hölder bound the initial sum of squared suprema by the coarse functional (4.13), with a fixed K^C endpoint cost allowed. Explicitly, in each coarse cube the sum of the local unit norms is bounded by the number of selected unit cubes to the power 1-2/P=b, times the squared L^P norm on a fixed enlargement. Schwartz reproduction supplies the required neighboring translates. Its measure parameter is Cm_0.

Iterate Lemmas 4.7 and 4.8 for J levels. After a rescalings the strip width is O(K^a/H), which is at most O(K^{-1}) before the last step. Thus the codimension-two decoupling hypothesis persists. The product of transport factors is H^{2-(n+2)b}, up to constants depending on the fixed J, while the final measure parameter is

Smooth frequency decompositions have bounded overlap, so final child input squared norms sum to C_J\norm{g}_2^2. The J decoupling losses can be made S^\eps by starting with a sufficiently small loss. Schwartz shifts may be truncated at a polynomial radius with negligible error, using bounded-frequency suprema and polynomial cube counts. Their weight sums cost fixed constants at each of the fixed number of levels.

The final time length is T=S/H^2, still a fixed positive power of S. The spatial region can have side S/H; to avoid paying for this aspect ratio, cover it by spatial boxes of side T and use Lemma 4.5 on the child time slab. Retained exact inputs have squared norms summable to the child energy. Their omitted errors have arbitrarily high negative powers of T, with summable local budgets. Each resulting spacetime box has side O(T).

Here is the terminal coarse-functional estimate on one such box:

Its exponent C_* is independent of J. Indeed, replace the field norm on each K^2 cube by its volume factor times a unit-cube witness for its supremum. Move the entire mass of that coarse cube to that witness. The resulting measure has ball parameter at most CK^{2n}m', and each moved mass is at most CK^{2n}m'. Since b\ge\beta, its bth power is at most (CK^{2n}m')^{b-\beta} times its \betath power. Apply Proposition 4.3 at scale T to the moved masses. The volume factor, the moved growth parameter, and this last comparison produce only K^{C_*}, proving (4.18). A bounded number of witness-grid choices and enlargements have the same cost. The inequality holds separately for every translated child measure, and translations are absorbed into modulations of the exact inputs. Consequently no count of spatial boxes or translation labels is introduced.

Combining the three scale contributions, apart from S^{n/(n+1)+\eps} and the fixed endpoint factor K^{C_*}, gives

Enlarge C_* to include the initial endpoint cost, still independently of J. Choose J so large that C_*/J<(b-\beta)/2. Then K^{C_*}H^{\beta-b}\le H^{-(b-\beta)/2}. Taking, for example, any c_0<(b-\beta)/2 absorbs the fixed rounding and proves (4.17). Constants may depend on fixed \alpha,J,\eps, but the saving exponent c_0 does not depend on \alpha. For disjoint subinputs the same construction is applied with their coefficient energies, whose sum is the original energy.□

Proof. Subdivide direction supports into neighborhoods of radius a sufficiently small fixed power of \nu. There are \nu^{-O_n(1)} choices of (n+1) neighborhoods. For each choice, send representative directions to the coordinate axes. The map, its inverse, and its determinant have size bounded by fixed powers of \nu^{-1}. The near-coordinate-axes multilinear Kakeya theorem [2, Theorem 1.15 and Remark 1.10], or its strengthened form [16], gives (5.3) after summing the choices. Weights follow by repetition and approximation.

Here is the finite induction giving the required uniform restriction constant. Let A_B(S,\nu) be the best normalized integral constant for inputs in a fixed box of size B. Decompose each input into exact packets at radius S, and on each \sqrt S-box retain the packets meeting its S^\theta\sqrt S enlargement. Their input squared norms are bounded by incident coefficient energy, and the discarded fields are O_N(S^{-N}) relative to their parent input norms. The enlarged supports remain \nu/2 transverse if S^{-1/2}\le c_B\nu. After rescaling by S, the padded tubes have width \delta=S^{-1/2+\theta}. Apply (5.3) with loss \sigma. Dividing its bound by the volume S^{-(n+1)/2} of a rescaled \sqrt S-box gives

Increasing N absorbs all errors into the same inequality for 1+A_B; when expanding products with exponent 2/n\le1, use subadditivity to bound terms containing a discarded factor. For a prescribed \eps, choose a finite J with (n+1)2^{-J}<\eps/2, and then \theta,\sigma with 2a<\eps/2. Iterating J times and using the terminal trivial bound O(S_J^{n+1}), where S_J\asymp S^{2^{-J}}, proves the integral estimate when S\ge(C/\nu)^{2^J}. In the complementary range the trivial bound is absorbed by increasing M_\eps. Thus the dependence on \nu^{-1} is a fixed power.

For the cube-supremum version the exponent p=2/n can be below one, so we include the submean argument. If F has Fourier support in the fixed compact region, choose a Schwartz reproducing kernel \psi and write F=\psi*F. For A>0 let M_AF(x)=\sup_y|F(x-y)|(1+|y|)^{-A}, which is finite because these extensions are bounded. In the convolution insert |F(w)|\le|F(w)|^p M_AF(x)^{1-p}(1+|x-w|)^{A(1-p)}. Peetre’s elementary inequality (1+|x-w|)\le(1+|x-z|)(1+|z-w|) then gives, uniformly in z,

using the boundedness of (1+|z-w|)^A|\psi(z-w)|. Taking the supremum and dividing by M_AF(x)^{1-p} proves the required Schwartz-average bound for M_AF(x)^p; the zero case is immediate. Choose Ap arbitrarily large. In particular this bounds (\sup_q|Ef_i|)^{2/n} by that average for each x\in q. Integrate the product over q and sum. For each independent list of shifts the translated fields are exact extensions with unchanged norms and supports. Apply the integral estimate and integrate the Schwartz weights. This proves (5.2).□

Proof. Sort the original unit cubes by their occupancy in L-boxes. A dyadic class retains L^{2n-o(1)} cubes, on which the squared supremum sum of each field is at least L^{2n-2e-o(1)}=L^{2n/(n+1)-o(1)}. The refined fractal estimate, Proposition 4.3, at radius L^2 improves this upper bound by a fixed power if the L-box occupancy is at most L^{n-\delta} for any fixed \delta>0. The counting hypothesis supplies the upper bound L^{n+o(1)}. Consequently the retained occupancies are L^{n+o(1)}. Divide coordinates by L^2 and use the occupied box centers. Counting the original cubes in an enlarged ball, then dividing by the minimum occupancy, proves (5.4).

Exact packet localization at radius L^2 gives coefficient weights with bounded total sum. On each retained physical L-box, keep trajectories in a padded enlargement. The padding exponent can tend sufficiently slowly to zero; rapid tails make the discarded field negligible. The retained exact input has squared norm at most Cw_i(x), where w_i(x) is its incident weight. Apply Proposition 4.3 at radius L to its L^{n+o(1)} selected unit cubes:

The counting version of (5.3) gives \sum_x\prod_iw_i(x)^{1/n}\le L^{o(1)}. If some weight exceeds L^{-e+\delta}, this and the other lower bounds allow only L^{n-\delta/n+o(1)} such points. Diagonal deletion leaves w_i(x)=L^{-e+o(1)} uniformly.

Let c_0 be the constant in the codimension-two strip gain, Proposition 4.9, and fix 0<c_1<c_0/2. For a finite list of rational \alpha, successively remove, at each x,i, strips of width L^{-\alpha} carrying remaining weight greater than L^{-c_1\alpha}w_i(x). At one scale there are at most L^{c_1\alpha} removals. Assign a packet to its first removal, so the corresponding subarray energies sum to O(w_i(x)). On the physical L-box the exact support enlargement is O(L^{-1}), smaller than the strip width. Proposition 4.9, followed by Cauchy over the removed strips, bounds their combined squared supremum sum by

This is negligible compared with the required local squared norm. The triangle inequality in that square-sum norm, followed by the unrestricted local fractal estimate, forces surviving weight L^{-e+o(1)}. On termination every affine strip has remaining weight at most the deletion threshold. Normalizing by the surviving weight costs L^{o(1)}. Diagonalize over finite rational lists to obtain (5.5). Constant or subpower enlargements of the strips are covered by applying the assertion at a slightly smaller rational exponent. The pruning is allowed to depend on x; it defines the conditional laws used next.□

Proof. This is the dimension-free lemma with this name in OpenAI [24, Section 3]. The short argument also explains its pointwise scope. If a variable has at most L^{C+o(1)} labels, then \mathbb P\{I_L(Y\mid V)>C+a\}\le L^{-a+o(1)}, uniformly in conditioning; integrating this bound gives uniform integrability. Writing p=\mathbb P(Y\mid V) and q=\mathbb P(Y\mid V,W) at the sampled labels, the conditional expectation of p/q is at most one. Thus

This proves the limiting conditioning inequality, without asserting finite-scale pointwise monotonicity. The restricted determination rule follows by summing probabilities over the permitted small label set. Tightness and a diagonal subsequence give the joint law. Include all rational tests, their costs, and the restricted-event indicators; open-set implications with slack justify their passage to the limit.□

Proof. A reference tube determines its off-axis coordinates to L^{o(1)} choices: its relative width and its direction-bin error are both at most L^{-t+o(1)}. Given (C,D), the block child label X_{s+t} and the coordinate label Z determine each other up to L^{o(1)} choices: the bin-defined change of basis has inverse norm L^{o(1)}, and grid-boundary ambiguity has only subpower multiplicity. Thus

In any ordering of the coordinates each chain-rule term conditions on a subset of Z_{\widehat i}, proving (5.7).

For a finite-valued label D, its Shannon entropy is H(D)=\mathbb E[-\log p_D(D)], with the natural logarithm. At finite scale the expected cost is the conditional mutual information of child and D given the parent, divided by \log L, with the same fixed exact tube additionally conditioned on if present. These mutual informations telescope along the spatial filtration. Filling gaps between disjoint blocks only adds nonnegative terms. The total is at most H(D)/\log L=O_n(b_*), since D has L^{O_n(b_*)} labels. Uniform integrability passes this to the limiting law. This is exactly the hypothesis of the “Block comparison and telescoping costs” lemma in OpenAI [24, Section 3]; the number of reference directions only changes the dimensional constant.

Partition a rational interval into blocks with b=b_*\downarrow0. The expected positive excess of the summed G_i increments over the F increment is O_n(b), so in the limit every interval has \sum_i\Delta G_i\le\Delta F. At [0,1], (5.6) forces equality. Nonnegative additive deficits must therefore vanish on every subinterval. Repeat this argument for all finitely many reference replacements and subtract the identities. This proves (5.8).

Finally choose any ordering that puts A first. Each chain-rule term is at least J_i-\ell_i, and their total surplus is \sum_i\ell_i-\ell\le\sum_i\ell_i by additivity. Summing just the terms for A proves (5.9). Equivalently, the lower bound for the complement subtracted from the total gives the upper bound for A. This is why a coordinate pair can be treated without fixing the other n-1 coordinates.□

Proof. Let r_\ell be the dual basis rows. Equations for ratios centered at (a_j,a_k) are the homogeneous equations (r_j-a_jr_i)\cdot v=0 and (r_k-a_kr_i)\cdot v=0. Their row map has smallest singular value L^{-o(1)}: the coefficients of r_j,r_k in any linear combination are the coefficients of that combination itself, and the primal and dual basis norms are subpower. Choose the center at a realized ratio in the ball, enlarging its radius by a fixed factor. Both rows then vanish at a bounded vector v_0=(2\xi_0,1). If a linear combination is (a,b), this implies b=-2a\cdot\xi_0, so \norm{(a,b)}\lesssim\norm a. Consequently the spatial restrictions of the two rows still have smallest singular value L^{-o(1)}. Their time-one slice is an affine (n-2)-plane, and approximate equations give distance L^{-u+o(1)} from it. For two small ratios one can center the equations at zero; they vanish at the ith reference direction, giving the same argument. This verifies both rank and its quantitative conditioning after restriction to the time-one hyperplane.□

Proof. In the exact reference basis, normalize the ith coefficient of the direction of T_i' to one. Replacement transversality and Cramer’s rule bound this normalization and all ratios by subpowers. Include in the joint limiting law the clipped inverse-size exponents

with value one when a ratio vanishes. At most one of these n exponents can be positive. Otherwise two coefficients are small at a fixed positive-power scale, placing T_i' in a codimension-two frequency strip by Lemma 5.6. Conditional on X and the exact references, the alternative still has its original law. Conditional independence and (5.5) make this event have probability tending to zero.

On \{r_j=0\} compare profile increments using the jth coordinate. For fixed rational 0<a<t/2, the finite event r_{j,L}<a/2 implies |a_j|>L^{-a/2}. The approximate basis fixed only by the reference bins changes this to a lower bound L^{-a} for large L. Given T_i',Z_j,C,D, the child position then has at most L^{O_n(a)+o(1)} possibilities, since that coordinate parametrizes the tube with inverse slope at most L^{a+o(1)}. This permitted set of child labels is determined by the displayed (T_i',Z_j,C,D), uniformly over hidden exact reference directions. Apply restricted determination before taking limits, and then let a\downarrow0. On \{r_j=0\} it gives

The comparison uses only the indicated bin-defined coordinates; exact reference directions have not been silently added to the conditioning.

On dyadic exponent partitions with block size b=b_*, distribute each cost uniformly over its block. Its expected integral is O_n(b). Dyadic summability makes all these cost densities tend to zero for almost every path and exponent. At simultaneous differentiability points, \dot G_i\le\dot G_j whenever r_j=0. These ratio events belong to the entire sampled path, independently of the current block. All endpoint integrals equal n/(n+1), so each such derivative inequality implies equality of the two profiles. Each profile thus agrees with at least n-1 others. Every equivalence class has at least n members, and two disjoint classes would need 2n>n+1 members. There is a single class, proving (5.10).□

Proof. This is He [21, Theorem 1] with ambient dimension 2, projection dimension 1, and set exponent 2d. Its Grassmannian nonconcentration condition is angular-ball nonconcentration in this case. The theorem gives the conclusion for all large subsets simultaneously, outside a direction set of probability a positive power of \Delta. Decrease the tolerance to obtain the stated version.□

Proof. Set K=L^b, and use the same reference-bin basis for D_0,D_1. Let Y_u=(Z_1,Z_j) at prefix u. Lemmas 5.5 and 5.7 give, with either bin conditioning,

Let \widetilde a_j be the alternative ratio determined by D_1, and let \Pi be the interval label at resolution K^{-1} of z_j-\widetilde a_jz_1. Given T_1',C,D_1, it has L^{o(1)} possible values: the central line is part of the tube label, the slope error is L^{-b_*+o(1)}, and the relative width is L^{s-1+o(1)}\le L^{-b+o(1)}. Moreover coordinate one parametrizes the full child position along this tube to subpower ambiguity. Thus

Since \Pi is determined by Y_b,D_1, the chain rule and (5.12) imply

Every statement so far conditions on parent and direction bins, not on the unused coordinates.

Pass the finite test to finite L with slack. It succeeds with probability at least a fixed p_*>0 along a subsequence. Fix parent and reference-bin labels whose conditional success probability p\ge p_*, and write \mu for that conditional law. No lower bound on the probability of this individual fiber is needed. Absorb fixed slack into \alpha=C_n\zeta+o(1). If A_{\rm bin} denotes the alternative bin, success implies the following bounds for the displayed labels whenever they have a successful realization:

Let A be the centers of all endpoint pair cells with a successful realization in this fixed fiber. The first line gives

The mass upper bound for a prefix divided by the endpoint mass lower bound counts the successful descendants in that prefix. A ball is covered by boundedly many cells at a comparable coarser tested prefix; the finite mesh costs K^{O_n(\zeta)}. Hence

The direct exponent is 2d_0; weakening it allows all grid and short prefix errors to be absorbed.

Retain alternative bins whose conditional success probability is at least p/2. Their successful mass is at least p/2. For each such bin, let A_a\subset A be its successful endpoint pair cells. The second and fourth lines of (5.14) give

Here initially \pi_a(z_1,z_j)=z_j-az_1; normalizing its projection vector can only decrease image distances.

We now construct one nonconcentrating direction measure without paying an inverse probability for an individual alternative bin. Restrict actual samples to success and the retained bins, normalize by their mass (at least p/2), and push them to the projection line spanned by (-\widetilde a_j,1). To bound a slope ball, first condition on the exact point and exact references. They determine the fixed parent and reference-bin fiber, and the alternative law is still the original one given the point. If the slope ball meets success, choose a successful alternative in it. Its centered-mass test (iii), at a fixed enlarged radius, bounds the original probability of the whole ball. Rounding to a coarser tested radius costs K^{O_n(\zeta)+o(1)}. Restriction only decreases this probability. Integrating over exact points and references, and finally normalizing the entire successful mass, gives

Subpower slope bounds, condition numbers, and the sine-angle formula justify the passage from slopes to projective lines. The approximate ratios have error K^{-1}L^{o(1)}. Rescale the coordinate range of A, of size at most L^{o(1)}, into a unit ball; covering numbers and all displayed bounds change by subpowers only.

Choose 0<\rho<\min\{d_0,\kappa_0\}. Lemma 5.8 applies to (5.15), (5.16), and (5.18) when \zeta is sufficiently small. For every direction in \supp\sigma, however, (5.17) provides a sufficiently large subset of the same A with projection smaller than the guaranteed gain. This is a contradiction. These are exactly the pair-cell, projection-cell, cost, and angular hypotheses of the “Projection block test” in OpenAI [24, Section 3]; the proof above verifies their dimensional transfer explicitly.□

Proof. Use the block-test tolerance \zeta_2 for angular slope c_1/4. Choose a positive rational a\ll c_1\zeta_2, and then a sufficiently reduced tolerance \zeta_1 for slope a/2. Test h_2(u)\ge au on a fine fixed relative grid from c\zeta_1b_0 to b_0. If every test passes, monotonicity fills gaps and nonnegativity handles smaller radii with the allowed additive error; choose j=2, b'=b_0. Otherwise a tested v obeys h_2(v)<av. By monotonicity and (5.11),

Choose j=3, b'=v. The choice a\ll c_1\zeta_2 makes the omitted smaller radii harmless for every block length in [v/4,v]. Reducing cutoffs and meshes leaves spare slack. The finitely many v/b_0 are fixed and bounded away from zero. This proves the “Finite angular alternatives” lemma of OpenAI [24, Section 3] from precisely its two monotone-marginal and positive-sum hypotheses.□

Proof. We use the arguments named “Binary derivative”, “Balanced subintervals inside a hill”, and “Finite alternation and a full-to-empty jump” in OpenAI [24, Section 3]. Their abstract hypotheses here are: G is 1-Lipschitz and nondecreasing; (5.11) holds; Lemma 5.9 holds with fixed positive tolerances for each d,d_0,\kappa_0; and costs on disjoint block lists total O_n(b_*) in expectation. All have been proved. The following details show that neither the number of coordinates nor the planar endpoint value enters these arguments.

If interior derivative values occur on a set of positive path-times-exponent measure, choose d\in(0,1) and 0<d_0<d so that a small neighborhood of d captures positive measure. At dyadic b_0\downarrow0, form a rational mesh of candidate blocks for every alternative in Lemma 5.10. Endpoints have spacing qb_0, with fixed rational q below all relative tolerances; lengths range between fixed positive multiples of b_0. Include each finite prefix mesh and use direction precision b_*=b_0. Each family has bounded overlap, so can be colored into finitely many disjoint lists. If \mathcal C_I is the sum of all costs for candidate I, then

For R_{b_0}(\omega,s)=\sum_{I\ni s}\mathcal C_I(\omega)/b_0, integration over s gives \mathbb E\int R_{b_0}\lesssim_n b_0. Dyadic summability forces R_{b_0}\to0 almost everywhere. At differentiability points, all nearby increments and tested prefixes have their linear approximation with error o(b_0). The angular alternative and the vanishing costs then give a successful block test. The rational meshes and test parameters form a countable family. For each fixed test, Lemma 5.9 makes its all-success event null; the preceding positive-measure interior-derivative event would force one of those null events to have positive probability. A countable cover of interior derivative values proves that the derivative is binary.

We also need a deterministic fact about this binary derivative. If a<b are respectively density points of E and E^c, then for every sufficiently small r>0 there is [x,y]\subset(a,b) with

To see this, set H(s)=G(s)-s/2. The density assumptions give a closed hill strictly inside (a,b), with equal endpoint heights and larger interior values. Among levels admitting a closed subinterval of length at least r with values above the level inside, choose the highest, by compactness. No component strictly above this level has length greater than r, since it would admit a higher such interval. The level set has no gap longer than r. Starting at the left endpoint, its first point at distance at least r lies at distance between r and 2r, and gives (5.21). Finitely many disjoint ordered density pairs give disjoint balanced intervals at every sufficiently small common r.

If a positive-probability set of paths had arbitrarily many disjoint ordered full/empty density pairs, apply the angular alternatives with d=1/2, d_0=1/4. For a successful angular size b'\asymp b_0, use (5.21) at a size yielding lengths in [b'/3,2b'/3], and round endpoints to the fixed fine relative mesh. Any prescribed number of disjoint hills gives that many spatially and angularly successful candidates for all sufficiently small b_0. Each must fail a cost test, by Lemma 5.9; such a failure costs a fixed positive multiple of b_0. Equation (5.20) bounds the expected number of failures uniformly in b_0. Fatou’s lemma contradicts its tending to infinity on a positive-probability set. Thus almost every path has finitely many alternations among density points. Their ordered types, together with the Lebesgue density theorem, make E a finite union of intervals modulo null sets.

Finally (5.10) gives |E|=n/(n+1)<1 and (n+1)|E\cap[0,s]|\ge ns. The latter prevents an initial empty interval and the former forces a later empty interval. There is therefore an interior full-to-empty jump. Choose a small block with midpoint within \eta b of the jump, round its endpoints rationally, and keep its halves inside the adjacent full and empty intervals except for this error. The countable rational choices cover almost every path, so one has positive probability. This gives (5.19) with b<1/8.□

Proof. Fix the rational block and put K=L^b. Endpoint and midpoint increments for F are both (n+1)b/2+O_n(\eta b); the conditional endpoint increment given T=T_1 is b/2+O(\eta b). Pass to finite scale with slack and fix a parent C=X_s=c having conditional success probability at least a fixed p_*>0. Write

Call (T,z) good when it has some successful realization. Let \mathcal Z be the good endpoint labels, represented by their centers in normalized parent coordinates. Since the bounds depend only on the displayed labels, goodness by existence preserves

The midpoint mass upper bound divided by the endpoint mass lower bound gives

Good incidences have central-line error \delta=K^{-1}L^{o(1)} in this bounded parent.

Choose small fixed \eps>0 and P=K^{1/2-\eps}. Polynomial partitioning in \R^{n+1} gives a nonzero polynomial of degree O_n(P) whose open complementary cells each contain at most

points; points on the zero set are assigned to the wall. This is the finite-point polynomial partition theorem [20, Theorem 4.1], whose polynomial ham-sandwich construction works in every fixed ambient dimension.

Take a wall of thickness a sufficiently large constant times \delta. At midpoint resolution r=K^{-1/2}, Wongkew’s hypersurface neighborhood-volume bound [33, Main Theorem], in a fixed ball slightly larger than the parent, gives the number of boxes meeting this wall as

Indeed \delta=o(r), so thicken to O(r), use the volume estimate, and divide by r^{n+1}. The equality follows from Pr=K^{-\eps}. The formula includes singular zero sets and all degree powers; the degree is allowed to grow here. Equations (5.22) and (5.23) bound the discarded good mass by

once \eta\ll_n\eps.

The wall contains the incidence error, so the nearby central-line point of each surviving endpoint belongs to the same open cell. A line not contained in the zero set crosses at most O_n(P) cells; a line in it has no off-wall incidence. Draw T with its original marginal given C=c, and then z,z' independently with law q_T. If g(T) is the q_T mass of good off-wall endpoints, then \mathbb E g(T)\ge p_*/2. Cauchy over the cells visited by T, followed by Jensen, gives

For |z-z'|\lesssim K^{-1/2} there are at most K^{O_n(\eta)} possible good second endpoints by (5.23); each costs K^{-1/2+O_n(\eta)} under q_T. Near pairs therefore contribute at most K^{-1/2+O_n(\eta)}.

For a farther pair, a line incident to both endpoints to error \delta has direction in one of two caps of radius K^{-1/2+o(1)}, hence in caps of radius K^{-1/3} for large L. Under the double sampling, the (T,z) marginal is the original joint law. Thus

An angular cap is contained, on bounded frequency sets, in a codimension-two strip of comparable width. Use (5.5) at exponent b/3<1/8 in each term of this mixture. Its cap probability is at most K^{-c_1/3+o(1)}. For each first endpoint, the same-cell list has at most K^{(n+1)\eps+O_n(\eta)} second endpoints by (5.24); this list is fixed before drawing T. On a good second incidence, bound q_T(z') by (5.22), then apply the cap bound to the remaining indicator and sum the list. The far-pair contribution is at most

No angular estimate conditional on the success event has been used. After division by (5.27), the near and far bounds are

Choose \eps\ll_n c_1, then \eta\ll_n\eps. Both tend to zero, contradicting (5.27).□

Proof of Proposition 5.1. A countersequence yields Lemma 5.3. Its information profiles coincide by Lemma 5.7; Lemma 5.11 supplies a full-to-empty block with any prescribed small fixed tolerance. Lemma 5.12 forbids it. Hence no countersequence exists, which supplies a fixed c>0 and a large-scale threshold as asserted.□

Proof. Choose one center \xi_0, and then greedily choose a center of largest distance from the affine span of the previous choices. If a maximum distance is at most C_0/K, all centers lie in that neighborhood of the current affine span, hence of a hyperplane. Otherwise the n Gram–Schmidt heights d_1\ge\cdots\ge d_n all exceed C_0/K. In the resulting orthonormal basis, the column matrix of \xi_j-\xi_0 is upper triangular with diagonal d_j and every entry in row j bounded by d_j, by the greedy choice. Factoring out these row heights leaves a unit upper-triangular matrix with bounded entries and bounded inverse, with constants depending only on n. Thus the inverse original matrix has norm O_n(d_n^{-1}). Perturbing the vertices by O(K^{-1}) changes its determinant by a relative O_n((Kd_n)^{-1}), which is small for sufficiently large C_0. Its determinant is at least \prod_jd_j\gtrsim K^{-n}. The exact support enlargements are smaller than these bin perturbations. The determinant of the spacetime directions (2\xi_j,1) is a fixed multiple of this simplex determinant. Common translation preserves it; unit normalization changes it only by bounded factors because the frequencies remain bounded.□

Proof. If no positive c_b works, normalize m_0=\mathcal E_0=1 and the fixed input norm bound to obtain a sequence with l\to\infty, K=l^{o(1)}, and broad contribution at least l^{2n/(n+1)-o(1)}. Polynomially small mass and norm levels are negligible by the polynomial number of cubes and uniform exact-field supremum bound. Pigeonhole a common active transverse tuple and dyadic levels \nu(Q)\asymp\chi, \norm V_{L^P(Q)}\asymp A_0 on N cubes. Tuple choices cost K^{O_n(1)} and level choices cost logarithms, all subpower here. We have

Each chosen active field has supremum at least K^{-A-(n-1)}A_0 on some unit cube in Q, since |Q|^{1/P}=K^{n-1}. Pigeonholing the n+1 relative integer offsets of these witnesses costs K^{O_n(1)}. Translate the fields separately by these fixed offsets so that their witnesses lie on a common unit cube associated to each retained Q. These cubes are distinct; translations modulate exact inputs and preserve supports and norms. Lemma 5.2 applies with determinant K^{-n}=l^{-o(1)}, and gives

In particular the inverse-transversality constant is controlled; no unspecified dependence is used.

For completeness, take limiting base-l logarithms (x,y,z) of (\chi,A_0,N) along a subsequence. The retained levels give polynomial bounds, so these subsequences exist. Equations (6.5) and (6.6) imply

They force equality throughout

Thus

The attached witness cubes have l^{o(1)}r^n counts for r\ge1: their full disjoint K^2-cubes, each of mass at least l^{-o(1)}, lie in a ball of radius r+CK^2 whenever their witnesses lie in a radius-r ball. This gives at most l^{o(1)}(r+CK^2)^n\le l^{o(1)}r^n witnesses. The translated fields and (6.7) now contradict Proposition 5.1. This proves a fixed positive gain; decrease it if necessary so the same c_b occurs in both the hypothesis and conclusion of the lemma.□

Proof of Proposition 6.1. It remains to make the positive gains dominate every loss uniformly in D. Bounded L is absorbed in the constant, and \tau>1 has no unbounded admissible scales, so assume 0<\tau\le1. Fix first

A broad leaf gains l^{-c_b}\le D^{-c_b/2}, and a terminal leaf gains, by (6.15),

Thus every leaf gains D^{-a_*} before its small losses and the displayed terminal K factors. In particular this preliminary gain is independent of \tau.

Choose v>0 sufficiently small, with fixed strict margins, that

These imply K\le l^{c_b} at all broad nodes, and permit the terminal K factors to consume at most D^{a_*/8} after fixed rounding constants are absorbed. The depth J=O(v^{-1}) is now fixed. Choose the decoupling and refined-fractal losses so that their total product along a path is at most D^{a_*/8}. Choose \eta>0 next, depending on J,M as well, so that all padding losses, including (6.16), cost at most D^{a_*/8}, and

This proves (6.12) at every existing child scale, because l'\ge D^{1/2}\ge L^{\tau/2}. Choose \gamma<\tau/8, so that the retained layer shift satisfies |k|/l=o(1) at every node. Finally choose the frequency-tail powers, spatial decay orders, and profile differentiability large enough to make all discarded errors negligible. Those last choices do not change c_b or c_s.

On a depth-j branch, (6.14) makes the transported quantity m_j^\beta l^{2n/(n+1)} equal, up to C^j, to the root quantity after multiplication by the narrow prefactor. At a fixed depth, angular bins partition energies, separate shifts have a uniformly summable total weight, finite neighbor choices cost a fixed constant, and spatial localizations have bounded energy overlap. Thus the total weighted leaf energy at that depth is at most C^j times the original energy. Summing over the bounded number of depths costs only a constant depending on \tau. The allocations above leave more than the saving D^{-c_s}.

For clarity, ignored errors are also uniform after the whole tree is summed. On each original spatial box normalize mass parameter and energy; there are polynomially many labels after localization, cells, shift choices after truncation, and branches at bounded depth. Their parameters are polynomially bounded. The tail orders were chosen last and therefore beat this entire fixed polynomial, relative to m^\beta L^{2n/(n+1)}D^{-c_s}\mathcal E. Minkowski over the integrable exact-layer weights, the Sobolev reduction (6.2), and the initial spatial energy sum now give (6.1), also for the prescribed derivatives.□

Proof. We give the geometric iteration, the measure transport, and the exponent calculation separately.

The narrow geometry. Assign every selected bin to one covering set. In a set of type (b), choose one approximating point for each assigned bin and sort the sizes of \abs{\nabla Q_H} there into dyadic classes m_1, down to 1/R; put all smaller gradients in the last class. There are O(\log R) classes. Each assignment concerns the whole bin, not a subset of its packet labels.

Consider one class through the J iterations of ratio K. If m_1 is a sufficiently small constant, the normalization of Q_H, its bounded coefficients, and its small value at an approximating point force its quadratic part to have norm bounded below. Some component of \nabla Q_H then has a linear part bounded below. The condition \abs{\nabla Q_H}\lesssim m_1 puts this class in a strip of width O(m_1+R^{-1}) within the O(R^{-1})-neighborhood of H. For parent inverse widths

this is a codimension-two strip at the next child precision.

For a dyadic class with a positive lower gradient bound, the other regime is

Choose an approximating point in the parent. Taylor expansion around it places all assigned descendants within

of the tangent affine (n-2)-plane to its level set in H. The last, small-gradient class never needs this second regime. If the gradient is bounded below by a fixed constant, only a bounded number of initial steps are outside the second regime. In general the gap between the two regimes is a factor O(K^2), so at most a bounded number of ratio-K steps need a crude estimate.

Use the functional \mathcal T_b of Lemma 4.8, with P=2/(1-b). Since b<2/n, we have P\leq q_{n-2}. At every good step Lemma 4.7 applies after rescaling the width in (7.8) to O(K^{-1}). The exceptional steps use Cauchy–Schwarz. Consequently the whole angular iteration, excluding the transport factors, costs

where the exponent C is independent of J. Constants depending on the fixed J are allowed. This distinction is essential: good-step losses multiply to R^\epsilon, rather than K^{CJ}.

For this nonrecursive calculation one may smoothly truncate compact profiles in native spacetime Fourier variables at radius L^{1/4}. With sufficiently many fixed derivatives, every relevant subfield and rescaling has error smaller than any prescribed negative power of L. There are only polynomially many labels of packets meeting the domain. The choice of \kappa makes the fixed hierarchy R=K^J small enough relative to L that the truncated supports satisfy the thickness and overlap hypotheses of the packet estimates at each step. Supremum estimates for these approximations use ordinary convolution. At the terminal step we return to the original compact child fields, so none of this truncation changes the induction’s profile class. The absolute error estimate is justified below, including when the tested mass is small.

Initial measure functional and recombination. Let \nu be the original measure restricted to the set on the left of (7.7), and write M_0=\nu(\R^{n+1}). For each cover/class index, pad the enumeration where needed and restrict \nu separately to the macro cubes. On a unit cube q, the density bound gives \mu(q)^{1-b}\lesssim(m\chi)^{1-b}. Hölder on the constituent unit cubes, followed by the weighted supremum estimate for the bandlimited approximation, bounds \lambda^2M_0 by the sum of the initial \mathcal T_b functionals, multiplied by

The sum of unit-cube convolution weights inside a K^2-cube is bounded by the enlarged Schwartz weight used for that cube. Thus the weighted versions of the transport inequality apply without a factor counting unit cubes. During each calculation the angular subset prescribed on the original macro is kept fixed, including in every shifted norm.

After J steps, the transport factors and (7.9) give

We explain why the macro restrictions can now be recombined. Write

For a fixed angular chain, class index, and complete grid-shift sequence, every step is a translated pushforward under A_{K,v_j}. The grids on the right of the transport inequalities are global and unrotated; rotations used to prove a strip inequality do not change these lists. Hence the composition is

with a translation z_{\rm sh} independent of the original macro cube. The inverse image of a terminal K^2-cube has diameter at most C_nR^2K^2\leq C_nR^4, regardless of the size of z_{\rm sh}. It therefore meets only boundedly many original macros. For their transported measures this gives, on every terminal cube Q,

The combined measure is dominated by one translated pushforward of \nu, has mass at most M_0, and has ball parameter

Infinite shift sequences are summed with their summable weights; neither the mass, the ball constant, nor the number of occupied cubes depends on their absolute translation. Since assignments were by whole original 1/R-bins, the terminal field is the same full compact bin field, or zero, on every contributing macro. Thus the outer induction will charge each original bin energy once.

The terminal induction. Put l=L/R. In each terminal K^2-cube with nonzero original child supremum, move its mass to a point arbitrarily close to that supremum and bound its P-norm by the supremum. This changes ball bounds by at most K^C. Sort the cube masses as \nu_{\mathrm{child}}(Q)\asymp m'd. Unit covering of the inverse image, whose volume scales as R^{n+2} while (7.12) scales as R^{n+1}, gives

The moved measure has ball parameter CK^Cm', relative unit density at most \min(1,Cd), and, after fixed constant inflation, admissible moment parameter D=1. Its relevant time range is that of the original compact child field. Spatial boxes can be partitioned as in the theorem.

Apply the outer induction at l and integrate the weak L^p bound against the trivial mass bound M_0. Since p>2, the resulting L^2 integral is bounded by

where E_{\mathrm{bin}} is that bin’s original coefficient energy. Indeed, integrating a distribution estimate \nu\{|V|>t\}\leq\min(M_0,At^{-p}) gives \int|V|^2\,\dd\nu\leq C_p A^{2/p}M_0^{1-2/p}. Multiplication by (m'd)^{b-1} returns to the mass-power functional; the fixed K-volume factor costs another K^C. The dyadic d sum converges at its lower end because

By (7.13), its total cost is C(K^CR\chi)^\delta.

To justify returning from the Fourier approximations even on cubes with zero original supremum, normalize m=1. There are polynomially many occupied cubes, and concavity gives

Choosing the fixed truncation order large enough makes the entire error, with all prefactors restored, at most L^{-N'}M_0^b for any prescribed N'. Either this absorbs into \lambda^2M_0, or, using \lambda\geq L^{-C} from (7.6),

The latter is the asserted absolute negligible mass after choosing N' large. Restoring homogeneity supplies the factor m.

The gain. Sum the disjoint bin energies. The factors of m and \chi before solving for M_0 are exactly m^{2/p}\chi^{2a}. Apart from K^CR^\epsilon and fixed logarithmic powers, the exponent of R is

At p=2, \delta=0, direct simplification gives \Gamma=\beta-b<0. Solving for M_0 multiplies this exponent by p/2>0. First choose \delta and p_{\max}-2 small, then J large so that K^C=R^{C/J} is paid by the strict margin, and finally choose the decoupling loss small. The remaining logarithmic factors are absorbed by increasing B for the fixed C_0. This proves (7.7) with a fixed c_2>0.□

Proof. For each bin restrict \mu|_Y to the macros where that bin is light. Since R^4\ll L, the weight in (7.17) is comparable at the macro center and at each point of the macro. Therefore the sum of its coefficient energies times their packet moments on this restricted measure is at most CWM_Y.

Put D_1=X^{P_1}. Split off the labels whose moment exceeds mL^{n+1}/D_1. In each bin their total energy is at most CD_1X^{-P_0}; summing over the O(R^n) bins only contributes a fixed power of R. The other labels satisfy the inner induction hypothesis at D_1, on the restricted measure, with the unchanged original profiles. Apply it at threshold cR^{-n}\lambda. Disjoint bin energies and p/2>1 give total exceptional mass at most

Choose P_1 so that 2D\leq D_1 and un p_{\max}-\sigma P_1+\sigma<0 with a fixed spare margin. The later choice of \kappa ensures D_1\leq L. If D_1 has already reached L^\kappa, the large-D case is available; otherwise this is a strict inner induction step.

For each split-off bin, use unrotated cap scaling and the outer induction at L/R. The ball parameter is CmR^{n+1}, the relative unit density is at most \min(1,C\chi R), and the scaled level parameter is at least gR^{-C}. Consequently all split-off fields together contribute at most

The fixed power R^C includes the bin count and threshold splitting; its exponent can use p_{\max}. Choose P_0 sufficiently larger than P_1 so that this expression, and (7.18), are small fractions of the lower bound in (7.16) once B is large. The thresholds of the O(R^n) light-bin fields sum to at most \lambda/2 when c is small. This proves (i).

On every adverse macro for (ii), the heavy-bin sum is exactly a prescribed-bin field of the type in Lemma 7.2. Its mass is at most the target bound times CR^{-c_2}D^\sigma, plus an absolute negligible error. Since \sigma<uc_2, increasing B makes this a small fraction. The construction may choose the adverse hyperplane and quadratic separately on each macro, as allowed by that lemma.

For (iii), the measure ball bound and the fixed time range give, by covering tube annuli,

where A_w can exceed P_0 by choosing C_w large (the fixed exponent M was chosen sufficiently large first). Sum with |c_\alpha|^2 and compare with WM_Y=X^{-P_0}mL^{n+1}. Markov’s inequality removes only a small fraction of Y and ensures the stated contribution in every heavy bin. Finally, successive maximal affine heights among the remaining centers give n robustly independent directions; failure would place all centers in a forbidden codimension-two strip. Choosing the strip constant large makes the volume bound stable throughout the bins.□

Proof. For a robust n-tuple, project its directions onto an n-dimensional space where their determinant is at least R^{-C}. Such a space exists by their n-volume lower bound. A further polynomially fine angular subdivision makes the projection common to the tuple’s families. Apply endpoint n-linear Kakeya to the projected tubes.

To relate its cube sum to \nu, divide into columns of side comparable to the tube width. Each tube meets only polynomially many width boxes over a projected column, uniformly in its label; its projected direction is bounded away from zero by the determinant condition. Each box has \nu-mass at most X^C. If a_{i,j}\geq0 denotes the incident weight of family i in the jth vertical box, Hölder with n factors followed by the monotonicity of finite sequence norms gives

The total coefficient weight in each family is at most one. The projected Kakeya bound, its polynomial transversality cost and the polynomial number of tuples therefore bound the mass seeing weight W from every family by X^CW^{-n/(n-1)}. This proves the first inequality in (7.21). Substitute (7.20) to obtain the second. We also have N_Y\lesssim L^n from the ball bound.

The separation into cells and a polynomial wall follows the Fourier-restriction partitioning framework of Guth [17, Sections 3.2–3.4]. For completeness, use polynomial partitioning with parameter P_2=X^{C'} on \nu|_Y. It gives a polynomial of degree O(P_2), O(P_2^{n+1}) sign regions, and mass at most CN_Y/P_2^{n+1} in each open region. The finite-set partitioning theorem follows from iterative polynomial bisection [20, Theorem 4.1]; its integrable-density form is Guth [17, Theorem 1.4]. For the present finite Borel measure, smooth it first and use that bisection argument. Normalize the coefficient vector of each of the finitely many factors separately and take a convergent subsequence. A compact subset off the limiting zero set has stable signs; its limiting mass obeys the same bound. Exhausting an open sign region by such compact subsets proves the assertion for the original measure. Mass on the zero set is already part of the wall. Connectedness of sign regions is not required.

A central line not contained in the wall meets only O(P_2) open sign regions. Let E_{\mathcal O} be the total coefficient energy of labels whose line visits \mathcal O. Then \sum_{\mathcal O}E_{\mathcal O}\lesssim P_2. Regions with E_{\mathcal O}>C_*P_2^{-n} have total mass at most CN_Y/C_* by the region mass bound. Choose C_* large. Away from a width-enlarged wall, a point and its nearby incident line point are in the same region. In every remaining region the preceding projected Kakeya argument with local energies bounds the off-wall mass of Y' by

The last inequality uses (7.20) and N_Y\lesssim L^n. Since

choosing C' large pays the number of regions. A fixed positive fraction of the mass must therefore be near the polynomial wall.□

Proof. Square-freeness implies that the singular set has dimension at most n-1. Decompose the bounded regular part into polynomially many pieces with a specified omitted coordinate j, on which the implicit graph x_j=\phi(\bar x) has slope bounded by a dimensional constant. Choose a finite set of base directions sufficient to polarize tensors through order three. Refine the pieces by the signs, including zero, of the corresponding fourth directional derivatives of \phi. Implicit differentiation makes these tests semialgebraic with polynomial complexity. Put all singular points, lower-dimensional pieces and frontiers into the exceptional set; use a strictly larger box so that its artificial boundary is harmless. The frontier dimension property gives the stated dimension bound.

At distance C\rho from this set, a straight base segment of length c\rho lifts in the same graph/sign piece. Indeed its lift has bounded speed. If maximal continuation stopped earlier, its Cauchy endpoint would be a frontier or a singular point within distance \rho, unless the implicit graph could be continued. Both alternatives contradict the stopping assumption. Global injectivity of the base projection is unnecessary.

On such a two-sided segment let g(t)=\phi(u+te), so |g'|\leq C. The fixed sign of g'''' makes g''' monotone. If |g'''(0)|=A, on one half-interval of length comparable to \rho its magnitude is at least A, with constant sign. The midpoint second difference of g' on that half-interval has magnitude at least cA\rho^2 and at most 4C. Thus A\lesssim\rho^{-2}. Apply the same argument on shorter interior intervals, then a difference quotient for g', to get |g''|\lesssim\rho^{-1}. Polarization bounds the full second and third tensors.

By property (a), the exceptional set is covered by X^C\rho^{-(n-1)} boxes. Each becomes a ball of radius O(\rho L) in tube coordinates and has \nu-mass at most C(\rho L)^n. Their total mass is X^C\rho L^n. Choose \rho=X^{-C_5} with C_5 large and require \Delta\ll\rho. By (7.21), this discards an arbitrarily small fraction of N_Y.□

Proof. Lift the projection of the nearby line position onto the local graph. The lift is O(\Delta) from p, and its first and second derivatives differ from those at p by O(\Delta X^C) by Lemma 7.5. For the macroscopic line \ell(s) define

At a relevant incidence, |H|\lesssim\Delta. Failure of (7.23) forces |H'| or |H''| to be at least cX^{C_6}/L, after increasing C_6 beyond the fixed derivative error powers.

Order the regular roots over the projected line parameter by cylindrical decomposition and refine by these inequalities and signs. This gives polynomially many smooth branch intervals. They are graph sections, not sectors, and their implicit derivatives are the actual derivatives of H. On an interval where |H|\lesssim\Delta and a first derivative of fixed sign has magnitude at least q, the length is O(\Delta/q). For a second derivative of fixed sign and magnitude at least q, a midpoint second-difference estimate bounds the length by O((\Delta/q)^{1/2}). Set q=cX^{C_6}/L and use (7.22). Increasing C_6 makes the sum of lengths at most X^{-A}. Thresholds change coefficients, not degrees or the number of quantified variables, so the interval-count exponent does not increase with A,C_6. Quantifying the existence of any eligible nearby point preserves this complexity bound and proves the last assertion as well.□

Proof. At a retained regular point p, let \mathbf n=(\mathbf n_x, \mathbf n_t) be a unit normal. The first line test gives

for a supplied bounded frequency, so |\mathbf n_x|\gtrsim1. The exact tangent-frequency hyperplane is consequently well conditioned.

Suppose the graph Hessian has norm greater than X^{C_7}/L. Normalize its quadratic form and restrict it via \omega\mapsto\overline{(2\omega,1)} to that exact hyperplane. The resulting polynomial on the hyperplane has coefficient norm comparable to one. To verify this, choose coordinates on the tangent space in which its time-one slice has bounded offset. Its direction space and one bounded point span the tangent space with bounded condition number because |\mathbf n_x|\gtrsim1. A homogeneous quadratic has the form

its restriction to t=1 records all of C,b,a. The bounded graph slope makes base projection another bounded isomorphism. This proves the coefficient comparison. Divide the restricted polynomial by its coefficient norm, so that it has norm exactly one as required in Lemma 7.2; this changes the following bounds only by fixed factors.

Move each supplied frequency to the exact tangent hyperplane. Its distance is O(X^{C_6}/L), and by the second line test its normalized quadratic value there is at most

Choose C_7 large relative to C_6,u and then make L dominant. This quantity is less than R^{-4}. Every heavy-bin center is within O(R^{-1}) of these frequencies, so all heavy bins are trapped in a forbidden quadratic neighborhood from Lemma 7.3. The contradiction proves

This is where the quadratic narrow estimate is essential: tangent directions on a nonflat ruled quadric may span an affine hyperplane, but their quadratic values still detect its curvature.

For each fixed graph coordinate j, partition the exact semialgebraic subset defined by regularity, the bounded slope condition and (7.25) into cells with property (b) above. Use paths inside these cells, not paths inside the larger regular wall. Even if the base projection has several sheets, the implicit gradient

is a single-valued function of the wall point. Along a piecewise smooth path p in this subset, with base coordinate u,

For path length D_0\leq X^C, these identities bound the gradient change by X^{C_7}D_0/L and the height error from the starting tangent plane by X^{C_7}D_0^2/L. A projected loop reaching a different sheet obeys the same identities. Thus every cell lies within X^C/L of one affine plane in macroscopic coordinates. Add the O(\Delta) attachment error and return to tube coordinates. The number of cells and the plate widths are both fixed powers of X, as claimed.□

Proof. A physical time interval of length h^2 has bounded length in the child time variable, even though a full child packet has duration b_1^2. At each such time, the Gram matrix of child atoms, also after the fixed derivatives in (7.27), has summable rows. Integration by parts gives decay in the scaled carrier separation b_1(\omega-\omega'), while compact spatial profiles localize intercepts. Transported phase-space counting bounds the number of labels at each scaled distance. All child frequencies are in a fixed bounded box, so differentiation introduces no positive power of b_1. Consequently

for the required derivatives. Unit-cell Sobolev, summed in space and over the bounded child-time interval, proves (7.29).

For the weighted bound apply Proposition 6.1, including its fixed derivative versions, to every full child array. The pushforward of \mu has ball parameter Cmh^{n+1} and its packet moment at scale b_1 retains the same D, since h^{n+1}b_1^{n+1}=L^{n+1}. The choice b_1\asymp(BD)^{C_{10}} ensures that D is at least a fixed positive power of b_1 once D is larger than a constant depending on B; the remaining bounded b_1,D cases cost only a constant depending on B,C_{10}. Also D\leq b_1 by the choice of C_{10}.

Convolve the pushed measure by the isotropic polynomial kernel used in (7.28), and restrict to the suitably enlarged time range. Its ball and packet-moment bounds are preserved: translation of a packet weight costs at most its fixed polynomial order, and P_*>M+n+2 makes that cost integrable. The convolved measure has mass at least a constant times mh^{n+1}d_{\theta,T} on the cells where the suprema are taken. The sparse estimate therefore gives (7.30), after division by the normalization in (7.27). Cluster energies are disjoint and enlarged physical time bins have bounded overlap, so this is a global sum.□

Proof. Partition the sheared normal spatial coordinate into slabs of width h, with bounded enlargements to accommodate witnesses. The projected measure in tangential spacetime has unit-cube mass at most Cm\chi h on each slab, by unit covering in the normal coordinate.

Only nearby normal and time labels of compact packets contribute in a fixed slab and time bin. Recenter at the time-bin origin. The residual normal frequency is O(1/h), so its drift during h^2 is O(h). Projected tangential packets consequently have bounded phase-space multiplicity. Rotating the lattice changes only bounded multiplicities and comparable polynomial weights. The omitted normal and time indices have bounded normalized displacements, so the projected envelopes are bounded by a constant times the full envelopes.

In the normal coordinate divided by h, expand each cutoff profile in a Fourier series. The residual normal carrier and time phase have bounded normalized derivatives and can be included in that profile. The exterior Fourier phase has modulus one. For mode m\in\Z, the tangential packet amplitude and envelope gain \rho_m\lesssim(1+|m|)^{-A_*}, with any required fixed A_* supplied by the original derivative order. Split thresholds by \gamma_m=c(1+|m|)^{-2}. The frame cutoff improves rather than deteriorates under the factor \gamma_m/\rho_m.

Witnesses need not be on the tested point’s projected unit cube. Translate the tangential spacetime packets by an integer offset, separately for each family, so that its witness lies in that cube. Each offset is O(R^4), so summing all choices costs R^C. Counting, transversality and envelope comparability are preserved.

Apply Theorem 3.1, normalizing family i by B_i, and multiply the resulting cube count by Cm\chi h. Before summing modes, the right side contains the decay factors

Every family has a positive decay exponent, even when it does not index the energy summand. All independent mode sums therefore converge for fixed large A_*. Sum slabs at this stage, keeping the same global B_i from (7.32) in every slab and putting only its nearby packet energies in the linear sum. Each packet occurs in boundedly many enlarged slabs. We obtain

Summing slabs after solving a nonlinear mass inequality instead would lose a multiplicity factor; (7.35) avoids it.

Substitute (7.32) and (7.31). Up to the displayed R and band-index factors, the result is

Raise both sides to k/(k+2)=1-\beta. The exponent of h becomes n+2n/(n+1) and the product exponent becomes \beta/n; the \lambda exponent is exactly -2. This is (7.34).□

Completion of the proof of Theorem 7.1. First sum the tuples with a tiny band. The per-bin bound E_i\lesssim h^{-n} gives

Here the polynomial number of clusters and the O((L/h)^2)=X^{2C_{10}} time bins may be counted explicitly. Their total power of X is fixed before C_{11}. To sum the density bands, extend the range to all nonnegative indices and use

Indeed assign a tiny index, bound the polynomial by a product of polynomials, and absorb each into half its exponential decay. The remaining series are geometric. There is no factor \log h in (7.36). Choose C_{11} large enough to pay all the previous cluster, time-bin and tuple powers and the desired negative power of D.

For a common non-tiny band, the energy expression in (7.34) is bounded by

Each packet has one density index, and can occur in only R^C coarse tuple and witness-offset choices. Normal slabs and enlarged time bins have bounded overlap. Cluster labels are disjoint coefficient subarrays. Hence this sum is charged directly to the global weighted bound (7.30), with no factor counting clusters or time bins. Also \ell\leq C+C_{11}\log_2X, so its band factor is only C(C_{11})(1+\log X)^C. The scale factors cancel as

The choices of u and then B absorb R^{C_f} and the fixed logarithmic powers into part of the sparse gain. Combining both types of tuples, and recalling that a fixed fraction of M_Y remained, gives

We now verify the promised polynomial level bound, before choosing p-2. From (7.21), M_Y\geq cmL^{2n}X^{-C_{\mathrm{geo}}}. Substitute this and \lambda=\chi^agL^{-e} in (7.37). Since 2n-2e=2n/(n+1), we obtain

Thus g\leq C_BX^{C_g} with C_g fixed by geometry, independently of \kappa,p-2,C_{\mathrm{ind}}. This confirms the noncircular order of choices in Section 7.1.

Finally (7.37) and \chi^{\zeta}\leq1 imply

provided

This requirement is compatible with the easy-range inequalities, since p-2 is chosen after all geometric exponents and after \kappa. All uses of induction removed small fractions of the hypothetical mass: their ratios were negative powers of X, or R^{-c_2}D^\sigma, fixed before choosing C_{\mathrm{ind}}. Every remaining constant C'_B is independent of that induction constant. Taking C_{\mathrm{ind}}>C'_B contradicts (7.16). Both inductions close, proving Theorem 7.1.□

Proof. Here \mathcal E g uses the same Fourier normalization as S(t); fixed normalizing constants below are harmless. Choose a smooth partition \sum_\theta\chi_\theta=1 on the frequency box, with bounded overlap and caps of side L^{-1} centered at \omega_\theta. In a fixed larger box B\subset\R^n form

extended by zero. Its Fourier-series coefficients c_{\theta,m} satisfy

by Parseval and bounded overlap. If \psi_\theta is a smooth bump equal to one on the support of this rescaled cap and supported inside B, then

as an L^2 identity, where the fixed lattice constant \gamma>0 depends only on B. In particular, no derivative of g has been taken. Writing

the corresponding extension terms are

For bounded \tau, integration by parts in \zeta gives arbitrary spatial decay for H_\theta and every fixed number of its derivatives, uniformly in \theta and L.

Take \rho\in C_c^\infty(\R^n) with \sum_{q\in\Z^n}\rho(u-q)=1, and a fixed time cutoff equal to one on the required native time range. In the qth summand replace the intercept \gamma Lm by \gamma Lm+Lq. With the new native spatial variable u, the profile is

Its C^J norm is at most C_{A,J}\braket{q}^{-A}. Extract this factor into the coefficient. The divided profile is compact and uniformly bounded in C^J, and (8.2) gives (8.1). A fixed q translates the entire intercept lattice by the same vector; hence phase-space multiplicity is unchanged.

For completeness, convergence holds in the pointwise form needed here. At any point of a fixed compact layer only O(1) intercepts for each cap can contribute, and there are O(L^n) caps. Cauchy–Schwarz therefore bounds the contribution of a coefficient tail by

This tends to zero uniformly. Taking A>n also gives uniform summation in q. Independently, the cap Fourier-series partial sums converge to g in L^2 on a bounded frequency set; Cauchy–Schwarz on that set makes their exact extensions converge uniformly. The two limits coincide.

Finally choose positive b_q with \sum_q b_q=1 and b_q\asymp\braket{q}^{-n-1}. The elementary implication

passes a weak p estimate to the sum, with cost \sum_q b_q^{-p}E_q^{p/2}. This is bounded by C\norm{g}_2^p after choosing the fixed A sufficiently large.□

Proof. Unitarity replaces f_N by S(t_0)f_N and puts the left endpoint of I at zero. Spatial translation also preserves its norm and frequency support. Set L=\sqrt N, X=Nx, and T=N^2t. The normalized bounded-frequency input is g(\eta)=N^{n/2}\widehat f_N(N\eta), up to the fixed Plancherel normalization. Thus

A measurable time graph T=T(X) over the dilated spatial ball carries the measure obtained by pushing forward \dd X. Its mass in any spacetime ball of radius r is at most C r^n, regardless of the regularity of the graph. Its native time range is bounded, its unit mass bound has \chi=1, and Definition 4.1 supplies the tube moment with D=1 and a fixed measure constant.

Apply Theorem 7.1 to the layers in Lemma 8.1 and use the summable threshold splitting there. On the scaled graph the result is

Returning to the original graph multiplies its mass by N^{-n}=L^{-2n}. The amplitude factor is exactly

This proves the level estimate for every measurable graph. For a finite list of times, choose the first time achieving the maximum; that graph is measurable. Exhaust a countable dense set of times and use the continuity of bounded-frequency extensions to obtain (8.4). At bounded N, the same statement follows directly from the Fourier L^1 bound, so no lower scale is omitted.□

Proof. Write m_I=|\{x\in\Omega:t(x)\in I\}| and let \mathcal D_a be the intervals at absolute depth a. The pairing kernel for the squared Fourier norm has phase, after \xi=N\eta,

If two intervals in \mathcal D_j have indices differing by a sufficiently large fixed constant, then |t(x)-t(y)|\ge C/N. Since |x-y| is bounded and \eta lies in an annulus away from zero, the time term dominates the gradient. Repeated integration by parts gives any prescribed negative power of N for the kernel. Integrate this bound once over the entire set of far pairs; there is no factor equal to the number of pairs of intervals. The other pairs have bounded neighbor multiplicity, and Hilbert-space Cauchy–Schwarz gives

Let J\subset I\in\mathcal D_j be at relative depth k, where 0\le k\le j, and put r=2^{-k}. Its time width is r/N. Dualizing Proposition 8.2 and integrating a weak L^p function over a set of mass m_J gives

Indeed \int_A|F|\le C_p|A|^{1-1/p}\norm{F}_{p,\infty} for p>1, and duality may be taken over unit L^2 inputs with the annular cutoff.

We also need the spatially localized bound

Partition space into cubes of side r. For pairs outside a fixed number of neighbor offsets, |x-y|\ge Cr whereas |t(x)-t(y)|\le r/N. The spatial term now dominates the phase gradient, giving

Its spatial integral is bounded. Thus the normalized contribution of these pairs is at most C N^{-2s}m_J\le C r^{2s}m_J, using Nr\ge1. For neighbor pairs, Cauchy–Schwarz reduces to single-cube energies. On a cube Q of side r, put

The frequency is Nr, the time interval has length 1/(Nr), and the squared Fourier norm acquires a factor r^n. If m_{J,Q}=|Q\cap\Omega\cap\{t\in J\}|, a fixed cover of the rescaled unit cube by unit balls lets Proposition 8.2 bound this normalized energy by

The last inequality uses m_{J,Q}\le r^n and 2-2/p>1. Summation proves (8.8). This remains valid at Nr=1: the scale-one instance of Proposition 8.2 is the elementary bounded-frequency estimate. We never descend beyond k=j. Taking the geometric mean of (8.7) and (8.8) yields

Starting at each I\in\mathcal D_j, follow its heavier child for j generations, breaking ties by a fixed convention. Let H(I,k) be the interval after k heavier-child steps, with H(I,0)=I, and let P(I,k) be the lighter child at the kth step; see Figure 2. Thus H(I,j) is the terminal heavy interval. These lighter children and H(I,j) partition I, including any atoms of t_\#\dd x, because intervals are half-open. Additivity of G_N, the triangle inequality, and weighted Cauchy–Schwarz with weights 2^{-sk/2} give

Here the terminal mass was bounded by m_I.

For child masses a\ge b\ge0,

For b>0, this follows by writing a/b\ge1 and differentiating (u+1)^\alpha-u^\alpha-1; the case b=0 is immediate. Sum the nonnegative deficits over a finite dyadic tree and telescope. Then pass to increasing finite trees. The result bounds the sum of the lighter-child \alpha powers over any set of distinct parents by m_{[0,1)}^\alpha/(2^\alpha-2).

Fix a relative depth k. As j\ge k and I\in\mathcal D_j vary, the parents of P(I,k) are distinct. Their absolute depth is j+k-1, so different j give different levels; for the same j, different initial I have disjoint descendants. This is precisely why the same time tree must be used for all frequencies. Hence

Also \sum_{I\in\mathcal D_j}m_I^\alpha\le m_{[0,1)}^\alpha. Sum (8.10) first at fixed k and then in k; both 2^{-sk/2} and the terminal factors 2^{-sj/2} are summable since s>0. Finally sum the errors in (8.6). This proves (8.5).□

Proof. Take a smooth inhomogeneous Littlewood–Paley decomposition f=f_{\mathrm{lo}}+\sum_{j\ge1}f_{2^j} with bounded frequency overlap. The low-frequency part satisfies \sup_{x,t}|S(t)f_{\mathrm{lo}}(x)|\le C\norm{f_{\mathrm{lo}}}_2. For a fixed measurable graph t(x) and phase w(x), frequency Cauchy–Schwarz and Lemma 8.3 give

Choose the annular cutoff in G_{2^j} equal to one on the corresponding Littlewood–Paley support. The first factor is the Sobolev square sum; no frequency logarithm or \ell^1 sum has appeared. For a finite time list, select the first maximizer of the full field and its conjugate phase (zero where the field is zero). These are measurable and are common to every frequency. The preceding bound is then the integral of that finite maximum. Increase the lists to a dense countable subset of [0,1) containing zero. Continuity of the bounded-frequency field and monotone convergence prove (8.12). All kernel estimates depend only on spatial differences, giving the asserted translation uniformity.□

Proof. For a fixed \rho>0, cover each annulus 2^\ell\rho\le|y|<2^{\ell+1}\rho by C(1+2^\ell\rho)^n unit balls. Uniformly in the center x,

The series is finite for every a>0, and for all sufficiently small a it is at most C_\rho A a^{-n/2}e^{-\rho^2/(8a)}. The integral over a bounded ball is finite by local integrability, which proves the first assertion.

At a Lebesgue point set H_x(r)=\int_{|y|<r}|M(x-y)-M(x)|\dd y=o(r^n). Given \eps>0, choose \rho so that H_x(r)\le\eps r^n for 0<r\le\rho. Radial integration by parts gives

The complementary integral tends to zero by (8.13), together with the elementary Gaussian tail bound for the constant |M(x)|. Letting \eps\downarrow0 proves the claim.□

Proof of Theorem 1.1. Fix n\ge3 and a complex-valued f\in H^s(\R^n). Choose f_j with \widehat f_j\in C_c^\infty and \norm{f_j-f}_{H^s}\le2^{-j}. The successive differences satisfy a summable H^s bound. By Theorem 8.4, on every unit ball

A countable covering of \R^n therefore gives a common full-measure set E_0 on which S(t)f_j(x) converges uniformly for 0\le t<1. Call the limit v(x,t) and set v(x,t)=0 off E_0. For every x, v(x,\cdot) is continuous on [0,1); it is jointly measurable as a pointwise limit on a measurable set. The measurability of E_0 and of all the suprema just used can equally be read from the countable dense set of times, by continuity.

Fix an arbitrary real t\in[0,1). Unitarity gives S(t)f_j\to S(t)f in L^2, whereas the construction gives pointwise convergence to v(\cdot,t) almost everywhere. Hence v(\cdot,t) is an L^2 representative of S(t)f. For each a>0 and x\in\R^n, the function K_a(x-\cdot) lies in L^2. Pairing it with this L^2 class and using Plancherel yields the everywhere-defined identity

The fixed t was arbitrary. Thus (8.14) holds for every triple (x,a,t) in the indicated ranges; no intersection of time-dependent exceptional sets has been taken. The convolution is absolutely convergent by Cauchy–Schwarz.

Let

These functions are measurable because the supremum equals the one over rational times. They tend to zero on E_0. Uniform convergence there and Fatou’s lemma applied to \sup_t|S(t)(f_\ell-f_j)| show that

All translated balls have the same constant by Theorem 8.4. The values on the null complement of E_0 do not affect this estimate. Intersect E_0 with the Lebesgue-point sets of the countably many M_j. By Lemma 8.5, at every point of this intersection K_a*M_j(x)\to M_j(x) for every j.

For every y and every t, the definition of M_j gives the simultaneous inequality |v(y,t)-S(t)f_j(y)|\le M_j(y). Combining it with (8.14) gives

where

This last convergence is uniform in x,t since \widehat f_j\in L^1. First let a\downarrow0 at a point of the common set. The limsup of the left side of (8.16) is at most 2M_j(x). Then let j\to\infty. This proves (1.2) for all real times in the interval at once.

Finally, intersect with the Lebesgue points of f and with the full-measure set on which v(x,0)=f(x), supplied by the fixed-time L^2 identity at zero. On the resulting E_f, v(x,0)=f^*(x) and time continuity gives v(x,t)\to f^*(x) as real t\downarrow0. The Gaussian limit in (1.2) is taken first, so these are exactly the asserted iterated limits.□