An L³ bound for the trilinear Hilbert transform

Proof. For distinct indices i,j, the map (x,t)\mapsto(y_i,y_j) has nonzero constant determinant. The estimate follows immediately when p_i=p_j=1 and the other two exponents are infinity. The estimate with all exponents infinity follows from the support and kernel bounds. The endpoints with just one exponent equal to one follow from the two-exponent bounds and normalized norm monotonicity. Interpolation, or log-convexity applied to this positive form, fills the polytope 0\le p_j^{-1}\le1, \sum p_j^{-1}\le2.□

Proof. We first remove the normalization. Start with disjoint roots at the largest interval size occurring in the finite collection. At a root J, put a_{j,J}=\norm{f_j}_{q,J}. A zero root size gives zero contribution below that root. Otherwise stop at the first strict descendants J' for which one average of |f_j|^q exceeds 16a_{j,J}^q. For each fixed j the union of its first stops has length at most |J|/16, so the union of all stops has length at most |J|/4. On the intervening tree, division of f_j by 16^{1/q}a_{j,J} makes all required local sizes at most one. Proposition 2.3 therefore bounds this tree by

Repeat at the stopped roots. The sets E_J=J\setminus\bigcup J' are disjoint over this recursive family, have measure at least 3|J|/4, and satisfy M_qf_j(x)\ge a_{j,J} for x\in E_J. Hence

Taking the supremum gives C_N\le C A+C cC_N, where C is bounded uniformly for 2<q<3. Choose c so that Cc<1/2 and absorb. This proves Theorem 2.2.

To recover the continuous kernel, take an odd compactly supported Gevrey function \phi with \phi^{(3)}(0)\ne0, and a compactly supported Gevrey function \omega such that \sum_{m\in\Z}\omega(X-m)=1. Set

Each factor is differentiation in the direction of integration on the corresponding line X=u-jS. Compact support therefore makes all four one-point marginals of W vanish. Spatial summation and a change of variables give

For the second equality one first takes t>0 and substitutes s=t/a; for t<0 use oddness of \phi^{(4)}.

Write a=b2^n, 1\le b<2. For each fixed b, a kernel a^{-1}W(x/a-m,t/a) can be put into the form r^{-1}w_I(x,t) with r=Da for a fixed sufficiently large dyadic D. To choose I, use a fixed finite family of shifted lattices. At scale r their boundaries may be placed at offsets 0,r/3,-r/3, with alternating signs between depths to preserve nesting. The four progression coordinates on the support occupy an interval of length O(a); for large D, one of these cells contains that interval with the fixed interior margin. Each chosen cell receives only a bounded number of translates. The rescaled derivative and support bounds are fixed, so choose c_0,C_0 once to include them. A bounded number of contributions at one cell can be estimated separately.

Theorem 2.2, summed over these lattices, gives a uniform bound for the absolute sum over any finite range of n and the relevant spatial translates. Integrate in b with measure db/b. For compactly supported bounded f_0 and Schwartz f_1,f_2,f_3, the resulting bound is

Here M_q is bounded on L^3 because q<3; use Hölder with the three exponents equal to three. Compact truncations of the other inputs justify use of the defining class for C_N: at a finite range of scales and with compact support in slot zero, every relevant progression coordinate lies in a fixed compact set.

After spatial summation the kernel is odd. Pair positive and negative t near zero; the difference of the two products of Schwartz functions is O(|t|) uniformly on the compact support of f_0. At large |t| the Schwartz bounds are integrable. The integrated scale kernels have absolute value at most C/|t|. Dominated convergence therefore permits the scale range to tend to all positive a and recovers a nonzero constant times the principal-value four-linear form. Reflection of t changes only its sign and gives the operator in Theorem 1.1. Duality against compactly supported bounded f_0 proves the inequality in Theorem 1.1. Density of Schwartz functions, trilinearity, and completeness of L^1 give the stated extension.□

Proof. The sum of the root lengths in generation n is at most \rho^n|J|, by induction. Summing this geometric series proves both assertions.□

Proof. The private component of the sum has squared norm \sum_v\epsilon_v|c_v|^2, giving the lower bound. The upper bound follows from Cauchy–Schwarz, since there are at most D columns and each has norm at most D.

Let E_y:\ell^2(\mathcal A)\to\mathcal H be the evaluation map E_yc=\sum_vc_v\Phi_v(y), and let

be the normalized Gram matrix. The bounds just proved give G\ge D^{-2}\operatorname{Id} and \|E_y\|\le D^{3/2}. Therefore the projection has the integral representation

This proves the supremum estimate. Finally, orthogonality to \Phi_v says that the physical pairing in the last assertion minus \epsilon_vc_v is zero.□

Proof. Put \Delta_0^i=P_0^i and \Delta_n^i=P_n^i-P_{n-1}^i for n\ge1. Within each family these are pairwise orthogonal projections. A prefix of the index set \{0,\ldots,T\} can be partitioned into dyadic index blocks, with at most one block of each length. For a block A put \Delta_A^i=\sum_{n\in A}\Delta_n^i. At any fixed block length, the operators \Delta_A^i are pairwise orthogonal projections.

Expand each factor of Z_{n(\nu)} using this binary decomposition. Fix a tuple of block lengths, one for each of the m factors. A term is zero if one of those lengths does not occur in the relevant prefix. Otherwise it has the form

Group rows with the same tuple of blocks. The row contraction bounds their sum of squared norms by \|\Delta_{A_m}^m\cdots\Delta_{A_1}^1g\|^2. Summing over the block tuples that occur is bounded by the sum over all tuples of those fixed lengths. Successive Bessel inequalities give

This argument uses orthogonality within each layer, not commutation between layers. There are at most L_T^m tuples of block lengths. The triangle inequality in the Hilbert direct sum of the row spaces proves the first assertion.

For the second, telescope in the order

Discard the left contractions and use Cauchy–Schwarz in the sum over i. For a fixed i, the rows B_n=\Delta_n^i, 1\le n\le T, satisfy the row contraction hypothesis. Apply the first assertion to the i-1 preceding families and the prefix index n-1. The resulting bound is

as claimed.□

Proof. For a nonempty product, we first divide the row cells into size segments and, on each segment, replace the input by a bounded function with the same first-layer moments. We then stop when the last-layer coefficient vector moves: nested-prefix estimates control the anchor products, and the pathwise birth-label budget controls the coefficient errors. Attempts are summed using both length and energy contraction. Finally, the size segments are summed using the bounds for their weighted averages and representative energies.

The assertion for an empty product is the row contraction hypothesis. Assume m\ge1, and abbreviate the ordinary L^2(J;\mathcal H) norm to \|\cdot\|_2. Add one final dyadic generation below all row cells, retaining columns and refitting there if necessary. This changes neither the hypothesis nor the quantity to be estimated.

Size segments and a bounded moment representative.. For every cell K put

Starting from J, stop at the first proper descendant cells for which the average is greater than 2a_J, and repeat inside each stopping cell. It is enough to run this stopping rule through the last possible row depth. A root with zero average contributes no rows and can be omitted. Denote the resulting stopping roots by \mathcal S.

Assign a row cell u to the deepest root K\in\mathcal S containing it. More generally, let \mathcal T_K be the full subtree of cells in K that are not contained in a proper stopping child of K, through the last row depth. Thus every assigned row cell lies in \mathcal T_K. Cut K at its next stopping roots and at final leaves on all other paths; denote this partition by \pi_K. Every assigned row cell is partitioned by cells of \pi_K. On a cut cell Q the average of \|z\| is at most 4a_K: its dyadic parent has average at most 2a_K, and a child average is at most twice its parent’s average.

For Q\in\pi_K, take all first-layer labels used by cells of \mathcal T_K containing Q, and project z|_Q onto their restrictions to Q. Let g_K be the direct sum of these projections, extended by zero off K. At most D labels occur on each such path. By Lemma 3.2, with

we have

The second identity follows from orthogonality on each cut cell. If u\in\mathcal T_K and v\in\mathcal V^1(u), then v was included on every cut cell inside u. Adding its moment identities over that partition of u gives

Let M_{\mathrm d}\|z\| be the dyadic maximal function within J. Stopping averages more than double on every path. Consequently

We also need the stronger bound for the moment representatives

Here is a proof that avoids any factor which is a power of D. At a fixed point y, split the roots containing y according as

The sum of a_K in the first class is at most twice its displayed upper threshold. Thus their total pointwise contribution to \sum_K\|g_K(y)\| is at most 2D^{-2}M_{\mathrm d}\|z\|(y). The second class has at most

members, since a_K\le M_{\mathrm d}\|z\|(y) and the averages double. Using the moment identity, pointwise Cauchy–Schwarz for the second class, and then Cauchy–Schwarz in y, we obtain

Solving this quadratic inequality in \sqrt S proves the asserted bound.

Anchors within one size segment.. Fix K\in\mathcal S, write g=g_K and a=a_K, and put

For each cell u\in\mathcal T_K, let c(u) be the unique last-layer coefficient vector of P_u^m\cdots P_u^1g, extended by zero to all labels.

We describe an attempt from a cell R\in\mathcal T_K, initially R=K. Declare R an anchor. From an anchor A, choose as its next anchors the first proper descendant cells u\in\mathcal T_K for which

Stop a path at its Tth transition and send that cell and its descendants to a new attempt, using the same g and a. Denote the disjoint new roots from R by \mathcal R(R).

To apply Lemma 3.3, complete each anchor generation to a stopping partition \pi_n of R, 0\le n\le T. The active atoms are the anchors of generation n. The other atoms are terminal cut cells on paths that have ended; leave those atoms unchanged in later partitions. When refining an active anchor, its next anchors together with the terminal cut cells not lying inside a next anchor partition it. Use the original masks on active anchors. On a terminal cut one can use an inherited local refit, for example with its original mask if that mask is already defined, and keep this local projection unchanged afterward. Hence these are refining partitions, and the resulting direct-sum auxiliary projection families P_n^i=P_{\pi_n}^i are nested. Put Z_n=P_n^m\cdots P_n^1 on L^2(R;\mathcal H).

For any R'\in\mathcal R(R), its ancestors in the anchor tree have generations 0,\ldots,T. On R' the restriction of (Z_n-Z_{n-1})(\1_Rg) is the difference of the constant-coefficient combinations at the corresponding two consecutive anchors. Inheritance makes all the older labels available at the newer anchor. The private coordinate bound in Lemma 3.2 therefore gives

Integrating over the disjoint new roots, summing in n, and applying Lemma 3.3 yields

Since \|g\|_\infty\le Ga, this implies both

Our fixed choice of T makes E<1/2. For completeness, b\ge8 and

The second inequality follows from m+1\le b/4 and \log_2b\le b/2. Thus

It follows by generation-by-generation summation that all attempt roots inside this size segment, denoted by \mathcal R_K, satisfy

Rows assigned to an attempt.. Assign a row in the size segment to the deepest attempt root containing its cell. For a row cell u assigned to the attempt from R, let A be its last ancestor anchor in that attempt, and let j be the generation of A. Here 0\le j<T. The cell A is an atom of \pi_j and contains u. Consequently the restriction of Z_j(\1_Rg_K) to u is exactly the product at the anchor, restricted to u. The stopping rule gives

The support condition on B_\nu now permits the decomposition of its input product into this anchor product and its coefficient error. The rows assigned to R, restricted to L^2(R;\mathcal H), still satisfy the row contraction hypothesis. Lemma 3.3 shows that their anchor contributions have squared sum at most

This is the step in which arbitrary, unequal physical depths of the row cells cause no additional cost: the prefix index is the row’s last anchor generation, and its cell lies wholly in that anchor atom.

For the error write e_{\nu,v}=c(u(\nu))_v-c(A(\nu))_v. It is supported on labels available on u(\nu) and has squared coefficient sum at most \tau^2a_K^2. Since u(\nu)\subset R\cap J_v whenever this coefficient can be nonzero, locality of B_\nu gives

Cauchy–Schwarz in v, followed by the row contraction for each fixed label, bounds the squared sum of errors in this attempt by

The last inequality uses the pathwise budget in its integral form:

In particular, no estimate of the total number of labels on different branches has been used.

Summation and the final constant.. Each row belongs to exactly one size segment and one attempt within it. The triangle inequality in the full row Hilbert space, the two packing bounds, and the estimates for S and the stopping averages give

Finally,

which is bounded by the stated expression after increasing its absolute constant. A complement is expanded as 1-P^i; the at most 2^m resulting ordered products obey the same form of bound, and their number is absorbed in that constant. For a fixed average of compositions, apply the estimate to each member and use the integral triangle inequality in the row Hilbert space. This proves the remaining assertions.□

Proof. If \mathcal E is empty there is nothing to prove. It is otherwise a finite set. We first bound coefficients in a minimal rational dependence. Suppose a_1,\ldots,a_k are minimally dependent vectors of the indicated height. Then k\le d+1 and their rank is k-1. For each i, clear the denominators in its d entries by a common denominator q_i\le B^d. The integer vector v_i=q_i a_i has each entry bounded by B^d. Select k-1 coordinate rows on which the vectors have rank k-1. Their signed maximal minors give a null vector (b_1,\ldots,b_k) with

None of these minors is zero: a zero component of a null vector would give a dependence on a proper subset, contrary to minimality. Returning to the original vectors gives

Set R=4dH. Suppose coefficient vectors have nonzero evaluation magnitudes

If these vectors were dependent, apply the preceding construction to a minimal dependent subset, ordered by evaluation magnitude, and let s_* be its largest magnitude. The coefficient of that term has absolute value at least one, whereas all the other terms together have absolute value at most

They cannot sum to zero. Thus vectors whose evaluation magnitudes are successively separated by a factor greater than R are linearly independent, and there can be at most d of them.

Choose the smallest element s of \mathcal E, cover [s,Rs], and repeat with the smallest still uncovered element. The selected elements are successively separated by more than R, so at most d intervals are needed. Their logarithmic lengths are \log_2R, and

Each logarithmic interval meets at most 2+\log_2R unit bins. This proves the first assertion.

For the second, consider one logarithmic value interval [s,s+\log_2R]. The condition \alpha\le r_n^\ell v\le\beta confines n to a real interval of length at most

That interval has at most two plus its length integer points. Summing over the at most d value intervals and using the preceding bounds proves the claimed scale count. The factor r_0 translates the interval and has no effect on its length.

For the third, partition \mathcal E by its occupied bins [2^j,2^{j+1}). For one such bin, a contribution at scale n is possible only if r_n^\ell2^j\le\varepsilon, and its maximum is at most r_n^\ell2^{j+1}. Let n_0 be the first integer for which the former inequality holds. The sum of these upper bounds for n\ge n_0 is at most

The maximum over all occupied bins is bounded by the sum of their separate maxima. Multiplying by the occupied-bin count proves the assertion.

Finally, the supremum norm of a nonzero evaluation vector is the magnitude of one of its nonzero coordinates, so the same set \mathcal E contains all such suprema. If there are at most h coordinates, its Euclidean norm lies between that supremum and \sqrt h times the supremum. Enlarging each of the d logarithmic intervals to the right by \tfrac12\log_2h covers all these Euclidean norms. The preceding bin, scale, and geometric-tail arguments then give exactly the additional costs stated in Lemma 3.5.□

Proof. For conjugation replace C,\lambda,\beta by their negatives and conjugate \chi. For products use the direct sum of the horizontal coordinates and the block diagonal sum of the matrices. The product rule for derivatives gives the required Gevrey bounds.

It remains to check a translated horizontal argument. Let that argument be \theta y+a. Choose a positive integer P such that every entry of PC is integral. The product of the denominators of the d^2 entries is an admissible choice, so P\le H^{d^2}. Write a=m_0+a_0, with m_0\in P\Z^d and \|a_0\|_\infty\le P, and replace the summation index by n+m_0. With w=\theta y-n, the new phase is

The two middle terms linear in \theta y change the unrestricted linear phase. The last term is constant. The term a_0^TC(w+a_0) can be included in the smooth amplitude \chi(w+a_0). The support and derivatives of that amplitude have quasipolynomial bounds because a_0, unlike m_0, is bounded by P. Translating an ordinary quadratic phase produces only a quadratic phase, a linear phase, and a constant. Rescaling simply rescales \theta,\lambda,\beta. All these calculations hold with \theta y replaced by Vz.□

Proof. We first show that every sufficiently small return has a bounded frequency increment. The lattice directions generated by these returns form a rational isotropic subspace with a bounded-height projection and bounded slow horizontal matrix. Returns in each coordinate direction then bound the remaining effective quadratic matrix. Let P clear the denominators of C, as in Lemma 4.2. Put F=\vartheta Q. The support of F, its supremum norm, its L^2 norm, and \int(1+\|z\|)|F(z)|\,dz are bounded by a quantity D quasipolynomial in H. The value of D can be enlarged finitely many times below.

Suppose

Translate z by h and replace the lattice index by n+m. On the support of the translated amplitude, Vz-n has norm bounded in terms of the chart complexity. Expanding the phase therefore gives

where c(h,m) is constant in z, and

provided \|h\|+\|d_h\|\le1. To verify that the error contains no unbounded coefficient, put u=Vz, w=u-n, and d=d_h. The nonconstant increment of the chart phase is

The first two terms, together with the ordinary quadratic increment, are exactly \Omega(h,m)^Tz; the term m^TCn discarded in this calculation is integral. The remaining d^TCw is bounded by D\|d\|. Translating the amplitude and \vartheta costs D(\|h\|+\|d\|). This proves the claimed estimate.

The Fourier transform of F is D-Lipschitz. Choose a point \xi_0 where its absolute value is at least 3\delta/4, and put \rho=\delta/(100D). Choose an integer

If necessary, increase D to cover the fixed volume constant in Plancherel’s theorem. Choose a>0 so small that

All of M,a^{-1},\rho^{-1} are quasipolynomial in H,\delta^{-1}. For a return with \|h\|+\|d_h\|\le a, the same translation calculation applies to \ell(h,m), 1\le\ell\le M. It produces Fourier values of absolute value at least \delta/2 at \xi_0-\ell\Omega(h,m). If \|\Omega(h,m)\|\ge1, the balls of radius \rho about those points are disjoint, and on each such ball the Fourier transform has magnitude at least \delta/3. Their contribution to its squared L^2 norm exceeds D^2, by the choice of M, contradicting Plancherel. Hence

For two returns satisfying this condition, direct antisymmetrization gives

The integer m'^TAm has absolute value less than one after decreasing a by a fixed factor. It is therefore zero.

We next find all directions generated by short returns. For d\ge1 let

This is a symmetric convex body with nonempty interior. Write \lambda_1\le\cdots\le\lambda_d for its successive minima for \Z^d, and \mu_1\le\cdots\le\mu_d for the successive minima of its polar body. We need the elementary transference bounds

For the lower bound, independent lattice vectors in subspaces of dimensions j and d+1-j cannot have all cross pairings zero. One of their integer pairings has absolute value at least one. For the upper bound, Minkowski’s second theorem [3, Theorem 1.3] gives

Here the last volume-product bound has a short direct proof. Choose maximal-determinant columns v_1,\ldots,v_d in \mathcal K. Their cross-polytope is contained in \mathcal K, and maximality places \mathcal K in their coordinate parallelotope. Taking polars gives an inscribed cross-polytope in \mathcal K^\circ with reciprocal determinant. The product of the two cross-polytope volumes is 4^d/(d!)^2. Since every opposite-minimum product is at least one, their product upper bound proves each individual upper bound.

Set L=16(d!)^2, and choose b_{\max}\le a/(100P(d+k+1)). The d+1 disjoint intervals

cannot all contain a successive minimum. Select one containing none, write its left endpoint as b_0, and put

The reciprocal of b_0 is quasipolynomial in the required parameters. Every n\in\Z^d\cap b_0\mathcal K has a representation n=P^{-1}Vh+w, with \|h\|_\infty,\|w\|_\infty\le b_0. Thus m=Pn, d_h=-Pw form a return satisfying the threshold a. The integrality argument above, applied to any two such n, shows that A vanishes on U\times U.

Let r_0=\dim U. If r_0<d, then \lambda_{r_0+1}>Lb_0; transference supplies d-r_0 independent integer vectors q_\nu in

Each has zero pairing with \Z^d\cap b_0\mathcal K, since that pairing is integral and has absolute value at most (d!)^2/L<1. They consequently span U^\perp. The inclusion [-1,1]^d\subset\mathcal K bounds their coordinates, while P^{-1}V[-1,1]^k\subset\mathcal K bounds each q_\nu^TV, by quasipolynomial quantities. If S has the q_\nu^T as rows,

Cramer’s rule, and the positive integral determinant of SS^T, give quasipolynomial rational height and norm bounds for this matrix and for P_sV. When r_0=d, take P_s=0; the same conclusions hold. Put P_f=1-P_s.

Finally we bound the effective ordinary quadratic part. For each coordinate vector e_j\in\R^k, sample P^{-1}V(te_j)\bmod\Z^d at equally spaced points t\in[0,b_0]. Partition the horizontal torus into cubes of side at most b_0/4. More than (1+4/b_0)^d samples give two in the same cube. Their difference yields a return

where M_0\le2(1+4/b_0)^d. Its lattice vector satisfies P^{-1}m\in b_0\mathcal K, so m\in U. In particular m-V_fh=-P_fd_h\in U. Isotropy gives

which is quasipolynomially bounded. Direct matrix algebra gives

The return bound for \Omega therefore bounds 2\Lambda_*t_je_j quasipolynomially. Since t_j^{-1} is also quasipolynomial, this bounds every column of \Lambda_*. This proves (2).

For d=0, the initial translation argument applies to every sufficiently small h, with \Omega=2\Lambda h; take h=ae_j/2 and U=\{0\}. This also proves the assertion in that case.□

Proof. Put u=Vz, u_f=P_fu, u_s=P_su, and w=u-n. For the symmetric part,

For the skew part, isotropy on U gives

The first term is precisely the extra quadratic included in \Lambda_*.

Choose an even integer P_0 clearing the denominators of C_{\mathrm{sy}}. Then -n^TC_{\mathrm{sy}}n/2\bmod\Z is determined by n\bmod P_0\Z^d. Also choose a common positive denominator q_0 for P_s. On the box under consideration, u_s=V_sz is bounded, and w belongs to the support of \chi. Thus P_sn=u_s-P_sw ranges over a bounded subset of q_0^{-1}\Z^d. The size of this set, P_0^d, and q_0 are all quasipolynomial in H,B.

Split the sum over n according to

For each pair (N,a), remove the fixed residue phase and the rational linear phase

What remains is a smooth amplitude in the two vector variables (u_s,w). To see explicitly that it admits a periodic Fourier expansion, let \eta(s) be a smooth cutoff equal to one on all values of V_sz in the box. Choose a smooth equality cutoff E with E(0)=1, supported in the coordinate cube of radius 1/(3q_0). Then the function

has the required value when s=P_su. Indeed its equality-cutoff argument then equals P_sn-N; the lattice spacing makes the cutoff exactly the indicator of the specified equality. It is periodic in u with period P_0 in each coordinate. By taking the support of \eta strictly inside a larger integer-period box, it extends smoothly and periodically in s, with a period P_1 quasipolynomial in H,B.

The resulting Fourier modes have the form

Substitution of s=P_su, and restoration of the rational linear phase for N, gives

These frequencies have the height asserted in the lemma. The sum of the two ordinary quadratic terms is z^T\Lambda_*z, proving the exact identity (3).

Here are the quantitative convergence details. On each of the bounded supports above, all coefficients in the smooth phases, cutoff scales, periods, and lattice-point counts are bounded by a quasipolynomial quantity D_0. The product rule and the chain rule imply, at order m, derivative bounds of the form

This estimate also bounds derivatives after rescaling the periodic variables to unit periods. It follows either by differentiating the quadratic exponential, whose m-th derivative is a polynomial of degree at most m times that exponential, or directly by induction on m. The cutoffs satisfy the same fixed-order Gevrey estimate.

For a nonzero integer Fourier index v=(p,\ell), select a coordinate with |v_j|\ge\|v\|/(2d)^{1/2} and integrate by parts m times in that coordinate. Thus

after increasing the constant to include the zero index. A dyadic shell has at most (CT)^{2d} indices. Taking m\ge J(d+1)^2+4d+4 gives the displayed shell bound. Since d\le\log H, the prefactor for this derivative order is quasipolynomial in H,B. Summing the shells proves absolute convergence and the bound for the full coefficient sum.

There are at most B^{C(d+1)^2} rational projection matrices of height at most B. For each such matrix there are quasipolynomially many choices (N,a). All derivative estimates are uniform in the bounded slow parameters and in amplitudes with the stated complexity bounds. Consequently we may replace every coefficient by its uniform envelope before summing over this finite list. If different indices give the same frequency, use the sum of their envelopes at that frequency. Choosing the derivative order larger by C(d+1)^2 absorbs, as well, the count (D_0T)^{C(d+1)} of bounded-height rational frequencies at an index scale T. This proves the pooling assertion in the form stated. Finally, the sum of the shells above a cutoff T_0 is at most C D_JT_0^{-J(d+1)^2}; choosing this to be at most \varepsilon gives the required quasipolynomial truncation.□

Proof. We first separate only \varphi_i. Temporarily suppress its linear phase, put \alpha_i=1, \alpha_k=0, and choose \alpha_a,\alpha_b so that

The four indices lie in a fixed set, so these coefficients are fixed rationals of bounded height.

Choose a nonnegative, compactly supported Gevrey function \rho on \R^d whose integer translates sum to one. Insert its partitions of unity at \theta y_a,\theta y_b,\theta t. Write the corresponding lifts as

For the original summation index m_i, put

Each k_j is integral because y_j=y_a+(j-a)t. The support restrictions bound the possible k_i,k_b quasipolynomially in H. Conversely, fixing these integers determines m_i from m_a,m, and the condition for k_b is enforced by a smooth equality cutoff in d_b-d_a-(b-a)d-k_b. Such a cutoff can equal one at zero and vanish at every other integer vector, so this insertion is exact.

Put S=\sum_jj^2\alpha_j. The two relations defining the \alpha_j’s imply the exact identity

Indeed the coefficients of the common lift d_a and of x cancel, and the coefficient of t in \sum_j(j-a)\alpha_jy_j is S. Solving this identity for the i-phase leaves the a- and b-phases with coefficients -\alpha_a,-\alpha_b, and the t-phase with coefficient S. The last displayed sum is linear in x,t, and hence is a sum of a linear function of y_a and a linear function of y_b. Restore the suppressed \beta y_i in the same way.

The remaining coupled amplitude is a smooth function of (d_a,d_b,d) on a bounded box. It is a product of the partition bumps, the equality cutoff, and \chi(d_a+(i-a)d+k_i). Its derivative bounds are quasipolynomial versions of the fixed Gevrey bounds. Choose slightly larger product bumps equal to one on its support, extend the amplitude periodically inside that larger box, and take its Fourier series. Every Fourier term factors into a function of d_a, a function of d_b, and a function of d. After the independent lattice sums, these are precisely three atoms of the form (1), with rational chart matrices -\alpha_aC,-\alpha_bC,SC. A Fourier index of size T contributes only derivatives of size T^m at order m to their smooth amplitudes, so its complexity is bounded by (DT)^C.

Apply the same construction to \varphi_k, now setting the coefficient of the i-slot to zero, and combine the two triples by direct sums of horizontal coordinates. The number of integral choices and the dimensions are quasipolynomially bounded in the original complexity. The integration-by-parts calculation in the proof of Lemma 4.4, with derivative order larger than the Fourier dimension by at least 110, bounds shell coefficient sums by DT^{-100}. The number of Fourier indices and integral choices in each bounded shell is also quasipolynomial. Normalize the three factors to have supremum norm one and put their bounded normalization factors into c. The bounds remain of the stated form. Absolute convergence justifies all rearrangements and proves the identity for every (x,t).□

Proof. Use (y_a,y_b) as coordinates; their Jacobian is |b-a| and t=(y_b-y_a)/(b-a). After normalization by r, the smooth kernel has a Fourier expansion on a fixed two-dimensional box whose coefficient sum is uniformly bounded. Each Fourier term merely modulates the two inputs. Insert \zeta(t/r), which is one on the relevant support. The remaining operator is convolution with Q(t)\zeta(t/r), after the fixed change of scale b-a. Its Fourier multiplier is bounded by C r A_r. Plancherel, the factor r^{-2} in the definition of H_I, and \|f\|_{L^2(I)}=r^{1/2}\|f\|_{2,I} prove the first assertion. The factors \psi_a,\psi_b are bounded multipliers and do not increase the two L^2 norms.

For the second assertion apply Lemma 4.3 to Q(r\,\cdot), with V=r\theta and \Lambda=r^2\lambda. Lemma 4.4, truncated to a sufficiently small uniform error, then gives terms

The formula for \Lambda_* in (2) gives exactly the asserted rational dependence of \lambda_*. Split the linear phase by

The remaining smooth two-variable kernel, w_I(x,t)e(\lambda_*t^2), has derivatives in normalized coordinates bounded by D^{C(m+1)}(m+1)^{C(m+1)} at order m. Its Fourier expansion, with fixed product cutoffs, has quasipolynomial coefficient sum and a quasipolynomial truncation giving the prescribed error. The resulting factors are a bounded one-variable smooth bump at scale r, a linear phase, and \psi_e.

For completeness, a smooth bump on I, restricted to I, is itself represented by a chart with zero matrix and one additional slow horizontal coordinate. Map I to an interval strictly inside one lattice cell, put the bump in the corresponding amplitude, and support that amplitude away from every other lattice translate. On I only the chosen translate contributes. Lemma 4.2 handles the affine offset and the product with \psi_e. Outside I a bounded periodic extension is harmless. This realizes all separated factors as bounded atoms. Choose the successive truncation errors divided by the already bounded coefficient sums. Only a fixed number of quasipolynomial budgets is composed, giving all the stated estimates.□

Proof. The usual cube Cauchy–Schwarz inequality is valid on this circle. One direct derivation writes a vertex as x_1^{\omega_1}+x_2^{\omega_2}+x_3^{\omega_3}, using three independent pairs of Haar variables. At the first Cauchy–Schwarz step the factors with the first coordinate equal to zero and one are put on opposite sides; repeating for the second and third coordinates gives

The same iterations for a single function express its eighth power as an integral of squared absolute values, proving nonnegativity. Expanding the cube of f+g, applying the displayed inequality to each mixed term, and summing the binomial coefficients gives the triangle inequality.

For the stated L^2 estimate, split the vertices into

Each four-tuple of corresponding linear forms is an integer linear automorphism of the four Haar variables, with determinant of absolute value one. Apply Cauchy–Schwarz to the product of these two four-factor groups, then change variables in each squared integral. Each squared integral factors into the four squared L^2 norms. This proves the estimate. Expanding the difference of two mixed cube forms proves their continuity, first for bounded functions and then for all L^2 functions by truncation. The fixed normalization between circle measure and interval measure proves the final assertion.□

Proof. Normalize I=[0,1). For L\ge1, define the cell averages

and extend them by zero to \Z/(16L\Z). Their discrete raw cube averages converge to the continuous raw cube average above. Here is a justification that needs no regularity of g. For a continuous bounded function the assertion is Riemann-sum convergence, because its cell averages differ from endpoint values by its modulus of continuity. For two functions bounded by one, telescoping the eight-factor product bounds the difference of their continuous cube averages by 8\|g-b\|_{L^1(\T_{16})}. The identical discrete bound holds for their cell averages, since

Approximation in L^1 by continuous functions bounded by one proves the assertion.

There is no wraparound contribution hidden in this comparison. For a nonzero supported cube choose x\in[0,1), and lift h_j as the difference between x+h_j\in[0,1) and x. Then |h_j|\le1, and every real vertex belongs to [-3,4]. Within that range, reduction modulo 16 lies in [0,1) exactly when the real vertex does. Thus the circle and interval zero-extension cube averages differ only by their fixed normalizing factors. The denominator defining U^3[L], namely the cube norm of the interval indicator, likewise converges to a fixed positive constant. For all sufficiently large L, therefore,

after shifting the integer labels by one if needed. No bound on this choice of L is required.

Apply Theorem 4.7. Write its dimension as d_0, its complexity as M_0, its Lipschitz bound as K_0, and its correlation as \varepsilon_0. All these quantities have the bounds asserted there. We next convert its nilsequence into chart atoms, tracking only quantities independent of the real orbit coefficients.

In second-kind Malcev coordinates adapted to the central filtration subgroup G_2, write a group element as (z,s), with z\in\R^p, s\in\R^{d_0-p}. Since [G,G]\subset G_2 and [G,G_2]=1, commuting the one-parameter coordinate factors into their prescribed order gives the group law

The coefficients of the bilinear map B are rational and have height bounded quasipolynomially in M_0 and e^{d_0}. Indeed, in step two every interchange contributes just its central commutator with coefficient z_i z'_j; all further commutators vanish. The rational structure constants in the Malcev basis are part of the nilmanifold complexity. Integer coordinate tuples represent exactly the lattice \Gamma.

Write the polynomial sequence in these coordinates as (z(n),s(n)). Its second group differences have no horizontal part, so z(n)=z_0+n\theta. For the unit right difference Dg(n)=g(n+1)g(n)^{-1}, the group law gives

The degree-two filtration makes D^3g=1. Thus s(n) is an ordinary vector polynomial of degree at most two. No bound is imposed on the coefficients of z(n),s(n).

Choose \rho as the product of fixed one-dimensional nonnegative Gevrey partitions supported in (-1,1), so that \sum_{m\in\Z^p}\rho(z-m)=1. Right multiplication by the lattice element (-m,0) gives

The function \widetilde F is periodic in every central coordinate. On a bounded horizontal box its coordinate Lipschitz constant is quasipolynomially bounded in K_0,M_0,e^{d_0}. To check this directly, the coordinate difference of two nearby group elements is

on a bounded box its coordinates are bounded by a controlled multiple of the coordinate displacement. The Malcev metric is bounded by the length of this coordinate displacement path. This proves the claimed coordinate Lipschitz estimate.

Multiply by a fixed horizontal cutoff equal to one on \supp\rho and supported strictly inside a larger period box. Periodize horizontally, and convolve in all coordinates with product Fejér kernels of order N_0. For a Lipschitz function on a circle the error is bounded by its Lipschitz constant times C\log(2N_0)/N_0, since this is an upper bound for the kernel’s first moment. Telescoping over the d_0 coordinates gives the same estimate multiplied by d_0. Choose N_0 so that this error is at most \varepsilon_0/4. It may be chosen quasipolynomially in K_0,M_0,e^{d_0},\varepsilon_0^{-1}. The resulting trigonometric polynomial has the form

where P_2 is a fixed integer horizontal period. Each coefficient has absolute value at most a controlled multiple of K_0, so its absolute coefficient sum is at most C K_0(2N_0+1)^{d_0}. Because the partition \rho is nonnegative and sums to one, the approximation error remains at most \varepsilon_0/4 after summing over m.

For a fixed central frequency k, define C_k by z^TC_kw=k^TB(z,w). The associated summand of this approximation is

Its outside phase is an ordinary quadratic in n. The matrix C_k is rational with controlled height, while the horizontal orbit is affine. Lemma 4.2 removes its constant offset, without putting an unbounded parameter in the amplitude. Thus each summand is an atom (1). The positivity of \rho shows that it has supremum at most one before the harmless normalization of the coefficients. Its complexity and the total absolute coefficient sum are quasipolynomial in \delta^{-1}. It follows that one such discrete atom a(n) satisfies

Here and below a shift of the polynomial argument incorporates the choice of integer labels.

We finish with an exact floor interpolation, which is necessary because the chosen atom may oscillate on the sampling scale. Write its parameters as (\lambda,\beta,\theta,C,\chi). Let P clear the denominators of C, and write \theta=\theta_0+Pq, with q\in\Z^d and 0\le(\theta_0)_j<P. At integer arguments n, reindexing the chart sum by Pqn replaces the ordinary quadratic coefficient by \lambda+(Pq)^TC\theta_0; the other extra term is integral. Reduce this new coefficient and \beta modulo one. The same sampled atom consequently has a representation with

Only this equality at integer arguments is needed.

Put \alpha=2\lambda+\theta^TC\theta, and define the enlarged horizontal vector and matrix

Choose 0\le\eta\le1, Gevrey of the fixed order at cutoff scale \epsilon, supported in (0,1), and equal to one outside endpoint strips of total length at most 2\epsilon. Choose a one-dimensional Gevrey partition \sum_{\ell\in\Z}\rho_0(r-\ell)=1. For lift variables (w,h,r), define

The atom

satisfies the exact identity

with value zero at integer v. Indeed the support of \eta forces m=\lfloor v\rfloor; the \ell-sum is one. With h=v-m and w=\theta v-n, the phase in the remaining summand is

which equals \lambda m^2+\beta m+(\theta m)^TC(\theta m-n). This proves the identity.

All parameters occurring in the new amplitude are now bounded: the reductions above bound \lambda,\beta,\theta and hence \alpha, while its support bounds h,r,w-\theta h. The cutoff \eta has Gevrey derivative bounds with scale \epsilon^{-1}. Thus the complexity of A is quasipolynomial in the old complexity and \epsilon^{-1}; its two extra horizontal coordinates and its new matrix entry -1 introduce no uncontrolled height. Moreover |A|\le1, because its values are a cutoff times sampled values of the bounded atom.

Take v=Ly. This changes only unrestricted real oscillation parameters. Exact cell averaging gives

Replacing the sampled atom by A(Ly) changes this integral by at most 2\epsilon, since the total measure of the removed cell endpoint strips is at most 2\epsilon. Choose \epsilon\le\kappa/8. The resulting correlation is at least 3\kappa/4, and its complexity is quasipolynomial in \delta^{-1}, independently of L. This proves the normalized-interval assertion. Lemma 4.2 transfers it to the original interval.□

Proof. All functions and inner products in this proof use normalized measure on I, and all functions are extended by zero for their \mathcal U(I) norms. Put \sigma=\tau/2. Let \mathcal A be the bounded atoms of the complexity supplied by Lemma 4.9 at threshold \sigma^2, restricted to I, and let \kappa>0 be the corresponding correlation bound. Choose M\ge\max(1,2\sigma/\kappa), quasipolynomial in \sigma^{-1}. Consider the convex balanced set

Here the absolute convex hull consists of finite combinations. This set contains an L^2 ball about zero.

Suppose f were outside the L^2 closure of \mathcal B. The real Hahn–Banach separation theorem, and the complex Riesz representation of a real functional as a real part, give h\in L^2(I), normalized so that

Balancedness of each summand then implies

The first inequality follows by testing bounded simple functions in the L^1 ball; no assumption that the L^1 dual is represented by an arbitrary measure is needed. Since \|f\|_{2,I}\le1, the first strict pairing inequality gives \|h\|_{2,I}>1. Testing the second bound with u=h yields

Thus \sigma h is bounded by one and has \mathcal U(I) norm greater than \sigma^2. Lemma 4.9 supplies \varphi\in\mathcal A with |\langle h,\varphi\rangle|\ge\kappa/\sigma>M^{-1}, a contradiction.

Therefore f belongs to the closure of \mathcal B. Choose an element of \mathcal B within \tau/2 in L^2(I). Its finite structured combination has coefficient sum at most M. Put the approximation error into its L^1 error term, using \|\cdot\|_{1,I}\le\|\cdot\|_{2,I}. The L^1 error is then at most \tau, and the uniformity error is at most \tau/2. All components lie in L^2(I), as required.□

Proof. Normalize I=[0,1). Extend the kernel smoothly to a sufficiently large fixed periodic (x,t) box, supported where its real arguments lie in I. The interior support margin permits this extension without changing the kernel on its support. Its Fourier coefficient sum is bounded by a constant determined by the fixed kernel bounds. Thus it suffices to prove the first estimate for each character-weighted progression average on the fixed circle.

Let R occupy slot j, g slot i, and the two bounded inputs slots k,\ell. Disintegrate Haar measure along y_j=x+jt and apply Cauchy–Schwarz in that variable. The first cost is \|R\|_2^2, and translation along the free direction (-j,1) introduces an independent Haar difference parameter h_1 in every remaining factor. Next disintegrate along y_k, keeping h_1 on the base. The repeated b factors are unchanged under translations in (-k,1); Cauchy–Schwarz therefore removes them at cost \|b\|_\infty^4 and introduces h_2. The corresponding step at slot \ell removes the four repeated c factors at cost \|c\|_\infty^8 and introduces h_3. After these three steps the eighth power of the original average is bounded by

Every coefficient i-j,i-k,i-\ell is a nonzero integer. Multiplication by it preserves Haar probability measure, and the three parameters remain independent. The last integral is exactly \|g\|_{\mathcal U}^8.

A Fourier character of (x,t) may be distributed among slots 0,1, since x=y_0, t=y_1-y_0. Multiplication by a circle character preserves the L^2 and supremum norms and the U^3 norm, so the same estimate applies to every Fourier term. The argument starts with bounded inputs; the mixed-cube estimate in Lemma 4.8 and L^2 approximation extend it to the stated class. Summing the kernel’s absolutely convergent Fourier series proves the first bound.

For the second bound, use the two distinct coordinates occupied by R and g. Their Jacobian is a fixed nonzero integer. Bound the other two inputs and the kernel in supremum norm. The double integral then factors into the two L^1 norms, giving the assertion after normalization.□

Proof. We keep track of the finite sequence of tolerances in the argument. Write q'=q/(q-1)<2, and truncate each opposing input at absolute value B\ge1, by putting its tail equal to zero. For the h tail,

For the u,v tails, their normalized L^2 bounds give

Use the local absolute form estimate with exponents (2,2,2,2) for the first tail and (2,q,q',2) for either of the other tails. In each case the reciprocal exponent sum is at most two. It follows, by replacing the inputs successively, that the total clipping error is at most C C_1B^{-(q-2)/q}. Choose

The clipped form has magnitude at least 3\delta/4. All clipped inputs have supremum norm at most B, and retain their original L^q or L^2 bounds.

Apply Lemma 4.10 to the clipped h/B with

On multiplying back by B, write its structured part as s_1=\sum_\nu a_\nu\varphi_\nu, with

Lemma 4.11 bounds the two error contributions by C C_1B^3\tau_1 each: the other two opposing inputs are bounded by B, and \|R\|_{1,I}\le C_1. The resulting form with s_1 in this slot still has magnitude greater than \delta/2.

Next apply Lemma 4.10 to the clipped u/B with

Its structured part s_2=\sum_\mu b_\mu\psi_\mu has coefficient sum

Sum the error estimates separately over the atoms of s_1. The two new error contributions are each at most C C_1A_1B^2\tau_2. Therefore, denoting the remaining clipped input by v_B,

By the absolute coefficient bounds, some bounded atom pair \varphi_i,\varphi_k satisfies

The complexities of this pair and \delta_0^{-1} are quasipolynomial in C_1,\delta^{-1}, with a q-dependent constant but an absolute log exponent. Notice also that \|v_B\|_{2,I}\le1.

Apply Lemma 4.5 to this pair. Let D_0 bound its absolute coefficient sum and its shell constants. For each term without its coefficient the absolute local form is at most C C_1, since its three atom factors are bounded and the two free inputs have the indicated L^2 sizes. Discard shells above T_0, with T_0 chosen so that C C_1D_0T_0^{-100}<\delta_0/4. The retained terms have quasipolynomial complexity. At least one such term has form magnitude at least \delta_1=\delta_0/(2D_0). The first part of Lemma 4.6 forces its Fourier supremum A_r to be at least \delta_1/(C C_1).

Use the second part of that lemma with a uniform kernel error at most \delta_1/(10C C_1). Pair independence bounds the resulting form error by \delta_1/10. The surviving form is a finite sum

with a quasipolynomial bound D_1 for \sum_\nu|c_\nu| and for all atom complexities. Fixed coordinate Jacobians have been included in the coefficients. Every second moment has magnitude at most one, by the L^2(I) bound on v_B and the boundedness of its atom. Consequently one of the first moments has magnitude at least \delta_1/(2D_1). Conjugating its atom if needed gives the stated normalized moment.

There are exactly two bounded-input decompositions, followed by the progression separation and the single-scale separation. Thus all budgets above involve a fixed number of quasipolynomial compositions. The only q-dependent exponent used before these compositions is the power defining B, whose logarithm is a q-dependent constant times 1+\log C_1+\log(1/\delta). This proves the final quantitative assertion.□

Proof. Set \mathcal K=1+m+\log(2H). Throughout this proof, D_1,D_2,\ldots denote quantities with logarithms bounded by a polynomial in \mathcal K. The single-scale part of Lemma 4.6 gives the bilinear bound CA_r before any expansion. When A_r>0, Lemmas 4.3 and 4.4 give an expansion into terms

The possible unscaled \lambda_* and \gamma are fixed rational-height evaluations of the original parameters. In particular, they do not change their underlying parameter vector when r changes.

We first justify a finite truncation in a norm appropriate for a scale sum. Use the pooled tail conclusion of Lemma 4.4 to make the sum of the worst-case absolute tail coefficients, over all the effective quadratic and linear choices, smaller than e^{-2m}(1+D_1)^{-2}, with any additional fixed polynomial reserve needed below. For each fixed tail pair (\lambda_*,\gamma), write e(\lambda_*t^2)=1+(e(\lambda_*t^2)-1). The linear part is estimated by the mean-zero square functions proved below, with a fixed modulation \gamma. The second part has smooth-kernel norm at most C|\lambda_*|r^2\operatorname{poly}(D_1); increasing the tail reserve handles this polynomial. At a fixed \lambda_*,

At each depth, contractions and Cauchy–Schwarz give the requisite product of ordinary L^2 norms; for an energy sequence the same argument costs at most 1+L. Thus the tail is acceptable in the absolute scale sum, not merely at an individual interval.

There is now a finite main catalog, with absolute coefficient norm and rational heights bounded by D_2. Choose a sufficiently small \epsilon>0 after that catalog, with \log(1/\epsilon) polynomial in \mathcal K. Mark as exceptional every scale where a possible nonzero |\lambda_*|r^2, or a possible nonzero r|\gamma-\gamma'|, belongs to a prescribed intermediate annulus between \epsilon and \epsilon^{-1}. Enlarge the annulus by fixed powers of the catalog bounds when necessary. The height-bin observation from Section 3 bounds the number of exceptional scales by a polynomial in \mathcal K. At these scales use the original single-scale bound CA_r\le Ce^{-m}, before summing coefficients. This gives the desired estimate for the exceptional scales.

At each remaining scale the effective quadratic is negligible, and the linear centers partition into clumps of diameter at most \epsilon/r, separated by more than \epsilon^{-1}/r. One way to arrange the stated diameter is to use a smaller tiny threshold than \epsilon by the main catalog count, and then take connected components of the tiny-distance relation. The enlarged intermediate annulus excludes all intermediate distances. Replace every effective quadratic by zero and every center in a clump by one representative. The height-bin observation gives

after readjusting \epsilon by catalog factors. Multiplication by the absolute coefficient norm therefore makes the total replacement error smaller than the right side of (5).

Every combined clump coefficient is O(e^{-m}). Indeed, test the original expansion of Q(r\,\cdot) against \zeta(u)e(-r\gamma_{\mathcal C}u), where \gamma_{\mathcal C} is the chosen representative. The coefficient of its own clump is multiplied by the nonzero constant \int\zeta. All other clumps give rapidly decreasing Fourier tails, and the quadratic, truncation, and replacement errors have already been made smaller than e^{-m} by the choice of \epsilon and the tail reserve. The defining Fourier supremum is at most e^{-m}.

For completeness, we prove the square-function estimate used for these linear clumps. Make the fixed change of coordinates (x,t)\mapsto(y_a,y_b). In these coordinates, w_I has zero integral in either variable while the other is fixed. Its double primitive has compact support in an interior rectangle of I\times I. Expand that primitive in a smooth Fourier series on a fixed larger rectangle, multiply by interior cutoffs equal to one on its support, and differentiate once in each variable. This expresses the kernel as a sum of tensor products of smooth, interior, mean-zero bumps. Their coefficients decrease faster than every fixed polynomial in the tensor indices, while their fixed-order derivative bounds grow at most polynomially. It is consequently enough to consider rows

where \omega_I is such a bump. The fixed bounded multiplier \psi_a or \psi_b can be incorporated into each row.

These rows are Bessel with squared bound polynomial in \mathcal K, jointly over the retained scales, intervals, and clump representatives. Here are explicit frequency estimates. Decompose the input for a row into the demodulated shells |\xi+\nu|\asymp2^\ell/|I|. At fixed scale and center, mean zero and the L^2 Poincaré inequality give row-operator norm O(2^\ell) when \ell<0. When \ell\ge0, two frequency antiderivatives and integration by parts give O(2^{-2\ell}). These estimates are summed over the disjoint intervals at that scale. For a fixed Fourier variable \xi and shell index \ell, the height-bin observation, applied to the original horizontal parameters, the fixed modulation, and \xi, bounds the number of possible scales by \operatorname{poly}(\mathcal K). At one scale, separation of the centers bounds their shell multiplicity by O(1+2^\ell). Thus the total frequency multiplicity is \operatorname{poly}(\mathcal K)(1+2^\ell). Taking its square root and summing the preceding row norms over \ell proves the Bessel estimate. Exact center frequencies form a null set and cause no additional term. A fixed bounded multiplier preserves the estimate.

For a projection sequence, apply Lemma 3.4 to these rows. Its cost is polynomial in \log(2D_{\mathrm h}). For an energy sequence, use instead the per-interval Bessel estimate over separated centers and then sum its given energy bound; no composition lemma is needed. Cauchy–Schwarz over the two sets of rows, together with the O(e^{-m}) bound for each combined coefficient, proves (5). If A_r=0, the cutoff Fourier transform is zero, so Q vanishes on the part of the t-axis relevant to the kernel.□

Proof. Expand the low pair by Lemma 4.5. Its dyadic shell coefficients have the stated rapidly decreasing absolute sums. Apply Lemma 5.1 to each shell and sum its Fourier-supremum bands. First discard sufficiently high separation shells; then, for each retained term, discard sufficiently small A_r bands. The shell and band thresholds have logarithms polynomial in 1+B+\log(2H_{\rm low}). The remaining scales admit the finite chart expansion from Lemma 4.4. Truncate its pooled tails in the absolute scale-sum norm exactly as in the proof of Lemma 5.1. Distribute the fixed modulation e(\gamma t) between positions a,b. The surviving rational-height choices of quadratic parts and linear shifts form a fixed finite list. All of its bounds have the asserted logarithmic size. The discarded errors have the required sum by (5).□

Proof. The predictor identity expresses the multiplier-weighted mean of the physical residual as \epsilon times the predictor’s column coefficient. Smooth-bump approximation below the delayed partition supplies the 2^{-K_0} term, while subtracting the coarse mean and summing private-coordinate increments gives the (1+K_0)\sqrt\epsilon term.

Let E_h denote conditional expectation on the depth-h partition of J, and write P_h for the direct sum of its local projections. On descendants where the predictor is present, put

If a_{h,p} is the predictor coefficient on a depth-h cell, orthogonality to \Phi_p gives

Consequently \mu_{h+1}-\mu_h is \sqrt\epsilon times the change of the p-th private coordinate of P_hF_a. This identity continues to hold after arbitrary further column insertions.

For a smooth normalized bump \omega_I on an interval of depth s, set H=s+K_0. The residual moment has the exact telescoping representation

To verify it, telescope \int_I D_h(\omega_I-E_h\omega_I) and use E_{h+1}D_{h+1}=\mu_{h+1}; the term involving \mu_h and E_{h+1}\omega_I-E_h\omega_I vanishes by conditional expectation. The final remainder tends to zero because D_h has uniformly bounded L^2(I) norm and the smooth bump’s conditional-expectation error tends to zero. Thus no convergence of P_hF_a to F_a is being assumed.

The smoothness of the bump gives a factor O(2^{s-h}) in each of the two infinite sums. Its other factor is a localized orthogonal projection increment, with an additional \sqrt\epsilon for the private-coordinate term. At a fixed depth offset h-s, sum squared row norms over intervals of depth s, then sum over s using orthogonality of the increments P_{h+1}-P_h. Minkowski’s inequality over the offsets gives an aggregate row norm

This estimate uses only the bounded multiplier \psi_a, and is unaffected by the size of the enlarged column catalog.

If \omega_I has mean zero, the first term can be written with \mu_H-\mu_s. Decomposing it into the K_0 successive private increments gives aggregate row norm at most

Apply the mean-zero tensor decomposition from the preceding proof to w_I, including the fixed distributed modulation in the fixed multipliers. Pair the resulting error rows in position a with the Bessel rows in position b. Compression handles a current projection in that position; an energy sequence is handled by its per-interval Bessel estimate. This proves (7) for the part of the kernel with e(\lambda_*t^2) replaced by one.

For the difference kernel w_I(e(\lambda_*t^2)-1), an ordinary smooth tensor expansion suffices. Its derivative costs contain the factor |\lambda_*|r^2\operatorname{poly}(D_{\rm pred}). The same telescoping argument estimates its expectation-free part. For the initial \mu_H term, which need not be tested against a mean-zero bump, use the per-depth bound C\sqrt\epsilon\|\1_JF_a\|_2 and sum |\lambda_*|r^2 geometrically over the scales satisfying |\lambda_*|r^2\le D_{\rm pred}. This accounts for the additional \sqrt\epsilon term in (7).

For sequential replacement, write a previously delayed projection as its current projection plus a rolling projection difference. The latter has a square-energy bound with L\le C(1+K_0), by telescoping its orthogonal increments. It is therefore an admissible opponent in the second application of the argument.□

Proof. Start with no columns at a segment root J and proceed in depth order. At a cell whose residual fails the desired inequality, Lemma 4.12 supplies a bounded atom \varphi with physical residual correlation at least \rho, where \log(1/\rho)\le k^{O(1)} and the atom complexity obeys the same logarithmic bound. Adjoin (\varphi,\mathbf e_v) with a fresh private coordinate and refit. The old residual has zero component in this fresh direction. The squared projection energy therefore increases by at least c\rho^2|I|. Repeated fits at one cell and refining fits at successive depths form one nested sequence of subspaces in L^2(J), so their total energy gain is at most \|\1_JF_j\|_2^2\le|J|.

Stop at the first cells where the cumulative number of insertions along a path would exceed C\rho^{-2}. Integrating the insertion count over the disjoint first-excess cells bounds their total length by C^{-1} times a fixed multiple of |J|. Choose C so this is at most |J|/4, and restart the same procedure on those roots. Iteration gives a geometric bound for the sum of segment-root lengths. On every retained cell the greedy fit has terminated with the required residual estimate.

With private weight one, the coefficient \ell^2 norm is bounded by the normalized projection norm. A catalog of at most \exp(k^{O(1)}) bounded scalar columns consequently gives the claimed physical supremum bound by Cauchy–Schwarz. Complexity and count bounds follow from detection and the insertion cap. The uniform polynomial-degree assertion is inherited from Lemma 4.12.□

Proof. For a finite terminal level, decompose each actual input into its increments up to that level and its terminal residual. In every term having at least two increments, record the earliest two levels u<t. Summing all later choices in either other position gives exactly the input minus its last update before t. This yields (8) for all such terms. The terms with at most one increment consist of either four terminal residuals, or one whole terminal projection and three terminal residuals.

We record the comparison that controls these remainders and the claimed pair estimates. If h>\ell are updates in distinct positions, replacing their pair of projections by the corresponding actual inputs has form-norm error at most

for an absolute d. Replace the more accurate projection first, using the physical supremum bound of the other as its opposing L^q norm, and then replace the less accurate projection against the actual L^q input. Choose A>d, with fixed reserve. The same reasoning applies when a projection is already replaced by an actual input. In particular, a pair of terminal residuals has form norm tending to zero. Their two remaining opponents have bounded L^2 size, proving the assertion about the terminal remainder on a finite interval collection.

For the tuple estimate, first suppose u<t-4. Whenever the early argument in position i is paired with a later difference or residual, compare that later argument directly with its actual input against each of the two earlier projections. The actual terms cancel in the later difference or residual. Its smallest accuracy index is at least t-4, and exceeds the relevant earlier indices; the stagger therefore defeats their supremum bounds. Pairs not involving this early position use only indices at least t-4, apart from the preceding projection in position j, whose index is exactly t-4.

If instead u\ge t-4, distinct positions force u\ge t-3; all nonzero projections in the tuple then have indices at least t-7. Expand a selected pair into projection/actual pairs and use the preceding comparison. At least one selected full argument has coefficient sum zero after replacing projections by their actual inputs, so the two-actual terms cancel. The same argument covers the pairs in the first case that do not contain the early position. Each error is bounded by a fixed multiple of \exp(-c_q k_{t-7}) for large starting log. Since k_{t-7}=k_t^{A^{-7}}, decreasing a fixed exponent below A^{-7} gives the stated e^{-2k^{\theta_0}} bound. The L^2 size assertion follows from contraction and the triangle inequality.□

Proof. We give the choices in their dependency order. On a common older segment, the low columns, their coefficients, and their path counts have a bound D_{\rm old}=\exp(k^{O(1)}). Apply Corollary 5.2 to every available pair of low labels, using a precision log B that is a sufficiently large fixed polynomial in k. Insert the conjugate predictor columns in positions a,b at the first joint availability of that pair, and retain them thereafter. Pairs inherited at the segment root have their predictors present from that root. Use private weight \sqrt\epsilon for these predictors.

Choose K_0 and \log(1/\epsilon) to be sufficiently large fixed polynomials in k, after D_{\rm pred} and the original low coefficient/count bounds have been fixed. The choices make both the error of Corollary 5.2 and the error of Lemma 5.3 smaller than e^{-2k} after all original-label sums. Only a polynomial in the logarithm of a subsequently enlarged high catalog occurs in these errors. Thus the choices remain valid under any later fixed-polynomial growth of that log catalog bound, by increasing the starting k.

Here is the summation over low labels in detail. Expand the two low projections into their original labels only in this application. A fixed pair is available on the subtree rooted at its deeper birth cell, or on the segment root if it was inherited. Its coefficients on a use cell are constant and bounded by a fixed power of D_{\rm old}. The number of such pair-birth roots on a path is also bounded by a fixed power of D_{\rm old}. Consequently, on a normalized segment root R,

Indeed Cauchy–Schwarz bounds this sum by the product of the square roots of the corresponding two sums of squared masses, and the path multiplicity bounds each by D_{\rm old}^C\|\1_RF_e\|_2^2. Apply Corollary 5.2 on both sides of each replacement, and then Lemma 5.3 to each retained term. A previously delayed opposing projection is its current projection plus a rolling square-energy sequence with L=O(1+K_0). This proves the absolute-sum error assertion, even after the extra fits now described.

The predictors alone need not give the pair estimates in the statement. We therefore add tuple-specific accurate fits. Proceed chronologically in projection depth h. At depth h, first admit any predetermined predictor columns becoming available there. A coarse decision cell at that step has depth s=h-K_0. Test the physical residual

on that coarse cell in the norm \mathfrak d_{e,I}. The normalized L^2(I) size of this residual is bounded, because P^{\rm new}_{e,h} is a contraction on the partition of I into its depth-h descendants. Require a precision whose log is a sufficiently large fixed power of k. On failure, detection gives a bounded scalar column with a coarse residual moment at least \rho', where \log(1/\rho')\le k^{O(1)}. Insert one new common label with private weight one on all the depth-h subcells of I, and retain that label on their descendants.

The joint gain from this insertion is at least c(\rho')^2|I|. To see this, let m_Q be the normalized residual moment on a depth-h subcell Q\subset I. The fresh private direction is absent from every old tested residual, so the gain on Q is at least c|Q||m_Q|^2. Cauchy–Schwarz and the coarse moment bound give

All ancestor extra requests affecting a coarse cell are common to its fine partition. Cap their cumulative number along a coarse path at a sufficiently large multiple of (\rho')^{-2}. First excess coarse cells have small total relative length, by the joint gains and the orthogonal energy bound for the entire chronological iteration. Send their coarse summands and descendants to fresh attempts. Predictor admissions have their separate path cap from the older lists and the finite predictor catalogs.

There is no conflict between a coarse cut and a delayed projection already used above it. If a coarse cell is cut when projection depth is h, all its strict ancestors have already had their delayed evaluations computed, since their projection depths are smaller than h. Continue the stopped estimator on the cut subtree by inheritance and refitting, with its predictor moments retained; its later residual tests are no longer needed for the summands sent to the new attempt. This extension supplies all finer projections needed in the telescoping proof of Lemma 5.3. A fresh attempt for the descendant coarse summands may use a new extra-label catalog. Intersect the resulting coarse segment families between the two high positions and the older lists. Their roots still have bounded total length, and each resulting piece uses restrictions of genuine nested projection sequences, including their extensions beyond its boundaries.

The accuracy order is as follows. Choose the first new high precision to dominate all previous projection and predictor costs. Detection fixes the first new catalog bound. Choose the second new high precision to dominate that first catalog’s physical supremum bound as well. For a fine subcell Q\subset I at depth s+K_0,

The private-coordinate coefficient estimate thus bounds the first new projection’s physical supremum on I by 2^{K_0} times a fixed power of its own count and conditioning bound. It does not involve the next high precision. The two choices are therefore successive fixed-polynomial log choices.

Apply the comparison argument of Lemma 5.5 to these two new approximants and the older projections. The new fits are more accurate than every older projection, and the second is more accurate than the first with sufficient reserve for its supremum bound. In each pair of full arguments in (9), the actual-input contributions cancel exactly as before. If the earliest low position has a much earlier index, the other selected full argument is a difference of later approximants and is tested directly against that earlier position. Otherwise all older indices involved are at least t-7. Decreasing the absolute exponent from Lemma 5.5 yields the claimed \eta bound. Contraction on the fine partition and on the older coarse cells gives the local L^2 bounds. Finally, the new high fits are specific to this tuple; their larger precisions do not enter the global update recurrence k_{t+1}=k_t^A.□

Proof. Fix a path point y, and let the fourth position be l. Writing the transpose at y+h, its squared L^2 norm is an integral in the scaled variables (h,t,t')/r, r=|I|, containing the fixed triple and its conjugate copy and the two smooth kernels at the shared anchor. The six atoms combine into the multivariable chart setting of Lemma 4.3. Recenter at the fixed y. All unscaled horizontal, linear, and quadratic parameters are now fixed along the path, before multiplication by r or r^2.

If \mathcal A_I>\delta, apply Lemma 4.3 to this integral at Fourier frequency zero, and then Lemma 4.4 with an error small compared with \delta^2. Every effective quadratic entry and every linear t-frequency in the truncated expansion is of the form r^2 a\cdot\chi_* or r a\cdot\chi_*, respectively, for a fixed finite parameter vector \chi_*. The coefficient vectors have rational height quasipolynomial in D_{\rm base} and \delta^{-1}; the dimension is polynomial in the original chart dimension. This description also covers the choices of rational splitting.

Choose tiny/large gap thresholds after the finite truncation, with reserve for its absolute coefficient norm. Outside polynomially many intermediate scales, the height-bin observation makes every effective quadratic entry that is bounded in the Weyl expansion tiny, and every retained t-frequency either tiny or large. Remove the effective quadratic by uniform approximation. Large t-frequencies integrate to a negligible amount by smoothness of the kernel. For a tiny t-frequency, replace its oscillation by a constant; the remaining t-integral is zero by the exact marginal

at fixed h,t'. The other kernel and the remaining linear oscillations are independent of this t variable. All errors, after coefficient summation, are smaller than \delta^2. This contradicts \mathcal A_I>\delta and proves the count.

Apply that count to the successive bands 2^{-n-1}\delta<\mathcal A_I\le2^{-n}\delta, using threshold 2^{-n-1}\delta. Their total contribution is bounded by

which is the asserted small-norm sum. The kernel may vary with I within its fixed class: only its uniform smoothness and exact marginal were used.□

Proof. Declare I active if some triple of available pure basic scalar columns in three positions has transpose norm greater than D_{\rm base}^{-C'}, with C' a sufficiently large fixed constant. Along a path there are at most D_{\rm base}^3 triples, and Lemma 5.7 gives only a logarithmic-polynomial number of active scales for each one. Enlarging the absolute power proves (10).

On an inactive interval, expand three of the B arguments into their basic columns. Their coefficient sums and the local L^2 size of the remaining argument cost only a fixed power of D_{\rm base}. For a fixed triple, the small-norm assertion of Lemma 5.7 bounds its path sum by D_{\rm base}^{-C'} \operatorname{poly}(\log D_{\rm base},C'). Sum over the pathwise triple count and integrate over path points. Choosing C' larger than all these fixed power costs, and using D_{\rm base}\ge e^k, gives the stated e^{-k}|R| error.

For each fixed leading level, all budgets and the multiplicity in (10) are fixed constants. Local absolute L^2 estimates and bounded sizes thus give a bound independent of the number of depths. For later levels, the pair estimate in Proposition 5.6 and the bounded sizes of the other two arguments give |H_I(Y)|\le C\eta. Decrease \theta once more, and increase the starting log, to absorb this fixed constant into the definition of \eta. Multilinearity expands H_I(Y) into the retained patterns and the already estimated term with both U arguments. For nonnegative quantities, \min(\eta,\sum_j a_j)\le \sum_j\min(\eta,a_j). The absolute errors and the two-U term may therefore be taken outside the minimum, leaving (11) separately for the remaining patterns.

This reduction retains the whole coarse arguments B_l. In particular, later filter removal in two nonside positions returns exactly these arguments; substituting Y_l=B_l+U_l changes the term only by terms with an additional basic side. The full complementary pair then has the pair-smallness already proved. All relevant current families remain inherited under restriction to a normalized subroot, and their rolling square estimates follow anew from orthogonal energy there. Hence a later stopping attempt can use the same base data.□

Proof. For the inner product, smooth the interval indicator with L^1 error much smaller than \tau_2 and derivative bounds polynomial in \tau_2^{-1}. The physical columns have bounded supremum norm. A larger inner product would therefore give a Fourier coefficient of their conjugate product above a fixed multiple of \tau_2 after recentering and rescaling. Lemma 4.3 gives (2) for the difference data. The fixed factor c_i in a same-slot product only changes the allowable height by a fixed power. Choosing H_{\rm ed} above this bound proves the first assertion.

For the second assertion, call the source slots i,j and the spectator slots a,b. In centered, rescaled spectator coordinates (z,w), the linear parts of the source coordinates are r(\xi_i z+\zeta_i w) and r(\xi_j z+\zeta_j w), where

All four coefficients for l=i,j are nonzero fixed rationals. The identities \sum_l c_l l^u=0 for u=0,1,2 give

First clip each spectator at a sufficiently large inverse polynomial in \tau_2. The discarded part has L^1 norm at most the inverse clipping height, since its L^2 norm is bounded. The absolute form estimate with exponents 1,2,\infty,\infty bounds this error. We can therefore suppose that the spectators have both bounded L^2 norm and supremum norm at most an inverse polynomial in \tau_2.

Use the return calculation from the proof of Lemma 4.3 for the displayed difference data. Choose an integer period clearing the chart and interpolation denominators. For a small return Vh=m+d_h, with m divisible by this period, translate z by h and translate the source lift indices by \xi_i m_v,\xi_j m_w. The amplitude and kernel errors are bounded by \abs{h}+\abs{d_h} times a quasipolynomial in the available complexity. The resulting modulation in w is

Indeed the mixed quadratic contribution in each source contains c_l\xi_l\zeta_l; factoring out \kappa leaves exactly the difference of the two curvature data. The modulation in z can be included in the first spectator and has no effect on its L^2 norm.

Suppose the original correlation is larger than \tau_2. Applying the calculation to \ell(h,m), for a sufficiently large quasipolynomial number of integers \ell, produces correlations of size at least a fixed multiple of \tau_2 between one fixed bounded kernel K_1(z,w) and tests f_\ell(z)g_\ell(w). Here the f_\ell have bounded L^2 norms, and

up to a common bounded multiplier. The error remains small by choosing the return threshold sufficiently small before choosing these multiples. If \abs{\Omega(h,m)} exceeds a fixed constant, the separated exponentials, multiplied by the clipped spectator, are Bessel on the fixed box, with bound controlled by that spectator’s squared supremum norm. Cauchy–Schwarz in z gives

The right side is bounded by the same Bessel constant times \norm{K_1}_2^2, while the left side grows with the number of returns. This is a contradiction. We have proved a uniform bound for \Omega(h,m) on every return below a quasipolynomially small threshold.

The final, algebraic part of the proof of Lemma 4.3 now applies to these difference data: the alternating return identity gives an isotropic lattice space, successive minima give a rational fast projection and a bounded slow component, and one return in each coordinate direction bounds the effective quadratic form. This part uses only the just-established return bound. It yields (2) at quasipolynomial height, contradicting d_s(v,w)>1 when H_{\rm ed} is chosen sufficiently large.□

Proof. List the path tags as v_0,\ldots,v_L, and put all their horizontal coordinates in one direct sum. For each edge choose its fast space U_e from the definition of an edge. Let U be the subspace of the joint coordinate space whose restriction to each consecutive pair belongs to U_e. These are rational linear equations. Clearing denominators and applying determinant bounds to one maximal independent system shows that the orthogonal projection onto U has logarithmic height polynomial in L, the dimensions, and \log H_{\rm ed}. The same determinant estimate bounds the norm of a right inverse on the range of this system.

Let V be the concatenation of the normalized horizontal vectors. Each defining equation has bounded error on V, because the edge slow component is bounded. The right-inverse bound therefore shows that V^F, the orthogonal projection of V onto U, satisfies \norm{V-V^F}\le H_1, where \log H_1\le\operatorname{poly}(P,k). On an edge, both its chosen fast component and the restriction of V^F lie in U_e, and their difference \Delta is bounded by this new budget. Since U_e is isotropic for the edge skew form, replacing its fast component by this restriction changes the effective correction by

which is bounded at the new budget. In particular, no unbounded fast coordinate enters this error.

Sum the edge effective quadratic forms using these common joint fast and slow components. The diagonal symmetric terms, the skew mixed terms, and the original \lambda terms telescope, leaving only the two endpoints. Let U' be the image of U under endpoint restriction. It is rational with the same type of height bound. For two vectors of U, summing their edge skew pairings cancels all intermediate coordinates; each edge pairing is zero. Hence U' is isotropic for the endpoint difference skew form. The restricted joint fast vector lies in U' and is within the new bounded error of the endpoint V. Replacing it by the orthogonal projection of endpoint V onto U' changes it by a bounded vector of U'. The preceding isotropy calculation again bounds the change in effective curvature. These are all the assertions of (2) in endpoint coordinates.□

Proof. At a present vertex a round’s top score lies in \{0,\ldots,d_0\}: the vertex itself is a competitor of distance zero. Individual primary competitor scores only increase with depth. At a fixed top score a marked winner can occur in at most one episode, with a fixed identity. Once a rival has destroyed its gap, that rival’s score cannot decrease; a different marked winner at the same top value would have the old winner as a rival of that value. Thus the first marked primary round, its score, and its winner have at most 5d_0^2 episodes.

Within a primary episode there are at most d_0 cohorts. Within one cohort all profile metrics are fixed. The vector of floor log-sizes of all the balls has at most 2d_0^2 epochs, including its initial value, since its coordinates increase and each has at most O(\log D) possible log-sizes. Within each such epoch the selected l,n are fixed. In that epoch the high metric is fixed, so high scores increase only by admission of competitors. The same marked-episode argument gives at most 5d_0^2 episodes of selected high data. Altogether there are at most 50d_0^7\le d_0^8 changes when k is large. Seven metadata coordinates, each with at most d_0+2 states, and the change counter require fewer than 256d_0^{15}<d_0^{30} ordered states. Decrementing the counter at each change makes the assigned color nonincreasing even if the other metadata increase.

For completeness, a fixed untruncated score situation is marked with probability at least 2^{-5}. Choose a canonical winner in every delay outcome, resolving ties deterministically, and increase only its delay by 5. The resulting outcome has a unique winner with gap at least 5. This map is injective, because its image identifies the increased coordinate; geometric probabilities decrease by exactly 2^{-5}. Truncation changes this probability by at most D2^{-d_0}. In particular it remains at least 1/64 for large k.

There are only polynomially many situations to check on a fixed spatial path. For the primary stage cap \lfloor d_s(z,v)/b_1\rfloor at d_0+5; larger values cannot affect any top or gap decision. The capped pair distances change only O(D^2(d_0+1)) times, by monotonicity. Counting the vertices to test gives O(D^3(d_0+1)) fixed primary situations. Conditional on all primary arrays, the primary, cohort, and ball epochs number at most O(Dd_0^5) across vertices. Each has at most D present prefixes, so at most O(D^2d_0^5) high situations require checking. These situations are fixed before revealing the independent high arrays. The probability that all d_0 rounds fail at any such situation is at most (1-1/64)^{d_0}. The union bound tends to zero with D, since d_0\ge1000\log(2D).

Integrate this pathwise failure probability over R. Some realization of the arrays has a small measure of failed paths. The first failure cells are disjoint, and their total length is at most that measure. This proves the claimed exceptional-root bound without a bound on the number of vertices across different spatial branches.□

Proof. At a marked primary center v at depth s, each rival’s true score is at most (g-5)b_1, since its quantized score is at most g-5. Consequently

The winning true score is greater than (g-1)b_1. If d_s(b,v)\le h, the winner score at b loses at most h, and f_b(s)\le(g-5)b_1+h<T_l. Both profiles are below the clipping range at every earlier depth. At all later depths the graph distance remains at most h, so their true profiles, and then their clipped profiles, differ by at most h. Thus both profile distances from b to v are at most h, which proves the first assertion with witness v_0=v.

Next compare a historical central witness v_0 with the current center v of the same group. Their common high winner a has delay at most d_0, and the marked top score g_2 is nonnegative. Hence

The earlier low ball at v_0, on the vertices present when it was chosen, is contained in the current high ball at v: use d^{(l+1)}\le d^{(l)} and R_l+2(d_0+1)b_2<R_{l+1}. This earlier low ball and the current low ball each have at least (3/2)^n vertices, whereas the current high ball has fewer than (3/2)^{n+1} vertices. The two low balls intersect. The triangle inequality in the low metric gives

For a pad-h vertex b it follows that d^{(l)}(b,v)\le A_h:=10R_lh+2R_l. Our constants give

For example the gap on the right before subtracting 10 is \tfrac12d_0^9b_2(H_{\max}+1), whereas A_{H_{\max}}\le10d_0(d_0+1)b_2(10H_{\max}+2). Since the clipped profile of v at the use is T_l, this proves the true rival bound

Let b' be a present graph neighbor of b at this use. Its winner score loses at most one. Both f_b and f_{b'} are below T_{l+1} at all depths up to s, and their difference is at most one at all depths after s. Thus d^{(l+1)}(b,b')\le1. For the low metric, after s the distance from b' to v_0 costs at most 10R_lh+1. Before s, its clipped profile is between T_l and T_l+A_h+1, while the clipped profile of v_0 is between T_l and T_l+2R_l. Their difference is at most A_h+1\le10R_l(h+1). All three predicates of (12) therefore hold at pad h+1. Finally, at admission the first predicate gives

which is polynomial in P,k and proves the path bound.

It remains to prove the usable-endpoint assertion. Let I,J be overlapping endpoints for distinct same-color groups G,G', with J the finer endpoint, and choose an actual-use certificate u for G' at or below J. Let b,b' be the respective mask vertices. Suppose, for a contradiction, that d_{\operatorname{depth}(J)}(b,b')\le1. This distance bound persists to u. The vertex b' remains in its mask at u. No future use of G on this same branch is required.

First suppose their primary winners z,z' differ. Their colors give the same primary round, score g, and equality case or separating length level. At a fixed length level, upper-half index cohorts on an increasing path either use the same dyadic block or increasing disjoint blocks. In either case the earlier winner belongs to the later frozen prefix W'. This is immediate also for equality cohorts. The earlier winner score at b persists, so adjacency at u gives

This contradicts the true rival bound for G' at u, which is less than T_{l+1}-10\le(g-5/2)b_1-10, since b_1 is much larger than H_{\max}.

Now suppose z=z'. Its fixed admission index and the common separating level determine the same cohort block; its prefix W is inherited on the overlapping path. The metrics coincide, and distinct groups must have distinct high winners a,a'. Historical witnesses v_0,v'_0 for the two masks lie on the common ancestor path of J and can be ordered along it. At the later historical witness the earlier high winner remains available in the same high round and metric. The earlier winner’s true score is greater than (g_2-1)b_2 at its own witness, whereas at the later witness it is at most (g_2-5)b_2. Therefore

On the other hand, at the certificate u the true rival bound puts f_{b'}(u) below T_{l+1}-10, and adjacency puts f_b(u) below T_{l+1}-9. Both profiles were below the high clipping range at all earlier depths, and they differ by at most one at all later depths. It follows that d^{(l+1)}(b,b')\le1. The two mask predicates then imply

a contradiction. This proves separation, including the case where the two endpoints have later certificates on different branches.□

Proof. For a fixed label v in G, combine all its occurrences into the scalar coefficient function

The private coordinate of label v in Z_G(y) is precisely \epsilon_v^{1/2}a_{v,G}(y). Thus, if W_0 bounds inverse square-root private weights,

This uses the combined coefficients, so any cancellation between repeated uses of a label is retained.

Fix labels v,w in distinct groups. Their tags have a deterministic last separating depth: by monotonicity, the depths where d_s(v,w)>1 form an initial interval. Cap this interval at the last depth in the finite calculation if needed, and call its endpoint s_*. If two supporting endpoint cells I,J overlap, their intersection is the finer cell. Lemma 6.4 says that its depth is at most s_*. Therefore every nonzero term in

is measurable on the grid at depth s_*. The whole product is measurable on that grid. Endpoint terms finer than s_* have no overlap with any term of the other label and contribute zero. If no separating depth occurs within the root, there are no overlaps at all.

Apply Lemma 6.1 on every grid cell. The two private labels cannot coincide on an overlap, since equal tags have distance zero. Hence

Pointwise Cauchy–Schwarz in groups and labels bounds the sum of all these cross terms by \tau_2 times a fixed power of the pathwise count and private-weight bounds, multiplied by \sum_G\norm{Z_G}^2. Taking \tau_2 smaller than the reciprocal of this factor proves the assertion. There is no factor counting repetitions or endpoint times.□

Proof. Fix a nonnegative compact smooth partition function \rho with \sum_{n\in\Z^d}\rho(u-n)=1. Its support and Gevrey bounds can be chosen within the basic budget. Choose a sufficiently large integer period P_0, fixed for this vertex, accommodating the supports of both its original amplitude and these partition functions. The templates have amplitude

where \ell ranges over residue vectors modulo P_0, in addition to an arbitrary external linear modulation. Their physical supremum norms are bounded by one because the partition weights are nonnegative.

Insert this partition in the other three slots and use lifts d_l=\theta_v y_l-n_l in all four slots. Extract the matching phase c_l(\lambda_vy_l^2+(\theta_vy_l)^TC_vd_l) from each input, putting its conjugate into the corresponding input multiplier. On the progression,

with bounded integer carries k_l. The common quadratic terms cancel because \sum_l c_l l^u=0 for u=0,1,2. The phase left by the carries is

which is a sum of linear modulations. There are quasipolynomially many possible carries within the fixed support bounds.

To separate the lifts, impose each specified carry equality by a smooth function of d_l-(1-l)d_0-ld_1 that is one at that integer and zero at all other possible integers. Such a function is exact here, since its argument on the progression is integer valued. Expand these coupled smooth functions in Fourier series on a box of period P_0 containing the bounded lifts. After multiplication by the partition functions, every nonsource factor has the required template amplitude, possibly with an unrestricted character index. Write an index as \ell=P_0q+r; then

since q^Tn is integral. Thus only the residue r enters the template, and the other part is an allowed linear modulation. The source factor, after cancellation of its phase, periodizes a smooth horizontal function; expand that function in ordinary torus characters. Its remaining dependence on y_i is only a linear modulation. Derivatives of sufficiently high order, polynomial in the dimension, bound all these absolute Fourier coefficient sums by a quasipolynomial in the fixed basic budget. For a template source, reducing its own character to a residue gives the same bound. There is no iterative increase of nonlinear complexity.

It remains to prove the assertion for one separated term. Express y_i,y_j in the two spectator coordinates y_a,y_b. Separate the smooth rescaled kernel by an absolutely summable Fourier expansion in these coordinates; source linear modulations can be absorbed into the two spectator tests. The target coordinate has the form

with both \alpha,\beta nonzero fixed rationals. Zero-extend the demodulated target, consisting of the physical residual multiplied by the conjugate of its bounded template, from I to a fixed larger interval and Fourier-expand it there. Its coefficients are normalized moments of the residual with same-tag templates and arbitrary linear modulations. If their supremum is M, the resulting integral is bounded by

Here A_m,B_m are the spectator coefficients at frequencies affine in m, with nonzero spacings proportional to \alpha,\beta. These families are Bessel on the fixed integration intervals. Multiplication of a spectator by its bounded demodulating template preserves its L^2 bound. The last display is therefore at most a fixed constant times M. Fourier expansions can first be finite; L^2 convergence and the same Bessel estimate pass to the limit. Summing the absolutely convergent separated terms gives the form bound H_0M. A form norm larger than t_0 consequently forces a target moment at least t_0/H_0, after enlarging H_0 by a fixed factor if necessary to avoid an unattained supremum.□

Proof. Hold the original graph vertices, masks, and named histories fixed. The numbers of masks and histories at one cell are \exp(k^{O(1)}): there are polynomially many original-vertex contexts, polynomially many ranks and colors, P pad choices per factor, and O(\log\mathcal L) factors. This count uses names of contexts and masks, not their numeric distance values or the number of inserted columns. The helper bases have the stipulated same count.

Upon a failed test, Lemma 6.6 supplies a bounded physical column with moment at least the threshold in (13) divided by H_0. Insert a fresh copy at the target rank. Its new private coordinate is zero in the old tested state and in the old projection. Thus that physical moment equals the vector inner product with the old orthogonal residual. If P_{\rm old},P_{\rm new} are the target projections before and after insertion and X is the old tested state, then

Indeed the fresh vector column has uniformly bounded norm, and projection onto its component orthogonal to the old range gives this inequality. If the base has zero norm no test fails.

Although X can change when other projections change, its squared increments can still be charged. Fix a named history, a target mask, and a trial sequence of at most T insertions. Every individual positive projection in that history is nested in the insertion index. Expand complements and use the binary-prefix argument of Lemma 3.4. The target increments are orthogonal rows; the right-hand history is evaluated at the old insertion index. The sum of their squared actions on this changing history is at most

This is precisely the row version of that binary argument, and it does not require treating a changing projected state as a fixed vector.

It follows that rank-r insertions at a cell are bounded by

The exponential factor includes the configuration count, the squared inverse tolerance, and H_0^2. It does not include the number of possible triggering source columns. For a fixed target history, any triggering source produces a fresh moment in that history’s old residual; the resulting target increment is charged to the same orthogonal row sequence regardless of that source’s identity.

There are at most M active cells on a path. Absorb this factor, and fixed factors for ranks and slots, in \exp(k^{C_1}). Starting with a basic bound n_0, choose

with C_2 sufficiently large, and set the global trial cap to the resulting n_{\max}, with spare fixed factors. Since there are only \mathcal L+5=k^{O(1)} ranks, \log n_{\max}=k^{O(1)}. With T=n_{\max} the binary-prefix factor is much smaller than the chosen ratio for large k. Thus no trial cap is attained. This argument also accommodates simultaneous changes of histories caused by insertions at other ranks: all the underlying positive projections remain nested along the common insertion sequence. It proves termination of the main closure, and the identical argument proves termination of the later helper closure on its fixed bases.

All constants used in this count were determined by the basic template bounds and the number of named graph contexts. They do not depend on H_{\rm ed}. We may therefore first fix these bounds, then choose \tau_2 sufficiently small for Lemma 6.5 and all pair-mismatch errors, and finally choose H_{\rm ed} by Lemma 6.1. This order is not circular.□

Proof. Maintain a range of colors [\ell,h], an upper primal guard slot, and when \ell>1 a lower residual guard slot. Initially the range is [1,\mathcal L]; use any of the at least three basic coarse arguments as the upper guard and take no lower guard. At a midpoint m=\lfloor(\ell+h)/2\rfloor, choose a third slot distinct from the guards and split it by the smeared cumulative filter at m and its complement. In the positive term that slot becomes the upper guard, the former upper guard is released, and the range becomes [\ell,m]. In the residual term it becomes the lower guard, the former lower guard is released, and the range becomes [m+1,h]. A guard is not postcomposed before it is released. Use a deterministic slot-choice rule for each branch of this binary search, even if some colors are empty at a particular cell. There are O(\log\mathcal L) splits per path and at most \mathcal L leaves.

Consider a leaf with color k'. Its upper guard is either the initial basic coarse argument or has outer cumulative rank \mathcal L-k'+1 and the corresponding pad. Expand this one outer primal support, keeping its average when present. For the initial basic guard, set the upper pad to zero. Call a source tag old if it lies within this upper pad of a vertex of color at most k'-1. If k'>1, the lower residual guard has rank \mathcal L-k'+2 and a base pad larger by at least 10P. Every old source therefore lies in all its residual masks and has lower rank. Equation (13) kills it, with arbitrary normalized L^2 spectators. For k'=1 there are no old tags.

Every remaining upper source is near a current vertex of color k'. In the initial basic case this is the source’s own original vertex; in the other case it follows from membership in the cumulative mask and exclusion of old tags. Let G be the current group of such a central vertex. The upper pad is smaller than the first ball pad, so Lemma 6.4 puts the source in every needed mask of that ball filter. The source cannot match two distinct groups: it would then violate same-color mask separation at the current cell. Moreover it is separated from the primal sources of every other current group’s ball filter.

Choose a third slot, distinct from the upper and lower guards, and insert its first-rank ball projection, summed over current groups. For the matching group, the difference from the identity is bounded by (13), using the upper source. For any other group, expand the new primal ball source and apply Lemma 6.1. Old-source terms on either side are still bounded using the untouched lower residual guard. This yields an identity, with small error, in which the third slot has pure support in one group’s first ball mask.

Keep that third slot as the source, and postcompose the other three slots with their own larger-rank, larger-pad ball filters in order. The first ball source has lower rank than each new target, and its mask is contained in all the new target masks. Thus (13) bounds each residual insertion. The third slot is left unchanged. The result is (14); each slot’s inner cumulative history depends only on the search path, and only its outer ball filter depends on the group.

We specify why the numerical errors are summable without spending a full coefficient bound in a main term. Each test using a rank-r residual expands only an outer lower-rank source. Its coefficient bound is at most D\sqrt{n_{r-1}} times the local input norm, canceled by the denominator in (13). Basic local sizes and all same-time contractions are bounded. The remaining counts of active cells on a path, leaves, current groups, and filter-insertion steps are fixed powers of D; choosing C_\tau sufficiently large makes their total length-weighted contribution at most e^{-k}|R|. Pair-mismatch errors can additionally contain fixed powers of the full main count, but \tau_2 was chosen after that count and makes the same bound hold. No expansion of a full column coefficient sum is retained in (14). Finally, \min(\eta,\sum_j a_j)\le\sum_j\min(\eta,a_j) for a_j\ge0, so the minimum can be distributed over the fixed leaves as claimed.□

Proof. Expand each complementary factor 1-\overline Q as a difference of identity and a positive average. The number of resulting products is at most 2^{O(\log\mathcal L)}=\operatorname{poly}(k). In any such product, remove the larger rank among the current outermost filters of c,d, regarding an unfiltered basic input as rank zero. Positive ranks are distinct: each cumulative rank was inserted in exactly one slot at one search split, and the four final ball ranks are distinct. Thus the other complementary source always has strictly lower rank than the filter being removed.

Suppose the next filter is on c, and use the projections and arguments defined in the statement. The A_l^p are orthogonal band projections at the current cell and sum to the identity.

On band p, the positive average P^{-1}\sum_{u=1}^P Q_{l,u} has weight (P-p+1)/P, including weight zero at p=P+1. Therefore the difference in the statement is exactly Equation (15), with the arguments written in role order. This proves the algebraic identity, including the terminal bands.

If p>q+2, the lower c band is a difference of primals, so expand its pure sources. Their rank is a main rank below the helper rank, and their tags belong to both helper masks needed to write the higher a band as a difference of residuals. The helper version of (13), applied to the history on the fixed base S_a, bounds the resulting terms. The two other arguments are arbitrary L^2 spectators.

If q>p+2, instead expand the outer primal source of x_d, using the basic expansion if it is already unfiltered. This is a main source of rank strictly below the outgoing rank on c. The lower a band is primal. Let \mathcal M_p be the outgoing vertex mask at its pad h+p. Classify a pure source of x_d as near if its tag is within one edge of some vertex in \mathcal M_p, and as far otherwise. For a near source, neighbor closure gives membership in \mathcal M_{p+1}, hence in both masks needed for the higher c residual band. Apply the main test with the entire helper band A_a^p x_a as one L^2 spectator. For a far source, expand the helper primal: every pair of source tags is separated, so Lemma 6.1 applies. Neighbor closure is the triangle inequality for a cumulative mask and Lemma 6.4 at the current actual use for a ball mask. The gap q-p>2 leaves at least this one-pad margin. The main version of (13), already imposed on the inner history x_c, then bounds it. Thus helper insertions need not alter any main catalog.

In the first direction the coefficient bound of the main source is canceled by the helper test’s denominator. In the second direction the lower-rank main source coefficient is canceled by the main test’s denominator; the helper primal serves only as an L^2 spectator. For pair mismatch, expanding the two primal supports can cost fixed powers of the helper and main counts. Those counts were anticipated before \tau_2 was selected. There are only O(P^2) pairs of bands and polynomially many transfer positions. Since \log P=O(k), these factors are absorbed by the prior accuracy choices. All filters are contractions, so the remaining norm factors are \norm{S_a}_{2,I}\norm{S_b}_{2,I} and bounded complementary basic sizes. This proves the stated far-error estimate. Averages in \mathcal D are bounded directly by their tested averaged histories and do not cost the number of pad choices.

Retain the near bands and forward only the transferred main term to the next step. Iterating deletes all complementary filters and gives the asserted final main term. The number of helper factors remains O(\log\mathcal L). Restoring B_c,B_d to Y_c,Y_d in that main term uses the full pair estimate; every resulting difference contains an additional basic side U. These are the three-side terms estimated in Section 7.□

Proof. Here v\in A_a means that v>0 almost everywhere and

We use the scalar weighted maximal theorem and the following family-of-pairs form of Rubio de Francia extrapolation: if a family \mathcal F of nonnegative pairs (U,V) satisfies, for some a_0\in(1,\infty) and every v\in A_{a_0},

uniformly in the pair, then the corresponding estimate holds at every a\in(1,\infty) and every v\in A_a. No operator relation between U and V is assumed; see [1, Theorem 3.9].

Initially let g\geq0 be bounded and supported in a bounded interval in y. All the weighted left-hand sides used in the following applications are finite: the inner norms are bounded by a constant times the scalar maximal function of the indicator of that interval. Thus the version of extrapolation with finite left-hand sides suffices. The scalar weighted L^t maximal inequality, integrated in w, gives the weighted L^t hypothesis for the family of pairs

Extrapolation gives their weighted L^a inequality for every a>1. Choose a=s and integrate this inequality in \xi. The resulting pairs

satisfy the weighted L^s hypothesis for every A_s weight. A second application of extrapolation, now with target exponent r, proves the first assertion. This argument never interchanges the three norms and requires no ordering among r,s,t.

For general nonnegative g, apply the result to \min(g,n)\1_{[-n,n]}(y) and use monotone convergence, including M_y g_n\uparrow M_y g. Absolute values treat general g. Finally apply the first assertion to g=|f|^q, with

and take the qth root. Each exponent is strictly greater than one, and the power identity for the iterated norms is exact.□

Proof. By homogeneity it suffices to take s_j=1. A zero size gives a zero input almost everywhere and hence a zero form. To estimate the expected absolute value, insert a common \sigma-dependent scalar of modulus one into the kernel signs. It is therefore enough to estimate the absolute value of the expected signed sum.

Rectangles of leaf depths.. For each slot write k_j(y_j) for the depth of its leaf. There are at most K+1 possible depths. Pad this list to an interval of consecutive integers of dyadic length at most 4(K+1) and use its dyadic subdivision. Partition all four-tuples of leaf depths as follows.

If the minimum t occurs in at least two slots, specify its set of attaining slots. Those slots have depth t, and all the others have depth greater than t. Designate two attaining slots a,b in a fixed order. If the minimum is unique, designate its slot a. The minimum and the second minimum have a unique smallest dyadic index interval whose lower and upper halves, denoted L,U, contain them respectively. Specify the nonempty set of slots other than a whose depth lies in U; all remaining slots have depth greater than every index in U. Choose a designated slot b in that nonempty set and put t=\min U. This gives a disjoint partition into rectangles of leaf masks.

There are O(1+\log(K+2)) length levels and a bounded number of slot assignments at each level. At a fixed length level and assignment, write E_{j,\iota} for the union of the leaves selected in slot j at position \iota. The sets E_{a,\iota} are disjoint as \iota varies, and so are E_{b,\iota}. There are O(K+1) positions. The nonunique-minimum cases are treated as an additional level with singleton positions.

Fix one rectangle and suppress \iota. Every slot has leaf depth at least t, except possibly slot a in the unique-minimum case. Call the former slots regular and the latter slot heavy. We may restrict the summation depths to s<t. Indeed, if s\ge t, one of the selected leaves has depth at most s and cannot meet the interior of the internal summation interval at depth s. This observation also excludes such leaves after the small coordinate enlargements below, by the fixed kernel support margin.

A slow lattice and a fast torus.. Write r_s for the interval length at depth s, and choose \delta=\beta r_t, where \beta>0 is a sufficiently small fixed dyadic number depending only on the support margin. Write

On a progression this gives

Take a fixed torus \T_0=\R/(T_0\Z) of sufficiently large length, with normalized Haar measure. In slots 0,1 define F_j^\sigma(n,w)=z_j^\sigma(\delta(n+w))\1_{E_j}(\delta(n+w)) on 0\le w<1 and set it to zero elsewhere on this torus. In slots 2,3 use the same formula on a fixed bounded lift interval containing all the required w_j, and set it to zero outside that interval. For example, T_0=16 and lift intervals contained in [-4,4] suffice. With fixed normalization factors, the fast Haar progression integral then reproduces the original integral: the first two factors restrict u,v to [0,1), and the other lifts do not wrap around the torus.

Replace the kernel at \delta(n_j+w_j) by its value at the slow coordinates

We use t' here for the real progression parameter, to distinguish it from the depth index t. The difference of the kernels is O(\delta/r_s), with the same fixed support enlargement. The corresponding normalized summand has the factor (\delta/r_s)^2 times a lattice sum and a fast progression integral.

Here and below the lattice form bounds follow from the same pair-coordinate argument as Lemma 2.1. For two different progression slots the integer pair map is injective onto its image, and the pair map on the connected torus is Haar-surjective. Thus the two-L^1, two-L^\infty estimates hold on the product of the normalized local counting measure and the fast torus. Interpolation gives the local absolute estimate whenever the four reciprocal exponents have sum at most two.

For clarity, all kernel-sampling errors can be summed as follows. Let M_{j,I} be the squared mass of the deterministic stack on the sampling windows relevant to I, including their bounded overlap multiplicity. The support margin and the choice of \beta place these windows inside I. Hence

The absolute L^2 progression estimate, followed by Hölder in \sigma, bounds the sum of the errors at depth s by

We used the normalized mass bounds in the other two slots. Since \sum_{s<t}\delta/r_s\le C, these errors cost at most C|E_a|^{1/2}|E_b|^{1/2} for the rectangle. Summing positions by the two designated disjointness properties costs C|R| at each length level.

Quadratic randomization on the fast torus.. Put c=(-1,3,-3,1). On Fourier coefficients in the fast variable define the unitary maps

The fast progression integral is unchanged when all four factors are replaced by U_{j,\gamma}F_j. Indeed its Fourier expansion is supported on

The nullspace of these two equations is spanned by (c_j)_j and (jc_j)_j, so m_j=c_j(h+jh') for real h,h'. Since \sum_j j^\ell c_j=0 for \ell=0,1,2, we have

Moreover, for each single slot and each fixed n,\sigma,

To see this, integration in w equates the sums of two Fourier indices, and integration over the common period 3 equates their sums of squares. The two pairs are consequently equal as multisets. The two possible pairings give the displayed bound. The identities are first proved for Fourier polynomials. The pairwise L^2 progression estimate and the fourth-moment bound, also applied to differences, justify passage to L^2 limits.

Let e_j(n) be the fast L^2 norm of the deterministic stack on the corresponding sampling window. Minkowski and the pointwise sign moment hypothesis give, for every needed finite a,

The windows overlap a bounded number of times, so

If slot j is regular, then e_j(n)\le C. In fact, all selected leaves have length at most r_t. Disjoint leaves meeting a window of length O(\delta) have total length at most O(\delta)+2r_t: all but at most two are contained in that window. Summing their complete leaf masses and dividing by \delta proves the bound, with a constant depending only on the fixed \beta. On the lattice indices relevant to any summation interval I, the local version of the preceding mass estimate is

in all four slots, including the heavy slot. All the required windows are inside I, whose leaves have normalized squared mass at most one.

Clipping the possible heavy slot.. Write G_j^{\gamma,\sigma}=U_{j,\gamma}F_j^\sigma. If a heavy slot is present, clip G_a radially at A=(K+2)^{10}; write the clipped function as G_a^A. For the normalized local counting measure in an interval I and Haar measure in w, unitarity gives, at every \gamma,

Its L^4 norm in the random variables is O(A^{-1}), by the local mass estimate and the eighth sign moment. For each regular slot, the local L^3 norm of G_j has bounded fourth moment in (\gamma,\sigma). Indeed the local measure is bounded, and the fourth-moment identity bounds its fourth power by a constant times

The absolute progression estimate with exponents (1,3,3,3) and Hölder of exponent four in the shared random variables therefore bound the expected tail error by C/A per normalized cell. Even the crude sum over O(K+1) positions and O(K+1) depths costs only C(K+2)^2/A times |R| per length level. This is bounded uniformly in K. Henceforth G_a denotes the clipped function when there is a heavy slot.

Return to the continuous form by bursts.. For fixed \gamma,\sigma,u,v, with u,v now ranging over the whole fast torus, set

where c>0 is a sufficiently small fixed constant. The torus argument in this formula is taken modulo T_0. If a real progression meets four such bursts, then the integer centers satisfy exactly n_j=(1-j)n_0+jn_1. For example, the discrepancies for j=2,3 are integers of absolute value less than one when c<1/20. For a fixed slow pair, the admissible offset polygon

has a fixed positive area. Thus, after replacing the continuous kernel by its value at the center, its integral on these bursts is that fixed area times \delta^2 times the desired sampled product.

This second replacement has the same summed error bound as the first one. On the relevant indices, all the original fast sampling windows still lie inside I. Unitarity and clipping bound the transformed local L^2 sizes by those of F_j^\sigma. The lattice-times-torus pairwise estimate, then Hölder in the random variables, therefore uses the same masses M_{j,I} and the same geometric factor \delta/r_s. Although Fourier evolution spreads the fast support on the torus, it does not create new slow indices, so this support argument remains valid.

We may now apply the definition of C_N(q) to the burst inputs with the original admissible kernels and original summation intervals. Only the comparison error used sampled kernels; no marginal cancellation hypothesis has been changed. It remains to bound

The exponents and the dependence on K.. Put

These four exponents are finite, strictly greater than q, and have reciprocals summing to one. Use fast exponents h_a=p_* for a heavy slot, and h_j=4 for every regular slot. The sum of their reciprocals is at most two. The fast progression Hölder estimate, followed by Hölder of exponent four in (\gamma,\sigma) and spatial Hölder with the exponents p_j, bounds the last display by

All three exponents in each slot exceed q. Apply Lemma 7.1 in exactly this norm order.

For a regular slot the fourth-moment estimate gives

For the heavy slot, clipping and unitarity give pointwise in (\gamma,\sigma)

Taking the fourth random moment and then the spatial p_* norm yields

No dimension or number of signs occurs in these bounds.

At a fixed length level and assignment there are J\le C(K+1) positions. Designated disjointness and Hölder give

The other masks have length at most |R|, so their factors contribute |R|^{1-2/p_*}. The resulting power of K+2 is at most

The number of length levels is logarithmic; the two sampling errors and the clipping errors have already been bounded at that cost. This proves Equation (16), with an absolute exponent C. All Fourier and measurability approximations are harmless: there are finitely many slow indices, the fourth-moment estimate controls differences, clipping is continuous in L^2, and the mixed-norm estimate gives convergence of the final bounds.□

Proof. For this estimate expand complements by linearity and couple each pad choice across all times in an average. Jensen permits us to treat a fixed choice. We have

where L_{G,s} is an individual nested ball projection and B'_s is a shared product of O(m) nested main projection families. Basic B inputs are fixed signed sums of at most two such current histories, which changes only a fixed factor. Split

For a fixed group, the rows \1_{E_{G,r,t}}(L_{G,t}-L_{G,r}) have a square Bessel bound, since increments of a nested orthogonal projection family on disjoint depth intervals are orthogonal. Their adjoints have ranges in sums of whole ball ranges at the two endpoints. A right-endpoint certificate also certifies the required earlier input cell when its mask is nonempty. Splitting at the r-cells if necessary therefore places these adjoint ranges in the usable endpoint spaces of Lemma 6.5. That lemma makes the rows jointly Bessel across groups. They are local to their r-cells, so Lemma 3.4 inserts the shared product B'_r at a cost G_0.

For the second term, the usable-range Bessel estimate at each common right endpoint first sums the outer ball projections over groups. Telescope B'_t-B'_r. In each telescoping term, discard the left contractions. The changed middle factor is an orthogonal increment on [r,t], and the factors to its right are evaluated at r. These middle increments form Bessel rows, and Lemma 3.4 inserts their right factors. Summing the O(m) telescoping terms is again absorbed in G_0.

An interval coloring treats bounded overlap. Expansions of the O(m) complements cost 2^{O(m)}, and averages are restored by Jensen, without a factor counting pad choices. The proof without an outer ball is the same middle-increment argument. Intervals crossing the start of the attempt occur with bounded overlap; the zero convention and the one-time projection bound treat them.□

Proof. For a sequence W_s, b\le s<b+K, write

where \mathcal P(b,s) is the usual disjoint binary decomposition of the prefix [b,s]. There is at most one interval of each length in this decomposition. For a fixed length, its intervals are disjoint as depth intervals, and the selected endpoint is at most s. Restrict each resulting fixed function to the union of the use cells selecting it. This preserves its value at every required use and only decreases its norm. The base position and the choices of lengths account for logarithmic factors.

For a basic-U input, use each depth s directly as a position. The current side square bound controls the sum of its energies; subsequent helper products are local contractions. On the banded side, paying at most P+1 indices is permitted in the first inequality of Equation (18).

For a fixed linear side V_{G,b}, let T_s be the product of its subsequent averaged projections. At a fixed band index, the outer band difference is a difference of two such products. The binary-prefix product estimate in the proof of Lemma 3.4, now on a sequence of at most K+1 times, gives

Individual projections are nested; averages are coupled across times and restored by Jensen. Sum the length levels and use Equation (17). Pay at most P+1 on the banded side.

For a rolling input write W_s=T_s(Z_{G,s}-Z_{G,u_b}). The base term is controlled by Lemma 7.3 on the intervals [u_b,b], whose overlap across blocks is bounded. Its increments split exactly as

The first term is bounded directly by Lemma 7.3. When estimating norms, keep the entire endpoint cells containing the selected use cells: these cells still have downstream certificates, and locality of T_t makes this enlargement legitimate.

For the second term, also expand [u_b,r] by binary prefix intervals in a common window of O(K) depths. Fix an outer length, an inner length, and an inner interval [a,v]. For all outer positions [r,t] using this same inner interval, use the fixed input

where \Omega is the union of the needed r-cells. This input is independent of the outer position. Since v\le r, locality on r-cells shows that it gives the required outputs of T_t-T_r. Its norm is bounded by keeping the union of the v-cells containing those r-cells; all of these have downstream use certificates. For this fixed input, the product-jump estimate bounds the sum over outer positions by (C\log(K+2))^{O(m)}\|X\|_2^2. Now sum the squared inputs over groups and inner intervals at the fixed inner length, using Lemma 7.3. The windows for different blocks overlap a bounded number of times. Finally sum the two length choices. This proves the rolling estimate with only small overhead, and with the allowed factor P for a banded side.

The number of search leaves, transfer steps, choices of side component, and length configurations is small overhead. Individual pad choices are not additional configurations: they remain inside the actual averaged operators throughout. Thus summing all required configurations preserves the asserted bounds after increasing G_0.□

Proof. The three-side power bound was proved above using the path square caps and 3p>2. Lemma 6.9 gives the decomposition into transferred main terms, numerical errors, and coupled near bands. Restoring the full complementary pair in a main term introduces a third basic side and leaves the pair-small estimate \eta on its two arbitrary side inputs. The frozen-complement argument accounts for every freeze error and every discarded large complement as a third strong factor. Lemma 7.4, the common maximal cuts, and the sparse comparison then give Equation (19) for the remaining terms.

All cuts were imposed on nonnegative partial ancestor quantities and removed the original summands at and below their first violation. The normalized base hypotheses, the basic square estimate, and the root-uniform hypothesis (17) are available anew when the construction is restarted on such subroots. The fixed functions in the parent attempt remain unchanged. The thresholds may be chosen to make the union of the bad roots an arbitrarily small fixed fraction of |R|. This proves the asserted uniformity and support convention.

Finally, the estimates have the required relation to the original cell minimum. The subadditivity of x\mapsto\min(\eta,x) removes absolute numerical errors and the critical unpowered terms. For the remaining cell sum use

The group sum and the prescribed coupled bands are formed before this inequality is used. Thus every power cost gains \eta^{1-p}, whereas Equation (19) is used exactly as an absolute estimate. This completes the conditional side argument.□

Proof. The branch cutoff makes the conjugating factor a smooth quadratic chart supported in a bounded branch cube. At I_0, the joint difference data of v and z obey the endpoint version of Equation (2), with the harmless fixed factor c_l. Recenter at I_0, choose one rational splitting there, and apply Lemma 4.4. Its tail estimate gives the stated uniform error and finite coefficient and count bounds. A single splitting gives a single effective quadratic coefficient for this primary. Recentring adds a common unrestricted linear shift; the differences of the remaining linear shifts are rational combinations of the two endpoint horizontal vectors. Clearing Q_0 gives Equation (21). All these choices are made once.

Here is an explicit lift, which will also explain its operator properties. Write the main atom as \Phi=(\varphi_\Phi,\epsilon_\Phi^{1/2}\mathbf e_\Phi), and set

It satisfies |q_\Phi|\le\delta_c on the matched subtree. For a main augmented input F, define the physical part of \mathscr R_{l,G}F by

Its mode coordinate m, belonging to \Phi, is

and is zero before the appropriate match. The main count and inverse weight bounds, followed by Cauchy–Schwarz, bound the physical correction’s pointwise operator norm by \delta_c D_{\mathrm m}^{C}. Choose \epsilon_{\mathrm m}>0 so small that the squared norm of the mode-coordinate map is bounded by one. The preceding count and coefficient estimates allow \log(1/\epsilon_{\mathrm m})\le\operatorname{poly}(P,k). Thus the whole lift has a uniform pointwise norm.

An outer ball output on a cell has a representation \sum_\Phi\beta_\Phi\Phi with constant coefficients, including when the projection is averaged. Its main private coordinate is exactly \epsilon_\Phi^{1/2}\beta_\Phi. Substitution in the displayed definition gives \sum_{\Phi,m}\beta_\Phi a_{\Phi,m}\psi_m, as required. The lift is pointwise and hence cell-local; it has not been commuted through any projection.□

Proof. Fix the primary, put r_s=r_0 2^{-s}, and write \tau=\log_2\mathcal T. For a nonzero evaluation set x(a)=\log_2(|a\cdot\Theta|r_0). Exception at block start b means

Lemma 3.5 places all these x(a) in at most d_{\max} packets of width W. A packet [\alpha,\alpha+W] can cause an exception only when

This interval has length at most 4K by our choice of K, so it contains at most five starts of the K-spaced block schedule. Zero evaluations are always tiny.

For the second assertion, choose a basis of tiny vectors from the defining set. Cramer’s rule expresses any height-\mathcal H vector in its span with coefficients of total absolute value at most (d_{\max}\mathcal H)^{C d_{\max}^3}. Its evaluation is consequently bounded by this factor times \mathcal T^{-1}/r_{v_b}. The choice of \mathcal T makes this strictly smaller than \mathcal T/r_{t_b}. The gap in Equation (23) then forces the tiny alternative.□

Proof. Abbreviate R'=R_*^{\mathrm{ht}}, and discard h<R'. If two modes of this primary are connected in CURRENT, sum the edge witnesses along a simple connecting path. There are at most D_* edges. Their quadratic parts cancel between the endpoints, because the primary has one effective quadratic coefficient. With a from Equation (21), this gives an integral m_h with |m_h|_\infty\le D_*E_h and

The vector a+(0,m_h/L_h) has height at most \mathcal H. Since D_*\Xi<\mathcal T, the gap implies

For each height i, let V_i be the rational span of the height-D_* vectors q' for which

Let U_i be the rational span of the integer vectors in K_B with size at most C_*^3D_*E_i. Both families increase. Discard every h for which the rank of either family changes between h-R' and h. There are at most 2d_{\max}R' such heights, since each rank can increase only d_{\max} times.

At a remaining height, a\in V_h=V_{h-R'}. Express a=\sum_jb_ja_j using a basis from the defining vectors at h-R', with witnesses m_j. The b_j and their common denominator are bounded by C_*. Subtracting the corresponding relations in K_{\mathrm{rat}} shows that

After clearing the common denominator this is an integer vector whose size is at most C_*^3D_*E_h; this follows directly from the witness bounds and the choice of A_0. It belongs to U_h=U_{h-R'}, which we denote by U.

The lattice U\cap\Z^{\dim\theta_z} is saturated: if a nonzero integer multiple of an integer vector belongs to U, the vector itself belongs to the rational space U. Thus the quotient \Z^{\dim\theta_z}/(U\cap\Z^{\dim\theta_z}) is free. In an integer basis of this quotient,

The right side has integral coordinates. The common denominator bound and R'>\log_2C_*+2\Delta show, coordinate by coordinate, that each is divisible by 6^\Delta: the valuations at both 2 and 3 are at least \Delta. Other denominator primes do not affect these divisibilities. Choose an integral lift m' of [m_h]/6^\Delta.

Then m'-6^{R'-\Delta}\sum_jb_jm_j\in U. Choose linearly independent integer vectors of size at most C_*^3D_*E_{h-R'} spanning the rational space U, from the definition of U_{h-R'}. Subtract integer multiples of these vectors from m', rounding the coefficients of the preceding difference. The result remains an integer lift and satisfies

The last inequality is one of the permitted fixed-power choices of A_0. This rounding argument uses a small integer spanning set of the rational space, not a lattice basis, and therefore incurs no covolume factor.

Finally, m_h-6^\Delta m'\in U\subseteq K_B, so

Its height remains at most \mathcal H. Lemma 8.2 makes its evaluation at most \mathcal T^{-1}/r_{v_b}, which is below the PRE threshold. Equation (21) gives the required direct edge.□

Proof. Write F=\pi(\mathcal F_h). Each predecessor of a good mode has more than |F|/1.1 primaries. Two such predecessor primary sets intersect in more than (2/1.1-1)|F|, and thus their intersection contains a nonexceptional height-good primary, since fewer than one tenth of the primaries are excluded. The witnessing modes of that primary belong to \mathcal F_h. Lemma 8.3 joins them directly in PRE at h-\Delta, so the two predecessor components agree.

Modes in the same minimal PRE component have the same components at every larger height and in CURRENT, so all their predicates agree. This proves the second assertion. For the last assertion, two core modes have at most 2\delta_*(J_*+1) bad heights between them. Our lower bound on J_* ensures that the interval of integer heights below 4\delta_*J_* contains more heights than this. They therefore have a common good height h_0 in that interval. Their CURRENT components at h_0 agree, since their minimal CURRENT components agree. The first assertion identifies their PRE components at h_0-\Delta. Nesting then identifies all later PRE and CURRENT components, which also identifies the corresponding predicates. Finally, |\mathcal H_{\mathrm{up}}|\ge(1-4\delta_*)J_*, so the full-range bad count \delta_*(J_*+1) is at most 2\delta_*|\mathcal H_{\mathrm{up}}| for our small \delta_* and sufficiently large J_*.□

Proof. Enumerate elements by first entry into the increasing sets, breaking ties arbitrarily. A containing set for the element of entry rank r has at least r elements. Its total contribution is at most A/r. Sum this bound over the elements.□

Proof. Write p_{i,h} and c_{i,h} for the two floor logs at height h. Their values are integers between 0 and L. For h\ge\Delta, the first failure indicator is bounded by

At fixed i, the sum of the first term over h is at most \Delta L, because each increment of the increasing sequence p_{i,h} is counted at most \Delta times. At fixed h, the temporal nesting gives

Including the initially bad heights, this contributes at most N\Delta(L+1)+(J+1)L bad incidences.

The second failure indicator is bounded by \beta^{-1} times the sum of the exceptional and height-bad fractions. At fixed i, Lemma 8.5, applied as h increases, bounds the height-bad fractions by AH_M. At fixed h, apply that lemma as i increases to bound the exceptional fractions by BH_M. The total bad count is therefore at most

On the other hand it is greater than \delta N(J+1). The assumed lower bound for J+1 absorbs NQ into half of this last quantity, proving the result.□

Proof. First verify the asserted oscillatory bound. The common factor e(c_lp^\rho) cancels between two physical atoms. Fourier-expand (\zeta^\rho)^2 on the \vartheta torus, retaining frequencies of size at most B_*, with absolute coefficient sum at most D_* and tail at most D_*^{-300}. At height zero these shifts are allowed because E_0=A_0 exceeds B_*. At a larger height the corresponding shifts are multiplied by L_h, and B_*L_h\le E_h, by the choice of A_0.

For each retained shift, nonadjacency gives a point of the relevant pre-cell where the absolute phase derivative exceeds its graph threshold. The affine-slope oscillation bound makes the absolute derivative at every point of that cell at least half the threshold. The derivative is affine and of constant sign, so the elementary first-derivative oscillatory estimate on any interval applies. For PRE on a pre-cell it bounds the normalized integral by C/\Xi. For CURRENT on an actual cell I, it gives the same bound, since |I|\ge r_{t_b}. The cutoff coefficient sum and tail give

after increasing the starting budget. Distinct labels have orthogonal private coordinates, so they contribute no other inner product. Earlier modes satisfy the same slope-variation estimate on the later pre-cell by inheritance.

For the Bessel conclusion it suffices, by duality, to bound the synthesis of arbitrary f_e in these projection ranges. Write f_e=\1_{J_e}\sum_{m\in\mathcal M_e}\beta_{e,m}\psi_m, and put b_e=(\sum_m|\beta_{e,m}|^2)^{1/2}. Private coordinates give

For an overlapping pair, constancy of these coefficients on their cells and the cross-pair estimate bound its inner product by

There are at most D_* participating rows at a path point. Consequently

Adding the diagonal proves the synthesis bound and hence the row Bessel bound. This calculation uses a pathwise label count; it does not count the number of unused depths between rows.□

Proof. Initially omit the shared histories. For one group, P_e\mathscr R_{l,G} is a Bessel family with the same type of bound, since \mathscr R_{l,G} is one fixed bounded operator. It depends only on J_e, and therefore also only on the containing decision cell u_e. Lemma 3.4 applies to the outer individual main ball projection at that cell. It permits arbitrarily many rows with the same decision cell. Its loss is a fixed power of \log(2D_{\mathrm m}). Averaging the pad choices uses the triangle inequality in the Hilbert sum of row outputs and costs no number-of-pads factor.

Let A_G be the resulting row operator for this group. For an individual outer projection its row adjoint has the order

Its range is thus in the main ball range on the entire cell u_e. With an averaged projection, it lies in the sum of these usable ranges. Lemma 6.5 gives

Duality proves the joint Bessel bound before the shared histories. This is the step for which full endpoint cells and their downstream certificates are needed.

Finally apply Lemma 3.4 again to these joint rows, with decision cells u_e, to insert the shared histories. Expand complements and the fixed signed sum in B_l, and keep the positive averages as averages. Their product lengths are O(\log\mathcal L), their column log-budgets are polynomial in k, and the number of signed terms is a small overhead. The compression loss is therefore G_0. The original input has squared norm at most |R|. All operators have remained in their indicated order.□

Proof. Fix l,G. An event is a triple (b,J,\mathcal E), where J is a retained v_b-cell and \mathcal E is a minimal PRE component of side modes. Such a component consists entirely of side modes by Lemma 8.4. Split block indices into their ten congruence classes. Within one class, link an event to a later event on a descendant path when some of their modes agree or are adjacent in the later minimal PRE graph.

For two linked events, every earlier CURRENT edge persists in the later PRE graph. Indeed the later pre-cell is contained in the earlier pre-cell, the mode lists persist, and the separation of block starts gives v_{\rm later}\ge t_{\rm earlier}. The minimal PRE link also persists at every height. Consequently, choosing any mode in each event of a linked chain gives

at every height.

Apply Lemma 8.6 with

The height multiplicity comes from Lemma 8.3; the exceptional-block multiplicity comes from Lemma 8.2. Every event mode has more than \delta_*(J_*+1) bad heights. Our choice of J_* meets the lemma’s lower bound. Thus every chain has length at most

For clarity, the inequality being absorbed is

Although R_*^{\mathrm{ht}} depends on the mode budget, it disappears from the final chain-length bound.

Give each event its longest-chain level. Linked events have different levels. Distinct overlapping events in one level therefore have no shared or adjacent modes in the later minimal PRE graph. Events on one cell are distinct components and have the same separation. Their label sets are pairwise disjoint on a path, with total count at most D_*. Lemma 8.7 applies to the orthogonal projection P_{\mathcal E,J} onto the augmented columns of \mathcal E, restricted to J. Summing the ten congruence classes and the bounded number of levels gives squared Bessel bound G_0 for the event rows.

The decision cell for a row is its containing u_b-cell. Apply Lemma 8.8 to obtain

Zero outputs from pre-depths before the attempt were omitted.

It remains to pass from projections to exact coefficient subtotals. Write T_{\mathcal E,J} for that subtotal. Every mode outside \mathcal E is PRE-nonadjacent to every mode inside it. The oscillatory estimate in Lemma 8.7, the private-coordinate lower bound, and \sum_m|\alpha_m|\le D_* give

For example, pairing the complementary subtotal against a unit vector in this range costs its coefficient \ell^1 norm, at most D_*^{3/2}, times the outside coefficient sum and the pair bound D_*^{-200}. Within one event level the number of occurrences on a path is at most D_*. The number of levels is G_0, and the budgets also include all groups present on these full-cell paths. Thus the squared additive errors summed over all events are negligible compared with |R|.

Finally, on each J the different minimal PRE subtotals are almost orthogonal by the same coefficient calculation. Their sum is V_{l,G,b}. Summing their squared norms in Equation (24) therefore proves the assertion.□

Proof. Say an earlier successful component links to a later one if a mode of the earlier component is equal or CURRENT-adjacent, at the later height-h graph, to a mode of the later component. Persistence of the earlier CURRENT edges then gives \mathcal F_{\rm early}\subseteq\mathcal F_{\rm late}. Write F=\pi(\mathcal F_{\rm late}). Later goodness and the common first type coordinate imply

Both predecessor primary sets lie in F. Their intersection therefore has more than (1/1.1+1/1.1^2-1)|F| elements, a proportion greater than one tenth. It contains a later nonexceptional height-good primary. Two witnessing modes of that primary lie in \mathcal F_{\rm late}, one from each predecessor. Lemma 8.3 joins them in the later PRE graph at h-\Delta. Earlier PRE edges persist, so

The same argument at one block reduces to uniqueness of the good predecessor.

Now process the considered pre-cells in block order. If a current predecessor contains an earlier successful predecessor of this type on its path, inherit its representative. Otherwise choose any mode of the current predecessor by a fixed ordering of labels. To prove consistency, take any two earlier predecessors contained in the current one. Equality of the second type coordinate gives each more than 1/1.1, and in particular more than one half, of the current mode count. They intersect. They lie on one path, so their shared mode is a link from the earlier to the later of them. The inclusion just proved makes them nested. By induction all their inherited representatives agree. Comparable pre-cell data occur on ancestors, and the list and graph are constant throughout each pre-cell, so this defines representatives consistently for all its actual descendants.

If two components on comparable uses with this type had equal or adjacent modes in the later CURRENT graph, the same inclusion argument would force inheritance of the earlier representative. Thus distinct representatives imply the asserted nonadjacency of every cross pair of modes. Suppose I' is the later, smaller actual cell, in a block with pre-cell J' and comparison depth t_{b'}. For each allowed n, nonadjacency gives

The slope oscillation on J' is negligible compared with this threshold, so its value at x_* is greater than \Xi/(2r_{t_{b'}}). Since |I'|\ge r_{t_{b'}}, this implies Equation (25).□

Proof. Modes in different minimal CURRENT components are nonadjacent in that graph. The oscillatory proof of Lemma 8.7 applies on I, because |I|\ge r_{t_b}. Its synthesis calculation on this single cell shows that for arbitrary subtotals f_{\mathcal D} in distinct components,

The constants are uniform, since the off-diagonal norm is at most a fixed small inverse power of D_* after the private-coordinate bound. In particular this is a two-sided estimate and applies to any subset of labels inside each component.

Let T_{\mathcal F} be the full coefficient subtotal on \mathcal F. Its core subtotal is A_{\mathcal F}=T_{\mathcal F}-V_{\mathcal F}. The preceding estimate bounds the sum of the squared core component norms in Equation (26) by C\|A_{\mathcal F}\|_{2,I}^2. The contribution of the coefficient floors is at most

Every mode outside \mathcal F is CURRENT-nonadjacent to the modes inside. Pairing against a unit vector of its augmented span, as in the proof of Proposition 8.9, gives

Combine these observations with \|T_{\mathcal F}-V_{\mathcal F}\|^2 \le2\|T_{\mathcal F}\|^2+2\|V_{\mathcal F}\|^2 to obtain the first bound, with its stated generous error.

Finally decompose V_{l,G,b} by all CURRENT components at height h, whether successful or not. They too are separated, so the two-sided almost orthogonality gives the second bound after discarding the nonsuccessful terms.□

Proof. First fix one height, type, and dyadic threshold D_*^{-1}\le\sigma\le D_*^2. For each group and representative, take the first actual-use cells on each spatial branch for which \sigma_{\mathcal F}\ge\sigma. Call them its first tops. First tops of the same representative do not overlap. For two overlapping first tops of different representatives, Lemma 8.10 gives CURRENT nonadjacency of all their component modes at the later block. The label sets are therefore disjoint on a path, with total mode count at most D_*. Lemma 8.7 proves a uniform Bessel bound for the rows P_{\mathcal F,I} at these tops.

The row is supported on its actual cell I, which is contained in the u_b-cell of its frozen list. This remains a valid decision cell for Lemma 8.8, even if many rows share it. All these endpoint cells have the top itself as a downstream actual-use certificate. That lemma gives

No restriction of the input ball projection to the smaller top cell is made in this step.

For the side contributions in Equation (27), sum first at one depth of one block. The actual cells there are disjoint; at each cell the component sum is bounded by \|V_{l,G,b}\|_{2,I}^2. Summing the at most K depths and then applying Proposition 8.9 gives

Since \sigma_{\mathcal F}\ge\sigma, the size comparison now yields

The final term can be absorbed, because \sigma^2\ge D_*^{-2}. Thus

There are only polynomially many choices in P,k of height, type, and threshold: the counts are controlled by J_* and powers of \log(2D_*). For each choice let N_\sigma(x)=\sum_{\rm tops}\1_I(x), summing also over groups. The displayed integral bound and Markov’s inequality give a pathwise bound N_\sigma(x)\le P^C\sigma^{-2} outside a set of as small a fixed measure as desired, after enlarging C and the constant in the stopping threshold. Summing these exceptional measures over all the choices still costs a small fraction of |R|. Implement the cut by stopping at the first actual cells where the corresponding partial top count exceeds its threshold. All the estimates are valid again on normalized subroots, so these summands are assigned to fresh attempts.

If a representative is used in the bin [\sigma_j,2\sigma_j) on a retained path, its first \sigma_j-top is an ancestor of that use. It too is retained; otherwise the descendant summand would already have been sent to a fresh attempt. Different representatives have different such tops on the path, so their count is at most N_{\sigma_j}(x). The full mode-label budget gives the additional bound D_*. For \sigma_j<D_*^{-1}, that full bound alone implies the minimum in Equation (28). Above the tested threshold range the bins are empty by the coefficient and atom bounds, after the allowed budget enlargement.

The side square bound is the pruning already made following Proposition 8.9. For amplitude control, Z_{u_b}=Z_s-(Z_s-Z_{u_b}), the bounded pointwise lift, and the side bounds give \|\widetilde Z_{l,G,b}\|_{2,I} +\|V_{l,G,b}\|_{2,I}\le P^C. Projection is a contraction on I, so Lemma 8.11 bounds \sigma_{\mathcal F} by P^C as well.□

Proof. We give the cardinality argument on a common fine grid. This entails no restriction on the real data. There are finitely many labels, scales, and shifts of size at most Q. Approximate their real coordinates on one grid, taking label and frequency approximants on a fixed multiple of its step so that the divisions by 1,2,3 below remain on the grid. Separation remains valid with, for example, \Sigma/2; the annular bounds remain valid with \mu/2 and 2\nu; and the wide pure-shift gap retains the narrower interval [\mu R_s/100,100\nu R_s] needed after relaxing the parameters. These assertions follow by choosing the step after all the finite data, and sufficiently small relative to the least R_s. Round interval widths upward by at most one grid step. All such changes cost fixed constants, absorbed in C_{\mathrm{res}} and C. No final estimate depends on the step. It therefore suffices to prove the grid version with these harmless fixed margins. We keep the original letters for the relaxed parameters.

A graph with many four-step paths.. We first record the precise graph observation to be used. Suppose that a bipartite graph has parts of size at most M, at least \eta M^2 edges, and a proper edge coloring with at most M colors, where 0<\eta\leq1. Pad the second part to m=\lceil M\rceil\leq2M points, choose a uniformly random point in that part, and let Y be its neighborhood. Then

Call an ordered pair in the first part bad if its codegree is less than c_0\eta^2M, with c_0>0 a sufficiently small absolute constant. Its probability of lying in Y^2 is its codegree divided by m. Summing over the at most M^2 ordered pairs gives an expected number of bad pairs at most c_0\eta^2M^2. It follows, by considering |Y|^2-100 times this number, that some Y has |Y|\geq c\eta M and fewer than |Y|^2/100 bad ordered pairs. Delete the points with more than |Y|/4 bad partners. Fewer than |Y|/25 points are deleted. For two retained points there are at least c|Y| points of Y that are good partners of both. Each such midpoint has at least c_0\eta^2M common neighbors with each endpoint. Choosing those two neighbors independently shows that any two retained points have at least

four-step paths between them, allowing repeated vertices. For fixed endpoints the ordered four-color tuples of these paths are distinct: the start vertex and a color determine the next edge uniquely, by proper coloring, and this reconstructs the whole path successively.

Counting blocks and difference covers.. Restrict to the scales with N_s\leq\delta M^2. The first hypothesis gives total mass at most LM^2. Choose parameters \delta\leq\alpha/2 and 0<\kappa<1, whose values will be fixed at the end. Order the retained scales by descending R_s, and form consecutive blocks of mass at most \alpha M^2, greedily. Every block except possibly the last has mass greater than (\alpha-\delta)M^2. The number J_0 of blocks is therefore at most 3L/\alpha. These are counting blocks, unrelated to the depth blocks used in the linearization.

Let R_{\mathfrak b}^+ be the largest reciprocal scale in a counting block \mathfrak b. With grid step h_0, set

taking h_0 small enough that \epsilon_{\mathfrak b}\leq17\nu R_{\mathfrak b}^+ for every block. For an integer t\geq1 define the finite symmetric neighborhood

These neighborhoods contain zero and satisfy U_{\mathfrak b}(t)+U_{\mathfrak b}(t') \subseteq U_{\mathfrak b}(t+t').

Fix a core role i and a block. As long as the mass whose i-label is still uncovered exceeds \kappa M^2, choose two other core roles j,k. Project these occurrences to a simple graph of (i,j) pairs, keeping one scale witness in the block for each pair. There are at least (\kappa/L)M^2 edges. Color a pair by its k-label. The coloring is proper, because a pair in the roles (i,k) or (j,k) also determines the entire tuple. Apply the preceding observation with \eta=\kappa/L to obtain a subset B_0 of role i, of size at least c\eta M.

For a four-step path between a,a'\in B_0, alternately add and subtract the four annular relations for the triple i,j,k. The intermediate i-label and both j-labels cancel. Division by \alpha_i\in\{\pm1,\pm2,\pm3\} gives

where q depends only on the four colors. The scalar error is at most 4\nu R_{\mathfrak b}^+ before division. The shift has size at most 4B before division, and 36/\alpha_i is an integer multiple of 6, which explains the choice of U_{\mathfrak b}.

Choose a maximal packing in B_0-B_0 whose distinct differences lie outside U_{\mathfrak b}(2). Choose one endpoint pair for each packed difference. Each has at least c\eta^5M^3 ordered four-color tuples, and two packed differences cannot use the same tuple, since that would put their difference in U_{\mathfrak b}(1)-U_{\mathfrak b}(1). There are at most M^4 color tuples in total. The packing thus has size at most C\eta^{-5}M, and maximality covers B_0-B_0 by that many translates of U_{\mathfrak b}(2). Remove B_0 and repeat. Each iteration removes at least c\eta M labels, so there are at most C\eta^{-1} iterations. We have proved that, except for \kappa M^2 uncovered occurrences in each core role and block, the incident labels are covered by at most

subsets B, each of whose difference set is covered by at most K_1M translates of that block’s U_{\mathfrak b}(2).

Nonfuture pairs.. Discard every occurrence missing an own-block covered label in any core role. This loses at most 4J_0\kappa M^2. A core role on a remaining occurrence is called future if its exact label belongs to a cover subset of that role in a later counting block. The sets of nonfuture allowed labels are disjoint over blocks, separately in each core role: appearance in a later cover would make the earlier one future. If m_i^{\mathfrak b} is the number of such labels, then \sum_{\mathfrak b}m_i^{\mathfrak b}\leq M. The side role is always nonfuture, and in that role set m_l^{\mathfrak b}=\sum_{s\in\mathfrak b}w_s, which has the same sum bound.

An occurrence with fewer than three future roles has at least two nonfuture roles. Assign one such pair to it. For a fixed pair the mass in block \mathfrak b is at most

For two cores this is pair multiplicity; for a core and the side it is the side incidence budget, with the harmless factor L\geq1. The sum over blocks is at most

There are at most six role pairs.

Every remaining occurrence has three future core roles. Assign one such triple and, for each of its labels, one future cover subset containing it, using fixed rules if there are choices. The number of resulting combinations is at most 4J_0(J_0K_1)^3. Fix a combination with mass \rho M^2>0. It consists of a current block, a fixed triple of roles, and subsets B_1,B_2,B_3 in those roles from strictly later blocks. The indices 1,2,3 in the next paragraphs enumerate these roles, and do not change their progression coefficients \alpha_i.

Four ranges with scale margins.. Set

where C_{\mathrm{gap}} is a sufficiently large absolute constant. If \rho\leq C\delta\ell, retain this bound and do no further work on the combination. Otherwise increase the absolute C so that all the following margins hold. Cut the ordered mass at its three quartiles and omit the individual scale bins crossing the cuts. At most three bins are lost, each of mass at most \delta M^2. Each of the four remaining ordered ranges has mass at least \rho M^2/8, and hence at least \rho/(8\delta) occupied dyadic bins. Write R^\# for the largest reciprocal scale of the second range.

The first range supplies, in each B_i, a set B_{i0} of distinct anchor labels with

Indeed a fixed label is incident to at most LM occurrences, by pair determination and multiplicity with any other core role. Each anchor has an occurrence at reciprocal scale at least R^\#, so distinct anchors are separated modulo the reserved shifts by more than \Sigma R^\#.

The last range supplies a set F of distinct actual triples with

choosing one scale witness for each. A triple has at most L occurrences. The third range contains at least \rho/(8\delta) occupied dyadic bins. Thus every second-range reciprocal scale exceeds every last-range reciprocal scale by a factor at least \Gamma, once \rho>C\delta\ell. Every future block lies below the entire current block in reciprocal scale, so the same assertion holds for every future-block scale used by this combination. Greedily thin the second range so that successive selected scales have ratio at least \Gamma. Each selection excludes at most 1+\lceil\log_2\Gamma\rceil\leq C\ell occupied bins, and therefore retains

scales. Choose one actual triple t_v at each selected scale R_v. Figure 2 shows the four ranges for this fixed combination.

We will compare the q|F| indexed outputs of the translated sets t_v+F with a cover of a common slab. The next two steps build covers of the future sumsets; the anchors will remove one factor M from the slab cover. The scale margins will then show that no cover box contains two of these outputs.

A small-growth scale without an exponential loss.. Write U_i and \epsilon_i for the neighborhoods and widths of the future block associated with B_i. Subdividing each coefficient interval and the scalar interval gives, for every integer t\geq1, a cover of U_i(t) by at most (2t+1)^{d+1} translates of U_i(1). Consequently there is an odd integer

Otherwise growth by more than 2 at each odd step would exceed the displayed polynomial-in-t bound by the time t is a sufficiently large absolute multiple of (d+1)^2. The exponential dependence on d of the preliminary cover is used only for this choice of H_i; it is not paid in the estimates below.

The translates a+U_i(H_i), a\in B_{i0}, are disjoint. An intersection would put their difference in U_i(2H_i), whose shift index in the \zeta lattice is O((d+1)^2B) and whose scalar width is at most C(d+1)^2\nu R^\#. The reserves on Q and \Sigma exclude this. With Z_i=B_{i0}+U_i(H_i), the difference cover for B_i-B_i gives

The minimal-growth subset argument.. Choose a nonempty X_i\subseteq Z_i minimizing r_i=|X_i-B_i|/|X_i|. Then r_i\leq CK_1L/\rho. We include the iteration argument, in the form used in [10]. For a finite family of translates C_0=\{c_1,\ldots,c_q\} let X_{i,j} consist of the points x\in X_i whose translate x+c_j is not in any earlier X_i+c_v. Thus |X_i+C_0|=\sum_j|X_{i,j}|. Every point of (X_i\setminus X_{i,j})-B_i+c_j already belongs to an earlier translate of X_i-B_i. The new points contributed at step j are therefore at most

by minimality; the empty-complement case is included. Summing and then iterating with C_0=-(m-1)B_i proves

where mB_i denotes the m-fold sumset.

One anchor translate contains a portion Y_i of X_i of size at least |X_i|/M, since there are at most M anchors. Its difference set is contained in U_i(2H_i). If \mathcal P is a packing of mB_i with distinct differences outside U_i(2H_i), the translates Y_i-p, p\in\mathcal P, are disjoint and belong to X_i-mB_i. Hence

Maximal packings give covers by the same neighborhood. We need only m=2,3.

A slab cover with only two label-count factors.. Put r_0=CK_1L/\rho, enlarging the absolute constant so that r_i\leq r_0 for every role. Consider the set of outputs in 2B_1\times2B_2\times2B_3 satisfying

Cover the first two sumsets by at most r_0^2M translates each of U_i(2H_i). Above a fixed pair of boxes, the possible third coordinate lies in 2B_3 intersected with a set of the form

Indeed substitute the first two coordinate boxes into the slab relation and divide by \alpha_3\in\{\pm1,\pm2,\pm3\}. The box shifts are 6n\cdot\zeta and the slab shifts are 36n\cdot\zeta, so the divided coefficients remain integral. Their sizes have the stated bound.

Let \mathcal P be a packing in this local third-coordinate set outside U_3(2H_3). Then B_{30}+\mathcal P\subseteq3B_3 is still such a packing. For an equal anchor this is the definition of \mathcal P. For distinct anchors, a collision would express their difference, modulo a reserved shift, with absolute value at most C(d+1)^2\nu R^\#, contradicting their separation. The global 3B_3 packing bound gives

Covering the local third-coordinate set by a maximal packing shows that the whole slab is covered by at most

translates of \prod_{i=1}^3U_i(2H_i). The use of all the separated anchors is what removes a third factor M.

All translated last-range triples are separated.. For each chosen second-range triple t_v, translate F by t_v. All outputs belong to the preceding slab, by adding their annular relations. A cover box contains at most one output. First, within one translation, distinct members of F differ in some label coordinate. Their last-range witnesses lie above every future scale, so hereditary separation is stronger than membership of that coordinate difference in U_i(4H_i).

Next consider different translations, and let R_{\mathrm{big}} be the larger of their two second-range scales. The annular residual of that triple has magnitude between \mu R_{\mathrm{big}} and \nu R_{\mathrm{big}}. The sum of the magnitudes of the other second-range residual and the two last-range residuals is at most \mu R_{\mathrm{big}}/10, by the factor-\Gamma margins. Thus the linear form of the output difference, after a shift of index O(B), has absolute value between 9\mu R_{\mathrm{big}}/10 and 2\nu R_{\mathrm{big}}. If the outputs lay in the same box, their coordinate differences would be in U_i(4H_i). The total scalar error in the linear form would then be at most

All shifts still have index O((d+1)^2B)\leq Q. We would obtain a pure shift of magnitude between 4\mu R_{\mathrm{big}}/5 and 3\nu R_{\mathrm{big}}, inside the forbidden gap. This proves the required separation.

The number q|F| of outputs is consequently bounded by the slab cover. Cancelling M^2 gives

Including the previously omitted case \rho\leq C\delta\ell, a convenient uniform consequence is

Choice of the auxiliary powers.. There are at most 4J_0^4K_1^3 combinations. Combining their mass, the uncovered mass, and the nonfuture-pair mass, we obtain

For sufficiently small absolute \delta, take \alpha=\delta^{1/1040} and \kappa=\alpha^2. Then \delta\leq\alpha/2, and the three powers of \delta on the right are 1/1040, 1/2080, and 1/10-52/1040=1/20. Since L,\ell\geq1, this proves the asserted estimate. For the remaining \delta, enlarge the absolute constant and use the total-mass bound LM^2.□

Proof. Identify a label a with its fixed value \omega_a(x_*). Different labels in one fixed role and type have distinct values by (25) with zero shift. We first establish pair determination. In the four-core case, subtract the two frequency tests for two incident tuples that have a given pair of labels in common. At a cost O(\nu\max(R_s,R_t)), the two remaining frequencies solve the linear equations for Q and T, modulo the bounded shifts. The inverse determinant divides by an integer of magnitude at most 3. Separation (25) forces the two alternative labels in each remaining role to agree. In the three-core case, the single equation for T_l similarly determines the third core label from the other two, with a divisor of magnitude at most 3. These conclusions hold across different scales, because the separation threshold uses the larger reciprocal scale. All shift differences and divisions fit the height reserve.

In addition, for any triple of core roles let l' be the excluded fourth role and put \alpha_i'=i-l' on that triple. The tests imply

For four cores this is t'-l'q', using both bounds in (31); for three cores it is precisely the nonzero T_l test. Fixing a pair fixes the whole core tuple, and hence the label values in this annulus. The height-bin observation, applied to those fixed values together with the coordinates of \xi, bounds the number of scale occurrences by P^C(1+\log(\nu/\mu))^C. We use L_0\geq1 for a bound of this form. This proves the first count for any pair of core roles.

For a pair consisting of a core role and the side, fix the core label and a scale. The windows in (32) associated with different incident triples are disjoint. If two windows overlapped, their Q values would be close modulo bounded shifts, with error O(\Lambda R_s). Combining this with the two T_l relations determines the two remaining core labels, again by a two-by-two system with divisor at most 3. Since \nu,\Lambda\leq CU_0\ll\Xi, separation forces those labels to be identical. Couple the fresh offset in all these windows. Parseval for each offset, followed by expectation, shows that the number of incident triples in the chosen side bin is at most

This remains valid on any subset of scales after summing. Summing over the fixed core labels proves the first inequality of (34) for a core/side pair as well.

We now apply Lemma 9.1 with M=M', L=L_0, d=\dim\theta_z, and

The fixed constants in (35) may be absorbed by decreasing \mu and increasing \nu by absolute factors. The pair and side-budget hypotheses were just verified. For separation, note that

Thus the shifts in the finite lemma are among those of (25), with Q=\lfloor E_h/6^{\Delta-2}\rfloor and \Sigma=\Xi/2. The choice of A_0 and B'_h=D_*^5E_{h-\Delta} ensures Q\geq C_{\mathrm{res}}(d+1)^2 CB'_h. Likewise \Xi=D_*^{1000} and \nu\leq CD_*^{250} ensure the separation reserve, with room for the fixed annular constants.

Finally, any successful component contains a nonexceptional primary, even when its representative is itself exceptional. For that primary the pure-horizontal cases of (23) apply to every shift in the finite lemma: their rational heights are smaller than \mathcal H. They are either at most \mathcal T^{-1}/r_{v_b} or at least \mathcal T/r_{t_b}. Since U_0^{-1}\leq\mu\leq1\leq\nu\leq CU_0 and the test scale lies between these depth scales, the prescribed choice of \mathcal T excludes the entire intermediate interval required by the pure-shift hypothesis. This checks every hypothesis of the finite lemma, uniformly over the current group.

It follows that, for an absolute c'>0,

For example, c'=1/2080 is admissible after changing the absolute constants. Indeed Lemma 9.1 gives the stronger coefficient C\ell L_0^{29}; the dimension and the logarithms in this expression fit the displayed budget.

Pair determination also gives N_s^G\leq(M')^2. Choose p_0\in(0,1) with 1-p_0<c', for instance p_0=1-c'/2. On the set of scales with

we may bound (N_s^G)^{p_0} by [2^{j+1}/(M')^2]^{1-p_0}N_s^G. Sum this inequality and use (36) with \delta=2^{-j}. The remaining geometric series has ratio 2^{1-p_0-c'}<1, and proves the second inequality of (34).□

Proof. Set

where \theta'>0 is a sufficiently small absolute number, to be fixed below. Hölder interpolation between the two inequalities in (34) gives, for any distinct roles i,j,

Order the amplitude bins from largest to smallest, temporarily calling their roles 1,2,3,4, so that \sigma_1\geq\sigma_2\geq\sigma_3\geq\sigma_4. Use the pair 1,2 in the first factor. At least one of this pair is a core role, because there is at most one side.

The path bounds (28) and the side square budget give

Expand the maximum M' into at most four choices of the role attaining it. The total exponent of the resulting product of M_i^G is 2p>1. For nonnegative sequences m_i^G and exponents \beta_i\geq0 with \sum_i\beta_i\geq1, one has

Indeed normalize each nonzero sum to one, apply Hölder with exponents (\sum_i\beta_i)/\beta_i, and use that the normalized sequences have \ell^{\sum\beta_i} norm at most one. This proves the needed group summation.

The largest size-to-count factor is \sigma_4^{-2}. In the core member of the first pair we may also use the count cap D_*. If this member is role 2, its factor \min(1,D_*\sigma_2^2) is bounded by the corresponding factor with \sigma_1. Multiplying by (\prod_i\sigma_i)^p from (33), the amplitude dependence is consequently at most a P^C factor times

Let the four displayed power exponents be e_1,e_2,e_3,e_4. Their sum is zero, and their three preceding partial sums are

All are strictly negative when \theta' is sufficiently small. Writing g_j=\log_2(\sigma_j/\sigma_{j+1})\geq0, the remaining power factor is 2^{\sum_{j=1}^3g_j(e_1+\cdots+e_j)}. Thus all three amplitude-gap sums are convergent geometric series.

Only a common shift of all the bins remains. The retained upper amplitude bound is P^C. Below \sigma_1=D_*^{-1/2} the prefactor \min(1,D_*\sigma_1^2)^{1-\theta'} decays geometrically; above it there are only O(1+\log D_*+\log P) possible dyadic bins up to that upper bound. The common-shift cost is therefore polynomial in the allowed logarithmic budgets. There are only 24 amplitude orderings, so this completes the amplitude summation.

It remains to sum the frequency bands. The first count loses only powers of 1+\log(\nu/\mu). The second count, after interpolation, loses at most \mu^{-C\theta'}\nu^{C\theta'}\Lambda^{C\theta'}. The powered kernel factor in (33) is \mu^p\nu^{-Lp}\Lambda^{-Lp}. First take \theta' small enough that p-C\theta'>0, and then take the fixed derivative order L sufficiently large. The sums over dyadic 0<\mu\leq1 and 1\leq\nu,\Lambda<\infty converge, including the logarithmic losses. Extending the actual truncated ranges to these ranges only increases the bound. The number of heights, types, role patterns, and retained leaves has an allowed polynomial cost. We have proved a bound C_qP^{C_4} for the path sum of the pth powers of all the principal core contributions, keeping each N_s^G inside its power.

To finish, let \mathcal C_I be the sum of their absolute principal contributions at a cell. Subadditivity for p<1 and \min(\eta,t)\leq\eta^{1-p}t^p give

Integrating the path estimate uses the exact identity \sum_I|I|F_I=\int_R\sum_{I\ni x_*}F_I\,dx_* for nonnegative cell data. It gives C_q\eta^\varepsilon P^{C_4}|R|. The previously separated absolute errors contribute O(e^{-k}|R|), which proves (37). All polynomial degrees in the finite counting lemma, and hence p_0, \theta', \varepsilon, and C_4, are absolute. Taking \theta' smaller if necessary gives p>2/3 as required by the side estimates.□