Triple ergodic averages with distinct integer slopes

1. Introduction

Let (X,\mathcal F,\mu) be a probability space and let T:X\to X be an invertible measure-preserving transformation with measurable inverse. Given bounded measurable functions f_1,f_2,f_3 and distinct nonzero integers \lambda_1,\lambda_2,\lambda_3, consider

The issue is convergence at almost every point, rather than only in norm. We assume that T is mixing, in the sense that

The probability space need not be standard or countably generated. Both positive and negative slopes are allowed. The proof depends on the analytic results of the companion manuscripts [H, V]; their role and the extension required here are described below.

1.1. Context and predecessors

Birkhoff’s pointwise ergodic theorem [Bir] treats one function. Products along several iterates arise naturally in Furstenberg’s ergodic proof of the multiple recurrence theorem and its connection with arithmetic progressions [Fur]. Multiple recurrence, norm convergence, and pointwise convergence are different questions. In particular, convergence in L^2 does not by itself establish convergence at almost every point along the full sequence.

For weakly mixing transformations, the product-of-integrals limit in L^2 is classical. Bergelson’s weakly mixing PET theorem [Ber, Theorem 1.2 and p. 340] includes distinct nonzero integer slopes of either sign. For general systems, Host and Kra [HK] and Ziegler [Zie] developed the norm-convergence theory using characteristic factors with nilpotent structure. These results explain the limiting structure without resolving the corresponding pointwise question.

Bourgain’s double recurrence theorem [Bou] establishes almost-sure convergence for two bounded functions evaluated at two powers of one transformation. Pointwise theorems for longer products also have substantial precedents under additional assumptions on the system. For example, Derrien and Lesigne [DL, Theorem 4.2] prove polynomial pointwise theorems for K-systems and exact endomorphisms; their invertible case includes the linear configurations considered here. Other structural hypotheses yielding pointwise convergence for all finite progression lengths include singular spectrum on the Pinsker factor of a weakly mixing system, by Assani [Ass], and the pairwise-independent-joining property for weakly mixing systems, by Gutman, Huang, Shao, and Ye [GHSY]. The significance of Theorem 1.1 is that its dynamical assumption is mixing alone, while the three slopes are arbitrary distinct nonzero integers.

The harmonic-analytic method separates convergence from identification of the limit. A uniform oscillation estimate gives pointwise convergence; mixing identifies that limit by an L^2 argument. Oscillation methods in ergodic theory are developed in [Bou, JKRW], and the passage from translation estimates to orbit estimates follows the transference principle of Calderón [Cal]. We give that passage directly, so it imposes no additional structure on the probability space.

1.2. Analytic input and the coefficient extension

The manuscript [H] proves an L^3 bound for the trilinear Hilbert transform through local four-linear forms. Its proof includes chart detection, localization by quadratic data, comparison on short depth blocks, and a separated-frequency counting estimate. The quantitative inverse theorem of Leng, Sah, and Sawhney [LSS] enters its chart detection argument. Arithmetic regularity and inverse-theorem methods also occur in Tao’s sublogarithmic cancellation theorem for truncated multilinear Hilbert transforms [Tao]; that cancellation theorem is distinct from the uniform local estimate used here.

The companion [V] adapts the local estimates to inputs that vary with depth and deduces pointwise convergence for the slopes 1,2,3. The local forms in both companions use the four coordinates x,x+t,x+2t,x+3t. A power of T and a permutation of the functions do not turn an arbitrary triple into this configuration. Thus their stated theorems cannot simply be invoked for the slopes in Theorem 1.1.

The new work is the extension of their local proofs to x+b_jt, 0\le j\le3, for fixed distinct integers with b_0=0. Three features are particularly useful beyond this application. First, an integer left inverse of the whole primitive configuration replaces a unimodular pair of coordinates; this removes extra lattice or torus configurations in discretization. Second, a Vandermonde relation provides the quadratic cancellation and the Fourier randomization for every set of four distinct slopes. Third, a height base divisible by all pair differences preserves the divisibility required in the separated-frequency argument. The last step must preserve a bound polynomial in the dimension: an exponential loss there would not fit the proof’s parameter choices.

These are extensions of the proofs in [H, V], not new proofs of their coefficient-independent projection and row estimates. We state the local conclusion completely, prove the coefficient-sensitive identities and estimates, and specify the results imported from the companions at their points of use. In particular, the technical part is a proof by reference to those manuscripts, whereas the ergodic deduction is presented in full.

1.3. Proof structure

Section 2 defines the admissible kernels and states the local estimate for inputs with controlled variation on blocks of depths. Section 3 proves Theorem 1.1 from that estimate: smooth scale differences are realized by cancellative kernels, then transferred to integer sequences and finite orbits. The resulting oscillation bound gives convergence for smooth profiles; moment-corrected approximations to interval indicators recover all integer lengths.

The remaining sections prove the local estimate. Section 4 establishes the coefficient modifications in detection and localization. Section 5 proves the short-block, height, and counting modifications. Section 6 closes the parameter argument, first for fixed inputs and then for depth-dependent inputs. All analytic constants may depend on the fixed slopes; no bound is required uniformly over triples of integers.

2. The local estimate and the coefficient algebra

The analytic estimate concerns four inputs on the real line. Its constant may depend on the slopes, but must be independent of the number of scales and of the number of changes of the inputs. This uniformity will give the oscillation estimate in Section 3.

2.1. Normalization and local forms

Set

Then S is invertible, probability preserving, and mixing, and T^{\lambda_j n}=S^{b_jn}. Henceforth the four distinct integers b_0,b_1,b_2,b_3 are fixed, and \gcd(\abs{b_1},\abs{b_2},\abs{b_3})=1. We make no assertion of uniformity as these integers vary.

Dyadic intervals are half-open. The depth of a descendant I of a dyadic interval J is s(I)=\log_2(\abs{J}/\abs{I}). A finite rooted tree contains every ancestor between any of its intervals and its root. The dyadic lattice may be translated.

For a nonempty integer block B=\{u,\ldots,v\} and functions z_s on a common domain, put

This is a pointwise bound on both size and variation within the block; no variation across block boundaries is included.

The schedules are prescribed at every depth, even depths with no summation intervals. This is exactly the hypothesis of [V, Theorem 3.2], except for the coefficients in the local forms. Sections 4–6 prove the theorem by extending that proof and its analytic inputs from [H]. Before turning to those technical arguments, Section 3 proves the implication from Theorem 2.2 to the ergodic theorem.

2.2. Integer reconstruction and quadratic cancellation

Two features of the coefficients recur in both the analytic proof and transference. Pairs of coordinates determine (x,t) over \R, but need not determine it over \Z. All four coordinates together do determine the integral step. Quadratic cancellation requires a separate identity, furnished by Vandermonde interpolation.

2.3. The fixed-input constant

The proof of Theorem 2.2 first establishes a fixed-input estimate. For 2<q<3, let M_q f=(M(\abs f^q))^{1/q}, where M is the uncentered Hardy–Littlewood maximal operator. Let C_N(q) be the least constant for which

for all finite collections \mathcal A of intervals in one translated dyadic lattice, whose lengths belong to at most N consecutive scales, admissible kernels, and compactly supported scalar L^q inputs. This is the definition in [H, Section 2], with our coefficients. Pair-coordinate estimates imply C_N(q)\lesssim_bN, so this constant is finite before any absorption argument. The analytic proof will show \sup_NC_N(q)<\infty for a suitable fixed q>2, and then use that bound in the scheduled-input argument. This order avoids using Theorem 2.2 to prove its own analytic input.

3. From the local estimate to pointwise convergence

We now deduce Theorem 1.1 from Theorem 2.2. This section is independent of the internal notation of the analytic proof. It follows the transference scheme of [V, Sections 2–3], including its use of signed smooth profiles. We give all the details, both to handle arbitrary integer slopes and to separate the two roles of the hypotheses: the local estimate gives almost everywhere convergence of smooth averages on every invertible probability preserving system; mixing identifies their limit.

Throughout, b_0=0,b_1,b_2,b_3 are the fixed primitive slopes, S is an invertible measure-preserving transformation, and e_j are the integers in (2.5). All constants in this section may depend on the slopes.

3.1. Smooth profiles and cancellation kernels

The derivative class in (3.2) is closed under fixed translations, dilations, differentiation, and finite linear combinations. It contains a nonnegative bump \rho of mass one supported in [-1,1]. For completeness, normalize

For |t|<1, put d=1-t^2. On a complex disk of radius cd about t, with sufficiently small absolute c>0, the real part of (1-z^2)^{-1} is at least c'/d. Cauchy’s estimate gives

For each n this tends to zero at the endpoints, and its maximum over 0<d\leq1 is at most C^{n+1}n!(n+1)^n. Thus the zero extension is smooth and satisfies (3.2) after enlarging G. Lemma 3.9 will construct averaging profiles arbitrarily close to interval indicators.

Fix a profile \varphi, and put

We use the block size and variation \mathcal V_B from Section 2. Block endpoints are always deterministic; the input values may depend on the spatial variable.

3.2. Integer sequences and finite orbit segments

The next step is where primitivity is useful. We do not assume that any pair of slopes has difference one.

For measurable functions |f_j|\leq1 on the probability space, put

All sums are finite. If \operatorname{supp}\varphi\subseteq[-B_\varphi,B_\varphi], with B_\varphi\geq1, then

Proposition 3.5, (3.10), and Lemma 3.6 already give almost everywhere convergence of V_k^\varphi. Mixing has not yet been used.

3.3. Identifying the limit by mixing

We include the norm argument to make explicit that it accommodates arbitrary distinct nonzero slopes and both signs of time. The set-mixing assumption on S implies, by uniform approximation by simple functions, that for bounded measurable g,h,

For each averaging profile, Lemma 3.8 gives V_k^\varphi\to p in L^2. Its almost everywhere limit is therefore p. More explicitly, if E_\varphi is the measurable conull convergence set, Fatou’s lemma gives

3.4. Removing the smooth weights

The moment restrictions need not obstruct approximation in L^1. The correction below, from [V, Lemma 2.8], moves a small amount of signed mass to a large scale. Uniform control of the supports or derivative constants is neither asserted nor needed.

4. Quadratic structure and localization for the new slopes

We begin the proof of Theorem 2.2 by establishing the quadratic detection and localization estimates for the configuration y_j=x+b_jt. These estimates supply the input to the short-block and frequency arguments of Section 5. The relevant constructions are those of [H, Sections 3–6] and [V, Sections 4–6]. We retain their interval projections and stopping constructions, and prove below the coefficient-dependent identities and estimates on which those constructions rely.

Throughout this section, the slopes are fixed. Constants may depend on them and on the admissibility parameters. A rational number has height at most H if its numerator and positive denominator in reduced form are at most H in absolute value; matrix heights are entrywise. We use the quantitative convention of [H, Section 3.1]: a quasipolynomial bound in parameters H_1,\ldots,H_m\ge2 is a bound of the form

The degree C is chosen independently of q, the number of depths, and the input schedules. A fixed number of such constructions may be composed. Constructions repeated a variable number of times retain the explicit count estimates in H and V.

4.1. Local bounds and the chart class

For a scalar or Hilbert-valued function, write \norm z_{p,I}=(|I|^{-1}\int_I\norm{z(y)}^p\,dy)^{1/p}, with the essential-supremum convention for p=\infty. Norms without an interval subscript use ordinary Lebesgue measure. The elementary bound used throughout the argument is unchanged except for its constant.

On the support of a local kernel, all four coordinates lie in an interval of length r=|I|. Consequently

In any pair of coordinate variables, the rescaled kernel thus has support in a fixed box and the same type of derivative bounds. Its integral in either coordinate with the other fixed is zero, by the corresponding one-coordinate marginal hypothesis. These facts also give C_N(q)\lesssim_b N and preserve the normalization-removal argument of [H, Lemma 2.4].

We recall the concrete class in which the quadratic structure is detected. Write e(u)=\exp(2\pi i u). A quadratic chart atom is a function

where \theta\in\R^d, \lambda,\beta\in\R, C\in\mathbb Q^{d\times d}, and \chi is smooth and compactly supported. An overall constant of modulus one is allowed. We use exactly the complexity convention of [H, Definition 4.1]: for complexity H\ge e^2, d\le\log H, the height of C is at most H, \operatorname{supp}\chi\subset[-H,H]^d, and

with a fixed chart constant c_{\rm ch}. The real oscillation parameters \lambda,\beta,\theta are unrestricted. A bounded atom has supremum norm at most one; any normalization factor is put in its external scalar coefficient.

The elementary operations, Weyl alternative, and chart expansion of [H, Lemmas 4.2–4.4] concern this class itself. Their proofs therefore apply without a coefficient change. We will need the precise splitting supplied by the Weyl alternative. In its multivariable version replace \theta y by Vz, and the ordinary quadratic phase by z^T\Lambda z, where \Lambda is real symmetric and z has at most three coordinates. Put

Let Q(z) be this multivariable chart. Suppose 0<\delta<1/2 and a Lipschitz function \vartheta, supported in a fixed box with supremum and Lipschitz norms at most H, satisfies

The Weyl alternative then supplies a rational subspace U, the orthogonal projection P_f onto U, P_s=1-P_f, and the bounds

The projection heights are at most H', and H' is quasipolynomial in H,\delta^{-1}. The subsequent chart expansion has the absolute coefficient and pooled Fourier-tail bounds of [H, Lemma 4.4]. These are the bounds used below, including when expansions must be summed over depths.

4.2. Separation along the configuration and local detection

The first coefficient-dependent step moves two structured inputs into the other two coordinates and the step variable. This is the form of separation needed for both detection and the later predictor estimates.

For one separated term, let

where \zeta is smooth, nonnegative, equal to one on [-1/3,1/3], and has \norm\zeta_1\le1, as in [V, Section 5.4]. The two-input form in the remaining roles has norm at most C_bA_r on normalized L^2(I) inputs. To see this, use

Fourier-expand the smooth normalized kernel in those two coordinates. Each term is a modulated convolution with a fixed dilation of Q(t)\zeta(t/r), so Plancherel gives the claimed bound. If A_r\ge\delta, the Weyl alternative and chart expansion further give the finite smooth separation in [H, Lemma 4.6]: for every 0<\varepsilon<1/2, the two-coordinate kernel has a separated approximation with uniform error at most \varepsilon, whose number of terms, atom complexities, and absolute coefficient sum are quasipolynomial in H,\delta^{-1},\varepsilon^{-1}. The retained phases are \lambda_*t^2+\gamma t, with |\lambda_*|r^2 bounded by the stated budget, and the linear phase splits as

Thus the fixed rational-height description of the effective quadratic and linear frequencies in that lemma is preserved.

4.3. The base reduction and inputs that vary with depth

Detection is used in H and V to construct finite approximation spaces on interval trees. We recall their meaning before verifying the scale estimates. A column with label v, born on an interval J_v, is an augmented function

where \phi_v is a scalar chart atom, \epsilon_v>0, and the \mathbf u_v are distinct orthonormal private coordinates. Only the first, physical coordinate enters H_I. A local projection on I is the orthogonal projection onto constant-coefficient combinations of the available columns restricted to I. Lists are inherited on descendants. Accordingly the direct-sum projections on successively finer interval partitions have nested ranges. The private-coordinate, compression, and height-bin estimates of [H, Section 3] depend only on these definitions and the column budgets, and apply unchanged.

There are two coefficient-dependent scale estimates in the base construction. For two fixed chart atoms, the separation above reduces the other two inputs to a kernel of the form w_I(x,t)Q(t), with fixed bounded one-coordinate multipliers. The proof of [H, Lemma 5.1 and Corollary 5.2] expands Q, groups nearby linear frequencies, and uses the zero marginals of w_I. In coordinates (y_a,y_l), its double primitive has compact support in an interior rectangle. Fourier expansion followed by one derivative in each coordinate expresses it as a sum of tensor products of mean-zero smooth bumps. In role p\in\{a,l\}, the resulting scalar rows have fixed bounded carrier functions

where \psi_p is the fixed separated multiplier in that role. These are the only coefficient replacements in the row estimate. They alter frequency separations and rational heights by fixed factors. H’s height-bin argument, ordinary Bessel estimate, and compression step therefore give its absolute scale sum and finite predictor list with the same quantitative orders, including the pooled-tail bound needed for summation over all depths. The finer partition comparison in [H, Lemma 5.3] uses the resulting one-coordinate predictor moments, so its projection-increment proof applies with the same replacement.

For the second estimate, fix three bounded chart columns and take the transpose in role l at y+h. Its other coordinates are

After squaring the normalized L^2 norm, the six atoms form a multivariable chart in (h,t,t'). At a fixed path point y, all parameters before multiplication by r or r^2 remain fixed as the depth changes. H’s Weyl alternative, chart expansion, and height-bin argument therefore apply. Away from the bounded number of intermediate scales, the bounded effective quadratic parameters are tiny and each retained t-frequency is tiny or large. Smoothness handles the large frequencies. Removing the tiny oscillations leaves exactly

This proves the active-scale count and the small-norm scale sum in [H, Lemma 5.7], with constants depending on b.

For scheduled inputs we use [V, Section 4]. Its row estimates concern Hilbert-space operators on refining interval partitions, evaluated on inputs at prescribed nondecreasing read depths. Their hypotheses are support on the input partition, representation of the adjoint ranges on the observation partition by the inherited columns, the stated order of successive partitions, and an ordinary Bessel bound before evaluation on the varying inputs. None involves the maps y_j. Restriction to a new spatial root preserves the pointwise block bound in Theorem 2.2. The original W_j and original depth schedules are retained; subsequences may have empty intersections with some blocks, as allowed in V’s row argument. Its nearest-endpoint extension is retained as well.

The smooth rows in [V, Lemma 4.6] require the ordinary Bessel bound just verified, with the fixed carriers above. Thus the scheduled scale-sum and predictor results [V, Lemma 5.3 and Corollary 5.4], and its delayed replacement Lemma 5.5, hold with the new coefficients. The remaining steps of its Section 5 use the same local detection, fixed-cell fits, projection energies, and staggered pair comparisons. A finer projection in a comparison is still applied to the input at the prescribed read depth, and its row estimate has no factor depending on W_j.

We now describe the output of the base construction. For the fixed-input argument, embed the scalar input as F_{j,s}=(f_j,0), independent of s, on a subtree satisfying \norm{f_j}_{q,I}\le1 on every cell, as in [H, Proposition 2.3]. For the scheduled argument, put F_{j,s}=(z_{j,s},0) with the inputs of Theorem 2.2. The adaptive fits and two-earliest-increments expansion of [H, Lemmas 5.4–5.5], or [V, Sections 5.1–5.2], express the original form, at any finite accuracy cutoff, in tuples with two differences of fitted projections and two original-input residuals, together with a remainder whose absolute length-weighted sum tends to zero on the fixed finite tree. Each tuple has an earlier and a later accuracy index, and a choice of the four roles. Their exact staggered ordering is retained from those results. High-position truncation, using the predictor estimates above, is [H, Proposition 5.6] or [V, Sections 5.5–5.6]. It assigns each original summation cell to a segment root R with inherited projection families; write \mathcal I_R for the cells assigned to R. Auxiliary projection cells are not additional summands.

Fix one tuple and let k be its later accuracy logarithm. The two positions called low in this construction are denoted i,j, and the other two by a,l. On a cell of depth s, the resulting arguments satisfy

All expressions are restricted to I; a projection indexed by a finer depth is the direct sum on that finer partition. The Q^{p,h} and P^{\rm new}_p are the inherited basic families just constructed, and K_0 is their positive integer delay. Their column sizes, inverse private weights, scalar complexities, coefficient sums, and path counts have a joint budget e^k\le D_{\rm base}\le\exp(C_{q,b}k^C). On every segment they give

One may take \Lambda_k to dominate C_{q,b}(1+K_0)\log(2D_{\rm base}). For the fixed-input argument the smaller bound in H also suffices.

For noninitial accuracy levels, every pair of full Y arguments has form norm at most \eta=\exp(-k^\theta), and |H_I(Y_I)|\le\eta. Here the pair norm is the supremum against normalized L^2(I) inputs in the complementary two positions; 0<\theta<1 is fixed before q. Define \mathcal A_R to consist of those cells for which some available triple of basic scalar columns has transpose norm greater than D_{\rm base}^{-C_{\rm act}}, with the fixed sufficiently large exponent in H or V. The active-scale estimate gives \sum_{I\in\mathcal A_R}\mathbf1_I\le D_{\rm base}^C.

There are three remaining patterns: z^\varnothing=B, z^a replaces only role a of B by U_a, and z^l replaces only role l by U_l. Put

The output of [H, Proposition 5.8] and [V, Proposition 5.6] is

Indeed the term with both U factors uses the full complementary pair bound and (4.6). The inactive triple scale sum bounds the omitted cells. On the others, apply \min(\eta,\sum x_m)\le\sum\min(\eta,x_m) to the full tuple before separating its patterns. No individual pattern is asserted to have the full-pair smallness.

For each fixed tuple the segment roots have total length bounded by a fixed multiple of the original root length, and the sum of the high-truncation replacement errors is at most C_{q,b}e^{-k} times that length. These are the segment-packing assertions in the cited base constructions. The finitely many initial accuracy levels use the active path bound and local sizes, without the factor \eta, and hence have depth-independent bounds. All these estimates are valid anew on a normalized subroot with its inherited lists and original input schedule. We have therefore reduced both input classes to estimating (4.7) uniformly over these concrete base patterns.

4.4. Curvature tags and separation of distinct tags

We now localize the base arguments according to their quadratic structure. Recall the nonzero integers c_j of Lemma 2.3. A column in role i has normalized curvature tag v=(\theta_v,C_v,\lambda_v) when its quadratic chart phase is written as

Its linear modulation is not part of its tag. This is H’s normalization with (-1,3,-3,1) replaced by c. Division by the fixed nonzero c_i preserves the chart budgets.

At length r, the difference data of two tags v,v' are

The ambient graph in [H, Section 6.1] joins two tags when these data admit (4.3) with bounds at most the chosen edge height H_{\rm ed}; equal tags have distance zero. This graph includes all hypothetical tags within the basic budget. Its distances depend on tags and depth, and do not increase when the interval length is halved.

4.5. Same-tag tests and the localization output

The complementary fact is that a column of one tag can detect a residual in another role using a template of that same tag. This keeps the approximation spaces finite without repeatedly increasing their nonlinear complexity. A pure source column means one scalar column label, rather than a linear combination of labels.

We record how these two tests enter the remainder of the imported construction. This also fixes the meaning of its output used in Section 5. Put P=\lceil\eta^{-\alpha}\rceil, where \alpha>0 will be chosen in the closing argument. The vertices are occurrences of the original columns and are ordered by admission. Added main and helper columns attach to those vertices and have their normalized tags; they create no new graph vertices. H’s short-path, color, and historical-mask constructions [H, Lemmas 6.2–6.5] form groups of these vertices and assign current groups to each summation cell. An actual use is such an occurrence of a current group at a cell. A mask is the subset of available vertices allowed in a specified local projection. A pad enlarges that subset by the prescribed graph radius; a positive smeared filter is the average of P nested pad projections. A history is the ordered product of such filters or their complements used in the binary localization. The number \mathcal L of colors is polynomial in k, and these main histories have O(\log\mathcal L) factors. Each group records a distinguished tag z, the primary winner in H’s color construction. Every vertex admitted to its masks has, already at admission, a graph path to z of length polynomial in P,k. The short-path lemma therefore gives the endpoint splitting (4.3) relative to z, with logarithmic height polynomial in P,k. This is the splitting used for the first matching of a column in Section 5.

We use the precise masks and historical-use convention of those lemmas. Its relevant properties are as follows. Masks are inherited on descendants; at an actual use, enlarging a pad by one includes every neighbor of the preceding mask. An endpoint is usable when its mask is nonempty and its cell has a downstream actual use of its group. Distinct groups of one color have tags separated by graph distance greater than one on overlapping usable endpoints. The endpoint bound along short paths follows from the rational splitting (4.3). The usable-space Bessel bound follows from same-slot separation in Lemma 4.4. These proofs involve only rational linear algebra, inherited interval projections, and the indicated splitting and separation bounds; thus their path, rank, and pad counts keep the quantitative orders in H.

For completeness, the closure condition imposed on these spaces has an exact form. At rank r, let n_{r-1} bound the number of labels through the preceding rank along a path, let D=D_{\rm base}, and let \tau_3=D^{-C_\tau}. For a basic argument a, each prescribed history \mathcal H, target mask \mathcal M, and pure lower-rank source whose tag belongs to that mask, the form norm against the residual (1-P_{r,\mathcal M})\mathcal Ha and two normalized L^2 spectators is required to be at most

On failure, Lemma 4.5 supplies a new column and hence a projection increment. The products-of-prefixes estimate of H Section 3 bounds the sum of those increments, exactly as in its Lemma 6.7. Its rank recurrence and main/helper catalog counts are therefore unchanged. This count depends on the names of masks and histories, including anticipated helper bases, but not on the numerical edge height or on later mode precision. As in [H, Section 6.4] and [V, Section 6.1], first fix these counts, then the mismatch tolerance, and finally the edge height needed in Lemma 4.4. The lower-rank count in (4.12) is retained, so no full source count is lost when these errors are summed.

It follows that the binary localization and outgoing-filter identities of [H, Proposition 6.8 and Lemma 6.9] hold for the present forms: they use (4.12), distinct-tag form separation, the local absolute bound, and the same nested projection algebra. In the color construction of [H, Lemma 6.3], the first failure of its marking conditions may remove a subtree. Increasing the fixed thresholds makes the sum of these roots’ lengths an arbitrarily small fixed fraction of |R|. Section 6 restarts the original summands on those roots. After these marking-failure cuts, let \mathcal A_R^{\rm main} denote the remaining active cells. For each base pattern and I\in\mathcal A_R^{\rm main} of depth s, the localization is

Here \lambda ranges over polynomially many fixed binary search leaves. The inner history \mathcal B^\lambda_{p,s} is independent of G, and \overline L^\lambda_{p,G,s} is the outer ball average. The minimum in (4.7) is distributed across the fixed leaves, while the group sum remains inside the original cell contribution. The filters act on the whole basic arguments in this identity. We next record their endpoint bounds, then the algebra that moves the complementary filters onto a side operand.

For varying inputs, this localization acts at each cell on its one actual tuple at that cell’s original depth. The insertion tests freeze that operand while the column list grows. The products-of-prefixes estimate therefore remains a fixed-vector estimate, and gives the same closure count; this is the argument of [V, Section 6.1]. After the operator families have been fixed, its Lemma 6.1 and Corollary 6.2 control their changes between endpoint depths. More explicitly, for one slot, search leaf, and color, write the full localized endpoint output as Z_{G,s}=\overline L_{G,s}\mathcal B_s B_s, where \overline L is the outer positive ball average and \mathcal B its group-independent inner history. If depth intervals [r,t] have edge-overlap multiplicity at most a, and E_{G,r,t} is a union of t-cells with downstream actual uses certifying the indicated group, those results give

The shared histories without an outer ball satisfy the corresponding estimate counted once across groups. Endpoints before the attempt root are zero, and all other endpoints use inherited continuations and local refits. The proof first establishes the ordinary Bessel bounds for operator changes, using usable endpoint ranges and compression, and then evaluates the rows at their prescribed depth inputs. Input changes use only the original block variation bound. Thus the constants in (4.14) are independent of the partitions and W_j.

4.6. Transferring the complementary filters

The side estimate will encounter two distinguished operands while the other two retain their whole localized histories. We state the projection identity for this situation explicitly. Fix an actual group use at a cell I, one search leaf, and four distinct roles a,a',c,d. Let S_a,S_{a'} be fixed Hilbert-valued L^2(I) operands in the first two roles. They are called side operands here because they are retained while the complementary histories are simplified; their particular construction and cross-depth bounds are given in Section 5. The other two roles have the whole localized arguments with basic inputs B_c,B_d from (4.13). Choose the side operands before closing the helper catalog. The helper version of (4.12) is imposed with these fixed states as bases and lower-rank main columns as sources, as in [H, Lemma 6.7].

Expand each complementary factor 1-\overline Q as the difference of the identity and a positive average. There are only 2^{O(\log\mathcal L)} resulting products. In one such product, remove the largest-rank outermost positive filter in roles c,d. The positive main ranks are distinct in the source construction, so the remaining source in the other complementary role has strictly lower rank. Suppose the filter being removed is on role c. Let Q_{c,p}, 1\le p\le P, be its individual pad projections, and Q_{a,p} the helper-rank projections in role a with the same masks and pads. For j=a,c, set

The A_j^p are mutually orthogonal projections and sum to the identity. Their final band is included, with weight zero in the positive average.

Let x_c be the inner history on B_c after deleting the outgoing filter, and let x_d be the entire remaining history on B_d. Write x_a=\mathcal D_aS_a, where \mathcal D_a is the product of helper averages already transferred, and x_{a'}=S_{a'}. In the next displays arguments are written in role order a,a',c,d; each still occupies its original slot in H_I. Multilinearity and (4.15) give

Indeed the coefficient of the pair of bands is the weight of band q in role c minus the weight of band p in role a. This proves H’s Equation (15), including its terminal bands, for the present form without a coefficient change.

The estimate accompanying [H, Lemma 6.9] permits only |p-q|\le2 to be retained: the other terms at this cell and group have total absolute value at most

for any prescribed fixed C_3, after the numerical accuracies are chosen. Its hypotheses are precisely the closure and separation estimates proved above. If p>q+2, expand the lower c-band into pure main sources and use the helper closure test on the higher a-band, written as a difference of residual projections. If q>p+2, expand the lower-rank source in role d. A source tag within one edge of the lower helper mask belongs, by neighbor closure, to the next outgoing mask; the main closure test then controls the higher c-band. For a tag farther away, expand the helper primal and use Lemma 4.4. The coefficient counts are canceled by the lower-rank denominator in (4.12), except for fixed count powers already included when the mismatch tolerance was chosen. The O(P^2) band pairs and polynomially many transfer positions fit those accuracy choices because \log P=O(k).

For each fixed offset h=q-p\in\{-2,-1,0,1,2\}, retain the coupled sum

The sum over p stays inside the cell contribution. When the current group is also summed, its sum stays there as well. This is the expression to which the short-block estimate is applied; taking absolute values of its individual bands would lose the matched-band summation used in that estimate.

Forward only the transferred main term in (4.16) to the next step. Removing the remaining positive filters in decreasing rank ends at

Restoring B_c,B_d to the full Y_c,Y_d uses their pair bound from the base reduction. Each correction has an additional basic U factor and is therefore a three-side term.

For the retained defects, the endpoint convention matters. If the outgoing filter is a ball filter, split A_c^{P+1}=1-Q_{c,P} into the shared identity and its positive ball endpoint. The other complementary band endpoints have usable ball ranges. If the outgoing filter is cumulative, its outer ball has already been removed and the entire complementary band expression is shared across groups. Role d likewise either has one positive outer ball filter or is shared. In the endpoint energy estimates, a shared identity is evaluated on the union of required cells and counted once; the original form still retains every actual use. These are exactly the endpoint classes controlled by (4.14) and [V, Corollary 6.2]; they give the side estimate in Section 5 its coupled band operands.

We have obtained the base minimum, the localized group expressions, their outgoing-filter identity, and the endpoint estimates with the same main-column budgets as H and V. The remaining work is to estimate the coupled near bands and the remaining localized quadratic contributions. Section 5 establishes the coefficient-dependent comparisons used for those two estimates.

5. Short blocks, rational shifts, and the core estimate

Starting from the localization proved in the preceding section, we now split the arguments into side terms and core terms. Side terms are controlled by square estimates for projection increments, together with a comparison on a bounded block of depths. Core terms are controlled by separation of the frequencies left after removing a common quadratic phase. We prove the coefficient-dependent parts of both arguments here. The projection, graph, and amplitude summations that we use from [H, V] are identified at their points of application.

Throughout this section, the primitive integer vector b=(0,b_1,b_2,b_3), the integers e_j, and the nonzero integers c_j are those of Lemma 2.3. Constants may depend on these fixed integers and on the admissible kernel bounds. We write e(t)=\exp(2\pi i t). For an interval I, the notation \|f\|_{2,I} denotes (|I|^{-1}\int_I|f|^2)^{1/2}, also for Hilbert-space-valued functions.

5.1. A comparison on a short block

We first establish the version of [H, Lemma 7.2] needed by the side argument. This estimate compares randomized inputs with the fixed-input constant C_N(q); in particular, the exponent of the block length must tend to zero as q tends to two.

This proposition supplies exactly the input used by the side estimates [H, Lemmas 7.3–7.4 and Proposition 7.5] and [V, Section 6.5]. Their remaining arguments concern nested projections, frozen stacks, and variation in depth. In addition to the local absolute form bound, they assume the square estimate H(17) for the linear sides. We identify its source below; no coordinate substitution is needed in those projection arguments.

5.2. A common gauge and a lattice of rational shifts

We describe the data from the mode construction of [H, Section 8] before checking its arithmetic. Fix a localized group G, with central tag z=(\theta_z,C_z,\lambda_z), and a role l. A primary is an available main column of this role’s outer projection. Its first matching cell is the cell where the finite expansion described below is chosen; that choice and its labels are then inherited on descendants. Thus later freezing does not resample a primary’s expansion.

Choose an integer Q_0 clearing the denominators of C_z, and put \vartheta=\theta_z/Q_0. For an offset \rho in the unit cube of horizontal coordinates, define

Choose \delta_c=\exp(-k^{C_{\rm cut}}) sufficiently small after the main budgets, with fixed exponent C_{\rm cut}. Let \zeta^\rho be the product Gevrey cutoff of [H, Section 8], vanishing within distance \delta_c of each branch cut and equal to one outside distance 2\delta_c. It is bounded between zero and one. For each role and group, one offset is kept fixed throughout its history. Its expectation is at least 1/2, and \mathbb E_\rho|1-(\zeta^\rho)^2|^2\le C d_{\max}\delta_c, where d_{\max} denotes an upper bound for the dimensions.

For a primary \Phi=(\varphi_\Phi,\epsilon_\Phi^{1/2}\mathbf e_\Phi) with normalized tag v, the endpoint splitting from the localization section and the chart expansion of [H, Lemma 4.4] give, at its first match,

The chart expansion uses this same error tolerance \delta_c, and all modes of one primary have the same quadratic coefficient. As in [H, Lemma 8.1], use mode atoms

The coefficient sum, mode count, normalized curvature, and rational heights have logarithms polynomial in the accuracy and localization parameters k,P. In particular, for modes of one primary,

Multiplication of the splitting by the fixed integer c_l changes only fixed constants in these bounds.

The resulting lift \mathcal R_{l,G} is one fixed pointwise bounded operator on augmented inputs. Indeed, set q_\Phi=\sum_m a_{\Phi,m}(\psi_m)_{\rm phys} -(\zeta^\rho)^2\varphi_\Phi, so that |q_\Phi|\le\delta_c. For an augmented input F, its lifted physical component is

and its mode coordinate belonging to (\Phi,m) is \epsilon_{\rm m}^{1/2}a_{\Phi,m} \epsilon_\Phi^{-1/2}F_\Phi. Choose \delta_c small and \epsilon_{\rm m} small after the main column budgets. Cauchy–Schwarz then gives a uniform operator norm. The private coordinate of an outer projection output records its column coefficient, so this operator gives exactly its mode synthesis. It is cell-local and independent of the freezing block.

There is also a common periodic multiplier for the four gauges. For four offsets, put d_j=d^{\rho_j}(y_j) and

The k_j are bounded integer vectors. Between branch cuts the lifts are affine and their affine parts cancel in this expression; hence the carries are locally constant. The three identities \sum c_jb_j^a=0, 0\le a\le2, give

For fixed carries the right side is an ordinary character on the torus with coordinates (\vartheta y_{l'},\vartheta t), for any anchor l': Q_0 clears C_z, and changing the anchor uses integer coefficients b_j-b_{l'}. Smooth cutoffs slightly larger than the original ones extend these character pieces by zero at the cuts. They give the smooth periodic multiplier, bounded by one, required in [H, Section 8.3].

The number of carries and the derivative bounds acquire only fixed factors per coordinate or derivative. Their logarithmic costs remain polynomial in P,k. Choose a mode budget D_*\ge e^k, \log D_*\le\operatorname{poly}_b(P,k), covering the mode lists, coefficient sums, heights, inverse private weights, and absolute Fourier sums, and large enough to dominate the dimensions and all fixed slope factors. Only then choose a Fourier cutoff B_* with tails at most D_*^{-300}. This preserves the order of choices in H; a coefficient-sum budget is not defined through its own truncation radius.

To control rational divisions at later frequency tests, fix an integer

The mode graphs will use shifts in the lattice generated by \vartheta/L_h, where

Here E_h bounds integer shift indices. The choice of D_{\rm s} is fixed independently of all depth and accuracy parameters. From this point, constants indexed by b may also depend on this fixed choice of D_{\rm s}.

We recall the scale geometry to make the graph conditions explicit. In a depth block [s_0,s_0+K), freeze coefficients at u_{s_0}=s_0-4K, compare on a cell J at depth v_{s_0}=s_0-2K, and put t_{s_0}=s_0+K. A mode has instantaneous frequency \omega_m(x)=2\lambda_mx+\gamma_m. On the list frozen at u_{s_0}, join two modes in the PRE graph if

The CURRENT graph has the same definition with threshold \Xi/r_{t_{s_0}}. Write \mathcal E_h(m), \mathcal F_h(m) for their components. Both graphs increase with height, because a witness n at height h becomes D_{\rm s}n at height h+1, provided A_0\ge D_{\rm s}. They also increase along successive depth blocks on inherited lists.

Here are parameter choices sufficient for the arithmetic. Let d_{\max}\ge2 bound all relevant dimensions, set \Delta=10, and put

Choose the largest height J_*, with 0\le h\le J_*, large compared with \delta_*^{-2}d_{\max}R'\log(2D_{\rm m}), where D_{\rm m} is the main-column budget and \delta_*>0 is a sufficiently small fixed number. Take A_0 to be a sufficiently large fixed power of C_*D_{\rm s}^{R'}B_*D_*, and \Xi=D_*^{1000}. Choose a rational height reserve and a frequency gap with

Finally choose K, polynomial in P,k, after these logarithmic budgets, so the height-bin packets of [H, Lemma 3.5] and the gap \mathcal T fit inside a block. Increase it so the pre-lag Taylor errors are as small as any prescribed fixed inverse power of D_*. All degrees can be fixed independently of q.

For a primary with horizontal parameters \Theta=(\theta_v,\vartheta), call a block nonexceptional when every rational vector a of height at most \mathcal H satisfies

The height-bin proof of [H, Lemma 8.2] gives O(d_{\max}) exceptional blocks per primary. At a nonexceptional block let K_{\rm rat} be the rational span of the tiny vectors, and set K_B=\{w:(0,w)\in K_{\rm rat}\}. Cramer’s rule and the choice of \mathcal T show that every height-\mathcal H vector in K_{\rm rat} is itself tiny.

We now specify the remaining output imported from [H, Section 8]. Write \pi(\mathcal D) for the primary labels represented in a set \mathcal D of modes. A mode is good at height h\ge\Delta when

and at least nine tenths of the primaries in \mathcal F_h(m) are nonexceptional and satisfy Lemma 5.2 at h. Heights below \Delta are bad. Call a mode core if it is bad at at most \delta_*(J_*+1) heights; call every other mode a linear-side mode. The increasing graphs and height stabilization give the coherence of these predicates, the bad-density count, and hereditary representatives by exactly [H, Lemmas 8.4–8.8, Proposition 8.9, and Lemma 8.10]. Those arguments use component inclusion, cardinality, projections, and the rational gap; their coordinate inputs have all been established above. A minimal PRE or CURRENT component means a component at height zero. Good modes in a CURRENT component have one PRE predecessor at height h-\Delta. Moreover, core modes in one minimal CURRENT component have identical good-height predicates throughout the upper range

Each core mode has bad fraction at most 2\delta_* in this range.

We also record the decomposition of the original localized outputs. Let Z_{l,G,s} be the current augmented output in role l. For a basic-B role in the block beginning at s_0, put

Let A_{l,G,s_0} and V_{l,G,s_0} be its exact core and linear-side mode subtotals, with all private coordinates retained. Define the full side by

In a basic-U role set S_{l,G,s}=Z_{l,G,s} and A_{l,G,s_0}=\mathcal E_{l,G,s_0}=0. On physical components these definitions give

For a basic-B role, the physical error is

The offsets can be fixed so that every term containing this physical error is negligible. The explicit lift, applied to Z_{l,G,u_{s_0}}=\sum_\Phi\beta_\Phi\Phi, gives

The main coefficients and original actual-use cells were fixed before the offsets. Their coefficient, size, and path-count bounds, together with the cutoff expectation, imply the error estimate from [H, Section 8, subsection on fixing offsets]:

With \delta_c chosen this small in the main budget, fix offsets satisfying this bound. All later selections only remove nonnegative terms; they need not be independent of the offsets. The linear-side square estimate is uniform in the offsets. The local absolute L^2 form bound and Cauchy–Schwarz therefore make the terms containing \mathcal E negligible, after the same size stops as in H. This explains why the subsequent estimates may use the exact core and full-side arguments.

Here are the quantitative conclusions used below. A successful CURRENT component contains a good core mode, whose PRE component at h-\Delta is its uniquely determined good predecessor. For fixed height, role, group, and the pair of floor logarithms of the predecessor’s primary and mode counts, choose a representative mode inherited under predecessor inclusion. On comparable actual use cells I,I', distinct representatives satisfy H(25), namely

for x_*\in I\cap I'. On an actual cell, a successful component \mathcal F has amplitude

Here \mathcal D runs over minimal CURRENT components, A_{\mathcal D} is the full core subtotal in the augmented mode space, and \alpha_m is its frozen mode coefficient. This is H(26); the added coefficient terms absorb uniform Taylor and Fourier errors.

The Bessel estimates and bad-density count give H(17):

This is the square hypothesis needed for the side estimates following Proposition 5.1. After the stopping construction of [H, Proposition 8.12], the representatives occurring in a dyadic amplitude bin \sigma_j along a retained path satisfy H(28):

Here \mathcal A_j^G is the set of distinct representative labels in that bin. All local side and component sizes are at most P^C.

For scheduled inputs, these statements are used in the form proved in [V, Section 6.3–6.4, especially Lemma 6.3]. The rows are cell-local projections composed with the fixed bounded lift \mathcal R_{l,G}. Their ordinary Bessel estimates come from the separated mode construction, and their adjoint main ranges lie in the usable endpoint spaces established by localization. V evaluates the depth input after the final main basic projection. Its row composition result therefore applies to the same size, variation, reading-depth, and endpoint hypotheses. In particular, the possibly large mode budget D_* is not inserted into the strong square constant G_0. This is the distinction required by V’s side estimate and by the final choice of q.

5.3. The side estimate and the near-band gain

We now estimate the contribution of the full sides to the cell minimum in (4.7). The input is the localized expression (4.13), its outgoing-filter identity, the endpoint variation estimate (4.14), and the linear-side square bound H(17). These are precisely the hypotheses of [H, Proposition 7.5]; for scheduled inputs their counterparts are assembled in [V, Section 6.5].

Expand each localized argument into its core, full side, and negligible physical error. Terms with at least three side factors have a length-weighted p-power bound P^C|R|, for any fixed 2/3<p<1. Indeed Cauchy–Schwarz first sums the group index in two side roles; writing the three resulting square sizes as a_s,b_s,c_s, their contribution is bounded using

This uses 3p>2 and the retained path caps; the fourth argument uses its P^C size bound. The group sum is formed before the power is taken.

In a term with exactly two sides, restore its two core complements to the whole current arguments Z_{c,G,s},Z_{d,G,s}. The corrections have a third side. The outgoing-filter identity then removes the complementary projections one at a time, transferring helper averages to one side, say role a. Its main term is

where \mathcal D_{a,s} is the resulting product of helper averages. Restoring the full base pair Y_c,Y_d introduces another third-side error. The pair-smallness from the base reduction bounds this main term by \eta times the two side sizes. Summing costs at most \eta P^C|R|.

At one transfer step, use the outgoing and helper bands defined in (4.15), with these two full sides as its fixed side operands. The exact outgoing-minus-helper defect is (4.16). Its far bands have negligible error. For each remaining offset h, |h|\le2, the retained expression (4.17) has coefficient -h/P=O(P^{-1}). Its paired sum over the band index must remain inside the cell contribution.

Freeze the complementary band stack and the other complementary argument at the block’s first depth. Their differences from current values have square budgets by (4.14). Freeze errors therefore supply a third strong factor and have the preceding p-power bound. The same argument handles cells where a frozen complement has large size: its difference from its bounded current value is then large. On the remaining ancestor hulls inside a block, both frozen complements have bounded local L^2 size.

The binary-prefix decomposition in [H, Lemma 7.4] now writes the two varying sides, at their uses, as selections from fixed vector stacks. The terminal complementary band 1-Q_{c,P} is first split into the identity and a positive ball term. Positive outer-ball ranges have joint group budgets, whereas group-independent identity and cumulative histories are counted once rather than once per group. This convention is needed for the following stack energies. One stack includes the band index. There are only G_0 overhead factors. Individual pad choices stay inside the actual positive averaged operators: a choice is coupled across depths, and Jensen’s inequality restores the average after estimating it. They are not additional stack configurations. The stack energies satisfy H(18):

Here \nu records the group, block, and fixed stack configuration. The functions remain fixed after subsequent cuts. Independent Rademacher signs select the two position indices, and a third family matches the band in \mathbf A_\nu with the frozen complementary band stack. Iterated Khintchine gives the moment hypothesis of Proposition 5.1.

For precision, the passage from these energies to a sum of local stack sizes uses the common hull cuts of [H, Section 7]. A hull consists of all ancestors between a use and its maximal use cell, including their full intervals. Write \Omega_\nu for the union of the original use cells. The main path and catalog counts give \sum_\nu|\Omega_\nu|\le M_0|R|, with \log M_0\le\operatorname{poly}_{q,b}(k). For a family of total stack energy at most E|R|, write \widetilde m_\nu for its full hull maxima. They obey

Stopping at level C_1E and integrating up to that level costs at most CE(1+\log(2+C_1M_0))|R|. Thus only a main-budget logarithm enters G_0. For the retained hull define

with value zero for an empty supremum; define m_{B,\nu} in the same way. The weak L^1 estimate for the dyadic maximal function, followed by truncation and integration, gives common individual cuts; Markov’s inequality for the truncated sum gives the joint cuts for which these squared hull maxima satisfy

The removed roots have arbitrarily small fixed total relative length. They are first crossings of partial ancestor quantities, so retained caps hold on whole intervals even in branches containing a later cut. A sparse decomposition of each retained hull into tree pieces gives disjoint major subsets of the piece roots. If a_Q,b_Q are the two normalized stack sizes at such a root, then

Complete each piece by its boundary children. The two side stacks then have the required leaf-size bounds, and the frozen complements have bounded leaf sizes. Proposition 5.1 applies to at most K depths of each piece. Multiplying by the transfer coefficient O(P^{-1}) yields the near-band bound

The logarithmic factor of the short-block estimate is included in G_0. Arbitrary signs at the original cells are allowed, so this is an absolute estimate for their coupled band contributions. No absolute sum over individual paired band indices has been used.

Thus [H, Proposition 7.5], with the coefficient-dependent comparison now proved, gives a complete conditional side bound. Third-side and freeze errors have p-power cost P^C|R|; transferred main terms have absolute cost \eta P^C|R|; the remaining near bands satisfy (5.10). Applying \min(\eta,x)\le\eta^{1-p}x^p only to the powered terms gives their contribution C_{q,b}\eta^\varepsilon P^C|R|, with \varepsilon=1-p>0. All cuts remove original summands at first crossings and the data are reconstructed on each normalized subroot. This is the root-uniform side estimate used in the closing argument. The scheduled version follows from the same fixed stacks and the row estimates already stated, as in [V, Section 6.5].

5.4. Frequency tests for three or four core arguments

Fix a retained path point x_*, an actual cell I=I_s with r=|I|, a group, and a common successful height h\in\mathcal H_{\rm up} for the core arguments. Put R_s=r^{-1} and

The representative a_j belongs to the good predecessor of the component in role j. Summing a PRE path and using the pre-lag Taylor bound expresses its selected physical argument, up to the controlled uniform error, as

The function g_j is a cutoff times a trigonometric polynomial. After collecting the common multiplier (5.4), all Fourier indices in any anchor coordinates lie in the box of radius B'_h, enlarged by fixed factors depending on b. This is H(29) with its complete coefficient-sum and error bounds.

To compare physical and torus amplitudes, take the horizontal tiny space K_B of a nonexceptional primary in a successful component, and let

This is a connected torus because the integer lattice being annihilated is saturated. The tiny/huge alternative makes physical interval averages of the relevant characters agree with Haar averages on \mathbb H, to the prescribed small error. The component floors above absorb those errors, giving H(30):

The selected sets are whole minimal CURRENT subtotals, by coherence of the height predicates on \mathcal H_{\rm up}. For distinct roles, u+(b_j-b_{l'})v and u+(b_i-b_{l'})v have independent Haar distributions on \mathbb H^2. Interpolating the six two-L^1 bounds gives

A bounded progression multiplier changes only the constant.

For four core roles put

Choose, by a fixed rule independent of the mode masks, a reference q'=Q+n\cdot\xi, t'=T+m\cdot\xi in the full Fourier box with \max(|q'|,|t'|)\le U_0R_s, where U_0=D_*^{250}. Introduce dyadic parameters by

Discard absent references and inner minima below U_0^{-1}. The four marginals of the normalized kernel make its Fourier transform vanish on t'=b_jq'. The mean value theorem near these lines and rapid Fourier decay give C_{L,b}\mu\nu^{-L} for every fixed L. The tiny/huge dichotomy puts all moderate frequencies in one pair of character cosets modulo K_B; summing that pair of cosets is a Haar integral bounded by (5.11). Thus its contribution is at most C_{L,b}\mu\nu^{-L}\prod_j a'_j.

For three core roles with side role l, anchor at y_l and put

Choose a moderate reference T_l+m\cdot\xi and write \beta=r(T_l+m\cdot\xi). Set \mu\asymp\min(1,|\beta|), \nu\asymp\max(1,|\beta|), with the same small and large discards. Partial Fourier integration of the kernel in the step variable gives a function W_\beta whose fixed-order derivatives are bounded by C_{L,b}\mu\nu^{-L}. At zero this uses the l-marginal; at large frequency it uses integration by parts. Duality in (5.11) bounds the remaining horizontal coefficient sequence in \ell^2.

More explicitly, average a fresh offset cutoff \zeta^{\rho_l}/c_\zeta in the side, where c_\zeta=\mathbb E_{\rho_l}\zeta^{\rho_l}\ge1/2 is constant. This insertion is an exact identity in expectation. Keep the bounded factor c_\zeta^{-1} outside the estimate and set S'=\zeta^{\rho_l}e(-c_lp^{\rho_l})S_{\rm phys} on I. Its normalized L^2 norm is at most that of the side. Write \widehat S'_I(k') for its normalized Fourier coefficients. The horizontal pairings decay in |k'+r(Q+n\cdot\xi)|. Schur’s estimate on a dyadic shell of radius \Lambda, followed by averaging the fresh offset, gives H(32) with

The contribution is bounded by C_{L,b}\mu\nu^{-L}\prod_{j\ne l}a'_j \sum_{1\le\Lambda\le U_0}\Lambda^{-L}b_\Lambda, where the sum is dyadic. Notice that the windows use the whole Fourier box and are independent of the mode masks.

The error estimates in [H, Section 9.1–9.2] now apply with these explicit tests. They use the absolute local form estimate, Fourier tails, the pre-lag, and the rational gap, all with reserves already chosen above. Per tuple the error is bounded by a fixed multiple of D_*^{-100} times the absolute core coefficient sums, and by the side size if a side is present. The common-height argument samples heights independently and uniformly from \mathcal H_{\rm up} until all roles are good; its first-success weights sum to one on each pure tuple. A core mode has bad fraction at most 2\delta_*, so the sum over failure assignments converges. Consequently these errors sum absolutely to O(e^{-k}|R|), and the principal cell terms have the form H(33):

Here the \sigma_j are dyadic component-amplitude bins, with the side bin belonging to b_\Lambda; N_s^G counts the representative tuples passing the tests. For four cores, \Lambda=1. We next bound these counts across scales.

5.5. Sparse mass for separated tests

The finite counting statement is useful independently of the mode construction. Its integer divisibility is the reason for the choice (5.5).

5.6. Application of the count and the resulting core bound

We verify the hypotheses rather than apply the original arithmetic form of [H, Proposition 9.2]. Fix a path, height, role types, frequency bands, and amplitude bins in (5.14). Identify each representative label with its value \omega_a(x_*), which is injective in a fixed role and type by (5.8). Define

Groups with a nonzero count have M'\ge1.

Two fixed core labels determine the others. For four cores, subtract the equations for Q,T of two candidate tuples. The two unknown label differences solve a system with determinant b_j-b_i, modulo bounded rational shifts and errors O_b(\nu\max(R_s,R_t)). For three cores the equation for T_l determines the last label, dividing by b_j-b_l. The shift reserve and separation force these differences to vanish. For every core triple with excluded role l', the tests also give the annulus

For four cores this is t'-b_{l'}q'; for three it is the T_l test. Once the labels are fixed, the height-bin lemma [H, Lemma 3.5] bounds their scale occurrences by L_0\le P^C(1+\log(\nu/\mu))^C.

If one role is a side, windows (5.13) for different triples incident to a fixed core label are disjoint. An overlap gives both a relation for their Q values with error O(\Lambda R_s) and their two T_l relations. Solving the resulting two-by-two system again divides by a b_j-b_i; separation then identifies the tuples. Couple the fresh offset across these windows. Parseval shows their incidence count is at most w_s=\sigma_l^{-2}\|S_{l,G,s}\|_{2,I_s}^2, giving the side-budget hypothesis.

Here are the promised reserve checks for all these divisions. In Lemma 5.3 take

Every divided \xi-shift is integral in this finer lattice, because each determinant divides D_{\rm s}. The fixed-power choice of A_0 ensures

Also \nu,\Lambda\le C_bD_*^{250}, while \Xi=D_*^{1000}, so the separation reserve holds. All rational shifts remain below \mathcal H. Every successful component contains a nonexceptional primary, even if its representative is exceptional. Applying (5.7) to its pure horizontal shifts excludes the entire intermediate interval required by the pure-shift hypothesis, since \mu\ge U_0^{-1} and the interval length |I_s| lies between r_{t_{s_0}} and r_{v_{s_0}}. This checks all five hypotheses.

Lemma 5.3, pair determination, and the side incidence bound therefore give the two counts H(34): for some fixed p_0<1,

Indeed the sparse-mass exponent c'=1/2080 allows any 1-p_0<c'. On the bins 2^{-j-1}(M')^2<N_s^G\le2^{-j}(M')^2, apply the sparse bound and sum the geometric series with ratio 2^{1-p_0-c'}<1.

Finally the amplitude and group summation of [H, Proposition 9.3] uses precisely (5.14), (5.9), and (5.16)–(5.17). Interpolate the two counts with a sufficiently small fixed weight \theta'>0, giving power p=(1-\theta')+\theta'p_0. The path square budgets sum the group factors, and the core count cap D_* controls the common amplitude shift. The three amplitude-gap sums are geometric, as in that proposition. The frequency losses after interpolation are at most \mu^{-C\theta'}\nu^{C\theta'}\Lambda^{C\theta'}; choose \theta' small and then the derivative order L large so they are summable against \mu^p\nu^{-Lp}\Lambda^{-Lp}. Applying the original cell minimum before subadditivity yields

for the terms with at least three core arguments, with fixed \varepsilon>0 and C_4 independent of q. The same calculation is the core step of [V, Section 6.6]: its frozen scheduled inputs provide exactly the amplitudes, representatives, and same-cell side incidence verified above. Together with Proposition 5.1 and the side square bound, this supplies the side and core estimates required to close the proof of Theorem 2.2.

6. Parameter choice and completion of the local estimate

The preceding sections provide the coefficient-dependent inputs to the completion arguments in [H, Section 10] and [V, Section 7]. We now explain the common parameter choice and the two distinct conclusions obtained from it. First we bound the fixed-input constant C_N(q) independently of N. Only after this absorption do we complete the proof for depth-dependent inputs.

6.1. The retained-cell estimate

We use the base reductions of [H, Proposition 5.8] and [V, Proposition 5.6], with the modifications proved above. At one noninitial accuracy level their target, on a normalized root R, is a sum of terms \abs I\min\{\eta,\abs{H_I(z_I)}\} over a specified subset of descendant cells. Here z_I is the four-tuple supplied by the base expansion at I, and \eta=\exp(-k^\theta) is its quality parameter, with k the accuracy log and 0<\theta<1 fixed. The constructions then decompose each of these four-tuples into the localized terms estimated in the preceding sections.

The stopping constructions in [H, Section 10.1] and [V, Section 7.1] are used in their stated order. Main columns and their uses are fixed first; mode expansions and the whole side functions are fixed next; helper projections are constructed after those side functions. Size and counting stops are then imposed on these fixed objects. A stop removes original summands from an attempt; it does not redefine a previously fixed stack function. In particular, the maximal functions in the short-block estimates remain defined on whole retained cells, including the portions below later stopping cells.

Let \mathcal A_R^{\mathrm{ret}} denote the cells not removed by these stops in the attempt rooted at R. The conclusions of the preceding sections give the same estimate as the retained-cell calculations in the two companions:

Here \varepsilon>0 is fixed, P is the transfer parameter, K is the depth-block length, and G_0 is the row and stack overhead. The constants may depend on b,\delta_0,C_0.

For clarity, the estimate has three sources. Terms with at least three core factors use the powered core estimate, and terms with at least three side factors use their square-energy bounds. With p=1-\varepsilon>2/3, both are converted to the first term of (6.1) by \min\{\eta,a\}\le\eta^{1-p}a^p. Terms with exactly two sides use the transfer estimate for the whole complementary arguments. Its main term is again included in the first term. The remaining coupled near-band sum is estimated at power one by the short-block comparison; it gives the second term, including its factor 1+C_N(q), by (5.10). Physical and numerical errors are estimated at power one as well. These alternatives exhaust the four-position expansion. The minimum is applied to the whole contribution of a cell before subadditivity is used; no smallness of a separately selected group is being assumed.

6.2. Restarts and the complexity hierarchy

The integrated size, path-count, and square-energy bounds allow the stopping thresholds to be enlarged by fixed factors so that the first stopped cells occupy at most \abs R/4. At each such cell send the original summands there and below to a new attempt. The local normalizations and column bounds are available on every new root. Linear-side energy is proved anew by the construction on that root, not by asserting a proportional restriction of a global energy bound. The resulting attempt roots satisfy

All the trees are finite, so the restart argument can equivalently be read as induction on their remaining depths.

In the scheduled-input construction, the original depths, functions, and block partitions remain attached to the summands. Restriction to a new root preserves the pointwise size and variation bounds with the same W_j. The row estimates of [V, Section 4] therefore apply at each restart with no factor depending on W_j.

The distinction between the two complexity scales is essential. The bounds supplied by the constructions are

Here D_{\mathrm m} bounds the main and helper catalogs, and D_* bounds the larger mode, Fourier, and rational-height data. The latter data enter the counting and mode estimates, but are not fed back into the main catalog construction. Strong stack estimates compress main projection families only; mode projections have already been included in the Bessel rows. This is precisely the separation in [H, Section 10.3] and [V, Section 7.3].

The coefficient modifications preserve this separation. Chart substitutions multiply heights and derivative bounds by fixed slope-dependent factors; the change of height base occurs only in the mode construction. The sparse-count estimate retains its dimension-polynomial reserves and its power saving. Thus all the degrees C,C_4,C_5,C_{19} can be fixed before choosing q. Their numerical values may be enlarged to cover both the fixed-input and scheduled-input constructions. Only coefficients such as C_q are allowed to depend on the later choice of q.

Choose q\in(2,3) sufficiently close to 2 that

for both constructions. This common choice is legitimate because local detection is available for every q>2 and short-block comparison for every 2<q<3. Next put P=\lceil\eta^{-\alpha}\rceil, choosing a fixed \alpha>0 so small that \alpha C_4<\varepsilon/2, as well as the analogous finitely many upper bounds already included in the powered costs. Finally choose the starting accuracy sufficiently large. Since P dominates every fixed power of k and G_0=P^{o(1)}, we obtain

Also P^{-1/4}\le\eta^{\alpha/4}, and e^{-k} is smaller than these quality powers for large k. Consequently, for some c_*>0, (6.1) and (6.2) give the full base-segment bound

6.3. Fixed inputs: accuracy summation and absorption

The base accuracy levels satisfy k_{t+1}=k_t^A for a fixed A>1. For tuples whose later level is t, there are O(t) choices of the earlier level and a bounded number of slot patterns. The segment-root packing in [H, Section 5], followed by (6.2), gives a bounded total root length for each tuple. The noninitial terms in (6.5) therefore have total coefficient controlled by

This series is finite, and its value is arbitrarily small when the starting accuracy is sufficiently large. The replacement and two-increment errors have the same summable multiplicity. The finitely many initial levels have depth-independent bounds from the active-scale reduction, as in [H, Section 10.4]. Thus the modified construction proves the following normalized conclusion.

The argument just given is the proof: take the coefficient in (6.6) below \tau and include its constant part, the initial levels, and the numerical errors in A_\tau. The base expansion is first taken at a finite accuracy cutoff. At each fixed cell the terminal residual tends to zero by the pair-comparison estimates of the base reduction; since there are only finitely many cells, the passage to all accuracy levels is legitimate. This is the concluding expansion step in [H, Section 10.4].

We include the outer normalization argument to make the absorption explicit. Start with disjoint roots at the largest scale in a finite collection from (2.7). At a root J, write a_{j,J}=(\abs J^{-1}\int_J\abs{f_j}^q)^{1/q}. If some a_{j,J} vanishes, all forms below J vanish. Otherwise stop at the first strict descendants J' on which an average of \abs{f_j}^q exceeds 16a_{j,J}^q. The union of first stops has measure at most \abs J/4. Dividing f_j by 16^{1/q}a_{j,J} normalizes the intervening tree, so Proposition 6.1 bounds its sum by

Repeat at the stopped roots. The sets E_J left after deleting their first stops are disjoint, have measure at least 3\abs J/4, and satisfy M_qf_j\ge a_{j,J} on E_J. Hence

Taking the supremum in the definition of C_N(q) gives

where C is independent of N and bounded for 2<q<3. Choose C\tau<1/2 and use the already established finiteness of C_N(q) to absorb the last term. We have proved

6.4. Depth-dependent inputs

Now perform the scheduled-input construction with the same q, using its own subsequent choices of \alpha and starting accuracy. Insert (6.7) into (6.5). The factor 1+C_N(q) is now a fixed constant. The scheduled base expansion of [V, Section 5] has the same accuracy multiplicity and segment-root packing, with the depth-row estimates supplying the uniform bounds. Its noninitial terms therefore sum by (6.6). Its finitely many initial levels are bounded by its active-scale reduction independently of the number of depths and of every W_j.

Finally, at a fixed original cell, all accuracy updates are evaluated on its same actual scheduled input z_{j,s(I)}. The per-cell pair comparisons therefore make the terminal residual tend to zero. There are finitely many cells, so the finite-cell sum commutes with this limit. The triangle inequality and the uniform summed bounds give (2.4), exactly as in [V, Section 7.4]. This proves Theorem 2.2, and hence, by Section 3, Theorem 1.1.