Fixed testing templates from one curvature tag

This chapter studies Lemma 6.6 in Section 6.4 of H, pages 46–47. The source is the fixed October 5, 2026 version of An $L^3$ bound for the trilinear Hilbert transform.

The aim is to turn a four-linear correlation into a large moment of the target residual against a template with the same tag. The nonlinear amplitudes come from one fixed finite family, while linear modulations may still be arbitrary real numbers. The same complexity bound applies whether the source is an original column or a template inserted later. Repeated insertions therefore do not cause the nonlinear complexity to grow.

Finite termination of the catalog under changing projection histories requires the separate argument of Lemma 6.7. This chapter establishes only the same-tag test used there.

Back to the guide to H

1. Fix the tag, the source column, and the required moment

Fix the normalized curvature tag of an original vertex,

and put $c=(-1,3,-3,1)$ and $e(t)=\exp(2\pi\mathrm{i}t)$. Let the basic complexity be $H\ge\mathrm e^2$, with

The original amplitude $\chi$ is supported in $[-H,H]^d$. The fixed Gevrey constant $c_0$ in Definition 4.1 of the source gives

There is no restriction on the sizes of $\theta,\lambda$ or the external linear modulation.

The source slot is $i$, the target slot is $j\ne i$, and the remaining two slots are $a,b$. The original source column has the form

It is normalized as a bounded physical column, as in the source. Write $F\in L^2(I)$ for the target physical residual. The two observer inputs $A,B\in L^2(I)$ satisfy the normalized norm bounds

The local form $H_I$ is the same as in the preceding chapters. Its kernel is smooth and supported where all four coordinates lie in the interior of $I$. Below, $H_I(\phi_i,F,A,B)$ means that the four functions occupy the specified slots $i,j,a,b$; their progression coordinates retain the original slot numbering. The proof uses only these smoothness and support conditions, not the four vanishing marginals of the kernel.

We will construct templates $\Phi_{j,r,\beta}$ with the same tag. If the four-linear form norm for this source and $F$ is $t_0>0$, then some template will satisfy

The constants may depend on the fixed kernel and Gevrey parameters. They do not depend on the interval, the insertion rank, or any real modulation.

2. Fix a smooth partition once

Choose a nonnegative smooth function $\rho_1$ supported in $(-1,1)$ such that

For example, take a nonnegative bump $\gamma_0$ in the fixed Gevrey class of the source, strictly positive on $[-1/2,1/2]$, and define

The function $q$ is positive and periodic, with a fixed positive lower bound. Only a fixed number of summands contribute at each point. The Leibniz recursions for products and reciprocals keep $\rho_1$ in the same fixed Gevrey order, with fixed constants.

Take the product

Then $\rho\ge0$, its support is contained in $(-1,1)^d$, and

If the one-dimensional derivatives satisfy $\|\rho_1^{(m)}\|_\infty\le C^{m+1}(m+1)^{s(m+1)}$, the product rule and $\prod_\nu\alpha_\nu!\le|\alpha|!$ imply a bound of the same type in higher dimensions:

Thus increasing the dimension incurs only an allowed quasipolynomial cost.

3. A finite amplitude family with arbitrary linear modulations

Put $R_0=H+1$, and choose the integer and residue set

For each slot $l$, residue $r\in\mathcal R$, and real number $\beta$, define

Every factor other than the nonnegative weight has modulus $1$. The integer partition therefore gives

There are only $P_0^d$ nonlinear amplitudes, and the logarithm of this number is $O(d\log(H+1))$. Since each coordinate satisfies $0\le r_\nu/P_0<1$, all amplitude derivatives have bounds uniform in $r,\beta$.

The external linear modulation $\beta$ still ranges over all real numbers. Thus the whole template family is not a finite catalog of columns. Only the nonlinear shapes form a finite family; Lemma 6.7 will determine which columns are actually inserted.

We also allow the source amplitude itself to be

Its support and derivative bounds are controlled uniformly by the same basic complexity. Below, $\chi_{\mathrm{src}}$ always denotes either the original $\chi$ or an amplitude of this template form. Neither the period nor the amplitude family is enlarged after an insertion.

4. Four lifts and finitely many integer carries

Along the progression $y_l=x+lt$, expand the source column over its integer lifts and insert the integer partition in the other three slots. For each slot, write

Here $d_l$ is a vector lift; $d$ still denotes the horizontal dimension. Every nonzero term satisfies $\|d_l\|_\infty\le R_0$.

Since $u_l=(1-l)u_0+lu_1$,

Here $k_0=k_1=0$, so only $k_2,k_3$ need to be enumerated. Moreover,

There are therefore at most $(12R_0+1)^{2d}$ possible carry pairs.

Define the matching phase in each slot by

In each of the other three slots, insert both $\Psi_l$ and its complex conjugate. The product of all four phases can be computed exactly. Using

and the carry relation, we obtain

Hence

Every remaining factor is a linear modulation in its own slot. This is exact, even when $\theta$ or $\lambda$ is large.

5. Select integer carries exactly with smooth functions

Take a fixed Gevrey function $\eta_1$ with $\eta_1(0)=1$, supported in $(-1/3,1/3)$, and put $\eta(u)=\prod_{\nu=1}^d\eta_1(u_\nu)$. Then

For a possible carry pair $\kappa=(\kappa_2,\kappa_3)$, use the convention $\kappa_0=\kappa_1=0$ and the product

At the actual lifts, both arguments are integer vectors. This product is therefore exactly the indicator of the specified carry pair. Summing over all allowed $\kappa$ gives exactly $1$. This is an identity at integer points, not an approximation to a discontinuous constraint.

To take a periodic Fourier expansion, multiply by a fixed smooth cutoff in each $d_l$, which on $[-R_0,R_0]^d$ equals $1$ and is supported in $(-2R_0,2R_0)^d$. Denote the product of the four cutoffs and the expression above by $F_\kappa(d_0,d_1,d_2,d_3)$. Its support lies inside the fundamental box of period $P_0$, so it can be periodized smoothly without changing its values at any actual lift.

We obtain an absolutely convergent expansion

The next section bounds the sum of the absolute values of the coefficients. Only after that estimate will we interchange these sums and integrals.

6. Why the absolute Fourier cost is quasipolynomial

We first establish an estimate that can be applied repeatedly. Let $F$ be a smooth function in $n$ dimensions, with period $P\ge1$ in each coordinate, and write

Rescale to the unit torus, apply $(1-\Delta)^n$, and integrate term by term. This yields

Count lattice points in cubical shells. The shell of radius $m$ contains at most $C^n(n+1)m^{n-1}$ points, so

Consequently,

In dimension zero there is only a constant term, which is treated separately by taking its absolute value.

Apply this estimate to $F_\kappa$ with $n=4d$ and $P=P_0$. Its derivatives come from fixed Gevrey cutoffs, linear combinations with fixed coefficients of absolute value at most $3$, and the product rule. Thus its derivatives through order $8d$ satisfy

This holds uniformly over all carry translations $\kappa$. Including the costs of the period $P_0$ and the number of carries, we obtain

This bound leaves room for all the multidimensional differentiations of the required orders. Since $d\le\log H$, it is quasipolynomial in $H$.

7. Reduce to residues before differentiating amplitudes

Fix one separated Fourier term. Each integer vector $\ell$ has the unique decomposition

The actual lift is $d_l=\theta y_l-n_l$, so

Here $q\cdot n_l$ is an integer. The possibly large vector $q$ changes only the external linear modulation; it no longer enters the nonlinear amplitude.

In a nonsource slot $l\ne i$, phase cancellation leaves

Writing $-\ell_l=P_0q_l'+r_l'$ rewrites this expression exactly as

Each $\beta_l$ may be any real number, so its size does not need to be estimated.

The source slot still contains a smooth horizontal periodization. Write $\ell_i=P_0q_i+r_i$ and define

This is a smooth function on the unit integer torus. At most $(2H+3)^d$ terms are nonzero at any point, and each coordinate of $r_i/P_0$ is less than $1$. The original amplitude bound and the preceding Wiener bound imply

The original source amplitudes and the template source amplitudes satisfy the same bound. The essential point is to reduce $\ell_i$ to its residue first. Differentiating an amplitude that still contains an arbitrary $\ell_i$ would not justify a uniform derivative bound.

Combining the external linear phase, the carry phase, and $e((q_i+m)\cdot\theta\,y_i)$ leaves the source slot with just a linear phase of modulus $1$, namely $e(\gamma_\sigma y_i)$.

8. The exact separated form

Collect all discrete indices into $\sigma$. The preceding calculation expresses the original form exactly as

The four slots of each $\mathcal B_\sigma$ contain, respectively,

Every $\Phi_{l,\sigma}$ belongs to the fixed template family of Section 3. Since the templates are bounded by $1$, the $L^2$ norms of the two observer inputs do not increase.

These identities can be justified first with finite Fourier sums, then by passing to the limit using absolute coefficient sums, uniform convergence, and continuity of integration. Both the source periodization and the carry expansion converge absolutely, while the original lift sum is finite at every point. Below we separately justify the limit for a target residual with only $L^2$ regularity.

9. Separate the smooth kernel in the two observer coordinates

Fix a term $\mathcal B_\sigma$. Write $r_I=|I|$ and let $x_I$ be the center of the interval. Set

The target coordinate satisfies

The four slots are distinct, so $\alpha,\beta\ne0$. Moreover,

The source coordinate is also a fixed linear combination of $y_a,y_b$. Its linear phase can therefore be absorbed separately into the two observer inputs without changing their $L^2$ norms.

The Jacobian is $r_I^2/|b-a|$, which cancels the scale factor $r_I^{-2}$ in the local form. The normalized kernel $K(u,v)$ is smooth and supported inside a fixed square; its derivatives of every order are controlled uniformly by the allowed constants of the original kernel. Expanding it on a box of period $8$ gives

This is the same Wiener estimate in two dimensions, with a constant independent of $I,\theta,\lambda$. Each kernel Fourier factor can also be absorbed into the corresponding observer input.

It therefore suffices to bound

Here $\|\mathcal A\|_2,\|\mathcal B\|_2\le1$, and $G$ is the normalized version of the target $F\overline{\Phi_{j,\sigma}}$, first extended by zero outside $[-1/2,1/2]$.

10. Target Fourier coefficients are allowed template moments

Write the supremum of all allowed template moments as

The template bound and Cauchy–Schwarz give $M\le\|F\|_{2,I}<\infty$.

Since $|\alpha|+|\beta|\le5$, the target argument in a separated term lies in $[-5/2,5/2]$. Regard the zero extension of $G$ as a function on $[-4,4]$ and take its Fourier expansion with period $8$:

The whole range of the target argument lies within one fundamental period, so no nonzero periodic copies enter.

Returning to the original interval,

The extra character merely adds $m/(8r_I)$ to the template's external linear modulation, together with a constant of modulus $1$. Therefore

Allowing every real linear modulation is essential here. When $r_I$ is small, the added modulation can be very large, but it introduces no new nonlinear complexity.

11. Two nonzero frequency spacings give two Bessel bounds

First truncate the target Fourier sum. Define

For the first sequence, take the period $p_\alpha=8/|\alpha|$. Since

the observer support of length $1$ can be extended by zero to a periodic interval of length $p_\alpha$. Parseval gives

The sign of the frequency only reorders the integer indices. The same argument for the second sequence gives

Thus the pairing with every finite target Fourier sum satisfies

No arbitrary $L^2$ observer function has been replaced by a pointwise bounded one, and no absolute sum of the target Fourier coefficients has been taken.

We now justify the limit. The target Fourier truncations $G_N$ converge on $[-4,4]$ in $L^2$ to $G$, and

For each fixed $v$, this follows by the change of variable $z=\alpha u+\beta v$; the range stays within the fundamental period. On the other hand,

The pairing is therefore continuous, and the uniform bound $3M$ for finite sums passes to the original $L^2$ target. Pointwise or absolute Fourier convergence is not needed.

12. Sum the bounds and choose an actual template

The absolute cost of kernel separation and Section 11 imply

Summing as in Section 8 gives

This holds uniformly over all normalized observer inputs.

If the form norm is $t_0>0$, first choose observer inputs whose correlation is greater than $t_0/2$. Then $M>t_0/(2C_{\mathrm{test}})$. Although $M$ need not be attained by a template, the definition of supremum still supplies an actual template whose moment is greater than $M/2$. Taking

proves the testing conclusion in Section 1, with

The period, residue family, partition functions, and all nonlinear derivative bounds are fixed from the outset. When a template becomes the next source column, its character is again reduced to a residue first, so the same bounds apply. There is no recursive loss that raises the complexity to a new power at every insertion.

In dimension $d=0$, all integer lift, carry, and residue indices are empty, and the partition is the constant $1$. The templates are ordinary quadratic phases with arbitrary linear modulations. The four-phase cancellation and the argument in the two observer coordinates still apply; only the horizontal Fourier expansions are omitted.

13. What this supplies for catalog closure

Lemma 6.6 now supplies the following conclusion: a failed four-linear smallness test forces the target physical residual to have a quantitatively large moment against a same-tag template of fixed complexity. This template can be assigned a new private coordinate and proposed as the next column to insert.

This does not yet bound the total number of failed tests. Other projections change the residual being tested, so one cannot repeatedly apply orthogonal energy bounds as if the residual were a fixed vector. The proof of Lemma 6.7 in H refers to the binary-prefix argument of Lemma 3.4. The next module will first prove in full and apply the first prefix-row estimate of Lemma 3.3 in Section 3.3 of H, then establish finite closure of the main and helper catalogs in Lemma 6.7 under the stated fixed inputs. Uniformity of the template family is an input to that argument, not a substitute for it.