Why three fixed basic columns leave only finitely many active scales

This chapter gives a close reading of Lemma 5.7 and Proposition 5.8 on pp. 37–38 of H. The fixed source is An $L^3$ bound for the trilinear Hilbert transform, October 5, 2026 version.

The catalog-termination argument in Chapter 9 needs two different counts: the number of original columns on a root-to-leaf path, and the number of active cells on that path. Having few original columns does not mean that the path is short. This chapter explains how oscillation of three fixed columns and the marginal cancellation of the kernel yield the second count, and states exactly where the first count comes from.

We start from the basic data already constructed on p. 37. Their construction and replacement errors in Proposition 5.6 on pp. 35–36 remain source inputs; this chapter does not infer that construction from the conclusions below. We develop the scale estimate for three fixed columns here. The three-variable analytic calculation is in the appendix to this chapter.

A companion proof of the source input above is now developed in H11: basic-column construction. Section 16 explains its original path-budget output. The proof retains the external discrete inverse theorem and explicit kernel conditions and does not use this chapter’s active-cell conclusion. The parameter note supplies a quantitative completion for the conventions on pp. 31 and 35. Actual later histories, whole helper bases, and global absorption are still not included.

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1. Original births, actual availability, and the whole-path budget

H source, p. 7 · H source, p. 37 · H source, p. 39

Fix a finite dyadic tree with root $R$. Each basic label $v$ has a support-birth cell $J_v$, a bounded basic scalar column $\phi_v$ used on that cell, and a private coordinate:

The scalar columns are quadratic chart atoms from Definition 4.1 on p. 15. Repeated occurrences of the same scalar function may still have different labels and different private coordinates. The slot is also part of the position at which an original column occurs.

Page 7 allows a projection to select only some of the columns already born. Let $\mathcal V_j(I)$ be the basic list actually available in slot $j$ on cell $I$. The required conditions are

whenever $I'$ is a child of $I$. Thus support birth and admission to a particular list may differ. Once admitted, a column is inherited by descendants. Refitting the coefficient of the same label on a descendant does not count as another birth. If an original label first becomes available farther down one branch, it must not be put into an ancestor's projection in advance.

For a complete root-to-leaf path $\pi$, take the union of the basic labels that have been available on its cells, rather than summing the list sizes over cells. The original budget is

If we charge by support births, the condition on p. 7,

directly yields a stronger upper bound. Here finitely many basic catalogs may be combined, with occurrences in different slots counted separately.

On p. 37, $D_{\mathrm{base}}$ is chosen to control these path counts, inverse private weights, atom complexity, and current coefficient sums simultaneously, and to satisfy

Page 39 also explicitly allows this basic budget to be enlarged to control the combined list of original vertices in all four slots. Denote the enlarged number by $D$. We may therefore take an integer $n_0\le D$, with

This budget comes from the existing basic construction. It is not a new consequence of the active-cell count below.

2. Current basic parts and delayed differences

H source, p. 4 · H source, p. 35 · H source, p. 36 · H source, p. 37

Fix two low slots $i,j$, and denote the other two by $a,b$. Page 37 writes

The depth projections here are direct-sum projections over the corresponding refined partitions. Each $B_l$ is the difference of at most two current basic projections. Its physical part on cell $I$ therefore has a constant-coefficient representation

The coefficients may vary with $I$. We will not treat them as constants along a path.

The source inputs also include bounded local normalized norms for all current $B,U,Y$ in $L^2$, and normalization of the original inputs at the root:

The latter condition follows from the original $L^q$ condition on the normalized tree on p. 4 and from $q>2$. It cannot be inferred just from the sizes of $B,U,Y$. This input normalization must also hold again when restricting to a new subroot. After increasing the starting parameter, $D$ can control these fixed constants as well. The later update layers also satisfy the following condition: any two full $Y$ arguments have form norm at most the quantity below against arbitrary normalized $L^2(I)$ test functions in the other two slots:

This condition comes from Proposition 5.6. It is not pair-smallness between pure basic columns, and does not say that every individual retained pattern is smaller than $\eta_0$.

3. The transpose norm of three fixed columns

H source, p. 4 · H source, p. 37

Fix basic scalar columns in three different slots, and let $l$ be the empty slot. On cells where all three are available, set

where $l$ is the only slot containing $f$. Each of the three fixed columns is bounded in absolute value by $1$.

The normalized form on p. 4 of the source is

Set $u=x+lt$. The transpose function is then

The kernel support implies that $T_I$ is supported on $I$, and

The last equality is $L^2$ duality for the bilinear pairing. If $T_I\ne0$, we may take $f=\overline{T_I}/\|T_I\|_{2,I}$.

The required estimate for three fixed columns has two parts. There are fixed $C,p$ such that, for $0<\delta\le1$ and every spatial path,

We sum only over cells where all three columns are already available. In fact, the proof controls the scales on their entire common-support subtree. Deleting unused cells cannot worsen these upper bounds.

4. Why we must fix a path point before rescaling

H source, p. 15 · H source, p. 16 · H source, p. 37 · This chapter: appendix, Section 1

Choose a point $y$ on the path, so that every cell under consideration contains $y$. In the square of the transpose norm, substitute

The factors from the three differentials exactly cancel $r^{-3}$, yielding

where

All the $\Theta_{I,y}$ are supported in one fixed box and have uniform bounds for derivatives of each fixed order. They may vary independently from cell to cell.

After recentering every $\phi_j(y+z)$ at the same $y$, set

The six atoms form one three-variable chart. Its unscaled horizontal matrix consists of the blocks

and the corresponding diagonal blocks of the chart matrix are $C_j,-C_j$. The ordinary quadratic matrix and linear vector are, respectively,

The recentered parameters are now fixed. Their sizes may be arbitrarily large. The scale $r$ only changes these data to

If we instead use each cell's own center without tracking the resulting changes in real parameters, there is no common parameter vector for the rational-height count below.

5. Why bounded rational height controls the number of scales

H source, p. 13 · H source, p. 14 · This chapter: appendix, Section 6

We first state the conclusion of Lemma 3.5 on pp. 13–14 that is actually used. Fix $\xi\in\mathbb R^d$, and consider all nonzero values produced by vectors whose entries have rational height at most $B\ge2$:

There is no bound on the size of $\xi$.

If $q_1,\ldots,q_m$ is a minimally rationally dependent family, then $m\le d+1$. Clearing denominators column by column and then taking the maximal minors of a full-rank submatrix yields integers $c_i$, all nonzero, such that

Minimal dependence ensures that the null vector obtained from these minors has no zero component.

Set $L_d=4dH_d$. If the absolute values of a sequence of nonzero evaluations are successively separated by factors greater than $L_d$, their rational vectors must be linearly independent. Otherwise, take a minimally dependent subfamily. In its integer relation, the term with the largest evaluation has absolute value at least that evaluation's absolute value, whereas the sum of the absolute values of all other terms is less than

times the largest value. Cancellation is impossible.

Starting with the smallest value of $\mathcal E$, cover it by $[s,L_ds]$, then continue from the first uncovered value. Thus at most $d$ such intervals are needed. For $r_n=r_02^{-n}$ and $\ell\in\{1,2\}$, the condition

allows only one depth interval for each such interval, of length

Hence

The sizes of the real parameters only move these intervals; they do not increase their number. Different rational coefficients may be chosen at each depth, because the set above includes all allowed coefficients from the outset.

6. A finite expansion of the three-variable chart at large-value scales

H source, p. 16 · H source, p. 17 · H source, p. 18 · H source, p. 19 · H source, p. 20 · This chapter: appendix, Section 2 · This chapter: appendix, Section 3 · This chapter: appendix, Section 4 · This chapter: appendix, Section 5 · This chapter: appendix, Section 6

Write

If $A_I>\delta$, the six-atom integral has absolute value greater than $\delta^2$ at Fourier frequency zero. The three-variable form of Lemma 4.3 on pp. 16–18 therefore yields a rational orthogonal splitting $P_f+P_s=1$ whose height and slow horizontal part are bounded by

and whose effective quadratic matrix satisfies

Here $C$ is the block matrix of the six atoms. Its symmetric and skew-symmetric parts, and $\operatorname{sym}$, all use the division-by-two conventions in Section 2 of the appendix. The appendix derives each step from a large Fourier value to small return increments, a rational splitting, and then this matrix bound. It does not simply treat the previous chapter's one-parameter conclusion as a three-parameter conclusion.

Truncating the periodic Fourier expansion on pp. 18–20 on the fixed box to error $\varepsilon_0$ yields

when $\varepsilon_0$ is a sufficiently small fixed multiple of $\delta^2$. The coefficients and the splitting may change with the scale.

The important point is the form of the frequencies. They satisfy

where the rational heights of $\sigma_\nu$ have a uniform quasipolynomial upper bound. Put the entries of $\Lambda_y$, $\beta_y$, and $V_y$, together with all pairwise products of entries of $V_y$, into one fixed real vector $\chi_y$. Then every entry of $M_{P_f}$ and every $(\omega_\nu)_2$ has the form

All possible projection matrices, truncated frequencies, and periodic groups are included in this height bound. The coefficients may change, but they cannot produce a new family of unbounded rational height. The appendix derives the finite expansion and this uniformity explicitly.

7. Why the integral must be small at nonexceptional scales

H source, p. 4 · H source, p. 37 · H source, p. 38 · This chapter: appendix, Section 6

After fixing the finite expansion and its absolute coefficient bound, choose

where $C_*$ depends only on the fixed kernel class and fixed box, and is sufficiently large.

Declare the following two types of scales exceptional: some allowed nonzero quadratic entry satisfies

or some allowed nonzero linear frequency in $\tau$ satisfies

The height bound in the preceding section and the scale bound in Section 5 imply that the total number of exceptional scales of both types is at most $CK^p$. We first count over the entire allowed family of rational evaluations, then choose the actual splitting on each cell. Thus a scale-dependent splitting has not been treated as fixed.

Suppose a nonexceptional scale still has $A_I>\delta$. The Weyl splitting already bounds the effective quadratic entries by $B_1$, so all of them must be smaller than $\varepsilon$. Replacing the quadratic exponential by $1$ on the fixed box costs at most $CB_2\varepsilon$.

In each remaining term, the $\tau$ frequency has absolute value either below $\varepsilon$ or above $L$. For the large-frequency terms, integrate by parts once in $\tau$. The other linear factors are independent of $\tau$, so only the fixed kernel amplitude is differentiated. The total contribution is at most

For the small-frequency terms, replace the $\tau$ exponential by $1$, at a total cost of at most $CB_2\varepsilon$. The resulting integral is exactly zero: for fixed $h,\tau'$, the other kernel and the remaining linear exponentials are independent of $\tau$, and

This is exactly the marginal condition in slot $l$ on p. 4.

Including the truncation error, the chosen constants ensure

contradicting $A_I>\delta$. Hence every large-value scale lies in the exceptional set, proving the large-value scale count. To respect the strict threshold range in Lemma 4.3, when $\delta\ge1/2$ we may use the count at the fixed threshold $1/4$. This changes only absolute constants.

8. From the large-value scale count to the small-value path sum

H source, p. 38

Partition $0<A_I\le\delta$ into the disjoint bands

The number of terms in band $m$ is at most the large-value scale count at threshold $2^{-m-1}\delta$. Their sum is therefore at most

Since $K\ge1$,

Zero terms contribute nothing. This proves the small-value path sum, including terms equal to the threshold.

Knowing only that $A_I\le\delta$ on every inactive cell is not enough. Even with $A_{I_m}=\delta/(m+1)$, every term is small but the path sum can still grow with depth. This section uses the same polynomial count at every smaller threshold.

9. Active cells and the root-to-leaf path count

H source, p. 38

Now fix a sufficiently large absolute constant $A$ and set

A cell $I$ is active if and only if there are actually available pure basic columns in three different slots whose transpose norm is greater than $\delta$. This definition applies to all retained patterns at once, regardless of whether a pattern's current coefficient is zero.

On a complete path, take the union of all basic labels, counting them by slot. Their total number is at most $D$. Thus there are at most $D^3$ triples from different slots. Each fixed triple is tested only after all three columns are available, and its number of active scales is at most

Taking the union over all triples yields

for a sufficiently large starting $k$. The last explicit exponent is not asserted to be optimal in the source. It is a convenient bound once the fixed parameters have been chosen; the source only needs $M\le D^C$.

If the tree has many disjoint branches, its total number of original labels may be large. The count is always taken on one path. Triples from other branches are not added to the current path.

10. Summing the error from deleting inactive cells by length

H source, p. 37 · H source, p. 38

Each retained pattern has at least three $B$ arguments. Fix three of their slots and expand their basic columns, leaving the fourth whole argument unchanged. The basic-column expansion and the definition of the transpose norm imply, on an inactive cell,

To avoid extra assumptions on the changing coefficients, we may use the following bounds term by term:

On a fixed path, first sum by triples. Every triple that occurs satisfies $A_{I;v_1,v_2,v_3}\le\delta$ on these inactive cells, so Section 8 yields

We spend an extra fixed power rather than incorrectly pulling the cell-dependent coefficients outside the whole path sum for a fixed triple.

This decomposition has three retained patterns in total. We may take $A=10$. After fixing $p,C$, increase the starting $k$ so that the total path error of all three patterns is at most $D^{-1}$. Finally, use the finite-sum Fubini identity

Summing all retained patterns, the error is at most

If the local sizes or basic coefficient sums were initially bounded by other fixed powers of $D$, simply increase $A$ after fixing these costs. The remaining parameter order is unchanged.

11. Why the two delayed differences can be estimated first

H source, p. 4 · H source, p. 7 · H source, p. 8 · H source, p. 37

For a high slot, write on the whole refined tree

Inheritance ensures that these are mutually orthogonal projection increments, and

For fixed $j$, first sum over disjoint cells at the same depth, then use the orthogonality of all $\Delta_h$. The total squared norm of the corresponding row vector is at most $\|1_RF_e\|_2^2$. Applying the triangle inequality over $j$ in the row Hilbert space yields

If $K_0=0$, the left side is zero. If the tree ends too soon, continue inheriting the existing columns and refitting locally, as prescribed on p. 7, to supply the required finite refinement. No new columns are needed.

The two low slots have no $U$, so the only pattern containing two $U$ arguments is

Full-argument pair-smallness and Cauchy–Schwarz yield

For the finitely many initial update layers where $\eta_0$ is unavailable, the same local absolute $L^2$ estimate still yields a bound independent of tree depth.

12. Why a minimum appears at the end

H source, p. 35 · H source, p. 36 · H source, p. 38

For the later update layers, the source input implies

Choose a smaller but still positive fixed exponent $\theta<\theta_0$ and increase the starting $k$, so that

Expand the full $Y$ into the three retained patterns and the two-difference pattern above. Apply the triangle inequality first, then use

Thus

On inactive cells, use $\min(\eta,x)\le x$ and the absolute error from Section 10. What remains is exactly equation $(11)$ of Proposition 5.8: estimate each sum

We do not claim that every $H_I(z)$ has the pair-smallness of the full $Y$. For the finitely many initial layers, it suffices to use the active path budget, the local absolute estimate, and the path-to-length identity. The constants may depend on those fixed layers, but not on the total depth.

13. What is preserved on restriction to a subroot

H source, p. 7 · H source, p. 38 · H source, p. 39 · H source, p. 43 · H source, p. 47 · H source, p. 48 · H source, p. 49

The active cells need not form an ancestor-closed tree. When a tree is needed, add the root and the cells connecting the active cells to it. The budget $M$ counts only active cells; there may still be many connecting cells. The original budget $n_0$ must cover all original births on each complete path, including columns born on connecting cells and first used at deeper active cells.

When restricting to a subroot $R'$ that already satisfies the required local normalization, keep the original functions, private labels, and lists. Labels already available above the subroot are inherited at the new root and are not charged again. Births below the subroot must not be used in the new root's projection in advance. Each new path extends to an original path, so the upper bounds for $n_0$ and $M$ remain valid. The orthogonal square-sum estimate in Section 11 can be proved again on $R'$.

When later removing filters from the two nonside slots, one must return to this tuple's original whole $B$ arguments. For these two slots $p,q$, the identity

shows that restoring the full $Y_p,Y_q$ allows the main term to use the source-input pair-smallness, while every other term contains an additional basic side argument. The whole $B$ cannot be replaced here by a selected subset of basic columns. That would not recover the original $Y$ through the same identity.

These are inheritance conventions for the spatial tree and basic data. The actual center witnesses in the historical ball masks on p. 43, and the projection words required for testing on pp. 47–49, are separately defined later. The active path budget itself neither lists those projection histories nor verifies their number.

14. Parameter order and the scope left open by this chapter

H source, p. 35 · H source, p. 36 · H source, p. 37 · H source, p. 38 · H source, p. 39 · H source, p. 49 · This chapter: appendix, Section 7

First fix the basic catalog, $K_0$, local sizes, and full-argument pair-smallness supplied by Proposition 5.6. Then choose $D$ to control these basic data. The exponent in the fixed-triple estimate depends only on the atom-complexity rules and fixed kernel class, not on real frequencies, tree depth, the number of branches, or the birth locations of the triple.

Next fix the active-threshold exponent $A$. Only then, for each required threshold $\delta$, choose the Weyl-splitting budget, Fourier truncation, and small/large frequency gap. The actual splitting and truncation may vary by cell, but their rational heights must lie in a uniformly fixed range. Finally increase the starting $k$ to absorb fixed polynomial factors and set $M$ and the original budget $n_0$ used in Chapter 9. New template insertions in Chapter 9 do not redefine activity in this chapter or add to its original vertices.

Under these explicit basic-data assumptions, this chapter proves the fixed-triple large-value scale count, small-value path sum, active path budget, and minimum reduction after removing inactive cells and the two-difference term. The appendix derives the three-variable Weyl estimate and finite chart expansion.

The parts not reconstructed here are the actual construction in Proposition 5.6 and of earlier original catalogs, the spatial compression in the full Lemma 3.4, the actual localization errors and history lists in Proposition 6.8, the whole helper bases in Section 7, and the global absorption argument for H. In particular, the numerical budget for $n_0$ still comes from the basic construction on p. 37. This chapter only aligns that source with the counting objects actually needed later. It does not establish closure of the whole paper H.