Historical ball masks and Bessel bounds for whole endpoint spaces

This chapter gives a close reading of Lemmas 6.4–6.5 in H, on pages 43–46 of the fixed October 5 version of An $L^3$ bound for the trilinear Hilbert transform. Starting with successfully chosen colour data, we prove the geometric properties of the masks and explain why the same endpoint and column label may be used repeatedly.

There are two explicit inputs: the successful marking data on pages 42–43 and the small-inner-product estimate on whole cells from Lemma 6.1. Existence of the colouring and curvature detection remain inputs to this chapter.

Back to the guide to H

1. Graph distances, birth labels, and fixed random arrays

Each occurrence of an original column defines a vertex. Two vertices with the same curvature tag may still have different private coordinate labels. For the current interval $I$, the vertex list $\mathcal V(I)$ only grows along a downward path. It is ordered by birth and has length at most $D$.

At depth $s$, denote the graph distance by $d_s(v,w)$; it may equal $+\infty$. The graph is first defined on all hypothetical curvature tags satisfying the basic complexity conditions. Equal tags have distance zero, and edges are defined by the rational splitting (2). We use the property

The distance depends only on the curvature tags and the absolute depth. It is independent of spatial location, the cell of birth, and current availability.

Fix one global realization of the main-round and high-round delay arrays. The delay assigned to a birth label in a given round is unchanged along inherited lists, and the high-round array is shared by all frozen contexts. Delays are not resampled for each centre, group, or depth. All required markings are assumed successful; Lemma 6.3 sends failed paths to exceptional roots separately.

2. What the colour data fix

Use the constants from the source:

Here $P$ is the scale parameter on page 39 of H. In a main round, the score at the current centre $v$ is

There is a unique winner $z$, with top score $g$ and a lead of at least $5$ over every competitor. The unrounded scores of the winner and its competitors therefore satisfy, respectively,

All delays lie in $[0,d_0]$.

If the winner has birth index $i$ and the current list has length $j\ge i$, then $j=i$ uses the equality stage and the prefix ending at $i$. Otherwise, take the unique smallest dyadic index block with $i$ in its lower half and $j$ in its upper half. Throughout this upper-half stage, fix the prefix through the end of the lower half as $W$. It contains $z$.

The colour records the main round and score $g$, the equality case or the length level of the stage, the parameters $l,n$ below, the high round and score $g_2$, and the change counter. A successful group $G$ also fixes $z,a,W$ and the exact stage block. It does not record the changing list length or the identity of the centre at every actual use.

3. Full historical profiles and frozen pseudometrics

Within a fixed main-round, winner, and prefix context, define, for every hypothetical tag $b$ and every absolute depth $s'$,

An empty maximum is $-\infty$. The definition includes depths before the tag is born. Each profile is nondecreasing in depth.

For $1\le l\le d_0$, define

In particular, set $C_l(-\infty)=T_l$. Then define

The finite clipped values make this expression well-defined in every case. It is a pseudometric, and

Taking the supremum over the entire absolute depth axis is essential: within a frozen context, these distances no longer vary with the current cell.

If, at some depth $s$, we have $d_s(a,b)\le h$, then for every $s'\ge s$ the triangle inequality and depth monotonicity give

whenever both profile values are finite. Finite graph distance also ensures that if one profile value is $-\infty$, then the other is $-\infty$; their clipped values are then equal at every level. In particular, if the competitor set $W\setminus\{z\}$ is empty, all profiles equal $-\infty$. To bound the frozen distance, the depths $s'<s$ must still be handled separately. Current graph adjacency does not by itself remove the historical part.

4. Counting two balls and choosing the high-round winner

Put

At every successful centre $v$, choose $l<d_0$ so that, among the currently present vertices, the low ball

and the high ball

have cardinalities with the same $\lfloor\log_{3/2}\rfloor$. Denote this integer by $n$. These balls increase with $l$, are nonempty, and have cardinality at most $D$. The stated choice of $d_0$ therefore ensures that some adjacent levels have this property.

The high round uses the distance $d^{(l+1)}/b_2$, again requiring a unique winner with a lead of $5$. For a fixed group, let the high-round winner be $a$ and its top score $g_2$. The centre itself is a competitor and the delays are nonnegative, so $g_2\ge0$. The winning delay is at most $d_0$. Hence, at every actual centre $v$ of the same group,

5. A mask retains historical centres

For an integer $1\le h\le H_{\max}$, a currently available vertex $b$ belongs to $M_{G,h}$ if and only if there is a historical centre $v_0$ on the same path, assigned the same colour and group at the current cell or an ancestor, such that

The mask is empty before the first centre assignment. Within one stage, the score condition improves with depth, the pseudometrics are fixed, and historical witnesses remain available. The masks therefore increase along the path.

After the stage ends, its qualifying sets may be inherited if nested operators are needed. But that stage has no further actual use on the path. A centre born on another branch cannot serve as a historical witness for this branch.

An endpoint is usable if its mask is nonempty and the group has an actual centre use at that cell or a descendant. This witness of actual use need not be at the endpoint itself. A usable endpoint is still within its own stage; a set retained only by inheritance after the stage ends has no future witness of actual use.

6. The graph ball about a centre lies in its mask

Let $v$ be the current centre, let $b$ be a currently present vertex, and suppose $d_s(b,v)\le h$. The main-round gap gives $f_v(s)\le(g-5)b_1<T_l$ at the current depth. Monotonicity extends this bound to every earlier depth.

At the current depth, $f_b(s)\le(g-5)b_1+h<T_l$, using $h\le H_{\max}\ll b_1$. Thus both profiles lie below the lower clipping threshold at every past depth. At future depths the graph distance is at most $h$, and the profile comparison, before and after clipping, has bound $h$. Consequently,

By the triangle inequality, the winner's score decreases by at most $h$. Taking the historical witness $v_0=v$ satisfies all three mask conditions.

7. The low distance between old and new centres

Let $v_0,v$ be a historical centre and the current centre of the same group. Section 4 gives

Form the old low ball using only the vertices present at the time of $v_0$. Since

this old low ball lies in the high ball about the current centre $v$. The current low ball also lies in that high ball. Each low ball has at least $(3/2)^n$ vertices, while the current high ball has fewer than $(3/2)^{n+1}$. Since $2>3/2$, the low balls must intersect. Hence

The old ball must be counted in the old vertex list. Vertices born later cannot be added to that earlier count.

If $b\in M_{G,h}$ has witness $v_0$, then

Our constants satisfy

For example, $A_h\le10d_0(d_0+1)b_2(10H_{\max}+2)$, whereas the right-hand side is $\frac12d_0^9b_2(H_{\max}+1)$.

The current centre has low-level clipped value $T_l$. Thus $C_l(f_b(s))\le T_l+A_h<U$. Because this is strictly below the upper clipping level $U$, we can remove the upper clipping and obtain the bound on the actual competitor score

A bound on the clipped value alone, without checking that it is below $U$, would not imply this conclusion.

8. Graph adjacency uses only one additional pad level

At an actual use, suppose $b\in M_{G,h}$, let $b'$ be a currently present vertex satisfying $d_s(b,b')\le1$, and assume $h<H_{\max}$. The winner's score at $b'$ decreases by at most $1$.

The bound on the actual competitor scores gives $f_b(s)<T_{l+1}-10$ and $f_{b'}(s)<T_{l+1}-9$. All past profile values are smaller, so their profiles after higher-level clipping agree there. At future depths the profiles differ by at most $1$. Therefore

The low-level distance needs a separate estimate. At future depths the triangle inequality loses at most $1$, giving a bound $10R_lh+1$. At past depths the current centre's clipped value is always $T_l$. The historical centre bound places the clipped values of $v_0$ in $[T_l,T_l+2R_l]$, while those of $b'$ lie in $[T_l,T_l+A_h+1]$. Their difference is therefore at most

The three conditions together give $b'\in M_{G,h+1}$.

Also, when a vertex first enters the mask, the first condition already ensures

The right-hand side is a fixed polynomial in $P,k$. Thus a quantitatively short path exists at entry, rather than only at some indefinitely later depth.

9. Overlapping endpoints need a future witness only for the finer group

Take distinct groups $G,G'$ of the same colour with overlapping usable endpoints $I,J$, where $J$ is the finer endpoint. Choose respective mask vertices $b,b'$. We claim

Suppose instead that the distance is at most $1$. Choose only a witness of actual use for $G'$ at or below $J$, denoted by $u$. The graph distance bound persists to $u$, and $b'$ remains in its own mask. We do not require $G$ to have further uses on the same branch leading to $u$.

First suppose the main winners satisfy $z\ne z'$. A common colour fixes the main round, the top score, and either the equality case or the length level of the stage. Along growing vertex lists, upper-half blocks at the same level are either identical or successive disjoint dyadic blocks. Hence the earlier winner $z$ belongs to the later frozen prefix $W'$. This also holds in the equality case. It is a competitor for the later group. The winning score at the earlier endpoint persists by monotonicity, and adjacency between $b,b'$ costs at most $1$. This gives

But the later group's bound on actual competitor scores requires

Since $b_1$ is much larger than $H_{\max}$, these inequalities contradict each other. Here $f'$ is defined using the later prefix.

10. A shared main winner forces different high-round winners apart

If $z=z'$, the fixed birth index and the common stage-length level determine the same exact stage block. The prefixes and all profile pseudometrics of the two groups are therefore identical. Distinct groups can then differ only in their high-round winners, $a\ne a'$.

The two historical centres $v_0,v_0'$ lie on the same ancestor path of $J$, so they can be ordered chronologically. At the later centre, the earlier high-round winner is still available. The high-round array, pseudometric, and top score $g_2$ are the same. The earlier winner has unrounded score greater than $(g_2-1)b_2$ at its own centre, but at most $(g_2-5)b_2$ as a competitor at the later centre. Cancelling its common delay and using the triangle inequality gives

This step would fail if high-round delays were resampled.

On the other hand, at the later use witness $u$, the actual competitor bounds and the assumed adjacency put both profiles of $b,b'$ below the lower threshold of the higher-level clipping. The same holds at all past depths. At future depths the difference is at most $1$. Thus

The two historical mask conditions now imply

contradicting the preceding bound. This proves endpoint separation, including when the future witnesses of actual use for the two endpoints lie on different branches.

11. Assumptions on whole-cell columns and private coordinates

Fix one slot and one colour. Each pure column $\Phi_v(y)$ consists of a physical function $\phi_v(y)$ and a private coordinate unique to label $v$, with coefficient $\epsilon_v^{1/2}>0$. Let

Assume the following small-inner-product estimate on whole cells: if $d_s(v,w)>1$, then at depth $s$ every whole cell $Q$ satisfies

This is the same-slot conclusion of Lemma 6.1. Its threshold is uniform in absolute depth, arbitrary linear modulations, and the original column complexity.

For each finite group $G$, consider the finite sum

Here $I$ is an entire usable endpoint cell whose mask contains $v$, and its coefficient is constant on all of $I$. Endpoint times may differ and labels may recur. We assume only that the numbers of groups and column labels on a spatial path, together with $W_0$, are bounded by $\exp(k^{O(1)})$.

12. Combine repeated uses before estimating

For each fixed $v,G$, first define

Then the private coordinate of $Z_G$ with label $v$ is exactly $\epsilon_v^{1/2}a_{v,G}(y)$. Orthogonality of the private coordinates gives

Here $\mathcal H$ is the Hilbert space consisting of the physical component and the private coordinates.

If the coefficients of the same label cancel between two endpoint terms, the left-hand side retains that cancellation. Taking absolute values separately at every occurrence would introduce an unnecessary repetition count, and could leave a positive quantity even when $Z_G=0$. The Bessel bound concerns the function after synthesis, with repeated coefficients combined.

13. The last separation depth makes the coefficient product measurable

Fix labels $v,w$ from different groups. By depth monotonicity, the depths satisfying $d_s(v,w)>1$ form an initial interval. Truncate it at the largest depth appearing in the present finite calculation, and denote the last separation depth by $s_*$.

If two supporting cells $I,J$ overlap, their intersection is the finer cell. Endpoint separation says that its depth is at most $s_*$. Consequently,

is constant on the grid at depth $s_*$. Any finer term is disjoint from every supporting term on the other side and contributes zero. If there is no separation depth below the root, there is no overlap at all.

Apply whole-cell detection to each cell at depth $s_*$ and multiply by the constant coefficient on that cell. This gives

This step uses the spatial independence of the distance determined by the labels and the absolute depth. The geometric argument used depth monotonicity of the graph. For this last synthesis estimate alone, one can instead choose the maximum finer depth among actual overlapping endpoints, so monotonicity is not asserted to be a minimal requirement for that final estimate.

14. Summation does not cost the number of endpoint uses

At a point of overlap, private labels from different groups cannot coincide: equal labels have curvature distance zero, contradicting endpoint separation. Thus the private part of an inner product between groups vanishes, leaving only the physical part. The preceding section gives

Suppose that at each point there are at most $M_G$ contributing groups, with at most $M_{\mathrm{lab}}$ contributing labels per group. Pointwise Cauchy–Schwarz and the combined-coefficient bound give

First fix the path counts and private-weight bounds, then choose

Expanding the squared norm and integrating now gives

No factor counts endpoint times, repeated occurrences of a label, or the depths of future witnesses. This constant is one explicit choice; the source only needs an absolute constant.

15. The adjoint preserves the whole endpoint cell

Let $P_{\mathcal U_I}$ be the orthogonal projection onto a finite column space on the entire endpoint cell $I$, and let $E\subset I$ be a union of finer cells of actual use. If the forward operator restricts its output after projecting,

then

The spatial restriction therefore acts on the input to the projection. The subsequent projection still outputs a constant-coefficient column combination on the whole of $I$, exactly as required here. One cannot freely treat $\mathbf1_E\mathcal U_I$ as a range space with the same Bessel bound.

The corresponding dual estimate also retains whole endpoint spaces. If $\mathcal Z_G$ denotes the finite synthesis space allowed for each group and the synthesis operator

has squared norm at most $3/2$, then its adjoint satisfies

This is the precise form in which D uses these whole endpoint spaces on page 44. It does not permit arbitrary changes to the spatial dependence of coefficients or arbitrary restrictions of the projection's output.

16. What has been proved

For successfully fixed marking data, the historical masks contain the centre balls, are closed under adjacency with an additional pad level, admit short paths at entry, and separate overlapping usable endpoints. Adding the whole-cell small-inner-product input and the bounds on path counts and private weights yields the Bessel bound for whole endpoint spaces.

The exceptional-set bound for successful marking, curvature detection in Lemma 6.1, finite closure of the column catalog, and the later two-sided estimates require separate arguments. The calculations in this chapter do not remove those conditions.