Why a column catalog still terminates with changing projection histories

This chapter closely reads Lemma 6.7 in H §6.4, pages 47–49, and fully develops the estimate it needs from Lemma 3.3, pages 8–9. The fixed source is An $L^3$ bound for the trilinear Hilbert transform, October 5, 2026 version.

The same-tag test in the previous chapter shows that whenever a four-linear smallness test fails, a new template has a large moment against the residual. This chapter proves that only finitely many such insertions can occur, and that the logarithm of their total number is still bounded by a fixed power of the original parameter. The difficulty is that the states being tested also change: one insertion can change several projections and hence all subsequent tested states.

The proof of Lemma 6.7 in the source cites the binary-prefix argument in Lemma 3.4. Here we directly prove and use the row form of Lemma 3.3, with insertion times as indices within one fixed cell. We do not first need the full compression conclusion of Lemma 3.4 for arbitrary spatial depth.

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1. Which data are fixed, and which can still change

Fix a finite dyadic tree and its active cells. Every downward spatial path from the root meets at most $M$ active cells, with the upper bound chosen so that $M\ge1$. This is an input from Proposition 5.8, page 38; we do not prove it again here.

Further reading: the original source above remains Proposition 5.8 on p. 38 of H. The required active-cell path-count argument under supplied basic data is now developed in H10, Section 9. Section 1 separately explains the original catalog budget, which remains a source input. The actual later history lists and whole helper bases in Section 7 must still be supplied separately.

Companion proof of the original budget: the basic-column construction in Proposition 5.6 on pp. 35–36 of H and its full-path budget are now developed in H11, Section 16. The detection appendix names the external discrete quadratic inverse theorem. This supplies a companion proof for the original-budget input of this chapter; it does not construct the actual later history lists or whole helper bases in Section 7.

The original graph vertices, birth records, colors, ball masks, and cumulative masks are all fixed. Each new column remains attached to an original vertex. It creates no graph vertex and changes no color or mask. Updating the catalog only adds columns to the column spaces allowed by the existing tags.

Write $\mathcal L$ for the number of colors and set the total rank count to

Original columns have rank zero. Fix an integer $n_0\ge1$ bounding the number of columns from the entire original catalog on any complete root-to-leaf path. Its logarithmic budget is also listed as an input below. This is the original path-count part of the basic catalog budget in §5.5, page 37. The main catalog uses ranks $1,\ldots,\mathcal L+4$, and the final helper catalog uses rank $R$. Fix a basic budget $D\ge2$ and the loss $H_0$ from the previous chapter's same-tag test. The small parameter in the source is

where $C_\tau$ is a sufficiently large constant chosen in advance.

For each cell and each slot, list in advance the names of the required target masks and projection histories. A name specifies the target rank, target mask, slot, base vector, the mask and rank used by each history factor, whether that factor is a projection or its complement, and the particular pad choice. It does not specify which columns will later be inserted.

The target projections and history factors of the main catalog use only the main ranks $0,\ldots,\mathcal L+4$, excluding the helper rank $R$. This is precisely the arrangement on page 47: the cumulative filters use ranks $\mathcal L-m+1$, and the four final ball filters use ranks $\mathcal L+j$, $1\le j\le4$. The separate helper rank $\mathcal L+5$ is introduced only on page 48. A helper history may contain projections of its own rank, but both the main catalog and all whole helper base vectors are fixed before that stage starts.

The history length is at most

Assume that the total number of such names for each cell and each rank is at most $S\ge1$. The inputs satisfy

Here and below, we may enlarge the fixed exponents and start at a sufficiently large $k$.

This count uses only the original vertex contexts, ranks, colors, pads, and names of factors in bounded-length histories. It counts neither numerical graph distances nor the pure source columns that actually appear later. If each factor has at most $\exp(k^{C_1})$ names and the length is $O(\log(2\mathcal L))$, then the total number of words is at most

which still satisfies the stated budget. The main-catalog names in the source come from histories needed for subsequent localization and transfer. The number of helper base vectors is supplied separately by the source conditions in Section 12.

2. Private coordinates and ordinary orthogonal projections within a cell

Fix an active cell $I$. Use the normalized Hilbert space

The first coordinate is the physical function; the others are private coordinates of the columns. A column has the form

Distinct private labels use distinct orthogonal unit vectors $e_v$. Every new template satisfies $|\phi_v|\le1$, and a new column is assigned $\epsilon_v=1$, so

Repeated curvature tags may still have different private labels.

For a target rank $r$ and a fixed vertex mask $\mathcal A$, the target projection is the ordinary orthogonal projection onto the span of all currently available columns of rank at most $r$ whose original vertices lie in $\mathcal A$. The coefficients in their linear combinations are constant throughout $I$.

Insert columns successively within the same cell. Time $t$ means that the first $t$ insertions have been completed. Denote the target projection by $Q_t$. Since the mask and rank restriction are fixed and old columns are retained,

Thus

is an orthogonal projection, and these increments have mutually orthogonal ranges for distinct $t$. Other uncomplemented projections with fixed names likewise increase with insertion time. Different projection families need not commute.

The base vector $g\in\mathscr X_I$ for each tested history is fixed before its stage begins. In the main catalog it is the specified basic $B$ or $U$ vector. Write the previous state as

where each $T_t^a$ is an increasing orthogonal projection $P_t^a$ or its complement $1-P_t^a$. The order of factors is part of the name. Every factor is a contraction, but their product need not be a projection.

3. The precise increment produced by a failed test

Prescribe integer budgets $n_r$ in advance to bound the total number of columns of rank at most $r$ on a spatial path. We choose them in Section 9. The threshold is not redefined using the current actual count.

For a target mask at rank $r$, the tested physical residual is

For every available lower-rank pure source column in the mask and any two normalized $L^2(I)$ observer functions, it must satisfy the form-norm bound

The denominator is $D\sqrt{n_{r-1}}$, not $\sqrt{Dn_{r-1}}$.

If a test fails, the same-tag Lemma 6.6 produces a bounded template $\phi$ in the target slot whose normalized moment against $F_t$ is strictly greater than

Attach it to the triggering source's original vertex and insert it at the target rank $r$; its tag lies in the target mask. Choose a completely new private unit vector and set

This private coordinate is zero in the old base vector, the previous history state, and the old projection space. Hence

Set $u=(1-Q_{t-1})\Phi$. The new private coordinate ensures $u\ne0$, and $\|u\|_{2,I}\le\|\Phi\|_{2,I}\le\sqrt2$. The new target space contains the old space and $\Phi$, so

If this step adds only this one column, the first inequality is in fact an equality with the projection magnitude on its right. The lower bound suffices.

If $g=0$, the history state is zero and no test can fail. Subsequent counts therefore consider only names with $\|g\|_{2,I}>0$.

4. Why orthogonal energy for a fixed vector is not enough

For a fixed vector $h$, the orthogonal increments satisfy

But Section 3 applies them to $X_{t-1}$, rather than a fixed $h$. That state changes with the other projections and the catalog.

This distinction cannot be omitted. In $\mathbb C^T$, let $Q_t$ project onto the first $t$ coordinates and freely choose $X_{t-1}=e_t$. Every state then has norm $1$, but

Thus a norm bound for each state alone cannot yield a total-energy bound independent of $T$.

Here we also know that each $X_{t-1}$ is obtained from the same fixed $g$ by finitely many increasing projections and their complements. This is exactly the structure used by the binary-prefix estimate.

5. The binary-prefix row estimate

Temporarily work in an arbitrary Hilbert space $\mathscr X$. For each layer $a=1,\ldots,m$, let there be increasing orthogonal projections

Also let there be finitely many bounded linear row operators

Each row may independently choose a prefix time $t(\nu)\in\{0,\ldots,T\}$. Set

The required conclusion of Lemma 3.3 is

When $m=0$, the product is the identity and the conclusion is the row contraction condition itself.

For each layer, define

Within a layer, these operators are mutually orthogonal projections. For a dyadic block $A\subseteq\{0,\ldots,T\}$ of integer indices, set

For a fixed block length, the blocks are disjoint, so their block projections are still mutually orthogonal.

Every integer prefix $\{0,\ldots,t\}$ decomposes into dyadic blocks, using at most one block of each length and at most $L_T$ possible lengths in all. For example, the binary expansion of the prefix length specifies the blocks to place from left to right. Thus

Now expand all $m$ factors in each row and first fix a tuple of block lengths $(\ell_1,\ldots,\ell_m)$. If a row's prefix has no block of one of the specified lengths, take its corresponding term to be zero. Otherwise, that term is

Group the rows sharing the same actual block tuple $(A_1,\ldots,A_m)$. By the row contraction condition, their sum of squared norms is at most

Sum next over the block tuples that actually occur, and enlarge the sum to all tuples with the specified lengths. Orthogonality within each layer yields

More explicitly, first fix the initial $m-1$ blocks and sum over the final layer. That sum is at most the squared norm of the product of the initial $m-1$ layers. Repeat backwards through the layers. No operators from different layers are interchanged.

Finally apply the triangle inequality in the Hilbert direct sum $\bigoplus_\nu\mathscr Y_\nu$. There are at most $L_T^m$ block-length tuples, and the vector corresponding to each has norm at most $\|g\|$. This proves the prefix-row bound.

6. Complements and previous times in the row bound

If each layer may be $P_t^a$ or $1-P_t^a$, expand the complements into at most $2^m$ signed words of uncomplemented projections. Each word has length at most $m$. Apply Section 5 to each word and then the triangle inequality in the row space to obtain

This bound requires no commutativity between different layers.

Return to the fixed cell $I$ and fix one target mask and one history name. Take the row operators and prefix times to be

Since the target increments have mutually orthogonal ranges,

It follows that

Restricting the sum to the times actually charged to this name can only decrease the left side.

Insertion steps at all ranks are counted together. Insertions at other ranks can change both the history and the target projection. These changes are already included in the corresponding increasing projection families. If a step does not change the target space, its row increment is zero. We need not first assume that the catalogs at other ranks have stopped changing.

The choice of a failed test may depend on the input and earlier insertions. To apply the row estimate, first fix the finite trial sequence actually obtained and all its projection families, then use the deterministic inequality valid for every such family. No probabilistic independence is required between the projections, row operators, and base vector.

7. Why the number of triggering source columns is not an extra factor

For each insertion, choose just one failed test as its witness and charge the insertion to that test's target mask and history name. Different lower-rank pure source columns may trigger the same name; all use the same increment lower bound from Section 3.

If, within a common sequence of at most $T$ insertions, a name is charged $N$ times, combining the one-insertion increment with the total energy for a changing history yields

Cancel the nonzero squared base-vector norm to obtain

There are at most $S$ names at each rank, so the number of rank-$r$ insertions in one cell is at most

This calculation groups insertions only by target name, not by the identity of the triggering source. For a fixed target name, the increments caused by all triggering sources belong to the same sequence of orthogonal rows and have already been counted together. Multiplying again by the current number of source columns would introduce an unnecessary loss and invalidate the simple rank budget used below.

8. From one cell to a path

A spatial path has at most $M$ active cells. Insertions take place separately in each cell, and child cells inherit columns already inserted by their ancestors. The same private label inserted by an ancestor is counted only once along the path.

Set

Then the number of new rank-$r$ columns born along the path is at most

Only the number of active cells along a path is used here, not the total number of cells in the tree. Different branches may have different new columns; the conclusion does not combine them into a global column budget.

The fixed-data budget lets us choose a fixed integer $a\ge2$ such that, for sufficiently large $k$,

This uses the original-column path budget $n_0$ supplied in Section 1. If the original count is zero, we may still take $n_0=1$ without changing the conclusion.

We next choose the rank budgets and prove that the algorithm stops before reaching them. This step must cover all ranks simultaneously; histories that will later change cannot be frozen in advance as constant states.

9. Choose every budget first, then exclude the first violation

Choose the integers

Then

Together with $m_*\le a\log(2k)$, this implies

Consequently, for a sufficiently large starting value of $k$,

This quantifies the source's choice of $n_r/n_{r-1}$ as the exponential of a sufficiently large fixed power of the parameter. The logarithmic prefix loss is much smaller than this ratio.

To handle original columns born at different depths correctly, define the trial counts on each complete root-to-leaf path $\pi$ of the entire finite tree. Reserve the whole original budget $n_0$ on each path at the outset. Whenever a new column of rank $s$ is generated, count it once, at its birth cell, on every complete path passing through that cell. Let

and define the cumulative counts by

This bounds the number of actual columns generated so far on the complete path. Later visiting the birth cell of an original column does not increase the count, since the original budget has already been reserved. Only generating a new column changes it, adding exactly one to every affected cumulative rank count on each relevant path.

These reservations are used only to prove an upper bound on the number of insertions. The actual available columns, tested states, and orthogonal projections in any cell still contain only columns already born there or inherited from ancestors. Future births do not enter the projections early. This definition is a concrete implementation of the source's a priori path bounds and trial cap; we do not claim that the source defines this counting object verbatim.

First construct the main catalog downwards from the root. Within an active cell, as long as a required test fails, choose any failed witness and insert the corresponding new column. To prove finiteness, temporarily impose the following rule: stop immediately when any cumulative trial count on any complete path first exceeds $n_r$. Initially the count is $n_0$, so no budget is exceeded.

Suppose an insertion in cell $I$ causes the first budget violation, and choose a witnessing complete path $\pi$. Before this insertion, its rank-$R$ cumulative count is at most $n_R$; the last insertion increases it by only one. Thus

The local insertion sequence in every cell on this path has length at most this number. Insertions on other branches do not enter the path count. Hence the bound from Section 8 applies at every main rank, and the budget margin yields

There are no helper columns yet in the main stage, so $a_R(\pi)=0$, and the same bound also holds for $s=R$. Sum the numbers of new columns at each exact rank directly. For every $r\ge1$,

The last two steps use $Q\ge4$ and $n_r\ge4n_0$. This sum also covers the case where one lower-rank insertion increases several cumulative counts. The rank-zero count is always the reserved $n_0$. Thus no rank can have a first budget violation.

Every failed test adds a new private label. Since the temporary stopping condition cannot be reached, a cell cannot undergo infinitely many insertions while its complete-path count stays at most $n_R$. All its tests must pass after finitely many steps. There are finitely many active cells in the tree, so the downward main construction also terminates. In the entire completed main catalog, each $a_s(\pi)$ with $s<R$ on every complete path still satisfies the same bound above.

Only after the entire main catalog is complete do we fix all whole helper base vectors and start the helper stage. Continue using the complete-path counts, which already include all main columns. Neither original births nor reading the existing main columns cell by cell changes them. Only new helper columns subsequently increase $a_R(\pi)$. If a first budget violation occurs in the helper stage, the same trial-length bound holds. In each cell, use the main catalog at the start of the helper stage as initial column data and the fixed helper base vectors as inputs. Restart the row estimate of Section 6 at time zero of this stage to obtain

Combine this new bound with the bounds for $a_s(\pi)$ already proved in the main stage. The same geometric sum again yields $\widetilde N_{\le R}(\pi)\le n_R/2$, a contradiction. Lower-rank counts do not change during the helper stage. Thus the helper construction also terminates after finitely many steps. The helper base vectors fixed later have not been treated retrospectively as fixed before the main stage.

New columns are inherited only by descendants and are not added back to their ancestors' available columns. Descendant constructions therefore do not change the cellwise tests already completed at ancestors. All main-history factors and targets use only columns of rank less than $R$, so helper insertions change neither main projections nor main-history states. After proving that the temporary stopping condition is unreachable, remove it. The actual failed tests, insertions, and inheritance steps are then the same as in the original construction. We have not substituted a stronger projection algorithm that already has future columns.

The resulting total path budget satisfies

The exponent is fixed and independent of spatial depth, the number of branches in the entire tree, and the precision of later linearization.

10. Smoothing averages add no extra pad-count factor

The positive smoothed filter in the source is the average of $P$ projections with the same rank and different pads:

Its complement can likewise be written as

A length-$m$ averaged history is therefore an average of all histories with specific pad choices, with nonnegative weights summing to $1$.

The catalog construction already assigns a name to each specific pad choice. For every term, the form test satisfies the same bound in source equation 13. The four-linear form is linear in the target residual, so the triangle inequality passes the same bound to the averaged history, with no extra factor $P^m$.

The number of specific pads is already included in the name count $S$ in Section 1. After proving the test for every term, we must not charge again by treating the average as an unnormalized sum.

The same argument applies to an average of target complement filters: prove the test for each specific target mask first, then average with the normalized weights.

11. How private coordinates explain the exact threshold normalization

Write the output of an ordinary projection using columns of rank at most $r$ as

Each coefficient is constant within the cell. The private coordinates are orthogonal and $\epsilon_v\ge D^{-2}$, so

If $Y$ is the orthogonal projection of an input $z$, contractivity and Cauchy–Schwarz imply

The physical columns need not be orthogonal; they may even be identical. The private coordinates keep the coefficients controlled.

Now expand a lower-rank source projection. Its coefficient sum is at most $D\sqrt{n_{r-1}}\|z\|_{2,I}$. If the target residual already satisfies source equation 13 for every pure source, linear summation yields

The four arguments are placed in the slots required by the application, without changing this coefficient estimate. This is precisely the role of the denominator $D\sqrt{n_{r-1}}$. Its two factors cancel the corresponding factors in the coefficient sum.

12. Fixedness conditions for the later helper catalog

After completing the main catalog, first construct and fix the whole side states needed in Section 7, and only then start helper insertions. The source requires that the number of these additional base vectors in each active cell be at most

including names for slots, groups, and modes. These are not the individual terms of a frequency expansion.

Helper tests use only pure source columns from the main catalog. Their base vectors are fixed at the beginning of the helper stage; subsequent helper insertions change only the projection histories. If these base vectors have additional private coordinates, choose all new helper coordinates from a set disjoint from them. The reason that physical and vector moments agree in Section 3 therefore remains valid.

Include the number of these fixed base vectors in $S$. At rank $R=\mathcal L+5$, use the same threshold, prefix-row estimate, and budget margin. This yields the finite termination of the helper catalog proved in stages in Section 9. The main catalog is already fixed at this stage, so this situation requires less than allowing other main ranks to change simultaneously.

This conclusion depends on the stated number of side states and on fixing them before insertions. We prove catalog closure under these inputs; the construction of these whole states in Section 7 remains a separate proof task. If the base vectors were freely chosen anew after every helper insertion, the counterexample in Section 4 shows that the present energy argument would no longer apply.

13. Parameter order for the catalog budget and graph threshold

The graph edge threshold $H_{\mathrm{ed}}$ determines which tags are adjacent and can change the specific masks. The insertion count above, however, uses only an upper bound on the number of names, not numerical distances. The complexity of each new column comes from Lemma 6.6 and is fixed from the outset; it does not grow with insertion rank.

We may therefore make the choices in this order:

1. Fix the budgets for basic template complexity, colors, and history length, and then fix all $n_r$ as in Section 9;
2. Using these path counts, private weights, and coefficient budgets, choose $\tau_2$ sufficiently small for Lemma 6.5 and the later mismatch errors;
3. Finally choose $H_{\mathrm{ed}}$ by Lemma 6.1 so that correlations between nonadjacent tags are below the required $\tau_2$.

Only for the graph finally chosen do we fix a suitable color realization and the actual masks, and then carry out the catalog construction. The counting estimates are uniform in these particular values. The same random color array is not required to work simultaneously for every possible $H_{\mathrm{ed}}$.

If template complexity increased with every insertion, $H_0$ in the first step would depend on the as-yet-unknown number of insertions, and this parameter order would fail. The previous chapter's distinction between a fixed amplitude family and arbitrary external modulation is exactly the input needed to avoid that problem.

14. What this chapter closes, and its limits

Under fixed original graph data, finitely many named histories, fixed base vectors, and an active-cell path budget, the main catalog terminates after finitely many steps, as does the subsequent helper catalog based on fixed whole side states. All required pure-source residual tests hold. The logarithm of the path column count is bounded by a fixed power of $k$, and each column's nonlinear complexity does not increase with insertions.

The proof develops three steps: new private coordinates turn a large physical moment into an orthogonal increment; the binary-prefix row bound handles changing projection histories of the same fixed base vector; and one simultaneous choice of all rank budgets rules out the first path-budget violation. Changing states are not treated as fixed vectors, and the energy budget is not paid anew for each triggering source.

The retained source inputs include the active-cell path count from Proposition 5.8, which finitely many named histories are required by later localization and transfer, and the construction and number of whole helper base vectors in Section 7. We have not proved the full Lemma 3.4 compression for arbitrary spatial depth, nor upgraded this local catalog closure to the full main theorem of H.

The actual later histories are now listed individually in H12, Section 6, and the history appendix. H12 uses them to develop current-group localization and both far-band directions. Construction and counting of complete side bases, linear side energy, and global near-band estimates remain later inputs. This companion argument does not change the conditional catalog-termination conclusion of the present chapter.