Technical appendix: from dyadic blocks of insertion stages to compression at arbitrary spatial depths

This appendix develops the parts of pp. 7–13 of H needed for the basic catalog construction. Chapter 9 only uses a row estimate for insertion stages. Here the rows may lie at different spatial depths, which requires a separate argument. Throughout the proof, all columns, birth cells, and inherited masks are fixed. We do not take a supremum over the adaptive choices themselves.

1. Column conditioning and the integral kernel

H, p. 7 · H, p. 8

Suppose each path has at most $D\ge2$ support-birth labels, and the columns satisfy

On a cell $I$, put $E_yc=\sum_vc_v\Phi_v(y)$. The private coordinates and Cauchy–Schwarz imply

The normalized Gram matrix

satisfies $G_I\ge D^{-2}\mathrm{Id}$. The projection therefore has the expression

It follows that

Without the lower bound from the private coordinates, the coefficient stopping argument below does not work within this budget.

2. A row estimate for one finite sequence of stages

H, p. 8 · H, p. 9

For each family, let $P_n^i$, $0\le n\le T$, $1\le i\le m$, be increasing orthogonal projections. Put

If the rows $B_\nu$ satisfy $\sum_\nu\|B_\nu h\|^2\le\|h\|^2$, then every choice of stages $n(\nu)$ satisfies

To prove this, put $\Delta_0^i=P_0^i$ and $\Delta_n^i=P_n^i-P_{n-1}^i$. The prefix $\{0,\ldots,n\}$ is a union of dyadic index blocks, with at most one block of each length. Fix the block length for each factor and group the rows using the same tuple of blocks. Row contraction bounds their squared sum by

Apply orthogonality within each family, successively in $A_m,\ldots,A_1$. The sum is at most $\|g\|^2$. There are at most $L_T^m$ tuples of lengths. The triangle inequality in the row space then proves $(\mathrm{A2})$. Projections from different families need not commute.

Telescoping one factor at a time,

Discard the contractions on the left, apply Cauchy–Schwarz in $i$, and then use $B_n=\Delta_n^i$ in $(\mathrm{A2})$ to obtain

We use a slightly weaker exponent to simplify the notation.

3. A bounded replacement with the same moments

H, p. 10 · H, p. 11

Now each row is local to an arbitrary cell $u(\nu)$ and satisfies $B_\nu=B_\nu1_{u(\nu)}$. There is no bound on the number of spatial depths. Add one terminal level to the tree without changing any original row.

Put $a_K=|K|^{-1}\int_K\|z\|$. Starting at the root, stop at the first descendant cells whose averages exceed twice the current root average. Iteration produces a root family $\mathcal S$. Roots with zero average contribute nothing. For $K\in\mathcal S$, form a partition $\pi_K$ from the next stopping roots and the remaining terminal leaves. Each cell $Q$ satisfies $a_Q\le4a_K$. The size segment consists of the cells in $K$ that are not contained in a proper stopping subroot. Assign each row cell to the deepest size-stopping root containing it. Rows on a stopping subroot belong to the new segment. The coefficient stopping below takes place within this segment, with terminal cells completing its boundary.

On $Q$, take all first-family labels used by ancestor cells of $Q$ within this segment, and project $z|_Q$ onto these columns. Their direct sum defines $g_K$, extended by zero outside $K$. Each path has at most $D$ labels, so $(\mathrm{A1})$ implies

The last inner product is real. For every cell $u$ in the segment, the moment against each first-family column is preserved on all partition cells inside it. Hence

This is not pointwise truncation. Ordinary truncation need not preserve the first-family moments.

The averages at least double along the stopping roots. If $M_d\|z\|$ is the dyadic maximal function within the root, a pointwise geometric sum and the $L^2$ maximal inequality imply

We also need a cumulative energy estimate stronger than $\|g_K\|_\infty\le Ga_K$.

Write $S=\sum_K\|g_K\|_2^2$. Fix $y$ and divide the roots containing it into

For the first class, the sum of $a_K$ is at most twice the threshold. The sum of the norms of these $g_K(y)$ is therefore at most $2D^{-2}M_d\|z\|(y)$. The second class contains at most

roots. By $(\mathrm{A4})$, apply Cauchy–Schwarz first pointwise and then after integration to obtain

Solving the quadratic inequality for $\sqrt S$ yields

Thus the bounded replacements cost only a logarithm, rather than a power of $D$.

4. Stopping on changes of the final-family coefficients

H, p. 11 · H, p. 12

Fix a size segment $K$ and write $g=g_K$ and $a=a_K$. Choose

Let $c(u)$ be the unique final-family coefficient vector of $P_u^m\cdots P_u^1g$, extended by zero to all labels.

In an attempt starting at $R\subseteq K$, the root is the zeroth selected cell. From a selected cell $A$, select as the next generation the first descendants satisfying

Hand the first cell reached after $T$ transitions to a new attempt. Let $\mathcal R(R)$ be the family of new roots.

Each generation of selected cells can be completed to a stopping partition of $R$: use terminal cells where no next generation exists, and keep their projections unchanged at later stages. Inheritance of the original masks makes the direct-sum projection families increasing. Thus the selected stages really satisfy the hypotheses of $(\mathrm{A2})$ and $(\mathrm{A3})$.

The path to each new root $R'$ has undergone $T$ coefficient changes. The private-coordinate lower bound shows that the vector norm of each change on this root exceeds $D^{-1}\tau a$. Therefore

Use $\|g\|_\infty\le Ga$ both to bound the right side and to bound the energy on the new roots. These two uses yield, respectively,

Both contractions follow from the same estimate $(\mathrm{A9})$. Length contraction alone does not imply energy packing.

From $b\ge8$ one can verify $L_T\le129(m+1)^2\log_2b\le b^4$. The second inequality follows from $m+1\le b/4$, $\log_2b\le b/2$, and $129b^3/32\le b^4$. Thus $(\mathrm{A8})$ ensures $E<1/2$. Summing over all generations of attempts,

Rounding up only changes fixed constants. Alternatively, if $b$ is not required to be an integer, one can first round $b$ up and keep the integer powers exactly as written.

5. The main term and coefficient error of each row

H, p. 12 · H, p. 13

Assign each row to the deepest attempt root $R$ containing its cell. Suppose the nearest selected ancestor of its row cell $u$ in this attempt is $A$, at stage $j<T$. The stopping rule implies

In the stage $j$ partition, the row cell lies entirely within the selected cell $A$. Hence the stage $j$ product on $u$ is exactly the restriction of the ancestor product. Estimate $(\mathrm{A2})$ bounds the main term, with squared sum at most

This stage is the index of the nearest selected ancestor, not the physical depth of the row.

Write the error as $\sum_ve_{\nu,v}\Phi_v$. If $e_{\nu,v}\ne0$, then $u(\nu)\subseteq R\cap J_v$. Put $\widetilde\Phi_v=1_{J_v}\Phi_v$. Locality implies

First apply Cauchy–Schwarz in $v$, and then use row contraction for each fixed label. The squared error sum is at most

The last step uses

Thus arbitrarily many labels on different branches do not increase the constant.

6. Summation and scope

H, p. 13 · Compression in the main text · Application to joint Bessel rows

Each row belongs to exactly one size segment and one attempt. In the Hilbert space of all rows, apply the triangle inequality to add $(\mathrm{A13})$ and $(\mathrm{A14})$. Then use $(\mathrm{A6})$, $(\mathrm{A7})$, and $(\mathrm{A11})$ to obtain

Since $L_T\le C(m+1)^3\log(2D)$, this proves $(11)$ of the main text. For an empty product, use the original row contraction directly. With complementary projections, expand each $1-P$; the resulting at most $2^m$ terms can still be absorbed into the same bound. For an average with respect to one finite measure fixed for all rows, suppose the projection products are measurable in the parameter and satisfy the stated budget uniformly. The integral triangle inequality in the row space adds only the total variation of the measure. This does not permit a separate measure or projection family for each row.

The argument proves depth-independent spatial compression for any fixed finite column system satisfying the birth-budget and conditioning assumptions. It does not prove the birth budget of an adaptive catalog. Sections 5, 8, and 12 of the main text establish the budgets for old columns, prediction columns, and additional high-slot columns, respectively. Neither result may be used as a premise for the other.