Unit 3, continued: representing depth-varying non-core functions by fixed vectors

This part develops D §6.5.1, pp.54–55, equation (6.19). Near differences require all matched indices to remain together on each original interval. To apply a fixed-function short-block estimate, the source places the required functions in a finite Hilbert direct sum, selecting one spatial-index coordinate at each use.

The target is the asymmetric pair of square-energy bounds

\sum_\nu\|\operatorname{stack}_{a,\nu}\|_2^2 \le P G_0|R|, \qquad \sum_\nu\|\operatorname{stack}_{b,\nu}\|_2^2 \le G_0|R|. \tag{Stack-energy}

The difference index is an additional vector coordinate only on the first side; it is not duplicated on the second. This distinction ultimately leaves the gain $P^{-1/2}$ .

1. Family indices versus vector coordinates

Fix a source projection-transfer step, a near offset $\ell$ , and a search branch. Write

\nu=(G,b,\chi),

where $G$ denotes the group, $b$ the analysis block, and $\chi$ the choices of components and dyadic lengths on both sides. Pair these auxiliary choices into one finite list, setting unused combinations to zero. After enlarging the permitted constant, the list has size at most $G_0$ .

On one side, define the fixed function

\operatorname{stack}_{b,\nu}(y) =(W_{\nu,i}(y))_{i\in\mathcal I_\nu} \in\bigoplus_{i\in\mathcal I_\nu}\mathcal H.

The particular dyadic interval index $i$ is a vector coordinate, not another part of $\nu$ . Hence

\|\operatorname{stack}_{b,\nu}\|_2^2 =\sum_i\|W_{\nu,i}\|_2^2.

The first side also carries the valid difference index $p$ , so its coordinates are $(i,p)$ .

When each complete coordinate is formed, restrict it to the union of intervals of use selecting it. Later removal of original summands does not alter this fixed function. Each use selects one dyadic interval index; the first side simultaneously retains every valid matched difference for the fixed $\ell$ .

The sum over $\nu$ in (Stack-energy) already includes every group and block. The auxiliary-list bound $G_0$ does not replace or recount the groups and blocks.

2. Nested projection products acting on one fixed vector

We develop the required H Lemma 3.3 (pp.8–9).

Let $\mathcal X$ be a Hilbert space. For each $j=1,\ldots,m$ , let

P_0^j\le P_1^j\le\cdots\le P_T^j, \qquad Z_n=P_n^m\cdots P_n^1,

be nested orthogonal projections, and set

L_T=1+\lceil\log_2(T+1)\rceil.

If finitely many row operators satisfy

\sum_\nu\|B_\nu x\|^2\le\|x\|^2,

then every choice $n(\nu)\in\{0,\ldots,T\}$ satisfies

\left(\sum_\nu\|B_\nu Z_{n(\nu)}x\|^2\right)^{1/2} \le L_T^m\|x\|. \tag{Prefixes}

Set $\Delta_0^j=P_0^j$ and $\Delta_n^j=P_n^j-P_{n-1}^j$ . Projection order here means inclusion of ranges. For each fixed $j$ , these differences are pairwise orthogonal. For an index block $A$ , define $\Delta_A^j=\sum_{n\in A}\Delta_n^j$ . Every prefix $\{0,\ldots,n\}$ is a union of dyadic index blocks, at most one of each length, with at most $L_T$ lengths.

Decompose each $P_{n(\nu)}^j$ accordingly. Fix $m$ lengths. For one such length tuple, group rows with the same tuple of index blocks. The joint row bound makes their square sum at most

\|\Delta^m_{A_m}\cdots\Delta^1_{A_1}x\|^2.

Sum over all index-block tuples, using orthogonality one layer at a time, to obtain at most $\|x\|^2$ . Projections from different layers need not commute.

There are at most $L_T^m$ length tuples. The triangle inequality in the Hilbert direct sum of rows proves (Prefixes).

2.1 Square sums of projection-product jumps

The exact expansion is

Z_n-Z_{n-1} =\sum_{j=1}^{m} P_n^m\cdots P_n^{j+1} \Delta_n^j P_{n-1}^{j-1}\cdots P_{n-1}^1.

For these $m$ terms, first apply $\|\sum_j u_j\|^2\le m\sum_j\|u_j\|^2$ , then discard the left-hand contractions in each norm bound. For fixed $j$ , the rows $B_n=\Delta_n^j$ satisfy the joint contraction bound. Apply (Prefixes) to the preceding $j-1$ layers and prefix $n-1$ . This gives

\sum_{n=1}^{T}\|(Z_n-Z_{n-1})x\|^2 \le m\sum_{j=1}^{m}L_T^{2(j-1)}\|x\|^2 \le m^2L_T^{2(m-1)}\|x\|^2. \tag{Product-jumps}

When $m=0$ , the product is the identity and every difference vanishes.

For ordered time intervals $[r,t]$ with disjoint interiors, list all endpoints in order, restrict each projection family to that nondecreasing list, and retain only the jumps $r\to t$ . The same type of estimate follows. On a time window of length $O(K)$ , its cost is

(C\log(K+2))^{O(m)}.

Each application requires one fixed vector $x$ throughout.

3. Dyadic prefix representations within a block

For a sequence $W_s$ in one block, write

W_s=W_b+\sum_{[r,t]\in\mathcal P(b,s)}(W_t-W_r),

where $\mathcal P(b,s)$ is the dyadic decomposition of the index prefix $[b,s]$ . At each fixed length, the intervals have disjoint interiors, and a use selects at most one interval of that length.

Take the base function $W_b$ and each difference as fixed vector coordinates. To represent the current $W_s$ exactly, select one coordinate for each of a few dyadic lengths and sum them. The length choices belong to $\chi$ ; the particular interval indices remain coordinates.

Three non-core components are needed: a basic $U$ output, the blockwise fixed non-core mode part $V_{G,b}$ , and the rolling part $Z_{G,s}-Z_{G,u_b}$ .

4. The basic $U$ component

Here the functions already formed at each depth can be used directly as coordinates. D equation (6.14) bounds their square energy over all actual uses. Subsequent auxiliary projection products are locally contractive on the corresponding current intervals. Restriction to the union of intervals selecting these coordinates only decreases norms.

Thus the energy of a basic $U$ family without an additional difference index is at most $G_0|R|$ . If the first side includes a difference $A_a^p$ , express it as two positive projection endpoints, estimate each, and sum over at most $P+1$ indices. The permitted cost is $P G_0|R|$ .

5. The function $V_{G,b}$ fixed within a block

Let $T_s$ be the subsequent auxiliary projection product, including a difference endpoint when needed. After fixing compatible individual projection choices, it consists of $m=O(\log L)$ nested projection families. Its operand

V_{G,b}

is one fixed function throughout the block.

For each fixed dyadic length, (Product-jumps) gives

\|T_bV_{G,b}\|_2^2+ \sum_{\substack{[r,t]\text{ required intervals}\\\text{of this length}}} \|(T_t-T_r)V_{G,b}\|_2^2 \le(C\log(K+2))^{O(m)}\|V_{G,b}\|_2^2.

Sum over dyadic lengths and use the established bound

\sum_{G,b}\|V_{G,b}\|_2^2\le G_0|R|,

to obtain the required energy. Restricting coordinates to their sets of use when they are formed only decreases the left-hand side.

The fixed-vector projection-product lemma applies because $V_{G,b}$ does not vary with $s$ inside the block. All actual depth-dependent functions cannot be substituted into one application of (Product-jumps).

6. The rolling part requires a second difference decomposition

Set

W_s=T_s(Z_{G,s}-Z_{G,u_b}).

The base value $W_b$ uses local-output variation over $[u_b,b]$ . These intervals have bounded overlap between blocks, so D Lemma 6.1 controls the energy of all base values.

For an outer dyadic interval $[r,t]$ , the exact identity is

W_t-W_r =T_t(Z_{G,t}-Z_{G,r}) +(T_t-T_r)(Z_{G,r}-Z_{G,u_b}). \tag{Rolling-jump}

6.1 The first term uses the proved variation estimate directly

For fixed outer length, the intervals $[r,t]$ have disjoint interiors within each block. Enlarge the union of required finer intervals of use to their containing whole depth- $t$ intervals. Each has actual descendant use.

The operator $T_t$ is locally contractive on depth- $t$ intervals, so the first term's restricted output norm is controlled by $Z_{G,t}-Z_{G,r}$ on whole endpoints. Apply D Lemma 6.1 over groups and intervals to get a square sum of the form $G_0|R|$ . This does not apply local projection contraction directly on arbitrary fine subsets.

6.2 For the second term, first fix the inner interval

For each block, fix one dyadic grid in a time window containing every $[u_b,r]$ and having length $O(K)$ . For example, use length $8K$ and left endpoint $u_b$ . Decompose every $[u_b,r]$ into prefixes of this grid. For one inner interval, define just one fixed input $X$ covering all its uses; the outputs $(T_t-T_r)X$ at different outer positions remain separate coordinates. Distinct intervals of a fixed length therefore have disjoint interiors. Fix the outer length, inner length, and one inner interval $[a,v]$ . Every outer position selecting it satisfies $v\le r$ .

Let $\Omega$ be the union of the whole depth- $r$ intervals actually needed by these outer positions. Define the fixed function

X=\mathbf1_\Omega(Z_{G,v}-Z_{G,a}). \tag{Fixed-X}

The depth $r$ may differ between outer positions. The set $\Omega$ is their entire union, fixed before proceeding.

By locality of $T_r,T_t$ , on every required depth- $r$ interval, using $X$ in place of $Z_{G,v}-Z_{G,a}$ leaves the required output unchanged. The whole interval lies in $\Omega$ , and both operators depend only on it and its descendants.

Apply (Product-jumps) to this one fixed $X$ and sum over all outer positions, at cost

(C\log(K+2))^{O(m)}\|X\|_2^2.

6.3 Summing the energies of the fixed functions

Enlarge $\Omega$ to a union of containing whole depth- $v$ intervals, which can only increase the norm. Each has some later actual use. D Lemma 6.1 therefore estimates $Z_{G,v}-Z_{G,a}$ on these complete endpoints.

At a fixed inner length, the windows $[u_b,r]$ have length $O(K)$ and successive block starts are spaced by $K$ . Their overlap is bounded, and same-length dyadic intervals within each window have disjoint interiors. Summing over groups, blocks, and inner positions thus introduces no fixed power of $K$ and gives only a square sum of the form $G_0|R|$ .

The two endpoints in (Fixed-X) may use $F_v,F_a$ . This does not obstruct the outer lemma: once $a,v,\Omega$ is fixed, their difference is a determined function, and every outer position acts on that same $X$ .

7. Why dyadic lengths and projection products cost only $G_0$

Restoring all dyadic length choices introduces only finitely many factors of $\log(K+2)$ . The nested projection-product cost is

(C\log(K+2))^{O(m)},\qquad m=O(\log L)=O(\log k).

The source parameters give $K\le\operatorname{poly}(P,k)$ and $P=\lceil\exp(\alpha k^\theta)\rceil$ . Therefore

\log\log(K+2)\le C_q+C\log(2k),

The logarithms of all these costs are bounded by a fixed power of $(1+\log(2k))$ and can be absorbed into

G_0=\exp\bigl(C_q(1+\log(2k))^C\bigr).

The paired list of auxiliary components and length choices has the same size bound.

Calculate first with compatible individual choices in every positive projection average, then restore the averages by Jensen's inequality. Shared inner choices remain common across groups and coupled across compared depths. The first side has at most $P+1$ extra difference indices, so it costs at most an additional factor $P$ compared with the side without differences. This proves (Stack-energy).

The strong energy bound contains no fixed power of $K$ . Earlier pathwise sizes and top counts could afford $P^C$ , but a crude $K^C$ cost here could erase the near-difference gain.

8. Why these fixed functions can be randomized

After this construction, each $\operatorname{stack}_{a,\nu}$ and $\operatorname{stack}_{b,\nu}$ is a fixed finite Hilbert-space-valued function. Its definition includes the original sets of use of all coordinates selected at that time. Later stopping removes summands, without restricting these fixed functions anew.

Specific dyadic interval indices and first-side difference indices remain vector coordinates. The next part uses independent Rademacher signs to select the current interval coordinates and the same difference sign to pair the first side with the frozen complementary difference at fixed $\ell$ . The resulting randomized scalar functions remain fixed across summation intervals.

Equation (Stack-energy) gives only $L^2$ energy. It does not automatically put each randomized scalar function in $L^q$ for $q>2$ . Invoking the local constant for fixed $L^q$ functions therefore requires the source's explicit short-block comparison, rather than passing directly between these function classes.

Source coverage

D pp.54–55, equation (6.19): fixed family indices, interval and difference indices as coordinates, and energy of the three non-core components.
H Lemma 3.3, pp.8–9: nested prefix products and fixed-vector square sums of jumps, with a full finite proof here.
D p.55: the two-level dyadic decomposition of rolling functions, fixing inner intervals before the outer fixed-vector estimate, and the $G_0$ cost.

The next part, D §6.5.2, states the short-block comparison precisely and combines matched randomization with whole-ancestor maximal estimates to turn (Stack-energy) into the near-difference bound with gain $P^{-1/2}$ .