Unit 3, continued: why frozen mode rows cost only the main-column bound

This part develops D §6.3 and Lemma 6.3, pp.47–49. The mode family may be much larger than the main column family. Before inserting functions from different frozen depths, the key is to assemble all mode rows into an ordinary Bessel operator. Composition with main projections then places the final adjoint range on the whole main endpoint interval.

We use the first-match expansion in H Lemma 8.1, the ordinary joint Bessel bound for the specified mode rows, separation of usable main endpoint spaces, and H Compression. Their respective roles are stated below. Deriving the mode-row Bessel bound from the frequency-graph construction still requires D §6.4 and the corresponding geometric arguments in H.

1. Three depths and two column-family sizes

Fix a late precision $k$ , a search branch, a position, and a color. As before, use

\log D_{\mathrm m}\le C_qk^C,\qquad G_0=\exp\bigl(C_q(1+\log(2k))^C\bigr)

to control the main column family and the permitted main-operator cost.

There is also a separate augmented mode space with constant $D_*$ , where

\log D_*\le C_q(1+k+P)^C.

This controls the number of modes, coefficients, relative rational frequency heights, and private weights. Later, $P$ may grow like $\exp(\alpha k^\theta)$ , so $\log D_*$ cannot be treated as having the same order as $\log D_{\mathrm m}$ .

In the choice made by H, the analysis-block length $K$ is a positive power of two satisfying

K\le C_q(1+k+P)^{C_5}.

Partition the depths into blocks $[b,b+K)$ and define

u_b=b-4K,\qquad v_b=b-2K,\qquad t_b=b+K.

The current depth of use satisfies $b\le s<t_b$ . Freeze the main output at $u_b$ , compare modes on common intervals at depth $v_b$ , and use the result at the later current depth.

The deterministic blocks of the original function sequence need not align with these analysis blocks. If $u_b$ precedes the current localization root, define the frozen output to be zero.

2. What remains fixed after the first match

D p.48 invokes H Lemma 8.1 (pp.65–66). For each main column $\Phi=(\phi_\Phi,\sqrt{\epsilon_\Phi}\,e_\Phi)$ , the source gives a finite, permanently labeled expansion on its first-match interval $I_0$ . Fix one candidate offset $\rho$ for each position and group, and use it for every first-match branch and analysis block. The corresponding modes are

\psi_m(y)= \left( \zeta^\rho(y)\mathrm e(\kappa_l p^\rho(y)) \mathrm e(\lambda_my^2+\gamma_my), \sqrt{\epsilon_m}\,e_m \right). \tag{Mode}

Here $\kappa=(-1,3,-3,1)$ and $l$ is the current position; $\zeta^\rho,p^\rho$ are the cutoff and phase functions defined in H. This part uses only the cutoff bound, the expansion accuracy below, and consistent mode labels.

The scalar coefficients $a_{\Phi,m}$ are fixed after the first match. All modes from one main column share the same $\lambda_m$ , and

|\lambda_m||I_0|^2,\quad \sum_m|a_{\Phi,m}|,\quad \#\{m\} \le D_1,\qquad \log D_1\le\operatorname{poly}(P,k).

Their relative linear frequencies satisfy the rational-height bound in the source statement.

Retain these labels, private directions, and coefficients on the relevant descendants; do not resample them at new freezing times. A main column with coefficient zero at one freezing time still retains its already available modes. Different first-match branches may use different private labels.

3. A bounded linear map independent of the analysis block

Besides approximating each main column, H Lemma 8.1 gives, for every fixed group $G$ , a pointwise linear map independent of the analysis block:

\mathcal R_G:\mathcal H_{\mathrm{main}}\to\mathcal H_{\mathrm{mode}}.

As a multiplication-type map on functions, it is uniformly bounded on $L^2$ and preserves spatial support.

Its explicit construction explains why the same operator works across blocks. Set

q_\Phi(y)= \sum_m a_{\Phi,m}(\psi_m)_{\mathrm{phys}}(y) -(\zeta^\rho(y))^2\phi_\Phi(y).

The first-match expansion gives $|q_\Phi|\le\delta_c$ on the relevant subtree. Set $q_\Phi$ and the mode coordinates from that first match equal to zero outside their subtree. The sums below include only labels already matched along the path through $y$ . Choose $\delta_c$ sufficiently small after fixing the main column constant.

For an arbitrary vector-valued function $F$ in the augmented main space, define the physical component of the map by

(\mathcal R_GF)_{\mathrm{phys}} =(\zeta^\rho)^2F_{\mathrm{phys}} +\sum_{\Phi\text{ matched}} \frac{F_\Phi}{\sqrt{\epsilon_\Phi}}q_\Phi.

For a mode belonging to main column $\Phi$ , define its private component by

(\mathcal R_GF)_m =\sqrt{\epsilon_m}\,a_{\Phi,m} \frac{F_\Phi}{\sqrt{\epsilon_\Phi}},

and set it to zero before the corresponding first match.

3.1 A uniform pointwise norm bound

The main pathwise index count and reciprocal private weights are controlled by $D_{\mathrm m}$ . Cauchy–Schwarz bounds the physical correction by

\left|\sum_\Phi\frac{F_\Phi}{\sqrt{\epsilon_\Phi}}q_\Phi\right| \le \left(\sum_\Phi|F_\Phi|^2\right)^{1/2} \left(\sum_\Phi\frac{|q_\Phi|^2}{\epsilon_\Phi}\right)^{1/2} \le \delta_cD_{\mathrm m}^{C}\|F\|.

Choose $\delta_c$ so that $\delta_cD_{\mathrm m}^C\le1$ to control this part.

Next choose positive private weights so that, for each main column,

\sum_{m\text{ belonging to }\Phi}\epsilon_m|a_{\Phi,m}|^2 \le\epsilon_\Phi.

For example, for a nonempty mode family choose one sufficiently small positive weight using the finite family size and common coefficient bound. Mode labels distinguish main columns, so the sum of squared private outputs is at most $\sum_\Phi|F_\Phi|^2$ . The reciprocal weights are still controlled by $D_*$ , since the logarithms of the mode count, coefficient bounds, and reciprocal main weights have the prescribed polynomial bounds.

Together with $|\zeta^\rho|\le1$ , this gives $\|\mathcal R_G(y)\|_{\mathrm{op}}\le C$ , uniformly in the group and candidate offset. Hence

\|\mathcal R_GF\|_{L^2}\le C\|F\|_{L^2}. \tag{Lift-bound}

3.2 Exact action on frozen main outputs

Consider a matched outer ball-projection output for this group. On a permitted endpoint interval $U$ , suppose

Z_G=\sum_\Phi\beta_\Phi\Phi

with constant coefficients $\beta_\Phi$ . Substitution into the definition shows that the physical correction adds precisely the terms $q_\Phi$ , while the private coordinates give exactly the mode coefficients. Thus

\mathcal R_GZ_G=\sum_{\Phi,m}\beta_\Phi a_{\Phi,m}\psi_m

is an exact identity. Its coefficients remain constant on the whole of $U$ .

In analysis block $b$ , the frozen mode output is

\widetilde Z_{G,b}=\mathcal R_G Z_{G,u_b}.

The coefficients $\beta_\Phi$ may vary between blocks. The map $\mathcal R_G$ , the mode list, and the expansion coefficients remain fixed. This permits the ordinary joint operator estimate to apply simultaneously at all freezing times.

4. Full hypotheses for the frozen mode rows

This is D Lemma 6.3, p.48.

For a fixed group $G$ , take finitely many mode projections $P_e$ . Each $P_e$ is the orthogonal projection from $L^2(J_e;\mathcal H_{\mathrm{mode}})$ onto the constant-coefficient span of its available restricted modes. On $R$ , it is implemented by restricting the input to $J_e$ and extending the output by zero.

Assume that, for every common mode-space function $H$ , all rows in this group satisfy

\sum_{e\text{ belonging to }G}\|P_eH\|_2^2 \le G_0\|H\|_2^2. \tag{Mode-Bessel}

Having each row individually be an orthogonal projection of norm at most one does not imply this joint bound. It is an explicit hypothesis that must first be established by the source mode construction.

Suppose

J_e\subseteq U_e,

where $U_e$ is a main endpoint interval with actual descendant use of group $G$ , permitted for every nonzero range in the outer main projection. The row's mode labels are already available on $U_e$ .

On $U_e$ , the basic function $B$ has the form

B_{U_e}=\sum_{\tau=1}^{h_0}\varepsilon_\tau Q^\tau_{U_e}F_{s(U_e)},\qquad h_0\le2.

Each nonzero component contains its specified underlying main projection $Q^\tau_{U_e}$ . The inner ordered product $\mathcal B_{U_e}$ is the group-independent combination of main projection averages and complementary operators prescribed by the search branch, of length $O(\log L)$ . Denote the outer positive projection average by $L_{G,U_e}$ .

The conclusion is

\sum_{G,e} \|P_e\mathcal R_G L_{G,U_e} \mathcal B_{U_e} B_{U_e}\|_2^2 \le G_0|R|. \tag{Frozen-bound}

The constant $G_0$ may be enlarged within its prescribed form. Endpoint depths may differ and need not be current depths of use for the group. Several rows may share one $U_e$ . The estimate is uniform in the candidate offsets fixed in advance.

5. An ordinary joint operator bound within one group

By (Mode-Bessel) and (Lift-bound) for the same fixed map,

\sum_{e\text{ belonging to }G}\|P_e\mathcal R_GF\|_2^2 \le C G_0\|F\|_2^2.

Each row depends only on $J_e$ , hence only on the containing decision interval $U_e$ .

First choose the same outer averaging index for every endpoint row in the group, so that the projection on each $U_e$ comes from one inherited family. Insert this individual outer main projection. The ordinary operator statement of H Compression applies with a cost polynomial in the logarithm of the main constant $D_{\mathrm m}$ . Restore the outer positive average by normalized averaging. For the fixed group this gives a row operator $A_G$ whose ordinary squared-norm bound still has the form $G_0$ .

The adjoint order for one row is

L_{G,U_e}\mathcal R_G^*P_e. \tag{First-adjoint}

Its range lies in the main group-projection range on the whole of $U_e$ , rather than only on the smaller mode-output interval $J_e$ .

6. The bound across groups uses main endpoint-space separation

For arbitrary groupwise row test vectors $w_G$ , (First-adjoint) shows that $A_G^*w_G$ is a finite combination of columns in permitted main endpoint ranges. The endpoint-space separation inequality from H, stated in the preceding part, therefore gives

\left\|\sum_GA_G^*w_G\right\|_2^2 \le C\sum_G\|A_G^*w_G\|_2^2 \le G_0\sum_G\|w_G\|^2.

By duality, all groups and mode rows together form an ordinary joint Bessel operator.

This step uses the complete main endpoint range on $U_e$ . Even if outputs are observed only on $J_e$ , the outer projection in the adjoint returns to the whole of $U_e$ , precisely as allowed by the space-separation statement.

7. Inserting the inner ordered product and underlying projection

Now that the rows across all groups have an ordinary joint bound, apply H Compression again on the decision interval $U_e$ to insert the common inner ordered product and one fixed underlying component $Q^\tau_{U_e}$ .

Expand complementary operators in the source order. Every projection choice in a shared inner average, and the underlying component $\tau$ , must be chosen compatibly across the row family for all groups. At each depth, use the corresponding projection from this same inherited family. Only after fixing these choices restore the positive averages and signed sums. Each fixed expanded term gives ordinary rows $T_e$ satisfying

\sum_{G,e}\|T_eH\|_2^2\le G_0\|H\|_2^2 \quad\text{for every }H\in L^2(R;\mathcal H_{\mathrm{main}}). \tag{Ordinary-frozen-rows}

We have not yet replaced $H$ by the row-dependent function $F_{s(U_e)}$ .

Each nonzero basic component has $Q^\tau_{U_e}$ closest to its input. Thus the last operator applied in $T_e^*$ is this underlying main projection, whose range consists of constant-coefficient combinations of main columns on the whole of $U_e$ . This projection remains even if every other factor is the identity.

The input support of $T_e$ is therefore $U_e$ , and its observation partition can be the complete partition at depth $s(U_e)$ . Arrange these depths in increasing rounds, with every row in a round using the same $F_{s(U_e)}$ . The complete partitions refine successively and the function indices are nondecreasing.

D Lemma 4.4 now turns (Ordinary-frozen-rows) into the varying-function estimate. Restoring the finitely many signed basic components, complementary-operator expansions, and normalized positive averages proves (Frozen-bound).

8. Why the final constant is $D_{\mathrm m}$ rather than $D_*$

The mode projections $P_e$ and map $\mathcal R_G$ have already been handled in constructing the ordinary joint Bessel operator. Their relevant hypotheses enter through (Mode-Bessel) and the uniform map norm.

For the varying-function row estimate, the relevant column representation is that of the final range of $T_e^*$ . This range is determined by the input-side underlying main projection $Q^\tau_{U_e}$ , so the column constant is $D_{\mathrm m}$ .

Ignoring the composition order and using $D_*$ merely because a mode projection appears in the middle would introduce extra losses that may contain a fixed power of $P$ . Conversely, if the underlying projection were removed, the final adjoint could no longer be asserted to have main-column range without a new check. This conclusion applies here only to the specified basic $B$ components.

The joint Bessel hypothesis for the mode rows must still be checked for the mode families actually selected later. Stating it as a hypothesis does not imply that arbitrarily large mode families satisfy the same bound $G_0$ .

9. The frozen depth determines use and observation

For a row from analysis block $[b,b+K)$ , its main endpoint interval has depth $u_b=b-4K$ . Thus

s(U_e)=u_b

is both the function index and the observation depth in the final varying-function row estimate. Neither the finer mode-output interval $J_e$ nor the later depth where the form is actually used changes this assignment.

The normalized sequence is already controlled on the whole depth- $u_b$ interval. No fine-interval norm needs to be enlarged back to an ancestor interval. Freezing $4K$ levels earlier therefore introduces no factor $2^{4K}$ here. The change in the actual output from depth $u_b$ to the current depth is controlled separately by the preceding depth-variation estimate.

Source coverage and the next step

D pp.47–48, equation (6.9): the different sizes of main and mode column families, the three analysis-block depths, and fixed modes after the first match.
H Lemma 8.1, pp.65–66: first-match expansions and a block-independent pointwise bounded linear map, with its construction and norm estimate developed here.
D pp.48–49, Lemma 6.3 and equations (6.10)–(6.11): ordinary joint bounds before varying endpoint functions are substituted, and the final adjoint's return to the main range.
H Lemma 8.8, pp.71–72: the corresponding fixed-function composition result, extended in D to varying endpoint functions.

The next part, D §6.4, must explain why the selected mode projections satisfy (Mode-Bessel), how these projections yield exact partial sums of mode coefficients, and how offsets are selected to bound the square energy of the non-core part. The current proposition does not replace those frequency-graph and counting hypotheses.