Unit 3, continued: smallness of the full quadruple and the three main patterns

This part develops D Proposition 5.6, pp.40–42, equations (5.29)–(5.31). The preceding part constructed the full quadruple $Y$ and proved small form norms for every pair and for the entire local form. We now split $Y$ into a current-projection part $B$ and a delayed increment $U$ , handle the term with two $U$ factors, and restrict the remaining three patterns to one family of active intervals.

We assume the precise statement of H Lemma 5.7: three fixed structured functions have only a controlled number of active scales along a path. Its proof using frequencies and rational heights is not developed here.

1. Roots, assigned original intervals, and the three types of functions

Fix a quadruple from the cyclic expansion with later precision level $t$ , and write $k=k_t$ . The preceding part produced a final root family $\mathcal R_{\mathrm{base}}$ , with each root $R$ assigned a family $\mathcal I_R$ of original summation intervals. These interval families partition the original summation set. Roots may be nested, but each original interval is assigned to only one root.

Fix one such $R$ . Positions $i,j$ are the low positions, and $a,b$ are complementary. For $I\in\mathcal I_R$ , always use the depth $s=s(I)$ measured from the original root $J$ . Define

Y_l=B_l+U_l, \quad B_i=X_i,\quad B_j=X_j,\quad U_i=U_j=0,

B_e=P^{\mathrm{new}}_{e,s}F_{e,s}-g_{e,\mathrm{prev},I}, \quad U_e=(P^{\mathrm{new}}_{e,s+K_0}-P^{\mathrm{new}}_{e,s})F_{e,s} \quad(e=a,b). \tag{BU}

This is an exact decomposition of the preceding definition of $Y$ . The functions retain their original depths of use.

Each $B_l$ is a signed sum of at most two current local projections applied to the same actual function. Distinct underlying projection families have distinct private directions, and their spatial sets of use are inherited from the original construction. Orthogonal projection is contractive, so the triangle inequality gives

\|B_{l,I}\|_{2,I},\quad \|U_{l,I}\|_{2,I},\quad \|Y_{l,I}\|_{2,I}\le C \tag{Local-sizes}

with a fixed constant $C$ .

2. Square energy of delayed increments over the root

Choose $D_{\mathrm{base}}$ to dominate the pathwise index counts, column sizes, reciprocals of private weights, structured complexities, and sums of absolute current coefficients for the finitely many underlying column families in this quadruple. A fixed enlargement gives

\mathrm e^k\le D_{\mathrm{base}}\le\exp(k^C).

Set

\Lambda_k=C(1+K_0)\log(2D_{\mathrm{base}})\le k^C.

Reapply the delayed-projection estimate of Corollary 4.5 to the complete descendant continuation of each projection family on the root, then restrict to the original intervals of use assigned to $R$ . This gives

\sum_{I\in\mathcal I_R}|I|\|U_{e,I}\|_{2,I}^2 \le\Lambda_k^2|R| \quad(e=a,b). \tag{U-energy}

The full row estimate is first proved on $R$ ; restricting the outputs can only decrease the square sum. This bound is not obtained by proportionally allocating an arbitrary parent-root energy bound.

3. The active-scale assumption for three fixed columns

H Lemma 5.7 (p.37) concerns three fixed bounded scalar structured columns in distinct positions, denoted by $\boldsymbol\phi=(\phi_{e_1},\phi_{e_2},\phi_{e_3})$ . For a dyadic interval $I$ in their common initial subtree, define

\operatorname{act}_I(\boldsymbol\phi) =\sup_{\|h\|_{2,I}\le1} \left|\mathcal H_I(\phi_{e_1},\phi_{e_2},\phi_{e_3},h)\right|, \tag{Activity}

Place each of the four functions in its original position.

The source asserts that, for $0<\delta\le1$ , along any fixed spatial path,

\#\{I:\operatorname{act}_I(\boldsymbol\phi)>\delta\} \le\operatorname{poly}\bigl(\log D_{\mathrm{base}}+\log(1/\delta)\bigr),

and

\sum_{\substack{I\text{ on this path}\\ \operatorname{act}_I(\boldsymbol\phi)\le\delta}} \operatorname{act}_I(\boldsymbol\phi) \le\delta\operatorname{poly}\bigl(\log D_{\mathrm{base}},\log(1/\delta)\bigr). \tag{Small-activity-sum}

These bounds are uniform in the path point and in the unrestricted real oscillation parameters.

The first bound counts large correlations; the second sums all small correlations. A counting bound at the single threshold $\delta$ is not enough. Group the positive small correlations into

2^{-n-1}\delta<\operatorname{act}_I\le2^{-n}\delta \quad(n\ge0)

and apply the first bound at threshold $2^{-n-1}\delta$ . Their sum is at most

\sum_{n\ge0}2^{-n}\delta\, \operatorname{poly}\bigl(\log D_{\mathrm{base}}+\log(1/\delta)+n+1\bigr) \le\delta\operatorname{poly}\bigl(\log D_{\mathrm{base}},\log(1/\delta)\bigr).

Zero correlations contribute nothing. Thus the second bound follows by summing the counting estimates at all smaller thresholds; it is not a logically independent additional assumption.

4. One active interval family for all three patterns

Choose a sufficiently large fixed constant $C_{\mathrm{act}}$ and set

\delta=D_{\mathrm{base}}^{-C_{\mathrm{act}}}.

Define

\mathcal A_R= \left\{I\in\mathcal I_R: \operatorname{act}_I(\boldsymbol\phi)>\delta \text{ for some triple available on }I\text{ }\right\}.

Along any path, there are at most $D_{\mathrm{base}}^3$ available column triples, multiplied by a fixed number of choices of positions. Apply the first bound in Section 3 to each fixed triple to obtain

\sum_{I\in\mathcal A_R}\mathbf1_I(y) \le M,\qquad M\le D_{\mathrm{base}}^C. \tag{Active-multiplicity}

Each triple uses the source statement on its own common initial subtree. Restriction to any subset of the original intervals can only decrease the left-hand side.

Integrating (Active-multiplicity) gives

\sum_{I\in\mathcal A_R}|I|\le M|R|.

This is the original active set used in the later localization. No additional projection-partition intervals are inserted.

5. Why only three of the four patterns remain

Since $U_i=U_j=0$ , the quadrilinear expansion of (BU) has only four terms. Define the three patterns with at most one increment:

z^\varnothing=(B_0,B_1,B_2,B_3),

z^a_a=U_a,\quad z^a_l=B_l\ (l\ne a), \qquad z^b_b=U_b,\quad z^b_l=B_l\ (l\ne b).

Write

\mathcal P=\{z^\varnothing,z^a,z^b\}.

Denote the fourth pattern by $z^{UU}$ . In positions $a,b$ it has $U_a,U_b$ ; in positions $i,j$ it has $B_i,B_j$ .

Then

\mathcal H_I(Y_I) =\sum_{z\in\mathcal P}\mathcal H_I(z_I) +\mathcal H_I(z_I^{UU}). \tag{Four-patterns}

6. The term with two $U$ factors is already summable over all scales

Suppose $t\ge12$ . The preceding part bounds each pair's form norm in the full quadruple $Y$ by

\eta=\exp(-k^\theta).

In the two low positions, $B_i=Y_i$ and $B_j=Y_j$ . This small form norm therefore applies to the double-increment term:

|\mathcal H_I(z_I^{UU})| \le\eta\|U_{a,I}\|_{2,I}\|U_{b,I}\|_{2,I}.

Apply weighted Cauchy–Schwarz over $I\in\mathcal I_R$ and then (U-energy) to obtain

\begin{aligned} \sum_{I\in\mathcal I_R}|I|\, |\mathcal H_I(z_I^{UU})| &\le\eta \left(\sum_I|I|\|U_{a,I}\|_{2,I}^2\right)^{1/2} \left(\sum_I|I|\|U_{b,I}\|_{2,I}^2\right)^{1/2}\\ &\le C\eta\Lambda_k^2|R|. \end{aligned} \tag{Two-U}

No new intervalwise smallness estimate is needed for either of the two $U$ factors.

7. Small total error from the three patterns on inactive intervals

Fix a pattern $z\in\mathcal P$ . It has at least three $B$ positions. For $z^a,z^b$ , choose its exactly three $B$ positions; for $z^\varnothing$ , choose any three positions.

On an inactive interval, expand these three $B$ factors into underlying scalar columns. The coefficients are constant on that interval, with total absolute mass bounded by a fixed power of $D_{\mathrm{base}}$ . Leave the fourth function unexpanded and use only its normalized $L^2$ bound from (Local-sizes).

For any fixed column triple $\boldsymbol\phi$ that occurs, inactivity gives

\operatorname{act}_I(\boldsymbol\phi)\le\delta.

Fix a path point $y$ . Summing over all inactive intervals on that path for this fixed triple, (Small-activity-sum) gives

\delta\operatorname{poly}\bigl(\log D_{\mathrm{base}},C_{\mathrm{act}}\bigr).

First bound the possibly interval-dependent coefficients by a uniform bound. Then sum over at most $D_{\mathrm{base}}^3$ column triples on this path and over the three patterns, yielding

\sum_{\substack{I\in\mathcal I_R\setminus\mathcal A_R\\y\in I}} \sum_{z\in\mathcal P}|\mathcal H_I(z_I)| \le D_{\mathrm{base}}^{C-C_{\mathrm{act}}} \operatorname{poly}\bigl(\log D_{\mathrm{base}},C_{\mathrm{act}}\bigr).

Integrate over $y\in R$ and interchange the finite sums. The left-hand side becomes a sum weighted by interval length. First choose the fixed $C_{\mathrm{act}}$ larger than all power costs, then choose the initial precision large enough that the remaining negative power absorbs the logarithmic polynomial. Since $D_{\mathrm{base}}\ge\mathrm e^k$ ,

\sum_{I\in\mathcal I_R\setminus\mathcal A_R}|I| \sum_{z\in\mathcal P}|\mathcal H_I(z_I)| \le\mathrm e^{-k}|R|. \tag{Inactive}

This calculation expands only three $B$ factors and leaves the fourth function in $L^2$ . Patterns containing $U$ are not fully expanded into large families of scalar columns.

8. Deriving a truncated pattern sum from smallness of the full $Y$

The preceding part also gives the intervalwise bound for the complete quadruple

|\mathcal H_I(Y_I)|\le\eta.

Equation (Four-patterns) gives a triangle inequality as well. Combining them,

\begin{aligned} |\mathcal H_I(Y_I)| &\le\min\left\{\eta,\, |\mathcal H_I(z_I^{UU})| +\sum_{z\in\mathcal P}|\mathcal H_I(z_I)|\right\}\\ &\le|\mathcal H_I(z_I^{UU})| +\min\left\{\eta,\sum_{z\in\mathcal P}|\mathcal H_I(z_I)|\right\}\\ &\le|\mathcal H_I(z_I^{UU})| +\sum_{z\in\mathcal P}\min\{\eta,|\mathcal H_I(z_I)|\}. \end{aligned} \tag{Truncation}

The last two steps use only the following inequalities for nonnegative numbers:

\min(\eta,A+B)\le B+\min(\eta,A), \qquad \min\left(\eta,\sum_j a_j\right)\le\sum_j\min(\eta,a_j).

When $\sum_j a_j\le\eta$ , the two sides of the second inequality are equal. When the sum exceeds $\eta$ , the right-hand side is at least $\eta$ , so the inequality still holds.

Define the remaining quantity to be estimated for each pattern by

M_R(z)=\sum_{I\in\mathcal A_R}|I| \min\{\eta,|\mathcal H_I(z_I)|\}.

Sum (Truncation) over the assigned original intervals. Apply (Two-U) to the double-increment term and (Inactive) to the inactive terms. This gives D equation (5.31):

\sum_{I\in\mathcal I_R}|I|\,|\mathcal H_I(Y_I)| \le C\eta\Lambda_k^2|R| +\mathrm e^{-k}|R| +\sum_{z\in\mathcal P}M_R(z). \tag{Active-reduction}

This deduction does not bound an individual pattern $\mathcal H_I(z_I)$ by $\eta$ , nor does it give arbitrary pairs in that pattern the small form norms of the full $Y$ . Individual patterns can be large. The subsequent analysis must control exactly $M_R(z)$ , with its minimum retained.

9. The finitely many initial precisions need no late-precision smallness

At the finitely many levels $t<12$ , the late-precision conclusion involving $\eta$ is unavailable. The local absolute estimate still bounds the total double-increment contribution by

C\Lambda_k^2|R|,

without the factor $\eta$ .

At these fixed precisions, $D_{\mathrm{base}},\Lambda_k,M$ are constants independent of total spatial depth. On active intervals, each pattern has a fixed local $L^2$ bound, and the local absolute estimate for four $L^2$ functions controls each form. Summing with

\sum_{I\in\mathcal A_R}|I|\le M|R|

gives the required bound. Section 7 still handles the inactive part. Thus the total cost of the finitely many initial precisions is independent of spatial depth.

10. The data that must be retained for localization

D §6 uses the complete functions $B,U$ , the same original active interval family, and the associated small pairwise bounds for the full $Y$ . The two low positions were expanded into scalar columns only for the preceding high-position replacement. Three underlying columns were expanded only for the inactive-term estimate here. No additional small correlation for a separated pattern can be carried into the next stage without justification.

On any new normalized subroot, restriction preserves the normalization of the sequence, the private coordinates of the underlying columns, and inheritance. Reprove the delayed-increment square-energy bound on that subroot using the full row estimate before repeating this argument.

The preceding replacement error for $X\to Y$ is a separate global estimate summed over all roots. By contrast, (Active-reduction) is proved anew on one fixed root. The global error cannot be rewritten as a proportional error on each individual root without proof.

Source coverage and the next step

D p.40, equation (5.29) and Proposition 5.6: the $B/U$ decomposition, total square energy, underlying column-function constants, and the active set.
D p.41: application of H's active-scale statement for three fixed columns, pathwise summation of inactive terms, and the double-increment term.
D pp.41–42, equations (5.30)–(5.31): deriving the three pattern estimates with minima from the full $Y$ , and handling the initial precisions.
H Lemma 5.7, p.37: active scales for three fixed columns and the sum of small correlations, still used as an assumption.

The construction in D §5 has now reduced each late-precision quadruple to three quantities $M_R(z)$ . The next stage is frequency localization in D §6. It must first establish ordinary joint Bessel bounds for local outputs across depths and frozen projections, before applying the varying-function row estimates proved in this unit.