Unit 3, continued: choosing offsets before bounding non-core square sums along paths

This part develops equations (6.13)–(6.15) in D §6.4, pp.50–51. Under the stated geometric assumptions from the source, the preceding part established square energy for the frozen non-core modes. We now combine it with the rolling variation of current outputs, choose an offset with small physical approximation error, and obtain pathwise square-sum bounds by stopping at the first threshold crossing.

The offset cutoffs from H have the averaged-error property stated below. The main localization data and actual sets of use must be fixed before choosing offsets. Later mode selection and stopping may depend on the selected offsets; no additional independence assumption is required.

1. The spaces containing the three components

For now, fix $\alpha>0$ and $0<\theta<1$ and use

\eta=\exp(-k^\theta),\qquad P=\lceil\eta^{-\alpha}\rceil =\lceil\exp(\alpha k^\theta)\rceil.

Thus $G_0=P^{o(1)}$ and $\log P=O(k)$ . The final common parameters will still be chosen in D §7.

Fix a position $l$ , group $G$ , and analysis block $[b,b+K)$ . The main output is $Z_{l,G,s}$ . Freeze it at $u_b=b-4K$ and apply the fixed linear map to obtain

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

Split its mode expansion according to the fixed classification into the exact core part $A_{l,G,b}$ and exact non-core part $V_{l,G,b}$ , so that

\widetilde Z_{l,G,b}=A_{l,G,b}+V_{l,G,b}.

The main and mode spaces share one physical scalar coordinate. Give their private coordinates disjoint labels and embed them in a common Hilbert direct sum. The vector sums and differences below are then well defined, and every norm retains the relevant private components.

At a basic $B$ position, define

S^{\mathrm{roll}}_{l,G,s}=Z_{l,G,s}-Z_{l,G,u_b}, \qquad S_{l,G,s}=S^{\mathrm{roll}}_{l,G,s}+V_{l,G,b}.

Also define the purely physical error

E_{l,G,b} =(Z_{l,G,u_b})_{\mathrm{phys}} -(\widetilde Z_{l,G,b})_{\mathrm{phys}}.

For $b\le s<b+K$ , this gives the physical identity in D equation (6.13):

(Z_{l,G,s})_{\mathrm{phys}} =(A_{l,G,b})_{\mathrm{phys}} +(S_{l,G,s})_{\mathrm{phys}}+E_{l,G,b}. \tag{Physical-split}

At a basic $U$ position, construct no core part: set $S_{l,G,s}=Z^U_{l,G,s}$ and take zero physical error. This position remains part of the non-core contribution throughout.

2. Data fixed before choosing offsets

Use the uniform parameter bounds and common analysis-block length from H. Before choosing offsets, fix:

The main column family, main projections, and search branch;
Every original actual main use $(l,G,I)$ and its frozen endpoint $u_{b(I)}$ ;
The coefficients of the frozen main output on the whole endpoint interval:

Z_{l,G,u_b}=\sum_\Phi\beta_\Phi\Phi.

We may therefore average errors over offsets for all these actual uses without redefining the summation set according to which modes are later retained.

The offset cutoffs in H satisfy

\mathbb E_\rho|1-(\zeta^\rho(y))^2|^2 \le C d_{\max}\delta_c,

where $d_{\max}\ge1$ is a uniform bound for the relevant dimensions. In dimension zero, use the source convention for constant cutoffs. The precision

\delta_c=\exp(-k^{C_c})

is chosen after the main column constant is fixed.

The construction of the fixed map gives the pointwise identity

E_{l,G,b} =(1-(\zeta^\rho)^2)(Z_{l,G,u_b})_{\mathrm{phys}} -\sum_\Phi\beta_\Phi q_\Phi, \quad |q_\Phi|\le\delta_c. \tag{Offset-error}

3. Coefficient bounds on whole frozen endpoints have no depth-lag loss

On an endpoint interval at depth $u_b$ , denoted by $U$ , the actual function $F_{l,u_b}$ has normalized norm at most one. The basic $B$ is a difference of at most two local projections. Inner current-depth projection averages and their complementary operators are contractive, as is the outer main projection. The frozen main output on $U$ therefore has a fixed normalized $L^2$ bound.

The private-coordinate lower bounds control the $\ell^2$ norm of its coefficients. The pathwise column count and column sizes then control their $\ell^1$ norm and the physical supremum. For a positive average, first extend coefficients by zero to a common index set, then average. The same conclusion holds:

\sum_\Phi|\beta_\Phi|,\quad \|(Z_{l,G,u_b})_{\mathrm{phys}}\|_{\infty,U} \le D_{\mathrm m}^C. \tag{Whole-endpoint-size}

The estimate is on the original whole depth- $u_b$ interval. No norm on a much finer interval of use is enlarged to an ancestor, so no factor $2^{4K}$ appears.

By (Offset-error), (Whole-endpoint-size), and the offset-averaging property,

\mathbb E_\rho|E_{l,G,b}(y)|^2 \le D_{\mathrm m}^C d_{\max}\delta_c.

The main bound also controls the pathwise number of actual main uses. Integrate and sum over all actual uses fixed before the selection to obtain

\mathbb E_\rho \sum_{l,G,I\text{ actually used}} |I|\|E_{l,G,b(I)}\|_{2,I}^2 \le D_{\mathrm m}^C d_{\max}\delta_c|R|. \tag{Average-error}

4. No new independence is needed after selecting offsets

Equation (Average-error) bounds the mean of a nonnegative random quantity. If its right-hand side is positive, Markov's inequality gives offsets for which the total error is at most twice that bound. If the mean is zero, the total error is zero almost everywhere. Fix such offsets and absorb the factor two into the common constant.

The preceding non-core mode energy bound

\sum_{G,b}\|V_{l,G,b}\|_2^2\le G_0|R| \tag{V-energy}

is uniform over all candidate offsets and therefore remains valid after this selection.

Later core classification, restriction, stopping, and removal of intervals may depend on the selected offsets. They only remove terms from the already fixed nonnegative error sum and cannot increase it. The required order is to fix the main uses before averaging; later choices need not be independent of the offsets.

After selecting offsets, the full augmented vector $S_{l,G,s}$ is fixed. D p.50 then invokes the auxiliary-column closure construction from H. Its test objects are these complete vectors and the originally specified ordered projection products; no new auxiliary basis is created for each mode, height, or size. The counts for this construction are already included in the main column constant. The closure result from H is not reproved here.

5. Integrated square sums of the three non-core components

This is D equation (6.14), p.51.

5.1 Rolling variation

For each depth of use $s\in[b,b+K)$ , the comparison interval is $[u_b,s]$ . A fixed depth edge can be covered only by blocks whose starting points lie in a range of length $5K$ . Block starts are spaced by $K$ , and each block has at most $K$ depths of use. The edge overlap of these comparison intervals is therefore $O(K)$ .

For each group, take the union of actually used depth- $s$ intervals as the right-endpoint set and apply the proved D Lemma 6.1. Intervals of use at one depth are disjoint, giving

\sum_{s,G}\sum_{I\text{ used at }s\text{ }} |I|\|S^{\mathrm{roll}}_{l,G,s}\|_{2,I}^2 \le CKG_0|R|. \tag{Rolling-square}

5.2 Reusing frozen non-core modes within a block

For fixed $G,b$ , the function $V_{l,G,b}$ is unchanged throughout the block. Intervals of use at each depth are disjoint, so the squared norms of its restrictions sum to at most its squared norm on the whole space. A block has at most $K$ depths of use. By (V-energy),

\sum_{s,G}\sum_{I\text{ used at }s\text{ }} |I|\|V_{l,G,b(s)}\|_{2,I}^2 \le KG_0|R|. \tag{Repeated-V}

This counts at most $K$ uses separately within each block. It does not assume that coefficients agree between blocks.

5.3 A basic $U$ position

At each fixed depth, apply the square-sum bound for outer group projections to the common function obtained by applying the shared inner ordered product to $U_s$ . Inner current-depth projections are local contractions, so they do not increase the total energy on the intervals of use at that depth.

The delayed-increment square-energy bound for the basic $U$ then gives

\sum_{s,G}\sum_{I\text{ used at }s\text{ }} |I|\|Z^U_{l,G,s}\|_{2,I}^2 \le G_0|R|. \tag{U-side}

The fixed polynomial in $k$ from the basic energy is absorbed into $G_0$ .

At a $B$ position, use

\|S^{\mathrm{roll}}+V\|^2 \le2\|S^{\mathrm{roll}}\|^2+2\|V\|^2.

Together with $K\le\operatorname{poly}(P,k)$ and $G_0=P^{o(1)}$ , this bounds the integrated square sum of the full $S$ in every position by $P^C|R|$ . The later stopping argument still requires the three component bounds separately.

6. From integrated square sums to retained pathwise bounds

We spell out the componentwise first-crossing construction cited in the source.

For one fixed component, give each original interval of use $I$ a nonnegative weight $a_I$ equal to the component's normalized squared size there, summed over the specified groups. We have

\sum_I|I|a_I\le B_0|R|.

At a path point $y$ , the total weight is

A(y)=\sum_{I\ni y}a_I,\qquad \int_RA(y)\,\mathrm{d}y\le B_0|R|.

Choose threshold $M B_0$ . Move down a path and cut out the first original interval where the accumulated weight exceeds this threshold, together with its original descendants. First-crossing intervals are disjoint. On each, the accumulated weight is constant and greater than the threshold. Hence

\sum_{I\text{ first crossing}}|I| \le\frac{|R|}{M}.

Transfer the first-crossing interval and its original descendant summands to a new local construction, retaining each strict ancestor in full. On every path, the prefix retained in the current construction has accumulated weight at most $M B_0$ . Even if the path later enters a cut interval, transferred terms do not count toward that prefix.

Perform this construction separately for the rolling part, repeated $V$ , and basic $U$ part. The search in D equation (6.5) has at most $\operatorname{poly}(k)$ leaves. The combined number of configurations of colors, positions, and these three components is also at most $\operatorname{poly}(k)$ . Enlarging the threshold by this number makes the total length of new stopping roots smaller than any prescribed fixed fraction, while every threshold remains a fixed power of $P$ .

On retained paths this gives D equation (6.15):

\sum_{s,G}\|S^{\mathrm{roll}}_{l,G,s}\|_{2,I_s(y)}^2\le P^C, \qquad \sum_{s,G}\|V_{l,G,b(s)}\|_{2,I_s(y)}^2\le P^C

at a $B$ position, and

\sum_{s,G}\|Z^U_{l,G,s}\|_{2,I_s(y)}^2\le P^C

at a $U$ position. Consequently,

\sum_{s,G}\|S_{l,G,s}\|_{2,I_s(y)}^2\le P^C. \tag{Path-squares}

Every sum includes only original actual uses. Each retained strict ancestor remains one complete original summand and is not split into new terms.

7. Local sizes of core and non-core parts

On a retained $B$ interval of use $I$ , the local size of the current output $Z_{l,G,s}$ is controlled by current projection contraction and the basic $B$ bound. Also,

Z_{l,G,u_b}=Z_{l,G,s}-S^{\mathrm{roll}}_{l,G,s}.

The fixed map $\mathcal R_{l,G}$ is pointwise bounded, so

\|\widetilde Z_{l,G,b}\|_{2,I} +\|V_{l,G,b}\|_{2,I} \le C\bigl( \|Z_{l,G,s}\|_{2,I} +\|S^{\mathrm{roll}}_{l,G,s}\|_{2,I} +\|V_{l,G,b}\|_{2,I}\bigr) \le P^C.

The exact core part $A=\widetilde Z-V$ satisfies the same type of local bound. The full $S$ is controlled by its components. Equation (Physical-split), or the frozen difference, gives the same type of crude local bound for the physical error.

This uses the current output and the controlled rolling difference to bound the frozen output. It avoids a crude enlargement from a much larger ancestor interval to the current interval.

8. Turning physical error into a small absolute error for the form

In the quadrilinear expansion, take a term containing $E$ and fix one physical-error position in it. On retained uses, the other three positions have local size at most $P^C$ in $L^2$ . The local absolute estimate and Cauchy–Schwarz give

\begin{aligned} \sum_{\text{actual uses}}|I|\,|\mathcal H_I(\cdots,E,\cdots)| &\le P^C \left(\sum|I|\|E\|_{2,I}^2\right)^{1/2} \left(\sum|I|\right)^{1/2}. \end{aligned}

The main bound controls the pathwise number of original main uses, so the final length sum is at most $D_{\mathrm m}^C|R|$ . Apply (Average-error) for the selected offsets to bound the right-hand side by

P^C D_{\mathrm m}^C d_{\max}^{1/2}\delta_c^{1/2}|R|.

Fix all polynomial degrees first. Then choose $C_c$ sufficiently large and increase the initial precision to bound this by $C\mathrm e^{-k}|R|$ . If a term contains several errors, use the small square sum for one of them and the preceding crude local bounds for the others. The expansion has finitely many terms.

Thus, within total error $O(\mathrm e^{-k}|R|)$ , subsequent arguments may retain only the core and non-core components, keeping every private coordinate in the norm estimates.

Source coverage and the next step

D p.50, equation (6.13): the physical decomposition, offset averaging, and fixing the complete non-core vector after selection.
D p.51, equations (6.14)–(6.15): componentwise integrated and pathwise square sums, and local core bounds.
H §8.7, pp.73–74: fixing main uses before averaging offsets; later offset-dependent restrictions only remove nonnegative error terms.

The next part, D pp.51–52, constructs core representatives inherited by descendants, proves their separation, and converts core sizes into energy and representative-count bounds at first threshold crossings. These conditions are needed before applying the source core-counting result in §6.6.