Unit 3, continued: constructing the two delayed projections

This part treats D §5.6, pp.37–40, equations (5.23)–(5.28). The preceding two parts fixed the predictor family and proved the delayed-replacement estimate. We now choose parameters in their order of dependence, construct the actual high-precision projections, and sum over the restarted local constructions.

Local detection from H, the fixed predictor families, and the previously stated operator results remain assumptions. The main checks are that the predictors are fixed before constructing projections in the complementary positions, that a coarse-interval request yields enough projection gain on fine intervals, that old projections remain available for old summands after stopping, and that the two high precisions can be chosen successively without circular dependence.

1. The fixed quadruple and the pairs of structured functions

Take a quadruple $X^{u,t}$ from the cyclic expansion in Unit 3, and set $k=k_t$. Positions $i,j$ contain the two projection differences and are called the low positions. The complementary positions $a,b$ contain

The terms low and high distinguish roles in this construction; they refer neither to spatial position numbers nor to function size.

First fix a common old part of the existing projection families, with root $R_{\mathrm{old}}$. The existing column functions, the number of indices along a path, the sum of the absolute current coefficients, and the structured complexity all have a bound

This notation means that the degree and constants are finite and fixed in advance. The exponent's degree does not grow with tree depth.

Only for this replacement in the high positions, expand the two projection differences in the low positions into their original scalar columns. Fix a pair of scalar columns, indexed by $p$, with common initial root $J_p$. For intersecting dyadic initial intervals, the common root is the deeper interval. If the pair is already inherited at the current root, record its starting point anew at $R_{\mathrm{old}}$. Keep the coefficients on each interval of use outside the form.

The number of low-position pairs that can occur along a path satisfies

Consequently,

No bound on the total number of indices across all branches is inferred from this pathwise bound.

2. Choosing predictors before the delay and private weight

For each $p$, use Corollary 5.4 to choose its fixed predictor family at a common precision $B\ge1$. The common complexity bound in the low positions provides one number $D_{\mathrm{pred}}$ controlling every family's size, coefficients, structured complexity, and quadratic-phase cutoff, with

Each family remains defined on its own $J_p$. In high position $e$, when the corresponding low-position pair first becomes jointly available, insert the conjugated predictor column

and inherit it thereafter. Its private direction is never reassigned. For pairs already inherited at the current root, their predictor columns are available from that root. The number of predictor columns along a path is at most $D_{\mathrm{old}}^C D_{\mathrm{pred}}$.

Only after fixing these families do we choose the integer delay $K_0$ and $\log(1/\epsilon)$ as sufficiently large fixed powers of $k$. Temporarily denote the numerical constant for the final high-position projection family by $D_h$. Set

For a complete high-position projection family, the preceding two parts give errors of the form

The second term already includes the finite sum over indices and coefficients of one fixed predictor family.

First choose $B$ to absorb fixed powers of the low-position coefficients and number of pairs. This determines $D_{\mathrm{pred}}$. Then choose $K_0,\log(1/\epsilon)$ to absorb fixed powers of $D_{\mathrm{pred}}$ and the same low-position costs, leaving an exponential margin stronger than the final $\mathrm e^{-k}$. For example, one can reserve a margin of $\mathrm e^{-3k}$.

This does not yet require the unknown $D_h$: all remaining dependence is polynomial in $\log(2D_h)$. Once we prove that $\log D_h$ is bounded by a fixed power of $k$, these costs are fixed polynomials in $k$ and can be absorbed into the reserved exponential margin by increasing the initial precision. We have not assumed that one parameter choice controls arbitrarily large final function families.

3. Coarse-interval tests and fine-partition projections

Fix a high position $e$ and one construction root $R_{\mathrm{fit}}$. Keep the original depths $s(I)$. Construct projections at increasing fine depths $h$. At this level, requests come from original coarse intervals at depth

Even when no original interval requests a column at the current level, refine the complete partition and insert predictor columns at their prescribed times.

On a candidate coarse interval $I$, use the same actual function $F_{e,s}$ and test whether

satisfies

The precision $\chi_e$ will be chosen in order below. The direct sum of fine projections on subintervals of $I$ remains a contraction on $L^2(I)$, so the residual has normalized $L^2(I)$ norm at most two.

After choosing $\chi_e$, use the uniform detection bound to fix a common lower bound $\rho'_e>0$ for all requests in this position and every subsequent restart; abbreviate it to $\rho'$. When the test fails, local detection from H supplies a bounded structured function $\phi$ whose normalized moment on the coarse interval is at least $\rho'$. Both $\log(1/\rho')$ and the logarithm of its complexity are controlled by a fixed polynomial in $k,\chi_e$.

4. The total fine-interval gain from one coarse request

This is D equation (5.24), p.38.

Insert the same new column $(\phi,e_v)$ throughout $I$, into the projections on every depth-$h$ subinterval $Q$, and inherit it on descendants. The physical function and private direction are the same, but projection coefficients may differ between fine intervals. Write

The new private direction does not occur in the old projection. The single-column calculation from the preceding part therefore gives a gain on $Q$ of at least $c|Q||m_Q|^2$.

Cauchy–Schwarz with weights $|Q|/|I|$ gives

Thus the total gain of all fine rows assigned to this coarse request is at least

This column counts as one index at each point of $I$. The pathwise count does not multiply by the number of fine intervals.

In a simultaneous insertion round at fine depth $h$, each coarse interval still failing its test may request one new column. It is assigned the increment rows on its fine subintervals in that round, with no row assigned to two requests. Both support and observation use the complete depth-$h$ partition, and each round uses $F_{e,h-K_0}$. Repeated rounds use the same function; as $h$ increases, the function index is nondecreasing.

Refinements and prescribed column insertions not assigned to requests remain in the global nested sequence. The assigned rows therefore satisfy the actual joint Bessel hypothesis of Lemma 5.1.

5. The request bound and the two projection families after stopping

The prescribed numerical bound $D_0$ includes every predictor column, including those inserted later at the current root, as well as the finite old data needed for a common column-function constant. Its logarithm is a fixed polynomial in $k$.

Additional fitting columns are inserted with private weight one. If there are at most $T$ additional successful requests along a coarse path, the completed constant satisfies

Lemma 5.1 chooses a bound with $\log T$ polynomial in $k,\chi_e$ and ensures that the original coarse request intervals where this bound is first exceeded have total length at most $|R_{\mathrm{fit}}|/4$. Only intervals from the original request tree can be cut out.

5.1 Continuing the old family for old summands

Suppose a coarse interval $I$ is cut out at depth $s$ and fine depth $h=s+K_0$. Its strict ancestors have smaller delayed projection depths, so the projection values used by their summands have already been fixed.

On the cut branch, keep the old family's collection of fine-interval projections at depth $h$ unchanged in subsequent rounds at that depth. At finer depths, only inherit these columns and form the corresponding local projections. Existing predictor columns and their private directions persist, so the residual telescoping identities for old ancestor intervals can still be continued to every required fine depth.

These old summands do not require the branch to pass any new residual tests afterward.

5.2 The new construction handles only the transferred original summands

Transfer the cut coarse interval and its original candidate descendants to a new construction. Retain the restrictions of the prescribed old data and predictor columns. Predictors already inherited at its root are immediately available; later pairs are still inserted when they first become jointly available, as originally prescribed.

The new construction starts with no additional fitting columns. Old additional fitting columns do not count in its new request history. Its new $D_0$ covers only its prescribed data, and its request bound counts only its own additional requests.

The normalized sequence still satisfies the hypotheses after restriction to the new root. Repeat the coarse-gain, fine-row ordering, and request-bound arguments. Stopping intervals again occupy at most one quarter of the new root's length. Recursion on the finite tree bounds the total length of construction roots by $4/3$ times the initial root length.

The old and new families apply to different original summands. This preserves old projection values while allowing depth-independent estimates to be proved again on each new root.

6. Choosing the two high precisions in order

This is the argument on D p.39.

Let

dominate the physical bounds for existing projections and the predictor-construction costs. First choose the high precision $\chi_1$ so that

For example, take

This is bounded by a fixed power of $k$. Complete all bounded-request constructions in the first high position.

The physical bound for the first projection must be fixed before choosing the second precision. On a coarse interval of use $I$, take a depth-$s+K_0$ subinterval $Q$. Then

The lower bound on private coefficients controls the $\ell^2$ norm of the projection coefficients. The pathwise index count and column-function sizes then bound the physical sum. Thus

times a fixed power of the completed first column-family constant gives a physical bound $S_1$. What matters is that

with no dependence on the second high precision.

Next choose $\chi_2\ge\chi_1$ such that

For example, take

which is again bounded by a fixed power of $k$. It also satisfies $\mathrm e^{-\chi_2}S_{\mathrm{old}}\le\mathrm e^{-2k}$. Both precisions and the logarithms of the two completed column-family constants are bounded by fixed powers of $k$, so there is a common numerical bound

Combining the finitely many old families and the two high families requires only a fixed enlargement of this bound.

These precisions are chosen for the current quadruple. They are not substituted into the global cyclic recursion $k_{t+1}=k_t^A$, so later complexity does not circularly alter the earlier precisions.

7. Assigning original intervals in overlapping local constructions

Take a common refinement of the construction trees in the two high positions and the finitely many old partition trees. The assigned projection family changes only at an existing construction root. The resulting root family $\mathcal S(R_{\mathrm{old}})$ therefore satisfies

These roots may be nested. Each original summation interval is still assigned to exactly one local part.

For an interval of use assigned to $S$, denoted by $I$, set

Each projection belongs to the family assigned to that original interval, with the family's complete continuation on descendants.

8. Reapplying the error estimate on intersections of pair roots and assigned roots

Fix a low-position pair $p$ and an assigned root $S$. If $J_p\cap S\ne\varnothing$, set

Dyadic nesting makes this intersection one of the two whole intervals. Retain only original coarse summation intervals assigned to $S$ on which the low-position pair is actually available.

The predictors and their scale coefficients keep their original choices on $J_p$. Each high position uses one projection family already fixed on $S$. Restriction to $K$ preserves its columns and private directions, with the same local reprojection. Every projection on an actual interval of use is consequently unchanged.

Before the first replacement in a high position, the opposing function is an original function minus an old current projection, hence has at most two components. After one delay, it is a current projection plus one energy component consisting of the entire delayed difference, minus an old projection. There are at most three components, with energy parameter $\Lambda$. These are exactly the permitted classes of the preceding two parts.

Apply the actual error identity for the finite predictor family before and after each replacement, then the delayed-replacement estimate to the retained predictor terms. Bound the low-position coefficients uniformly outside each interval's form. The error for this pair and assigned root is at most

Summing over intersections does not require a constant pointwise overlap bound for $\mathcal S$. Set

By (Pair-roots),

Together with (High-size), the exponential margin reserved in Section 2 absorbs all fixed polynomial costs. This gives D equation (5.26):

The estimate is proved anew for the actual local projections on each intersection root, then summed according to the assignment of original intervals. No parent-root error is allocated in proportion to subroot lengths without proof.

9. Small pairwise correlations survive replacement in the high positions

Write the new high approximation as $H_e=P^{\mathrm{new}}_{e,s+K_0}F_{e,s}$. Its residual passes detection at precision $\mathrm e^{-\chi_e}$ on the coarse interval of use and has the controlled physical bound.

For the two high approximations, replace the second, more accurate one first, then the first. The pairwise comparison from Unit 3 gives

The position subscripts $1,2$ here indicate only the construction order of the two high positions.

For a new high approximation paired with an old approximation, first test the new residual against the old physical bound $S_{\mathrm{old}}$, with error at most $\mathrm e^{-2k}$. Then handle the error between the old approximation and its actual function. If all relevant old levels are at least $t-7$, the earlier comparison bounds the total error by

If the low position $i$ is very early, do not replace its old residual. The other, late factor cancels after replacement by the actual function, so test its late residual directly against the existing projections in position $i$, as in the very-early-level case of the earliest-two-increments proof. The new high residual is controlled by $\mathrm e^{-\chi_e}S_{\mathrm{old}}$; the old late residual is controlled by the separation of precision levels.

For $t\ge12$, the functions $Y_j,Y_a,Y_b$ all cancel when their existing approximations are replaced by the same actual function. Every pair therefore contains such a late factor. Since $k_{t-7}=k^{A^{-7}}$, choose an absolute exponent $\theta_{\mathrm{av}}>0$ so that the form norm of every pair from $Y$ is at most

provided the initial precision is sufficiently large.

Orthogonal projection remains contractive on the fine partition of a coarse interval of use, so every function in $Y$ has local normalized $L^2$ norm bounded by a fixed constant $C_{\mathrm{size}}$. Choose

Increasing the initial precision again gives both a form-norm bound $\eta$ for each pair and

The latter follows by multiplying one pair's form norm by the $L^2$ norms of the other two functions. At this point, it applies only to the complete quadruple $Y$.

10. The resulting local data

For the current quadruple on the original normalized root $J_0$, denote the final family of local roots by $\mathcal R_{\mathrm{base}}$. For each root $R$, denote its assigned original summation intervals by $\mathcal I_R$. The families $\mathcal I_R$ partition the original summation set, and every $I\in\mathcal I_R$ lies in $R$. Summing (Replacement-sum) over the old roots gives

This is D equation (5.28). Only original summation intervals are assigned to the roots. The additional intervals in the projection partitions do not enter the original sum.

Source coverage and the next step

• D p.37, equation (5.23): common roots and pathwise counts for low-position pairs; fixing predictors before choosing $K_0,\epsilon$.
• D p.38, equation (5.24): total projection gain from distributing a coarse moment among fine rows; the distinct roles of old continuations and new constructions.
• D p.39, equations (5.25)–(5.26): successive high precisions, fresh estimates on intersections with assigned roots, and error summation.
• D p.40, equations (5.27)–(5.28): small correlations for the complete $Y$ and replacement over all roots.

The next part, Proposition 5.6, splits $Y$ into a current-projection part $B$ and a delayed increment $U$. The total square energy of two $U$ factors controls the double-increment term. The truncated sum must be deduced from the smallness of the complete $Y$; (Full-Y-small) cannot simply be assigned to an individual $B/U$ pattern.