Unit 3, continued: cyclic approximation in four positions and the earliest two increments

This part develops D Lemma 5.2, pp.29–30, equations (5.7)–(5.9). The preceding part constructed projection approximations at each precision, assuming the local detection result from H. We now prove a finite quadrilinear identity, estimate the correlation of each pair of functions at late precision levels, and verify that the terminal remainder tends to zero.

On each original interval $I$ , every precision level acts on the same actual function $F_{e,s(I)}$ . Increasing the precision does not change the depth index of the function being approximated. A comparison on one interval is not automatically summable over all scales. The limit over intervals below is taken only on the original finite tree.

1. The estimates available at each precision

Fix $q\in(2,3)$ . The precision levels $t=1,2,\ldots$ update the positions cyclically:

\iota(t)=(t-1)\bmod4\in\{0,1,2,3\}.

Choose an absolute constant $A>1$ and a sufficiently large $k_1$ , and set

k_{t+1}=k_t^A.

On an original summation interval $I$ , write $s=s(I)$ . Denote the approximation at level $t$ in position $\iota(t)$ by

g_{t,I}=P_I^tF_{\iota(t),s}.

Projections at different precisions need not be nested. The inheritance property proved in the preceding part applies to the spatial-depth projections within each fixed precision. For $t\le0$ , set the corresponding $g_{t,I}$ equal to zero.

The preceding construction gives

\|g_{t,I}\|_{2,I}\le1, \quad d_{\iota(t),I}\bigl(f_{\iota(t),s}-(g_{t,I})_{\mathrm{phys}}\bigr) \le\mathrm e^{-k_t},

and

\|(g_{t,I})_{\mathrm{phys}}\|_{\infty,I} \le M_t,\qquad M_t\le\exp(C_q k_t^{d_{\mathrm{det}}}). \tag{Fit-bounds}

If necessary, enlarge the absolute exponent $d_{\mathrm{det}}$ in the detection estimate once, so that it controls both the degree bound and the physical bound for the projections. This exponent is independent of $q$ , whereas $C_q$ may depend on $q$ .

Give the private coordinates from different precisions and restarts disjoint labels, and embed them in one Hilbert space consisting of the physical scalar coordinate and the direct sum of all private coordinates. Extend each old projection by zero on the other private coordinates. The difference $g_t-g_{t-4}$ and the augmented-space norms below then have a common ambient space. The form still uses only physical components.

From now on, fix $I$ and suppress the subscript $I$ . All forms use only physical components. The normalized actual functions satisfy $\|f_{e,s}\|_{\infty,I}\le1$ .

2. A form norm for two specified positions

For functions in distinct positions $i,j$ , denoted by $a_i,a_j$ , define

\|[a_i,a_j]\|_{\mathcal H,I} =\sup_{\|z_l\|_{2,I},\,\|z_m\|_{2,I}\le1} |\mathcal H_I(z_0,z_1,z_2,z_3)|,

Here positions $i,j$ are fixed at $a_i,a_j$ , and $\{l,m\}$ are the two remaining positions. When comparing two such bilinear tests, take the supremum of the difference of the corresponding forms.

The quantity $d_{j,I}$ from the preceding part permits any one opposing position to have an $L^q$ normalization. Thus, if a residual in position $j$ , denoted by $r$ , satisfies $d_{j,I}(r)\le\varepsilon$ , then every function $h\in L^q(I)$ in another position satisfies

\|[r,h]\|_{\mathcal H,I} \le\varepsilon\|h\|_{q,I}. \tag{Residual-test}

The zero-function case is immediate. Otherwise, divide $h$ by its normalized norm.

3. Replacing projections at two precisions by the actual functions

This is the comparison in D equation (5.9), p.30.

Suppose $v>w$ and $\iota(v)\ne\iota(w)$ , and denote their respective positions by $j,i$ . Bilinearity gives

[g_v,g_w]-[f_{j,s},f_{i,s}] =[g_v-f_{j,s},g_w]+[f_{j,s},g_w-f_{i,s}].

In the first term, use the later residual and place $g_w$ in the opposing $L^q$ position. In the second, use the earlier residual and place the bounded actual function in the $L^q$ position. Equation (Residual-test) gives

\bigl\|[g_v,g_w]-[f_{j,s},f_{i,s}]\bigr\|_{\mathcal H,I} \le \mathrm e^{-k_v}M_w+\mathrm e^{-k_w} \le C_q\bigl(\mathrm e^{-k_v}\exp(C_qk_w^{d_{\mathrm{det}}}) +\mathrm e^{-k_w}\bigr). \tag{Pair-replacement}

If one function is already an actual function, a single residual test suffices. For example,

\|[g_v-f_{j,s},f_{i,s}]\|_{\mathcal H,I} \le\mathrm e^{-k_v}.

The actual function has $L^q$ norm at most one, so it contributes no further complexity factor.

Choose $A>d_{\mathrm{det}}$ with a fixed margin. Since $v\ge w+1$ ,

k_v\ge k_w^A, \quad -k_v+C_qk_w^{d_{\mathrm{det}}}\le-\frac12k_v

for a sufficiently large initial $k_1$ , simultaneously for all subsequent $v>w$ . The right-hand side of (Pair-replacement) is therefore at most

C_q\exp(-c_q k_w)

with uniform constants. This comparison explains why the precision recursion requires power growth, rather than only $k_t\to\infty$ .

4. The exact expansion at a finite terminal level

Fix a terminal level $n$ . For position $e$ , let $g_{e,n}$ be its last approximation at or before $n$ , or zero if that position has not yet been updated. Define the terminal residual by

r_{e,n}=F_{e,s}-g_{e,n}.

The increments in a fixed position are $g_t-g_{t-4}$ , so the exact identity is

F_{e,s} =\sum_{\substack{1\le t\le n\\\iota(t)=e}}(g_t-g_{t-4}) +r_{e,n}. \tag{Finite-telescope}

Substitute these four identities into the quadrilinear form. In an expanded term containing at least two increments, let $u<t$ be the earliest two selected levels. Their positions must be distinct, since the linear expansion in each position contributes only one summand. Set

i=\iota(u),\quad j=\iota(t),\quad \{a,b\}=\{0,1,2,3\}\setminus\{i,j\}.

Define the quadruple

X_i^{u,t}=g_u-g_{u-4},\qquad X_j^{u,t}=g_t-g_{t-4},

X_a^{u,t}=F_{a,s}-g_{a,\mathrm{prev}},\qquad X_b^{u,t}=F_{b,s}-g_{b,\mathrm{prev}}, \tag{Two-earliest}

Here $g_{e,\mathrm{prev}}$ is the last approximation in position $e$ strictly before $t$ , or zero if there is none.

In the remaining positions $a,b$ , the permitted terms are increments later than $t$ or terminal residuals. Summing these possibilities gives exactly $F_{e,s}-g_{e,\mathrm{prev}}$ by (Finite-telescope). Each original expanded term therefore belongs to precisely one group determined by its earliest two levels, yielding

\mathcal H_I(F_{0,s},F_{1,s},F_{2,s},F_{3,s}) =\sum_{\substack{1\le u<t\le n\\\iota(u)\ne\iota(t)}} \mathcal H_I(X^{u,t})+\operatorname{Rem}_{n,I}. \tag{Expansion}

The term with no increments is the form evaluated on the four $r_{e,n}$ . When an increment occurs in exactly one position, summing over all levels in that position gives $g_{e,n}$ . The remainder is therefore exactly the following sum of five terms:

\operatorname{Rem}_{n,I} =\mathcal H_I(r_{0,n},r_{1,n},r_{2,n},r_{3,n}) +\sum_{e=0}^{3} \mathcal H_I(r_{0,n},\ldots,g_{e,n},\ldots,r_{3,n}).

The ellipses mean only that position $e$ is replaced by $g_{e,n}$ ; the other three positions retain their terminal residuals.

5. Why the terminal remainder tends to zero

At a terminal level $n\ge4$ , all four positions have been updated. Take terminal residuals in two distinct positions. Their last update levels $v,w$ are distinct elements of $\{n-3,n-2,n-1,n\}$ ; suppose $v>w$ . The later residual passes detection at precision $\mathrm e^{-k_v}$ , while the physical $L^q$ norm of the earlier residual is at most $1+M_w$ . Hence

\|[r_{\iota(v),n},r_{\iota(w),n}]\|_{\mathcal H,I} \le\mathrm e^{-k_v}(1+M_w)\longrightarrow0.

The limit follows from the same choice $A>d_{\mathrm{det}}$ as above.

Each terminal residual has normalized $L^2$ norm at most two, and each projection approximation has normalized $L^2$ norm at most one. Each of the five remainder terms contains at least three residuals. We can therefore apply the preceding form-norm estimate to two residuals and control the other two functions by their $L^2$ norms. It follows that

\operatorname{Rem}_{n,I}\longrightarrow0

on every fixed original interval.

If $\mathcal U$ is a fixed finite family of original summation intervals, then

\sum_{I\in\mathcal U}|I|\,|\operatorname{Rem}_{n,I}| \longrightarrow0.

This exchanges a limit with a finite sum only. It does not establish direct summability over an arbitrary infinite interval family. The individual $(u,t)$ terms still require separate estimates.

6. Small form norms for every pair at a late precision

D Lemma 5.2 also asserts that one can choose an absolute constant

0<\theta_0<A^{-7}<1,

such that, for $t\ge12$ , any two functions in distinct positions of the quadruple $X^{u,t}$ satisfy

\|[X_l^{u,t},X_m^{u,t}]\|_{\mathcal H,I} \le\exp(-2k_t^{\theta_0}). \tag{Small-pairs}

6.1 Cancellation after replacement

Replace every projection approximation in a factor by the corresponding actual function, leaving any part that is already an actual function unchanged.

In position $j$ ,

X_j=g_t-g_{t-4}

becomes $F_{j,s}-F_{j,s}=0$ . Both updates exist because $t\ge12$ .

For positions $a,b$ , the last update strictly before $t$ lies among $t-1,t-2,t-3$ , so

X_e=F_{e,s}-g_{e,\mathrm{prev}}

also becomes zero. Only the early position $i$ can lack a preceding update when $u\le4$ ; its replacement need not cancel.

Any two positions include at least one member of $\{j,a,b\}$ . Thus, once the termwise replacement errors are controlled, the term with both factors fully replaced by actual functions is zero.

6.2 A very early position $i$ requires a separate estimate

First suppose $u<t-4$ , and consider a pair containing $X_i=g_u-g_{u-4}$ . Denote the other function by $V$ . Every existing projection level in it is at least $t-4$ , hence strictly later than $u$ and also than $u-4$ when that update exists.

Since the coefficients of $V$ sum to zero after replacement by the actual function, it can be rewritten exactly as

V=\sum_v\varepsilon_v(g_v-F_{\iota(v),s}),

with at most two terms, $|\varepsilon_v|=1$ , and all these levels $v$ belonging to the same position as $V$ . Therefore

[X_i,V] =\sum_{\substack{w\in\{u,u-4\}\\w\ge1}}\sum_v \epsilon_w\varepsilon_v[g_w,g_v-F_{\iota(v),s}].

Test each term directly with the later residual. This introduces no $\mathrm e^{-k_w}$ error from the earlier residual:

\|[g_w,g_v-F_{\iota(v),s}]\|_{\mathcal H,I} \le\mathrm e^{-k_v}M_w \le C_q\mathrm e^{-c_qk_v}.

All levels satisfy $v\ge t-4$ , so the total error is at most $C_q\mathrm e^{-c_qk_{t-4}}$ .

This avoids treating the fixed small error at a very early level $u$ as though it decayed with $t$ .

6.3 The remaining pair comparisons involve only the last seven levels

Still assuming $u<t-4$ , consider a pair that does not contain position $i$ . Every existing projection level in it is at least $t-4$ . Expand into finitely many pairs of projections or actual functions. Apply (Pair-replacement) to a projection–projection pair and a single residual test to a projection–actual-function pair. The terms containing only actual functions vanish by Section 6.1, leaving an error of at most $C_q\mathrm e^{-c_qk_{t-4}}$ .

If $u\ge t-4$ , then $\iota(u)\ne\iota(t)$ rules out $u=t-4$ , so $u\ge t-3$ . The preceding update level in the early position, $u-4$ , is at least $t-7$ , and the existing projection levels in the other positions are no earlier. The same pairwise replacement therefore gives

\|[X_l,X_m]\|_{\mathcal H,I} \le C_q\exp(-c_qk_{t-7})

for every pair of distinct positions. Each factor has at most two terms, so the number of error terms has a fixed absolute bound.

6.4 Expressing the level gap as a fixed power of the current precision

The recursion gives

k_{t-7}=k_t^{A^{-7}}.

Choose an absolute $0<\theta_0<A^{-7}$ . Then take the initial $k_1$ sufficiently large that

C_q\exp(-c_q k_t^{A^{-7}}) \le\exp(-2k_t^{\theta_0})

for all subsequent $t\ge12$ . This proves (Small-pairs).

The constant $\theta_0$ is independent of the interval, depth, and number of input pieces. It can also be chosen before $q$ , using the absolute degree bounds and $A$ ; the dependence on $q$ is absorbed into the initial precision.

7. Function bounds and a common local construction

Each of $X_i,X_j$ is a difference of two projections of norm at most one, and each of $X_a,X_b$ is the difference between an actual function and a projection. Thus

\|X_e^{u,t}\|_{2,I}\le2 \quad(e=0,1,2,3).

This bound holds in the augmented Hilbert space and therefore also controls the physical components used in the form.

A fixed quadruple involves spatially partitioned constructions at only boundedly many precisions. In a common refinement, a nonempty intersection of dyadic root intervals equals the deepest of those intervals. The required common roots thus come from finitely many existing root families, and their total length is bounded by a fixed constant times the original total root length. This does not reset the original spatial depth $s(I)$ .

Source coverage and the next step

D p.29: the cyclic update rule $\iota(t)$ , the precision recursion, and the requirement that every projection act on the fixed function $F_{e,s(I)}$ .
D p.30, equations (5.7)–(5.8): the finite expansion by the earliest two increments and the five-term terminal remainder.
D p.30, equation (5.9) and the second half of Lemma 5.2: comparison of two precisions, the separate treatment of a very early position, $t-7$ , and $\theta_0<A^{-7}$ .

The next step is D §§5.3–5.6: replacing the two residual positions by delayed projections while retaining the small pairwise correlations proved here. A finite family of predictor functions must first be determined by the two fixed structured functions from the projection-difference positions; only then are the projections in the complementary positions chosen. It is the projection depth that is delayed: the operand must remain the same $F_{e,s(I)}$ . This step does not follow automatically from the interval-by-interval comparisons proved here.