Unit 3, continued: Local detection, projection gain, and stopping intervals

This part studies the first part of D §§5.1–5.2, pp.27–29, principally Lemma 5.1 and its first application. A large local correlation is detected by a new bounded function. Adding a new private coordinate then increases projection energy by a fixed amount. The varying-function row estimate bounds how often this can happen along a spatial path.

We assume the precise local detection statement of H Lemma 4.12. What follows is the deduction of projection gain, the length bound for stopping intervals, and the sum over finitely many restarts. The inverse theorem and detection proof in H are not proved here.

1. The local form and the correlation being tested

Fix the original root interval $J$ . The spatial depth of an original interval is always $s(I)=\log_2(|J|/|I|)$ . A later construction root $R\subseteq J$ only restricts the spatial domain of the original functions. Restarting does not reset spatial depths or function indices.

Fix a temporary exponent $q\in(2,3)$ . The entire local construction uses the same $q$ . D §7 must later check that it can have the same value as in H's fixed-function local estimate.

On an interval $I$ , write $\|f\|_{p,I}$ for the normalized $L^p$ norm. The local quadrilinear form in D Definition 3.1 is

\mathcal H_I(z_0,z_1,z_2,z_3) =|I|^{-2}\int_{\mathbb R^2}w_I(x,t) \prod_{j=0}^{3}z_j(x+jt)\,\mathrm{d}x\,\mathrm{d}t.

The kernel has fixed interior support, cancellation along four directions, and smooth derivative bounds, with all parameters fixed throughout. For a variable with added Hilbert coordinates, the form uses only its first, physical scalar component.

At position $j$ , for a physical residual $r$ , define

d_{j,I}(r) =\max_{l\ne j}\sup |\mathcal H_I(z_0,z_1,z_2,z_3)|,

where $z_j=r$ and $\|z_l\|_{q,I}\le1$ , and the normalized $L^2$ norms of the other two positions are at most one. The supremum ranges over all such measurable scalar functions.

This measures whether the residual has a detectable quadrilinear correlation with two $L^2$ functions and one $L^q$ function. Passing the accuracy- $\delta$ test means $d_{j,I}(r)\le\delta$ .

2. The quadratic structured functions used

D p.27 invokes H Definition 4.1, p.15. To specify the detected class, write $\mathrm e(v)=\exp(2\pi\mathrm i v)$ . Its functions have the form

\phi(y)=\mathrm e(\lambda y^2+\beta y) \sum_{n\in\mathbb Z^d} \chi(\theta y-n) \mathrm e\bigl((\theta y)^T C(\theta y-n)\bigr). \tag{Chart}

where $\theta\in\mathbb R^d$ , $\lambda,\beta\in\mathbb R$ , $C$ is a rational $d\times d$ matrix, and $\chi$ is smooth and compactly supported. Multiplication by a constant of modulus one is allowed. The case $d=0$ includes ordinary quadratic phases.

The height of a rational number is the larger of its absolute numerator and positive denominator in lowest terms. Matrix height is the maximum entry height. Complexity at most $H\ge\mathrm e^2$ means

d\le\log H, \quad\operatorname{height}(C)\le H, \quad\operatorname{supp}\chi\subseteq[-H,H]^d,

and, for every multi-index $\alpha$ ,

\|\partial^\alpha\chi\|_\infty \le H^{c_{\mathrm{at}}(|\alpha|+1)} (|\alpha|+1)^{c_{\mathrm{at}}(|\alpha|+1)}.

The smoothness constant $c_{\mathrm{at}}$ is fixed in advance for this function class. H denotes it by $c_0$ ; the notation is changed here to distinguish it from D's kernel-support constant. The real oscillation parameters $\lambda,\beta,\theta$ are not bounded by $H$ .

A bounded atom in the original text also satisfies $\|\phi\|_\infty\le1$ . A scalar of modulus other than one is either placed in an external coefficient or absorbed into $\chi$ subject to the complexity bounds. We call the normalized object a bounded quadratic structured function. H Lemma 4.2 permits conjugation, with complexity increasing to at most the exponential of a fixed power of the logarithm of the original complexity.

The projection-gain argument uses only $|\phi|\le1$ and the detected correlation lower bound. The quadratic structure itself enters later separation and frequency counting for two fixed functions.

3. The precise local detection assumption from H

The detection statement in H Lemma 4.12, p.27, uses fixed $q>2$ and $C_1\ge1$ and assumes

\|r\|_{2,I}\le C_1, \quad d_{j,I}(r)>\delta, \quad0<\delta<1/2,

The strict inequality $d_{j,I}(r)>\delta$ permits an actual test whose form has absolute value greater than $\delta$ ; attainment of the supremum is unnecessary. Detection supplies a bounded quadratic structured function $\phi$ and $0<\rho\le1$ such that

\left|\frac1{|I|}\int_Ir(y)\overline{\phi(y)}\,\mathrm{d}y\right| \ge\rho,

and

\log(1/\rho),\quad\log\operatorname{complexity}(\phi) \le C_q\bigl(1+\log C_1+\log(1/\delta)\bigr)^{d_{\mathrm{det}}}. \tag{Detection}

The exponent $d_{\mathrm{det}}$ is absolute and can be fixed before $q$ . The coefficient $C_q$ may depend on $q$ . If the original detection is written with a bilinear moment and no conjugation, closure of the class under conjugation converts it to the inner-product convention used here.

Here $k\ge1$ is an accuracy parameter, distinct from spatial depth $s$ , so $\mathrm e^{-k}<1/2$ . When $C_1=2$ and $\delta=\mathrm e^{-k}$ , choose in advance a single $\rho$ for all candidate intervals with

\log(1/\rho)\le C_q(1+k)^{d_{\mathrm{det}}}.

Every detection then uses this smaller common lower bound. Global parameters are not reselected for each interval or residual.

4. Why a new private direction increases projection energy

Source: D p.29, equation (5.6).

Fix a candidate interval $I$ and the current function $F=(f,0)$ . Let $P$ project onto the local constant-coefficient span of the existing columns. The physical residual is

r=f-(PF)_{\mathrm{phys}}.

If $\|f\|_{2,I}\le1$ , contractivity of the projection gives $\|PF\|_{2,I}\le1$ , hence $\|r\|_{2,I}\le2$ .

After detecting $\phi$ , choose a previously unused orthogonal private direction $e_v$ and add the column

\Phi=(\phi,e_v).

The new coordinate vanishes in $F$ and every old column. Therefore

\langle F-PF,\Phi\rangle_{L^2(I)} =\int_Ir\overline\phi, \quad \|\Phi\|_{L^2(I)}^2\le2|I|.

Set $v=\Phi-P\Phi$ . It is orthogonal to the old range, and

\langle F-PF,v\rangle=\langle F-PF,\Phi\rangle, \quad\|v\|_2\le\|\Phi\|_2.

The nonzero detected moment implies $v\ne0$ . If $P^+$ is the projection after adding the column, its range is the orthogonal sum of the old range and $\mathbb Cv$ . Hence

\begin{aligned} \|(P^+-P)F\|_{L^2(I)}^2 &=\frac{|\langle F-PF,v\rangle|^2}{\|v\|_2^2}\\ &\ge\frac{\rho^2|I|^2}{2|I|} =\frac12\rho^2|I|. \end{aligned} \tag{Gain}

The physical functions $\phi$ need not themselves be orthogonal. The private coordinate preserves the same nonzero pairing with the old residual, while orthogonal projection makes the energy calculation exact.

5. Insertion requests, a count limit, and a length bound

Source: D Lemma 5.1, p.28.

5.1 The full hypotheses

Take a root $R$ and normalized functions $(F_s)$ satisfying the preceding block-variation condition. Candidate intervals come from one fixed finite original request tree below $R$ . Process them in nondecreasing depth, allowing repeated requests on the same interval.

Fix $0<\rho\le1$ , $D_0\ge2$ , and an integer count limit $T\ge1$ . If the current request would raise the successful-request count on its spatial path above $T$ , stop before executing it and remove that interval and its candidate descendants from the current construction.

For each successful request $v$ on interval $I_v$ , specify a set $\mathcal R_v$ of row indices responsible for its gain. Require the sets $\mathcal R_v$ to be pairwise disjoint and satisfy

\sum_{\nu\in\mathcal R_v}\|\Delta_\nu F_{s(\nu)}\|_2^2 \ge c\rho^2|I_v|. \tag{Charged-gain}

Here $c>0$ is a fixed absolute constant. All rows must belong to one actually constructed finite nested sequence of full-partition projections, with joint Bessel constant one and every support, observation, and sampling-index condition of D Lemma 4.4. The completed fixed-column family must also have a constant satisfying

D_T\le C D_0(T+1)^C. \tag{Completed-size}

The quantity $D_0$ includes all columns prescribed for this construction in advance, even those added to the projections only later.

Let $\mathcal C_T$ be the first intervals where the count limit would be exceeded. The conclusion is

\sum_{I\in\mathcal C_T}|I| \le C T^{-1}\rho^{-2}\bigl(\log(2D_T)\bigr)^2|R|. \tag{Cut-length}

5.2 Fix the constructed rows before applying the operator estimate

After processing all requests, fix the resulting rows and columns. Their construction may depend on $(F_s)$ , but the joint Bessel inequality must hold for arbitrary functions. Orthogonality of nested projection increments supplies exactly that property.

Apply the varying-function estimate with Bessel constant one. Since gain rows are never reused,

c\rho^2\sum_v|I_v| \le\sum_{\nu\text{ used for a gain}}\|\Delta_\nu F_{s(\nu)}\|_2^2 \le C\bigl(\log(2D_T)\bigr)^2|R|. \tag{Total-gain}

This does not sum separate fixed-function Bessel bounds for different current inputs.

5.3 The successful-count function and first stopping intervals

Define

n(y)=\#\{v:y\in I_v\}.

If a previously processed request interval meets the current candidate, nondecreasing depth and dyadic nesting make it contain or equal the current interval. The previous success count is therefore constant on that candidate.

Every first over-limit interval lies in $\{n\ge T\}$ : just before the next request is refused, $T$ successes have already occurred. Removing all candidate descendants makes the first over-limit intervals pairwise disjoint. Thus

T\sum_{I\in\mathcal C_T}|I| \le\int_Rn(y)\,\mathrm{d}y =\sum_v|I_v|.

Combining this with (Total-gain) proves (Cut-length). The integral identity only sums finitely many indicators; no extra probabilistic assumption is used.

6. Choosing the count limit in advance for a one-quarter length bound

Source: the second half of Lemma 5.1, D p.28.

Set

a=1+\log(2D_0)+\log(1/\rho), \quad T=\left\lceil C_*\rho^{-4}(1+a)^8\right\rceil.

By (Completed-size),

\log(2D_T)\le C'(a+\log C_*).

The coefficient of the root length in (Cut-length) is therefore at most

C C_*^{-1}\rho^2(1+a)^{-8}(a+\log C_*)^2.

where $0<\rho\le1$ and $a\ge1$ . For example, the inequality

(a+\log C_*)^2 \le2a^2+2(\log C_*)^2

bounds this by a fixed constant times $C_*^{-1}(1+\log C_*)^2$ . Choose a sufficiently large fixed $C_*$ in advance to make it at most $1/4$ , while retaining

\log(2T)\le C(a+\log C_*).

The count limit is chosen before any request is processed. The eventual $D_T$ need only obey the prescribed polynomial bound. One cannot wait for an arbitrarily large final column family and then choose $T$ retrospectively.

7. First application: every retained interval passes its accuracy test

Source: the construction following Lemma 5.1, D p.29.

7.1 Complete the partition before inserting columns

Each construction starts with empty projections on its root $R$ . At a spatial depth, first partition the whole root to that depth and recompute the local projections of retained columns on every cell. Next add the columns prescribed at that depth, and finally perform simultaneous insertion rounds. Intervals without requests keep their projections.

The next depth again refines the entire partition, including branches not tested again in this construction. Refinements, column additions, and identity transitions all remain in one finite nested projection sequence. This full sequence is needed for the joint operator bound.

7.2 Row conditions after a failed accuracy test

An original candidate interval $I$ has depth $s=s(I)$ . Test

r=f_{j,s}-(PF_{j,s})_{\mathrm{phys}}.

If $d_{j,I}(r)>\mathrm e^{-k}$ , (Detection) supplies a new $\phi$ with the common lower bound $\rho$ in (Gain). Each request is assigned its projection-increment row on $I$ , so no responsible row index is repeated.

At one depth, use the full depth- $s$ partition for both supports and observations. Every insertion round uses $F_{j,s}$ . Repeated rounds at the same depth are allowed. At the next depth the partition refines the previous one and indices remain nondecreasing. Split each increment over whole partition cells; the orthogonality argument in Corollary 4.5 preserves Bessel constant one.

Each path receives at most $T$ bounded columns with private coefficient one, yielding a constant satisfying (Completed-size). Lemma 5.1 bounds the total length of first over-limit intervals by $|R|/4$ .

7.3 Restarting on the stopping intervals

Take each first over-limit interval as a new root. Transfer only the original summation intervals and descendants belonging to it in the original request tree. Restriction of the original function sequence retains the normalized block-variation condition. Begin a new empty-projection construction; detection, gain, and row estimates apply again, so its stopping intervals occupy at most one quarter of its length.

A new root cannot immediately stop at itself: that would make the stopped length equal the root length, contradicting the one-quarter bound. Every new root is therefore a strict descendant in the original finite request tree. Recursion terminates after finitely many steps.

The total root length at generation $l$ is at most $4^{-l}|R|$ . Hence all attempted roots satisfy

\sum_{R'\text{ a construction root}}|R'| \le\sum_{l\ge0}4^{-l}|R| =\frac43|R|. \tag{Root-sum}

Every original summation interval is finally assigned to exactly one construction that retains it. Projections needed by old ancestor summands continue through inheritance and local reprojection. New constructions may use different column families. Auxiliary partition cells do not thereby become new summands.

8. The resulting local approximation and size bounds

On a finally retained interval $I$ , let $g_I=P_IF_{j,s(I)}$ . Its physical residual has passed the test:

d_{j,I}\bigl(f_{j,s(I)}-(g_I)_{\mathrm{phys}}\bigr) \le\mathrm e^{-k}, \quad\|g_I\|_{2,I}\le1.

Within a retained part, the sets of used columns are inherited by descendants, forming a genuinely nested full-partition projection family.

For fixed $q$ , (Detection) bounds $\log(1/\rho)$ and the logarithm of each atom's complexity by a fixed power of $k$ . Section 6 then bounds $\log T$ by a fixed power of $k$ . Both the number of columns along a path and the structured-function complexities are at most $\exp(C_q(1+k)^C)$ .

With private coefficients equal to one, if $g_I=\sum_vc_v\Phi_v$ , the private component gives pointwise

\sum_v|c_v|^2\le\|g_I\|_{2,I}^2\le1.

Each physical structured function is bounded by one, so

\|(g_I)_{\mathrm{phys}}\|_{\infty,I} \le\sum_v|c_v| \le\sqrt{\#\{v\text{ available} \}}\left(\sum_v|c_v|^2\right)^{1/2} \le\exp(C_q(1+k)^C).

Every new root obtains these bounds from its own construction. No parent family's total energy is assigned to arbitrary subroots merely in proportion to length.

Scope and the next step

D p.27, equations (5.1)–(5.4): local size and variation of the original functions, fixed-column representations, and the varying-function row bound.
D p.28, Lemma 5.1, equation (5.5): the count limit, total gain, stopped length, and parameter choices.
D p.29, equation (5.6) and the subsequent construction: private-direction projection gain, complete partitions, finite-tree restarts, and local residual bounds.
H Definition 4.1 and Lemma 4.12, pp.15 and 27: the structured-function class and the assumed local detection statement.

The next part treats D Lemma 5.2, pp.29–30. Accuracy is increased successively at four positions, always using the same actual function on each fixed original interval. The quadrilinear form is expanded by its two earliest increments, with pairwise estimates for the remainders and late-accuracy correlations. This finite algebraic expansion and quantitative comparison determine which local terms remain to be studied.