Unit 3, continued: common parameters and completion of the local estimate

This part develops D §7, pp.58–61. The core and side-function estimates still contain several parameters and a fixed-function constant $C_N(q)$ depending on the original number of depths. We explain the order of choices and the return to Theorem 3.2.

The completion is conditional on the stated localization, counting, short-block comparison, and fixed-function proof from H. In particular, H's proof must work at the same prescribed $q$ . A theorem asserting only the existence of some $q>2$ does not suffice.

1. The local statement to be proved

1.1 Full hypotheses on kernels and functions

Take a root interval $J$ in any dyadic grid, a finite rooted tree $\mathcal T$ with maximal original depth $d$ , and a summation family $\mathcal I\subseteq\mathcal T$ . Depth always means

s(I)=\log_2(|J|/|I|).

Each interval $I$ has length $r=|I|$ . Its kernel $w_I\in C_c^\infty(\mathbb R^2;\mathbb C)$ satisfies the following conditions for fixed parameters $c_0>0,C_0\ge1$ :

On its support, for $j=0,1,2,3$ , the point $x+jt\in I$ lies at distance from $\partial I$ at least $c_0r$ .
For every $v\in\mathbb R$ and every $j$ ,

\int_{\mathbb R}w_I(v-jt,t)\,\mathrm dt=0.

For all nonnegative integers $a,b$ , with $n=a+b$ ,

\|\partial_x^a\partial_t^bw_I\|_\infty \le r^{-n}\big(C_0(n+2)\big)^{C_0(n+2)}.

Define

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

In position $j$ , specify measurable functions $f_{j,s}:J\to\mathbb C$ at every depth, including depths with no summation intervals. Partition the depth set $\{0,\ldots,d\}$ in position $j$ into $W_j\ge1$ deterministic nonempty consecutive integer blocks. On each block $B=\{a,\ldots,b\}$ , assume almost everywhere that

\max_{s\in B}|f_{j,s}(y)| +\sum_{s=a}^{b-1}|f_{j,s+1}(y)-f_{j,s}(y)| \le W_j^{-1/2}. \tag{Block-condition}

The block partitions may differ between positions. Extend every function by zero outside $J$ .

D Theorem 3.2 seeks the bound

\sum_{I\in\mathcal I}|I|\, \left|\mathcal H_I (f_{0,s(I)},f_{1,s(I)},f_{2,s(I)},f_{3,s(I)}) \right| \le C_{\rm loc}(c_0,C_0)|J|. \tag{Local-target}

The constant is independent of the tree, depth range, four block partitions, and $W_j$ .

1.2 The target left by the precision expansion

The earlier finite precision expansion reduces the noninitial terms to a target on a basic segment $R_0$ :

\sum_{I\in\mathcal A_{R_0}} |I|\min\{\eta,|\mathcal H_I(z_I)|\}, \qquad \eta=e^{-k^\theta},\quad 0<\theta<1. \tag{Segment-target}

Here $k=k_t$ is the later precision parameter. On each original cell, $z_I$ is a complete quadruple. The minimum comes from the small average of the full $Y$ before splitting into the prescribed $B/U$ patterns. No new cellwise smallness is assumed for each pattern.

2. Assigning the same original intervals to successive local constructions

2.1 What is fixed in one construction

A local construction starts from $R\subseteq R_0$ . First fix main projections, color histories, ball masks, and actual uses. Choose offsets from these existing uses. Once the complete side functions are fixed, perform auxiliary closure and the size and counting stops that depend on them.

After its functions and operators are fixed, later stopping changes only the assignment of original summation intervals to constructions. It does not restrict the parent construction's functions, projections, or already defined whole vector functions anew.

Roots where the initial marking fails are first bounded by a separate measure estimate from H's marking construction, giving a sufficiently small fixed fraction. These roots do not come from adjusting thresholds for the nonnegative accumulated quantities below. Their original summands also pass to a new construction.

The remaining stopping quantities are pathwise accumulated nonnegative counts, squared sizes, or ancestor suprema. Their integral bounds permit fixed enlargements of thresholds. Combine their first-crossing roots with the initial marking-failure roots, discard contained roots, and retain the first layer. Its total length satisfies

\sum_{R'\text{ a first stopping root of }R\text{ }}|R'| \le\frac14|R|. \tag{Quarter}

The joint-vector indices $(G,b,\chi)$ were already summed together in the total energy. No additional factor for all groups and blocks is allowed here. Remaining auxiliary-choice costs in the strong estimate enter $G_0$ ; configuration counts polynomial in $P,k$ occur in the power estimates.

Ancestor maximal functions remain defined on complete retained ancestor intervals, including portions later below stopping roots. Excessive frozen complementary sizes first contribute to the power-error sum. A new construction starts only when the accumulated error first exceeds its threshold.

2.2 Which original terms remain at a stop

At a first stopping cell $R'$ , transfer $R'$ itself and its original descendant summands to a new construction. Its strict ancestors remain complete original terms in the old construction. Every original summation cell is eventually assigned to exactly one construction.

Depth remains measured from the original $J$ by $s(I)$ ; function indices and deterministic block partitions are not reset. An auxiliary output before the new root is zero by convention. Function indices outside the original range use the previously specified constant extension from the nearest endpoint.

2.3 Reproving energy bounds on each new root

On $R'$ , restricted original functions still satisfy (Block-condition), so

\|F_{j,s}\|_{L^2(R')}\le |R'|^{1/2},

and the basic varying-row test has length $|R'|$ on its right-hand side. Inherit basic columns with their private-coordinate constants and perform the actual local approximation on the new root. Keep the old continuation needed by the parent construction unchanged.

Reapply the proofs of rolling square energy, main-projection variation, mode-row bounds, and linear-side energy on the new root. They give the same constants times $|R'|$ . A parent-root estimate

\sum_\alpha\|u_\alpha\|_2^2\le C|R|

alone does not imply energy on every subroot smaller by the factor $|R'|/|R|$ . Arbitrary function families have no such property.

2.4 Geometric summation of construction-root lengths

Organize the recursion into generations. Generation zero contains only $R_0$ . By (Quarter), the total root length at generation $n+1$ is at most one quarter of that at generation $n$ . Hence

\sum_{R\text{ a construction root}}|R| \le |R_0|\sum_{n=0}^\infty4^{-n} =\frac43|R_0|. \tag{Root-packing}

The original tree is finite, and each new root lies strictly deeper in the part awaiting assignment. The actual recursion is finite. The infinite geometric series merely supplies a uniform depth-independent bound.

3. Combining the three position patterns

On a retained original cell, the localization identity is first a sum over fixed search leaves and current groups. The number of leaves is bounded by a fixed power of $k$ . Apply the minimum to the cell's complete contribution, then separate fixed search leaves by the permitted subadditivity. Within each leaf, initially retain the group sum on the original cell.

Expand each position into core and side parts. The four positions give three cases:

At least three core positions: use the preceding bound

C_q\eta^\varepsilon P^{C_4}|R|+O(e^{-k}|R|).

At least three side positions: use the three-side-function $p$ -power estimate with exactly the exponent $p=1-\varepsilon>2/3$ supplied by the core calculation.
Exactly two side positions: use the transferred main term, freezing errors, excessive complementary-size errors, and restoration of the full $Y$ . Keep the complete matched near-difference sum within each interval and apply the preceding strong estimate.

For nonnegative contributions $x$ requiring a power estimate, use

\min(\eta,x)\le\eta^{1-p}x^p=\eta^\varepsilon x^p.

Near differences and all physical and numerical errors are estimated directly at power one. Absorb the finite search choices and remaining smaller costs into constants. D equation (7.1) becomes

\begin{aligned} \sum_{I\in\mathcal A_R^{\rm ret}} |I|\min\{\eta,|\mathcal H_I(z_I)|\} \le{}& C_q\eta^\varepsilon P^{C_4}|R|\\ &+(1+C_N(q))C_qP^{-1/2}G_0 (K+2)^{C_{19}(q-2)}|R|\\ &+C_qe^{-k}|R|. \end{aligned} \tag{One-accuracy}

Here $N=d+1$ is the original number of depths, and $C_N(q)$ is the preceding constant for fixed compactly supported scalar $L^q$ functions. It has not yet been treated as independent of $N$ .

Sum over constructions using (Root-packing). The same right-hand side holds with $|R|$ replaced by a fixed multiple of $|R_0|$ .

4. Different parameter growth rates

The source equation (6.9) gives

\log D_{\rm base},\ \log D_m\le C_qk^C, \qquad \log D_*\le C_q(1+k+P)^C,

K\le C_q(1+k+P)^{C_5}, \qquad G_0=\exp\!\big(C_q(1+\log(2k))^C\big). \tag{Growth}

The fixed exponents may be chosen before $q$ .

The distinction matters. Main-projection rows cost only $\log D_m$ and finite history products, which are absorbed into $G_0$ . The larger $D_*$ remains in mode construction, Bessel estimates, and power counting. Replacing the required $D_m$ by $D_*$ in the strong row estimate would no longer allow (Growth) to absorb the cost into $G_0$ .

Next choose

P=\left\lceil e^{\alpha k^\theta}\right\rceil, \qquad \alpha>0,\quad 0<\theta<1. \tag{Power-parameter}

For fixed $\alpha$ ,

\frac{\log G_0}{\log P} \le \frac{C_q(1+\log(2k))^C}{\alpha k^\theta} \longrightarrow0.

so

G_0=P^{o(1)}.

This means that, for each fixed $\delta>0$ , sufficiently large $k$ gives $G_0\le P^\delta$ . The initial threshold may depend on the already fixed $q,\alpha$ .

5. Why H and D need the same $q$

5.1 An existence statement alone is insufficient

D's short-block cost is

(K+2)^{C_{19}(q-2)}.

To make this smaller than $P^{1/8}$ using (Power-parameter), choose $q$ sufficiently close to two. At that same $q$ , we also need

C_H(q):=\sup_{N\ge1}C_N(q)<\infty. \tag{Fixed-uniform}

A uniform local theorem in H asserting only the existence of some $q\in(2,3)$ would not ensure that its $q$ meets D's extra smallness condition. D p.60 therefore invokes the order of parameter choices inside H's proof.

5.2 Required features of H's proof

The source lists:

H's local detection holds for every $q>2$ , with absolute logarithmic exponents; $q$ affects only coefficients.
H's short-block comparison holds for every $2<q<3$ , with the explicit block-length exponent $11(q-2)/(q-1)$ .
H's other exponents, separated scales, and mass exponents are fixed before $q$ . Its power parameter and initial precision may be chosen after $q$ .

Let the corresponding absolute exponents in H be $C_{19}^{H},C_5^{H}$ . Choose $q\in(2,3)$ satisfying

\max\{C_{19}^{H}C_5^{H},\,C_{19}C_5\}(q-2)<\frac18. \tag{Common-q}

This is possible because the quantities in the braces are finite and fixed before $q$ .

Run H's fixed-function parameter selection, normalized absorption, and external reductions at this chosen $q$ to obtain (Fixed-uniform). H may choose its own power parameter and initial precision; they need not equal D's later choices.

This is D's dependence on the adjustable parameter range of H's proof. We verify the quantifiers required by D, without completing H's entire fixed-function absorption argument anew.

6. Turning the two main costs into positive powers of the smallness parameter

After (Common-q), choose

\alpha>0,\qquad \alpha C_4<\frac{\varepsilon}{2}. \tag{Alpha}

The earlier $C_4$ already includes every side-term power cost.

6.1 Core terms and side terms estimated by powers

For large $k$ , the bounds

e^{\alpha k^\theta}\le P\le2e^{\alpha k^\theta},

give

\eta^\varepsilon P^{C_4} \le 2^{C_4} e^{-(\varepsilon-\alpha C_4)k^\theta} \le C\eta^{\varepsilon/2}. \tag{Powered-decay}

6.2 The strong near-difference estimate

By (Growth), after fixing $q$ ,

(K+2)^{C_{19}(q-2)} \le C_q(1+k+P)^{C_5C_{19}(q-2)}.

The parameter $P$ eventually exceeds every fixed power of $k$ . Since (Common-q) is strictly below $1/8$ , increasing the initial $k$ gives

(K+2)^{C_{19}(q-2)}\le C_qP^{1/8}.

Use $G_0=P^{o(1)}$ again to ensure $G_0\le P^{1/8}$ . Then

P^{-1/2}G_0(K+2)^{C_{19}(q-2)} \le C_qP^{-1/4} \le C_qe^{-\alpha k^\theta/4} =C_q\eta^{\alpha/4}. \tag{Strong-decay}

At this point, $1+C_N(q)\le1+C_H(q)$ is a fixed constant and can be absorbed into $C_q$ .

6.3 Numerical errors

For any fixed $c>0$ , since $\theta<1$ , sufficiently large $k$ gives

k\ge ck^\theta,\qquad e^{-k}\le e^{-ck^\theta}=\eta^c.

For example, take

c_*=\min\{\varepsilon/2,\alpha/4\}>0

and enlarge the initial precision. Equation (One-accuracy) and the sum of root lengths yield

\sum_{I\in\mathcal A_{R_0}} |I|\min\{\eta,|\mathcal H_I(z_I)|\} \le C_q\eta^{c_*}|R_0|. \tag{Quality-bound}

This is D equation (7.4). Its constant is now independent of the original tree depth $N$ .

7. Summing all precision levels

The precision sequence satisfies

k_{t+1}=k_t^A,\qquad A>1,

with fixed initial $k_1>1$ . For later level $t$ , there are only $O(t)$ choices of earlier level and a fixed number of position patterns.

For a fixed precision tuple, intersections of the finitely many basic segment families have total root length at most a fixed multiple of $|J|$ . Local constructions on each intersection add only the fixed factor in (Root-packing). The noninitial terms therefore have total cost bounded by

C_q|J|\sum_{t\ge12}t\,e^{-c_*k_t^\theta} \tag{Accuracy-series}

as claimed.

Convergence of this series follows directly from the recursion:

\log k_t=A^{t-1}\log k_1, \qquad k_t^\theta=\exp(\theta A^{t-1}\log k_1).

Thus $k_t^\theta/t\to\infty$ , and for all sufficiently large $t$ ,

c_*k_t^\theta\ge2t.

The tail is bounded by the summable sequence $t e^{-2t}$ . High-precision replacement errors and double- $U$ errors have the same multiplicity, at most $O(t)$ , and the same type of decay.

For the finitely many initial levels, use the depth-independent bounds from the basic active-scale reduction. Summing them crudely over every depth would reintroduce the unwanted factor $N$ .

8. The final precision limit involves only finitely many original cells

Fix the original tree and finite summation family first. At a finite precision cutoff, apply the exact earliest-two-increments identity. On each original cell $I$ , every precision update acts on the same actual function

F_{j,s(I)}.

The previous cellwise pair comparison makes the cutoff remainder tend to zero as precision tends to infinity.

There are only finitely many original cells, so

\lim_{T\to\infty} \sum_{I\in\mathcal I}|I|\,|\operatorname{Rem}_{I,T}| = \sum_{I\in\mathcal I}|I| \lim_{T\to\infty}|\operatorname{Rem}_{I,T}| =0.

No limit is interchanged with a sum over an infinite tree. The remaining precision-expansion terms are uniformly summable by (Accuracy-series), and the triangle inequality yields (Local-target).

The final uniformity comes from the basic cross-depth estimates D (5.2), (5.4), (5.21), and (5.12), together with the one-depth bound (5.1). Their constants are independent of the four block partitions and $W_j$ . Localization resets neither depth nor block partitions, so this uniformity persists to the end.

9. The precise scope of completion

This part completes the final deduction under the following inputs:

D §5's basic expansion, cross-depth energy estimates, and fixed complete-function comparisons.
The stated localization, side-function estimates, core counts, and original-cell minima in D §6.
H's geometric localization and separation, scalar kernel calculations, finite sparse counting, and short-block comparison.
H's fixed-function proof permits the common choice $q$ in (Common-q) and gives $\sup_NC_N(q)<\infty$ at that value.

Under these assumptions, reassignment of original terms, fresh estimates at every new root, parameter choices, and precision summation prove D Theorem 3.2. The earlier chapters on C, from local estimates to oscillation and then pointwise convergence, explain how estimates of this kind enter ergodic averages.

These conditional deductions do not certify the complete correctness of H or the four main theorems. Fixed-function closure and the localization setup in D §6.1 are developed in the supplement on main-projection closure and localization. The geometric, detection, and analytic assumptions from H that have not been checked step by step retain their separate conditions; this parameter calculation does not replace them.