Unit 3, continued: inherited core representatives and first threshold crossings

This part develops the end of D §6.4, pp.51–52, equations (6.16)–(6.17). The later core count needs two kinds of data: frequency separation between distinct representatives along a path, and a bound on the total number of representatives across groups in each size class. These come respectively from inherited representative selection and the square energy of mode projections.

The source uses H Lemmas 8.4, 8.10–8.11, and Proposition 8.12. Under the stated graph-stability and small oscillatory inner-product assumptions, we develop the set-theoretic, almost-orthogonality, and stopping-sum arguments. The rational-height stability theorem from H is not reproved here.

1. The common predecessor of good modes

Retain the PRE components $\mathsf E_h(m)$ , CURRENT components $\mathsf F_h(m)$ , and main-column set map $\pi$ . Beyond counting bad heights, height stability includes this substantive assertion: if a main column is nonexceptional and stable at height $h$ , and two of its modes lie in one CURRENT component, then those modes are directly adjacent in the PRE graph at height $h-\Delta$ . This is the conclusion of H Lemma 8.3.

Suppose two good modes lie in the same $\mathsf F_h$ , whose main-column set is $F$ . Each good PRE predecessor contains more than $|F|/1.1$ main columns. Their main-column sets therefore intersect in more than

(2/1.1-1)|F|>|F|/10.

At most one tenth of the CURRENT component's main columns fail the nonexceptionality and stability conditions. The intersection therefore contains a main column satisfying both. Choose its witness mode in each predecessor. Height stability provides a PRE edge between the witnesses, so the two PRE predecessors are identical.

Thus all good modes in one CURRENT component have the same predecessor at height $h-\Delta$ . This is the unique object meant below by the predecessor of a successful component.

2. Upper heights select complete core partial sums

Set

H_{\mathrm{up}}=\{h\in\mathbb Z: \lceil4\delta_*J_*\rceil\le h\le J_*\}.

If two core modes belong to one minimal CURRENT component, together they have at most $2\delta_*(J_*+1)$ bad heights. The source chooses $J_*$ large enough that more than this many integer heights precede the upper range. Both modes are therefore good at some earlier height $h_0$ .

They share a CURRENT component at every height. Section 1 gives the same PRE predecessor at $h_0-\Delta$ . Graph nesting then gives the same components at all later required PRE and CURRENT heights. They consequently have the same good/bad classification on $H_{\mathrm{up}}$ and share one predecessor whenever they are good.

Selecting upper good heights therefore retains or removes the entire core partial sum in a minimal CURRENT component, rather than selecting only some of its core modes. On $H_{\mathrm{up}}$ , each core mode has bad proportion at most $2\delta_*$ , by

|H_{\mathrm{up}}|\ge(1-4\delta_*)J_*

and the bound on its bad heights over the full range.

3. Successful components and two-part types

Fix a position, group, and upper height $h$ . A CURRENT component $F=\mathsf F_h$ is successful if it contains a core mode that is good at that height. Write its unique good PRE predecessor as $E=\mathsf E_{h-\Delta}$ .

Its type consists of two integers:

\left( \lfloor\log_{1.1}|\pi(E)|\rfloor,\ \lfloor\log_{1.1}|E|\rfloor \right). \tag{Type}

The first counts main columns and the second counts modes. Both are used below; the first alone is insufficient.

A representative is a permanent mode label from $E$ . It is not a newly rounded numerical frequency, and its current coefficient need not be nonzero.

4. Connected predecessors of the same type are nested

This is H Lemma 8.10, pp.74–75.

Fix the height, position, group, and type. Call an earlier and a later successful component connected if some mode in the first coincides with, or is adjacent in the later CURRENT graph to, a mode in the second.

Old edges and modes persist, so the earlier CURRENT component lies in the later one. Denote the latter's main-column set by $F$ . Its good condition gives

|\pi(E_{\mathrm{late}})|>|F|/1.1.

The predecessors have the same first type coordinate, so

|\pi(E_{\mathrm{early}})| >|\pi(E_{\mathrm{late}})|/1.1 >|F|/1.1^2.

Both sets lie in $F$ . Their intersection therefore contains more than

(1/1.1+1/1.1^2-1)|F|>|F|/10.

main columns. One of them is nonexceptional in the later block and stable at that height. Its two witness modes lie in the same later CURRENT component and are directly adjacent in the later PRE graph at height $h-\Delta$ . The earlier PRE edges persist as well. Hence

E_{\mathrm{early}}\subseteq E_{\mathrm{late}}. \tag{Predecessor-inclusion}

This is set inclusion, which does not follow merely from comparable component sizes. The direct PRE edge comes from the height-stability assumption.

5. Consistent representatives inherited by descendants

Process predecessors of one type in analysis-block order. If the current predecessor contains an earlier successful predecessor on the path, inherit its representative. Otherwise choose the first label in the current predecessor according to a fixed mode-label ordering.

We must check that several earlier predecessors contained in the current one do not require different representatives. Equality of their second type coordinates makes each earlier predecessor contain more than $1/1.1$ of the current predecessor's modes, hence more than half. Any two earlier predecessors therefore intersect.

They lie on one path, and their common mode connects them. By (Predecessor-inclusion), they are nested in time and, inductively, already share an inherited representative. The rule is therefore consistent.

If representatives on two comparable intervals of actual use differ, their components can contain no coincident modes or modes adjacent in the later CURRENT graph. Otherwise, (Predecessor-inclusion) would force the later component to inherit the earlier representative.

6. From graph nonadjacency to a frequency gap at one point

Let distinct representatives be $a,a'$ on comparable intervals of actual use $I,I'$ , and choose $x_*\in I\cap I'$ . Take $I'\subseteq I$ as the finer interval, allowing equality. Its analysis block $b'$ may be the same as the block containing $I$ . For every permitted integer shift

n\in\mathbb Z^{\dim\vartheta},\quad\|n\|_\infty\le E_h,

CURRENT nonadjacency makes the supremum on the later PRE interval $J'$ greater than $\Xi/r_{t_{b'}}$ .

The source oscillation bound for frequency slopes also applies to restrictions of earlier modes to the later comparison interval, giving

\operatorname{osc}_{J'}(\omega_a-\omega_{a'}) \le\frac{2C D_*2^{-4K}}{r_{v_{b'}}} \le\frac{\Xi}{2r_{t_{b'}}}.

The last inequality follows from the source choice of block length $K$ . An integer shift does not change oscillation, so the affine difference at $x_*$ has absolute value greater than half the original supremum threshold. The later interval of actual use satisfies $|I'|\ge r_{t_{b'}}$ , hence

|\omega_a(x_*)-\omega_{a'}(x_*)+n\cdot\vartheta/L_h| >\frac{\Xi}{2}\max\{|I|^{-1},|I'|^{-1}\}. \tag{Representative-gap}

This is D equation (6.16).

The comparison uses the same function value $\omega_a(x_*)$ attached to the permanent label. Frozen coefficients may change between blocks, but the representative's phase function is not reselected with them.

7. Why core size includes a small coefficient term

This is the definition on D p.52 and H equation (26).

For an interval of actual use $I$ and a successful CURRENT component $F$ , let $D\subseteq F$ range over minimal CURRENT components. Let $A_D$ be the exact partial sum of all core modes in $D$ , retaining augmented coordinates. Define

\sigma_F^2 =\sum_{D\subseteq F} \left( \|A_D\|_{2,I} +D_*^{-30}\sum_{\substack{m\in D\\m\text{ core}}}|\alpha_m| \right)^2. \tag{Component-size}

The coefficient term is small but necessary. It controls later uniform linearization and Fourier truncation errors without replacing the core's actual $L^2$ size by the full sum of absolute coefficients.

Let $P_{F,I}$ be the orthogonal projection onto the span of all modes in $F$ on $I$ , and let $V_F$ be the exact non-core partial sum over these labels. The comparison in H Lemma 8.11 is

\sigma_F^2 \le C\left( \|P_{F,I}\widetilde Z_{G,b}\|_{2,I}^2 +\|V_F\|_{2,I}^2+D_*^{-50}\right), \tag{Size-comparison}

and, on the same interval of actual use,

\sum_{F\text{ successful}}\|V_F\|_{2,I}^2 \le C\|V_{G,b}\|_{2,I}^2. \tag{Side-subtotals}

8. The almost-orthogonality calculation behind the size comparison

Distinct minimal CURRENT components satisfy the source small oscillatory inner-product bound on the actual interval $I$ . The private-coordinate coefficient calculation from the preceding part controls all absolute off-diagonal inner products, as well as the synthesis norm. Thus, for arbitrary coefficient partial sums $f_D$ ,

\left|\left\|\sum_Df_D\right\|_{2,I}^2 -\sum_D\|f_D\|_{2,I}^2\right| \le\varepsilon_*\sum_D\|f_D\|_{2,I}^2

where $\varepsilon_*$ is a fixed small negative power of $D_*$ . This gives almost orthogonality in both directions.

Let $T_F$ be the complete coefficient partial sum over $F$ . Then

A_F=\sum_{D\subseteq F}A_D=T_F-V_F.

Almost orthogonality gives $\sum_D\|A_D\|_{2,I}^2\le C\|A_F\|_{2,I}^2$ . The total squared contribution of the coefficient correction is at most

C D_*^{-60}\left(\sum_m|\alpha_m|\right)^2 \le C D_*^{-58}.

Modes outside $F$ are CURRENT-nonadjacent to modes inside $F$ . The same projection-versus-partial-sum calculation as before gives

\|P_{F,I}\widetilde Z_{G,b}-T_F\|_{2,I} \le D_*^{-190}.

Combine these estimates with $\|T_F-V_F\|^2\le2\|T_F\|^2+2\|V_F\|^2$ . This gives (Size-comparison), with room in the error allowance $D_*^{-50}$ .

Partition all non-core modes by their CURRENT components at height $h$ , apply the same two-sided almost orthogonality, and discard unsuccessful components. This proves (Side-subtotals). The estimate concerns one actual interval throughout and does not compare coefficients from different blocks.

9. The first interval where each representative crosses a threshold

Fix a position, upper height, type, and dyadic threshold

D_*^{-1}\le\sigma\le D_*^2.

For every group and representative, take on each spatial branch the first interval of actual use satisfying $\sigma_F\ge\sigma$ . Call it a first-threshold interval, or top. A group and representative may have several disjoint tops on different branches.

First fix a group $G$ and prove an ordinary operator bound. One representative has at most one top along a path. For different representatives in the same group with overlapping tops, Section 5 excludes coincident or later-CURRENT-adjacent modes between their full component label sets. These top-projection rows therefore have pathwise disjoint labels, and the source separated-mode row result gives a uniform ordinary Bessel bound for each fixed group. No frequency separation between different groups is asserted here.

Each row is supported on its current actual interval $I$ , while its main decision endpoint remains the ancestor at frozen depth $u_b$ . The top itself establishes actual use. D Lemma 6.3 combines the within-group bounds through main endpoint-space separation and gives

\sum_{\mathrm{tops}} \|P_{F,I}\widetilde Z_{G,b}\|_2^2 \le G_0|R|. \tag{Top-projections}

Even when several tops share one frozen ancestor, the main input projection cannot first be restricted to the smaller tops. The complete main projection on the ancestor is used.

10. An energy bound for total top length

Multiply (Size-comparison) by $|I|$ and sum over all top events. Actual intervals at one depth are disjoint, and successful components on one interval satisfy (Side-subtotals). A block repeats over at most $K$ depths, so

\sum_{\mathrm{tops}}|I|\|V_F\|_{2,I}^2 \le CK\sum_{G,b}\|V_{G,b}\|_2^2 \le KG_0|R|.

Combining this with (Top-projections) gives

\sigma^2\sum_{\mathrm{tops}}|I| \le C(1+K)G_0|R|+C D_*^{-50}\sum_{\mathrm{tops}}|I|.

Since $\sigma^2\ge D_*^{-2}$ , increasing the initial bound makes the last term at most half the left-hand side. Absorb it to obtain

\sigma^2\sum_{\mathrm{tops}}|I| \le C(1+K)G_0|R|. \tag{Top-length}

This directly absorbs the same sum of top lengths. No crude bound on the total number of tops is needed.

11. From top lengths to pathwise representative counts

Set

N_\sigma(y)=\sum_{\mathrm{tops}}\mathbf1_I(y),

counting events by group and representative. Distinct top events using the same interval are still counted separately. Equation (Top-length) gives

\int_RN_\sigma(y)\,\mathrm{d}y \le C(1+K)G_0\sigma^{-2}|R|.

The number of choices of heights, types, and thresholds is a fixed polynomial in $P,k$ . Heights are controlled by $J_*$ , both type coordinates range logarithmically, and there are $O(\log D_*)$ dyadic thresholds. These polynomial bounds are part of the source parameter choices.

For each choice, cut out the original actual interval where the accumulated pathwise top count first exceeds a threshold. The same finite first-crossing argument as before allows threshold

P^C\sigma^{-2},

while keeping the total length of stopping intervals from all choices below any prescribed small fixed fraction. Here we use $K\le\operatorname{poly}(P,k)$ and $G_0=P^{o(1)}$ , enlarging the fixed degree and initial precision to absorb the finite number of configurations.

On a path retained in the current construction, if a representative is actually used in size class $[\sigma,2\sigma)$ , its first $\sigma$ -top is an ancestor of that use and is still retained. Otherwise the descendant original summands would already have passed to a new construction. Distinct group–representative labels give distinct top events, so their number is at most $N_\sigma(y)$ .

Let $\mathcal A_\sigma^G$ denote the distinct representative labels actually used along a retained path, for fixed position, height, type, and size class. This gives D equation (6.17):

\sum_G|\mathcal A_\sigma^G| \le\min\{D_*,P^C\sigma^{-2}\}. \tag{Representative-count}

The first bound comes from the total pathwise mode count. If $\sigma<D_*^{-1}$ , then $P^C\sigma^{-2}\ge D_*$ , and that total mode bound already gives the minimum. Size classes above the tested range are empty after the fixed enlargement of the source bounds.

12. Local core-size bounds and subsequent use

The preceding part gives, on retained paths,

\|\widetilde Z_{G,b}\|_{2,I}+\|V_{G,b}\|_{2,I}\le P^C.

The projection $P_{F,I}$ is contractive on $I$ . Together with (Size-comparison) and (Side-subtotals), this gives

\sigma_F\le P^C

at every retained use.

The data now available for the later core count are:

Upper good-height selection retains complete core partial sums of minimal CURRENT components;
For fixed position, height, and type, representatives are inherited by descendants and satisfy (Representative-gap) for every permitted integer shift;
Within a size class, the number of actual representatives across all groups satisfies (Representative-count);
Core sizes and full local non-core sizes are bounded by $P^C$ .

All these statements concern the same actual intervals of use and permanent labels. The quadrilinear sum with at least three core positions has not yet been estimated. D §6.6 applies the separated-frequency count from H to these data. The case with two non-core positions requires a separate treatment of matched difference sums.

Source coverage

D pp.51–52, equation (6.16): inherited representatives of fixed type and pointwise frequency gaps.
D p.52: complete core size, size comparison on one interval, energy of frozen top rows, and equation (6.17).
H Lemmas 8.4, 8.10–8.11, and Proposition 8.12, pp.69, 74–77: source good-height and predecessor structure, representative selection, almost-orthogonality comparisons, and stopping at tops.