Unit 3, continued: counting core tuples and interpolating amplitudes

This part develops D §6.6, pp.57–58, equations (6.23)–(6.24). The corresponding source argument is H §9 in the October 5, 2026 version, especially Proposition 9.2 (pp.87–88) and Proposition 9.3 (pp.89–90).

The preceding part treated two side-function positions. Here at least three positions are core. One interval may contain many representative tuples; taking a power below one separately for every tuple would lose the counting gain. We therefore retain each group's full count and sum successively over depths, groups, amplitudes, and frequencies.

The scalar kernel calculations in H §§9.1–9.2 remain inputs. The finite combinatorial proof of Lemma 9.1 is given in Finite counting on sparse scales. We state the conclusions and hypotheses used here, check what changes with D's varying functions, derive power-count bounds from sparse mass, and carry out the interpolation and summation.

1. Fixed data and the quantity being counted

Fix a point $x_*$ on a retained path, a group $G$, an upper height, the types in each position, frequency parameters $(\mu,\nu,\Lambda)$, and four amplitude classes $\sigma_j$. Here

take dyadic values. If all four positions are core, set $\Lambda=1$.

At depth $s$, write the path interval as $I_s$, its length as $r_s$, and its reciprocal scale as $R_s=r_s^{-1}$. The real number assigned to representative label $a$ is

The same label uses this value at every depth.

Let $N_s^G$ count the tuples of core representative labels passing the source frequency test for these fixed data. A tuple is counted at most once at each depth. These are the tuples retained by the actual scalar calculation, not an arbitrary set of tuples with roughly nearby frequencies.

For core position $j$, denote the label set by $\mathcal A_j^G$ and define

If there is a side position $l$, define

The normalized local norm is

Omit groups with no nonzero count. Every remaining group satisfies $M'\ge1$. For a nonzero use in the side position, the Fourier-window size below is at least $\sigma_l$, and Parseval gives the same conclusion.

2. The scalar quantities and their sources

2.1 Common heights, complete components, and errors

For a core tuple, H §9.1 samples common heights and takes the first at which every core position succeeds. Failures are assigned only to positions. Both success and failure conditions select complete core partial sums within minimal CURRENT components, which may also contain side modes. A core partial sum is not cut anew according to the size of one Fourier mode inside its component.

Let the number of core positions be $m_0\in\{3,4\}$. Among the preceding $n$ failures, let $n_j$ be assigned to position $j$. The source core size $\sigma_{F_j}$ and torus-function size $a_j'$ satisfy

Use the deterministic bound to raise the second moment to the $m_0$-th moment, then apply Hölder:

At length $n$, there are at most $m_0^n$ failure-position records. Thus, for a sufficiently small fixed $\delta_*$,

The failure records therefore introduce no additional factor growing with depth.

Scalar approximation, Fourier tails, and discarded small terms are bounded absolutely before taking powers, with total $O(e^{-k}|R|)$. This error is not subsequently raised to the $p$-th power and resummed.

2.2 The frequency tests

Retain H's horizontal frequency vector and height scales, and write

The frequency vector $\vartheta_{\rm freq}$ is distinct from the small scalar interpolation parameter $\vartheta$ used later. The retained frequency range is

At the end we may enlarge these finite ranges when summing.

For four core positions, set

Choose reference vectors $n,m$ in the complete finite shift range so that

lie in the permitted intermediate-frequency range. Use a fixed rule for these references, independent of later failure records or component selections. The parameters satisfy

For three core positions and side position $l$, set

Choose a permitted $m$ and classify by

Very small nonzero parameters and cases without an intermediate-frequency reference have already been included in the absolute error.

Use the actual side function $S_{l,G,s}$ on the current interval. Insert a newly sampled translated cutoff and remove its fixed quadratic phase to obtain $S'$, with

The corresponding window size is

The shift vector $n$ ranges over the full Fourier range, so the window does not change with the failure record. Different tuples on the same cell are tested using the same new shift $\rho_l$. The side amplitude $\sigma_l$ is the dyadic class containing $b_\Lambda$.

2.3 The scalar input used here

H §9.2 states that, after removing the absolute errors above, the interval contribution for fixed data is bounded by

where $L$ is any sufficiently large fixed integer.

The factor $\mu$ comes from the kernel's vanishing marginals; $\nu^{-L}$ comes from its smooth far-frequency decay. With three core positions, $\Lambda^{-L}$ comes from decay in pairing the side-function Fourier windows. Four core positions use a four-factor $L^2$ estimate on the torus; three core positions use its dual form and Parseval. These require the full frequency-shift range, pure-shift gap, and height margins retained in H's construction. Equation (Cell-core) is a specified input here, not a consequence of representative separation alone.

3. The complete finite sparse-counting statement

H Lemma 9.1 (pp.81–87) uses the following finite data.

There are four positions $0,1,2,3$, three or four of which are core, forming the set $\mathcal J$. Given

each core position has a finite real-label set $\mathcal A_i$ with $|\mathcal A_i|\le M$. Within a position, the assignment of labels to real numbers is injective. Also fix $\zeta\in\mathbb R^d$.

At finitely many distinct integers $s$, take

Assume:

(1) Labels in two distinct core positions determine at most one complete core tuple, and that pair of labels appears at at most $L$ scales. (2) If there is a side position $l$, there are weights $w_s\ge0$ with $\sum_s w_s\le M$ such that at depth $s$ at most $w_s$ tuples contain any specified core label. (3) If distinct labels $a,a'$ in one core position appear at scales $s,t$, then for every integer vector satisfying $|n|_\infty\le Q$,

(4) For every three-core subset $\mathcal T\subseteq\mathcal J$, let $l'$ be the omitted position. Every tuple at scale $s$ has an integer vector satisfying $|n|_\infty\le B$ such that

(5) At every scale $s$ with a tuple, for all $|n|_\infty\le Q$,

Also require

Then for $0<\delta\le1$,

The constant is independent of the number of scales, the number of labels, and the particular real frequencies.

We use this conclusion below; its four-step-path and slab-covering proof is developed in the linked extension. The annulus and pure-gap assumptions are substantive and cannot be replaced by uniqueness from two labels.

4. Conditions to recheck for D's varying functions

4.1 Two core labels determine the remaining labels

With four core positions, suppose two tuples share labels in positions $i,j$, leaving positions $a,b$. Subtract their $Q,T$ tests. The unknown differences satisfy an approximate two-variable linear system with matrix

Its determinant is $b-a$, with absolute value in $\{1,2,3\}$. The unknown differences therefore lie in ranges described by finite shifts and an error $O(\nu\max(R_s,R_t))$. The separation margin allows these subtractions, integer divisions, and shifts, forcing equal labels in both remaining positions.

With three core positions and two labels fixed, one equation for $T_l$ determines the remaining label. Its coefficient is the nonzero integer $j-l$ of absolute value at most three, and the same separation margin gives uniqueness.

The separation scale is $\max(R_s,R_t)$, so this also compares different depths. The number of scales at which a complete tuple passes the test is controlled by H's height-grouping result:

This step uses the frequency-annulus test as well as label uniqueness.

4.2 Incidence counts between core and side positions

Fix one core label and one scale. If two distinct three-core tuples have overlapping windows in (Side-window), their $Q$ frequencies differ, after finite shifts, by $O(\Lambda R_s)$. Combining this with their two $T_l$ equations again gives a two-variable system with determinant of absolute value at most three. Source separation is much larger than the allowed $\nu,\Lambda$, so the other two labels must agree.

The windows of distinct tuples are therefore disjoint. Every counted tuple satisfies $b_\Lambda\ge\sigma_l$. Apply Parseval for the same newly sampled shift, then take expectation, to obtain

Summing over scales gives exactly $\sum_s w_s=M_l^G$.

This calculation requires the actual side function on the current cell. Older frozen coefficients from different blocks may be unrelated and cannot be treated as one common Fourier sequence.

4.3 Inherited frequency gaps

To apply the finite lemma, H sets

Equations (Four-test) and (Three-test) supply the annulus test for every three-core subset. To check the finite lemma's two numerical requirements, the source uses

Here $Q_{\rm res}$ is the parameter $Q$ in the finite lemma. The quantities $E_h,L_h$ are the shift range and denominator scale in the original height construction. The equality uses $L_h/L_{h-\Delta}=6^\Delta$. H's choice of the height parameter $A_0$ gives

Moreover, $\nu\le C D_*^{250}$, the horizontal dimension satisfies the previously fixed bound, and $\Xi=D_*^{1000}$. The source separation parameter therefore ensures

Fixed constants in the annulus bounds are absorbed by decreasing $\mu$ and increasing $\nu$ by fixed factors. The margins above remain valid.

The pure-shift gap uses at least one nonexceptional main label in every successful component. The representative itself may be exceptional; a suitable main label in the same component is what is needed. The source rational-height bound $H$ covers all pure horizontal shifts in (Count-reserves). For them, the source frequency dichotomy gives

The reciprocal scale of a tested cell lies between $r_{v_b}^{-1}$ and $r_{t_b}^{-1}$. Together with $\mu\ge U_0^{-1}$ and $\nu\le CU_0$, the source choice of $T$ makes the first alternative strictly less than $\mu R_s/1000$ and the second strictly greater than $1000\nu R_s$. This verifies (Pure-gap).

These numerical relations are hypotheses inherited from H's construction. We check their correspondence to the finite lemma's conditions without reproving the entire mode-graph and height construction.

The parts affected by D's new functions are thus the fixed representative values, positionwise counts, and actual side-function square sizes. The other frequency tests and gaps remain those of H.

5. Two counting bounds

For any distinct positions $i,j$, the incidence counts above give

For two core positions, this counts label pairs times their scale multiplicity. For one core and one side position, it follows from (Incidence), and remains true after multiplication by $L_0\ge1$.

The finite sparse-mass lemma gives an absolute $c'>0$, for example $c'=1/2080$, such that

The coefficient includes the dimension, logarithms, and $L_0$ cost from (Sparse-mass).

Uniqueness from two core labels gives

Choose

On scales satisfying

the inequality $p_0-1<0$ gives

Apply (Small-count-mass) and sum over $h\ge0$ to obtain

Here $1-p_0-c'=-c'/2<0$. This is the second estimate in D (6.23). Zero-count scales contribute nothing.

6. Interpolating the two counts

Choose a small absolute number $\vartheta\in(0,1)$, to be fixed later, and define

Hölder gives directly

The count $N_s^G$ remains intact inside its power; it has not been expanded into $N_s^G$ unit terms.

7. Summing over groups using total degree greater than one

The preceding component-size and top counts, together with the side-function pathwise square sums, give for each position

A core position also satisfies

The source logarithmic growth condition absorbs later factors of $\log D_*$ into permitted polynomials in $P,k$. It does not treat $D_*$ itself as a small power of $P$.

We need the elementary fact that, if $m_i^G\ge0$, $\beta_i\ge0$, and

then

To prove it, omit zero-exponent positions. If a required sum vanishes, both sides vanish. Otherwise normalize by

Hölder with exponents $B/\beta_i$ gives

using $0\le a_i^G\le1$ and $B\ge1$ in the last step.

In (Interpolated-count), split $M'=\max_iM_i^G$ into at most four cases according to where the maximum occurs. Each term has total degree

We will choose $p>2/3$, so $2p>1$ and (Group-sum) applies.

8. Three amplitude ratios and one common scale

Temporarily order the positions by amplitude and relabel them $1,2,3,4$, with

This labels only amplitude order, without changing positions in the original ergodic average. Use the pair $1,2$ in (Interpolated-count). Since there is at most one side position, at least one member of this pair is core.

By (Role-total), the worst amplitude cost from $M'^{2\vartheta p_0}$ is bounded by

The core member of the pair can also use (Core-cap), giving the factor

If that core position is $i=2$, then $\sigma_2\le\sigma_1$ allows the same bound with $\sigma_1$. Constants and powers of $P$ are absorbed into $P^C$.

Multiply by the $p$-th power of the amplitude factor in (Cell-core) to obtain

Denote the four exponents by $e_1,e_2,e_3,e_4$. Substitution of $p=1-\vartheta+\vartheta p_0$ gives

and

A sufficiently small fixed $\vartheta$ makes all three values strictly negative.

Set

Since the total exponent is zero,

Each amplitude gap $g_j$ therefore contributes a convergent geometric series.

It remains to sum over simultaneous multiplication of all amplitudes by one dyadic factor. On retained cells, $\sigma_1\le P^C$. When

the prefactor in (Amplitude) is

and decays geometrically as $\sigma_1$ decreases dyadically. In the range $D_*^{-1/2}\le\sigma_1\le P^C$, there are only

dyadic amplitudes. The common-scale cost is therefore only the permitted logarithmic polynomial. Finally sum over at most $24$ amplitude orderings.

9. Frequency summation and the order of choices

The frequency loss in (Count-one) is only a fixed power of $1+\log(\nu/\mu)$. After interpolation, the second count costs at most

The $p$-th power of the kernel coefficient is

First choose the absolute parameter $\vartheta$ sufficiently small that

and all three strict negative inequalities in (Partial-exponents) hold. Then choose a sufficiently large fixed integer $L$ so that

We then have

Strict geometric decay absorbs the fixed logarithmic power. Enlarging the actual finite frequency ranges to these infinite ranges only increases the bound.

The numbers of heights, types, position patterns, and retained search leaves are absorbed into a fixed power of $P$. Thus, on every retained path,

The parameters $p_0,\vartheta,\varepsilon,C_4$ can all be fixed before deciding how close $q$ must be to two.

10. Returning to the minimum on each original interval

Let $C_I$ be the sum of the absolute main contributions on an original cell. First take the cell's minimum, then use subadditivity for $0<p<1$:

The sum may be split across groups and configurations, but each term's $N_s^G$ must remain intact. Splitting earlier into unit contributions from individual label tuples would lose the gain in (Count-small-power).

For nonnegative cell data $F_I$, finite summation and integration give the exact identity

Inserting (Path-power) yields

Add the errors previously controlled separately in absolute value to obtain D (6.24):

Source coverage and remaining work

• D pp.57–58: the three checks for applying the core count to varying functions, and equations (6.23)–(6.24).
• H §§9.1–9.2, pp.78–81: common heights, complete-component selection, torus kernel calculations, and actual side-function windows. The results used are stated here; torus approximation and all Fourier errors are not fully reproved.
• H Lemma 9.1, October 5, 2026 version, pp.81–87: the full hypotheses are stated here, and the combinatorial proof has been developed and independently reviewed in the extension. The source frequency gaps and height grouping required by D retain their own assumptions.
• H Proposition 9.2, pp.87–88: pairwise uniqueness, Parseval for actual side functions, and the deduction of $p_0$-power counting from sparse mass.
• H Proposition 9.3, pp.89–90: the group inequality, amplitude-gap sum, frequency sum, and cellwise minimum developed in full here.

The next part combines the three types of position patterns, handles reassignment of original intervals on restarts, and explains why H and D require the same $q>2$. Only after these parameter choices is the local estimate independent of original tree depth.