Unit 3, extension: finite counting on sparse scales

This part gives a step-by-step proof of H Lemma 9.1, October 5, 2026 version, pp.81–87. It supplies the counting gain at a power below one in D §6.6. Its connection with amplitude interpolation is explained in Sections 5–6 of “Counting core tuples and interpolating amplitudes.”

The proof concerns only the finite proposition under the five stated hypotheses. The source version and its citation in D have been checked against each other. D's application must still supply the frequency gaps, positionwise counts, and scalar kernel estimates.

1. The finite proposition and its five hypotheses

The four positions are $0,1,2,3$. Three or four are core, forming the set $\mathcal J$. Given

each core position has a finite real-label set $\mathcal A_i$, with distinct labels within the position and $|\mathcal A_i|\le M$.

Fix $R_0>0$. For finitely many distinct integers $s$, specify

The quantity $R_s$ is reciprocal scale. A tuple appearing at different scales contributes a separate occurrence at each.

Assume:

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

(4) For every three-core subset, let $l'$ be the omitted position and set $\alpha_i=i-l'$. Every tuple at scale $s$ has an integer vector satisfying $|n|_\infty\le B$ such that

Each coefficient $\alpha_i$ belongs to $\{\pm1,\pm2,\pm3\}$. (5) At every occupied scale, for all $|n|_\infty\le Q$,

Also assume that a fixed sufficiently large absolute constant $C_{\rm res}$ satisfies

We will prove that, for $0<\delta\le1$,

The constant is independent of dimension, the number of scales, and the number of labels.

2. Placing the real data on a common fine grid

The labels, scales, and shifts with $|n|_\infty\le Q$ form finite sets. Approximate all real data on a common grid of spacing $h_0>0$, placing original coordinates at a fixed integer multiple of $h_0$ so that later division by $1,2,3$ stays on the common fine grid.

Choose $h_0$ after all the finite data. Making every error a sufficiently small multiple of the smallest $R_s$ preserves:

• The separation constant $\Sigma/2$;
• The annulus lower bound $\mu/2$ and upper bound $2\nu$;
• A narrower interval strictly inside the original pure-shift exclusion interval, still with a fixed margin.

Specifically, use working parameters $\mu=\mu_{\rm old}/2$, $\nu=2\nu_{\rm old}$, and $\Sigma=\Sigma_{\rm old}/2$, leaving fixed-factor margins in the original $C_{\rm res}$. The grid proof needs only

This interval lies strictly inside the original wider forbidden interval, so sufficiently fine approximation of the finite data preserves it. Rounding is not asserted to preserve the entire thousandfold forbidden interval for the renamed parameters.

Rounding a width upward to the grid adds at most $h_0$. The absolute margins in (Reserves) absorb these fixed factors. Below, reuse the original letters for the gridded data, but use only (Grid-gap) in the pure-shift contradiction. The interval obtained in Section 10 lies strictly inside this narrower forbidden interval. Every estimate is independent of $h_0$ and ultimately applies to the original real data.

3. A bipartite graph with many four-step colored paths

Suppose each side of a bipartite graph has at most $M$ vertices, with at least $\eta M^2$ edges and $0<\eta\le1$. There are at most $M$ edge colors, and edges incident to one vertex have distinct colors. We seek a left-side set of size at least $c\eta M$ such that every pair of its vertices has at least $c\eta^5M^3$ four-step paths between them.

3.1 A random neighborhood and bad pairs

Add isolated vertices to the right side until it has $m=\lceil M\rceil\le2M$ vertices. Choose one uniformly and let $Y$ be its left neighborhood. Then

Call an ordered left-side pair bad if it has fewer than $c_0\eta^2M$ common neighbors. The probability that both endpoints lie in $Y$ is their common-neighbor count divided by $m$. Thus the number of bad ordered pairs in $Y^2$, denoted by $B_Y$, satisfies

For sufficiently small $c_0$,

Hence some $Y$ satisfies $|Y|\ge c\eta M$ and $B_Y<|Y|^2/100$.

3.2 Removing a few vertices

Within $Y$, remove vertices with more than $|Y|/4$ bad partners. Fewer than $|Y|/25$ vertices are removed. The remaining set $B_0$ still satisfies

For $a,a'\in B_0$, at least $|Y|/2$ middle vertices $z\in Y$ are good partners of both. Each pair $(a,z)$ and $(z,a')$ has at least $c_0\eta^2M$ common right neighbors, chosen independently. The number of four-step paths is therefore at least

Repeated vertices are allowed. A starting vertex and four ordered edge colors uniquely determine each successive step. For fixed endpoints, distinct paths therefore give distinct ordered color quadruples.

4. Blocking the counting mass and covering difference sets

First treat sufficiently small $\delta>0$. Larger $\delta$ will be handled at the end by the crude total-mass bound. Retain only scales with $N_s\le\delta M^2$. Uniqueness and scale multiplicity for any pair of core positions give

Choose parameters, to be fixed later,

Order retained scales by decreasing $R_s$ and greedily divide them into consecutive counting blocks of mass at most $\alpha M^2$. Every block except the last has mass greater than $(\alpha-\delta)M^2$. Their number satisfies

These blocks are defined by counting mass, unlike the consecutive-depth analysis blocks used earlier.

4.1 Finite shift neighborhoods for each block

Let $R_b^+$ be the largest reciprocal scale in block $b$, and set

For a sufficiently fine grid, $\epsilon_b\le17\nu R_b^+$. For integers $t\ge1$, define the finite symmetric set

It contains zero and satisfies

4.2 Constructing a graph from uncovered occurrences

Fix a core position $i$ and a block. As long as occurrences involving its uncovered labels have mass greater than $\kappa M^2$, choose two other core positions $j,k$.

Project these occurrences to a simple bipartite graph on the labels in positions $(i,j)$, keeping one scale witness in the block for each pair. Since each pair appears at most $L$ times, there are more than

edges. Color each edge by its third-position $k$ label. The coloring is proper: two incident edges of the same color would share two core labels and hence be the same complete tuple and edge.

Set $\eta=\kappa/L$. Section 3 gives

and (Four-paths).

4.3 Four annulus relations eliminate the intermediate labels

A four-step path has successive left vertices $a,a_1,a'$, right vertices $b,b'$, and colors $c_1,c_2,c_3,c_4$. Add and subtract its four three-position annulus relations alternately. The intermediate labels $a_1,b,b'$ cancel. Division by $\alpha_i$ gives

where $q$ depends only on the color quadruple.

The color term is explicitly

The shift coefficient $36/\alpha_i$ is an integer multiple of six. Combining four shifts gives, in the $6\zeta$ grid, an index of size at most $24B\le N_0$. The scalar error is at most $4\nu R_b^+$ and is contained in $\epsilon_b$.

4.4 A maximal separated subset of the difference set

Choose a maximal subset of $B_0-B_0$ in which differences of distinct elements lie outside $U_b(2)$. Choose endpoints for each selected difference. Each endpoint pair has at least $c\eta^5M^3$ color quadruples.

Two selected differences cannot share a color quadruple: by (Color-difference), their difference would lie in

a contradiction. There are at most $M^4$ color quadruples altogether, so the number of selected differences is at most

By maximality, translates of $U_b(2)$ centered at these differences cover all of $B_0-B_0$.

Remove this batch of labels $B_0$ and repeat. Each step removes at least $c\eta M$ labels, so there are at most $C\eta^{-1}$ steps. Thus, for every core position in every block, apart from uncovered occurrences of mass at most $\kappa M^2$, the labels are covered by at most

sets $B$. Each difference set $B-B$ is in turn covered by at most $K_1M$ translates of $U_b(2)$.

5. Removing occurrences with fewer than three future positions

First discard occurrences with an uncovered label in any core position in the current block. The total loss is at most

For a remaining occurrence, call a core position future if its label also belongs to a covering set in that position in a later counting block. Otherwise call it nonfuture.

For a fixed position, permitted nonfuture labels from different counting blocks are disjoint. A label covered in a later block would be future in an earlier block. Let $m_i^b$ count the nonfuture labels in each block. Then

A side position $l$, when present, is always nonfuture. Define

An occurrence with fewer than three future positions has at least two nonfuture positions. Assign it a pair $i,j$ by a fixed rule. In block $b$, its class has mass at most

For two core positions, this follows from label pairs and scale multiplicity. For one core and one side position, use the side incidence bound; the extra factor $L\ge1$ is harmless.

By $\min(x,y)\le\sqrt{xy}$ and Cauchy–Schwarz,

At most six pairs are possible, costing only an absolute constant.

6. Three fixed future covering sets and four scale segments

Every remaining occurrence has at least three future core positions. Use a fixed rule to select three positions and a later-block covering set for each label. The number of configurations is at most

Fix a configuration of positive mass $\rho M^2>0$. It specifies the current counting block, three core positions, and covering sets $B_1,B_2,B_3$ from strictly later blocks. The indices $1,2,3$ merely enumerate these positions; the original coefficients $\alpha_i$ remain unchanged.

Set

If $\rho\le C\delta\ell$, the required small-mass bound already holds. Assume $\rho>C\delta\ell$ below, with a sufficiently large absolute constant.

Order this configuration's mass by decreasing reciprocal scale, split it into four quartiles, and discard the scale cells crossing the three boundaries. Each cell has mass at most $\delta M^2$, so each segment retains at least

mass and contains at least $\rho/(8\delta)$ occupied dyadic scales. Let $R_\#$ be the largest reciprocal scale in the second segment.

6.1 Separated anchors from the first segment

A fixed core label is incident to at most $LM$ occurrences altogether. The first segment therefore provides, in each $B_i$, an anchor set $B_i^0$ with

Every anchor has an occurrence witness at reciprocal scale at least $R_\#$. By (Separated), distinct anchors remain more than $\Sigma R_\#$ apart under permitted shifts.

6.2 Many actual triples in the last segment

A complete triple occurs at most $L$ times. Choose distinct actual triples $F$ from the last segment, with one scale witness per triple, so that

6.3 Scale gaps from the second and third segments

The third segment contains at least $\rho/(8\delta)$ distinct dyadic scales. Since $\rho>C\delta\ell$, every second-segment reciprocal scale is at least $\Gamma$ times every last-segment reciprocal scale. Strictly later blocks have still smaller reciprocal scales and satisfy the same gap.

Greedily retain second-segment scales with successive reciprocal-scale ratios at least $\Gamma$. Each choice removes at most $1+\lceil\log_2\Gamma\rceil\le C\ell$ scales, leaving

scales. At each selected $R_v$, choose an actual triple $t_v$. We will consider the $q|F|$ indexed outputs $t_v+f$ with $f\in F$.

7. One small-growth step avoids an exponential dimension cost

Let $U_i,\epsilon_i$ denote the neighborhood and width for the counting block containing the future covering set $B_i$.

Partitioning the integer-coefficient intervals and scalar interval covers the larger neighborhood by at most

translates of $U_i(1)$, namely a cover of $U_i(t)$. Consequently, some odd integer

satisfies

Otherwise cardinality would double every two steps. By $t\asymp(d+1)^2$, exponential growth would exceed the stated bound in $t$ of degree $d+1$.

The dimension-dependent exponent is used only to find one small-growth step. Subsequent bounds use the fixed factor two; they do not include $(2t+1)^{d+1}$ in the final constant.

7.1 Anchor neighborhoods are disjoint

The sets

are pairwise disjoint. An intersection would put an anchor difference in $U_i(2H_i)$, with shift index at most $C(d+1)^2B$ and scalar width at most $C(d+1)^2\nu R_\#$, contrary to (Reserves) and anchor separation.

Set

Then $|Z_i|=|B_i^0||U_i(H_i)|$. The cover of $B_i-B_i$ gives

8. A minimal-growth subset and covers of double and triple sumsets

Choose a nonempty $X_i\subseteq Z_i$ minimizing

over all nonempty subsets. By (Small-expansion),

8.1 Full proof of the iteration inequality

Temporarily write $X=X_i,B=B_i,r=r_i$. Given a finite shift set $C_0=\{c_1,\ldots,c_q\}$, define

Then

Every point of $(X\setminus X_j)-B+c_j$ already belongs to an earlier $(X-B)+c_v$. The number of new points at step $j$ is therefore at most

By minimal growth of $X$, a nonempty $X\setminus X_j$ satisfies

The empty-set case also holds. Thus there are at most $r|X_j|$ new points; summation gives

Successively taking $C_0=-(m-1)B$ proves by induction

8.2 Separated subsets of sumsets

The set $Z_i$ consists of at most $M$ anchor neighborhoods. The part of the minimizing set in some one neighborhood,

satisfies $|Y_i|\ge|X_i|/M$ and

If distinct elements of $P_i\subseteq mB_i$ have differences outside $U_i(2H_i)$, the translates $Y_i-p$ for $p\in P_i$ are disjoint and lie in $X_i-mB_i$. By (Iterated-growth),

A maximal separated set therefore gives a cover of $mB_i$ by the same number of neighborhood translates. Only $m=2,3$ is needed here.

9. Anchors remove the third label-count factor

Set

Consider outputs in $2B_1\times2B_2\times2B_3$ satisfying

as stated.

Each of the first two coordinates is covered by at most $r_0^2M$ translates of $U_i(2H_i)$. Fix one pair of covering boxes and solve (Slab) for the third coordinate. It lies in the intersection of $2B_3$ with

Division by $\alpha_3$ introduces no noninteger shift indices. The neighborhood shift coefficient is six and the annulus coefficient is thirty-six; both are divisible by $1,2,3$.

In this local third-coordinate set, choose a maximal subset $P$ separated by the neighborhood $U_3(2H_3)$. The set

is still separated. For one anchor this follows from the definition of $P$. A collision from different anchors would, by (Third-location), make their difference after a permitted shift only $O((d+1)^2\nu R_\#)$, contradicting anchor separation.

Thus (Sumset-packing) with $m=3$ gives

Using all separated anchors removes the third factor $M$.

The entire slab is therefore covered by at most

translates of $\prod_iU_i(2H_i)$.

10. At most one indexed output per covering box

Every $t_v+f$ satisfies (Slab), by adding the second- and last-segment annulus relations. We must also show that these indexed outputs are separated at the covering scale.

10.1 Distinct triples under one translation

If $f,f'\in F$ are distinct, at least one coordinate label differs. Their last-segment occurrence witnesses precede every future block, so their reciprocal scales are larger. The separation in (Separated) therefore exceeds the corresponding shift and scalar widths of $U_i(4H_i)$.

Coordinate differences within one covering box must belong to $U_i(4H_i)$. Thus $t_v+f$ and $t_v+f'$ cannot lie in the same box.

10.2 Outputs from different translations

Consider $t_v+f$ and $t_{v'}+f'$, and let $R_{\rm big}$ be the larger second-segment reciprocal scale. The absolute annulus residual of its triple lies in

The other second-segment residual and the two last-segment residuals have total absolute value at most $\mu R_{\rm big}/10$, by the $\Gamma$ gap.

The linear form of the output difference, plus a shift with index $O(B)$, consequently has absolute value in

If the outputs lie in one covering box, their coordinate differences belong to $U_i(4H_i)$. The box width introduces additional scalar error at most

The combined shift index is still at most $C(d+1)^2B\le Q$. This would produce a pure shift of absolute value in

contradicting the narrower forbidden interval in (Grid-gap).

Each covering box therefore contains at most one indexed output. This excludes coincidences both within a single translation and between different translations.

11. Comparing the output and covering counts

By (Translators) and (Last-triples),

Compare with (Slab-cover) and cancel $M^2$ to get

Since $\ell,K_1,L\ge1$, we may weaken this slightly to

The previously omitted case $\rho\le C\delta\ell$ satisfies this bound as well.

12. Choosing the two auxiliary small parameters

Equation (Configurations) controls the number of configurations. Add their mass, the uncovered mass, and (Nonfuture), obtaining

The factor $L^{29}$ in the last term comes from

corresponding respectively to configuration mass, the four counting-block choices, and four difference-cover costs.

For sufficiently small $\delta$, choose

Then $\delta\le\alpha/2$. The three powers of $\delta$ are

Since $0<\delta\le1$, the smallest exponent $1/2080$ gives a common bound. With $\ell,L\ge1$, (Before-choice) becomes (Sparse-count).

For the remaining $\delta$, use total mass $LM^2$ and enlarge an absolute constant. Rounding preserves the discrete label and tuple indices and every $N_s$. The working ratio $\nu/\mu$ is four times the original ratio, so

The grid estimate therefore returns the same factor $\ell$ in the original proposition, changing only an absolute constant. All bounds are independent of $h_0$. For the given finite data, one sufficiently fine grid suffices; no infinite-scale limit is required.

13. Applying this gain in D

Set

Group scales by dyadic sizes of $N_s$ relative to $M^2$. The factor $\delta^{c'}$ in (Sparse-count) offsets the amplification of small counts by $p_0<1$, giving

D then interpolates this bound with the first-power incidence count and uses cross-group positionwise counts to sum amplitudes. This finite lemma alone supplies neither the scalar kernel estimate, the positionwise counts, nor the pure-shift gap. Their hypotheses retain their separate sources.

This finite proof has been independently reviewed. The checked scope is the stated finite combinatorial proposition and its correspondence with D §6.6, excluding H's other analytic and geometric proofs and the multiple-mixing result from R used by A and B.