Technical appendix: an all-scale prediction catalog for two fixed low columns

Starting from the chart expansions on pp. 18–22 of H, this appendix proves the finite prediction catalog needed for Lemma 5.1 and Corollary 5.2 on pp. 29–32. The essential order is to fix the unscaled labels first and then prove an absolute scale sum. A uniform approximation at each scale is not enough. We use the October 5, 2026 version; all page numbers below refer to that version.

Fourier analysis first treats fixed inputs and square-energy sequences. Projection inputs also require the full-depth estimate in the spatial compression appendix. Rational splitting and finite expansion use the analytic results in the technical appendix to Chapter 10. We do not assume the conclusions of Lemma 5.1 or Corollary 5.2, or use the later basic-catalog or active-cell results.

1. The target and its inputs

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Take a birth interval J and two fixed low columns in distinct slots i,j, with complementary slots a,b. For any finite collection of dyadic descendants, we seek fixed triples

such that the original low-column product in the local form can be replaced by

The catalog and its atoms are fixed on J. Only the coefficients and enabled labels may change with the scale; when a label is enabled, $|\lambda_\nu|r^2\le D_{\rm pred}$. The catalog size, absolute coefficient sum, and atom complexity are all at most $\exp(\operatorname{poly}(1+B+\log(2H_{\rm low})))$. The error must satisfy an absolute length-weighted scale-sum bound, not merely a bound at each scale.

We use the following inputs.

• The kernel assumptions on p. 4 of H: interior support, vanishing single-slot marginals in all four slots, and normalized derivative bounds.
• Rational splitting comes from Lemma 4.3, pp. 16–18 of H, and finite expansion on a fixed box from Lemma 4.4, pp. 18–20. The forms needed here are developed in the technical appendix to Chapter 10. In particular, we use the rational-projection height bound and the bound for the scaled quadratic part. The all-scale pooled tails are proved in this appendix.
• Elementary Fourier expansion, Plancherel, the interval Poincaré inequality, and cofactor estimates for finite matrices. Their uses are specified below.
• Lemma 3.4, pp. 9–13 of H, is used only for projection sequences. Its proof is in the spatial compression appendix. It fixes the column system before proving the full-depth estimate and does not follow from the prediction catalog.

We do not use Lemma 5.1, Corollary 5.2, Proposition 5.6, or any later conclusion as a black box.

2. Lemma 4.4: which labels are genuinely fixed?

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Source: pp. 18–20 of H. The atom and height conventions are on p. 15.

First fix a separated term Q with unscaled parameters $(\lambda,\beta,\theta,C,\chi)$. At scale r, the inputs to Lemma 4.3 are

If the Fourier lower bound is $\delta$, we obtain a rational orthogonal projection $P_s$. Its height and the sizes of $rP_s\theta$ and $r^2\lambda_{P_s}$ share a quasipolynomial upper bound R, and

The right side contains no r. Different values of r may select different $P_s$, but all admissible $P_s$ form a finite set of size at most $R^{O(d^2)}$.

Following p. 19, let $P_0$ be an even integer clearing the denominators of the symmetric rational matrix, and let $q_0$ clear the denominators of $P_s$. The labels also include

and the Fourier index $v=(p,\ell)\in\mathbb Z^{2d}$. Since $rP_s\theta$ is uniformly bounded over all admissible scales, $N$ lies in one fixed bounded set. The slow-variable cutoff and its period can be fixed in advance, rather than chosen anew at each scale. The corresponding unscaled linear frequency is

Thus the rational coefficients of $\gamma$ have height at most $D(1+\|v\|)$, whereas $\lambda_{P_s}$ is a fixed rational combination of $\lambda$ and $\theta_j\theta_k$. The period $P_1$ and the data $N$, $h$, $P_s$ can all be included in the total label in advance.

2.1 A sufficient pooled envelope

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There is no need to merge equal frequencies. Keeping repeated labels makes the estimate more direct. For each label $\alpha=(P_s,N,h,p,\ell)$, take a uniform coefficient envelope $b_\alpha$, including the supremum over the allowed amplitudes and bounded slow parameters. For any fixed J, we can arrange

Indeed, each Fourier series has $2d$ indices. Integrating by parts m times yields $(1+\|v\|)^{-m}$, while all derivatives of order m on the uniform support and period are bounded by

Choose $m\ge J(d+1)^2+4d+4$ and absorb the number of projections, lattice points, and residue classes into $D_J$. Since $d\le\log H$, this constant is still quasipolynomial. The same derivative bound holds uniformly for every admissible amplitude. We may therefore take a uniform upper envelope for each Fourier coefficient first, and then sum these envelopes.

If labels with the same $(\lambda,\gamma)$ are to be merged, sum $b_\alpha$ within each fiber. The absolute sum is unchanged. The source’s further absorption of the frequency count is possible, but is not needed here.

The uniform catalog therefore consists of unscaled labels. A separate numerical frequency list for each r cannot replace this construction.

3. What Lemmas 4.5 and 4.6 supply

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Source: pp. 20–22 of H.

In Lemma 4.5, choose $\sum\alpha_j=\sum j\alpha_j=0$ and put $S=\sum j^2\alpha_j$. The phase identity separates the original two columns into

The integer lift differences $k_i,k_b$ are bounded by the original support conditions. Their extra terms are fixed linear phases, whose sizes do not enter atom complexity. The remaining coupled amplitude is smooth in three independent lift variables. Its Fourier expansion separates into three factors. The two-column separation is applied only twice, so the dimension and quasipolynomial budget remain controlled. On shell T, the absolute coefficient sum is $DT^{-100}$ and the complexity is $(DT)^C$. None of these steps depends on the interval scale.

The first part of Lemma 4.6 follows directly from Fourier analysis and Plancherel. In $(y_a,y_b)$ coordinates, the normalized smooth kernel has an absolutely summable Fourier expansion. After removing that kernel, the remaining convolution kernel is $Q(t)\zeta(t/r)$. Its multiplier bound is $CrA_r$, which becomes $CA_r$ after canceling the form normalization and the L² scaling. The fixed bounded factors $\psi_a,\psi_b$ can be treated as multipliers.

The second part of Lemma 4.6 supplies a separable approximation at one scale, not an all-scale catalog: its general smooth bump has an additional scale r. The catalog construction instead stops at $e(\lambda_*t^2+\gamma t)$ in Lemma 4.4, distributes only $e(\gamma t)$ between the two slots, and keeps the original kernel $w_I$ and the quadratic phase. This is the form needed for Corollary 5.2.

4. Scale sums for a fixed linear frequency

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We first work without any height-bin result for several frequencies.

By the four marginal cancellation conditions, in $(y_a,y_b)$ coordinates the integral of $w_I$ in either variable is zero. Its double primitive is compactly supported in an interior rectangle of $I\times I$. Expand the normalized double primitive in a Fourier series and differentiate in each variable. This yields an absolutely convergent tensor expansion

The one-dimensional bumps on both sides have mean zero. Their normalized derivative bounds grow as fixed powers of p,q, while the coefficients decay to any fixed polynomial order. All constants are independent of the number of scales.

Fix a frequency $\nu$. For the mean-zero bump rows

decompose by relative frequency $|\xi+\nu|\asymp 2^\ell/r$. At a fixed scale the intervals are disjoint. If $\ell<0$, subtract the interval average and apply Poincaré; the row-family norm is $O(2^\ell)$. If $\ell\ge0$, take two frequency antiderivatives and integrate the bump by parts twice; the norm is $O(2^{-2\ell})$. For fixed $\xi\ne-\nu$ and $\ell$, only a bounded number of r occur. Plancherel and summability in $\ell$ prove

The constant is independent of $\nu$, the depth, and the chosen subcollection. A fixed $\psi$ is absorbed through $G=\psi F$. We do not claim that $\psi\omega_I$ still has mean zero.

Consequently:

• For a fixed input F, use the displayed estimate directly.
• For an energy sequence $Z_I$, use $|W_{I,\nu}Z_I|\le C\|Z_I\|_{L^2(I)}$ for each row and then sum the given energy bound, obtaining the L² factor.
• For a projection input $P_IF$, the row family $W_{I,\nu}M_\psi$ is local and Bessel. This is the only case in which Lemma 3.4 is used, at a cost $\operatorname{poly}(\log(2D_h))$.

Summing the tensor indices and applying Cauchy–Schwarz between the two slots bounds the absolute scale sum for the fixed linear phase $e(\gamma t)$ by

For fixed inputs and energy sequences, omit $D_h$.

4.1 Absolute scale sums for pooled tails

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Fix one tail label with unscaled phase $(\lambda,\gamma)$. Its coefficient may change with scale, but its absolute value is at most $b_\alpha$. This label is enabled only on scales satisfying $|\lambda|r^2\le R$.

Write

Use the preceding argument for the linear part. For the quadratic error, the kernel support condition $|t|\le Cr$ directly implies

Thus only the local absolute L²×L² estimate is needed, with no new smooth Fourier expansion. At a fixed depth, intervals are disjoint, projections are contractions, and the energy of an energy sequence at that depth is bounded by its total energy. The resulting cost is $C(1+L)^2\|F_a\|_2\|F_b\|_2$. When $\lambda\ne0$,

When $\lambda=0$, the error is zero. The estimate permits any subset of scales. Hence the total loss over all tail labels is at most

Choose the pooled tail mass below $e^{-2m}(1+R)^{-2}$ to obtain an acceptable absolute scale-sum error. The logarithmic budget for this truncation is still $\operatorname{poly}(1+m+\log H)$.

This is noncircular. The fixed-frequency linear Bessel estimate was proved above. The geometric sum uses a fixed unscaled $\lambda$; an unrelated $\lambda_r$ cannot be selected at each scale. Pooling ensures precisely this fixedness.

5. How bounded height controls scales without a catalog-size cost

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Source: Lemma 3.5 on pp. 13–14 of H. We explain its elementary basis and the form used here.

Given a finite-dimensional real parameter vector $\Theta\in\mathbb R^d$, consider all nonzero evaluations $|a\cdot\Theta|$ with rational coefficient height at most R. Clear denominators in any minimally linearly dependent family of coefficient vectors. Integer cofactors yield nonzero integer dependence coefficients of size at most $d!R^{d^2+d}$. Thus, if successive nonzero evaluation magnitudes differ by a factor greater than $4d\,d!R^{d^2+d}$, their coefficient vectors are linearly independent. There can be at most d of them.

Greedily select the smallest uncovered evaluation. The logarithms of all nonzero values then lie in at most d intervals of length $O(d^3\log(2R))$, and occupy at most $O(d^4\log(2R))$ dyadic magnitude bins. It follows that:

• The number of scales satisfying $\alpha\le r^j v\le\beta$ is at most $\operatorname{poly}(d,\log(2R),\log(2\beta/\alpha))$, for $j=1,2$. A more precise count keeps a factor $d\log(\beta/\alpha)$; only this polynomial version is needed here.
• $\sum_r\max\{r^jv:r^jv\le\tau\}\le C d^4\log(2R)\tau$. This follows directly by a geometric sum within each occupied magnitude bin.

This is not a statement about an arbitrary finite frequency set. The frequencies must arise from the same parameter vector of controlled dimension, with controlled rational height.

For one fixed separated term, we can take

The dimensions are $O((d+1)^2)$ and $d+1$, respectively, and $d\le\log H$. All quadratic parameters and all differences of linear centers in the finite catalog are bounded-height evaluations of these vectors.

6. An estimate for one Fourier-supremum band

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Fix $e^{-m-1}<A_r\le e^{-m}$ and put $K=1+m+\log(2H)$. If the fixed cutoff has integral greater than 1, merge the bounded number of top bands into m=0; this only changes absolute constants. Lemma 4.3 and Sections 2 and 4 first provide a main catalog satisfying:

• The rational heights of the projections and frequencies, the quantities $|\lambda|r^2$ for enabled quadratic terms, and the main absolute coefficient sum A are all at most $\exp(\operatorname{poly}K)$.
• The discarded tails have absolute scale-sum error $e^{-2m}\operatorname{poly}(\log(2D_h),1+L)\|F_a\|_2\|F_b\|_2$.

Choose $0<\tau\ll1$ such that $\log(1/\tau)=\operatorname{poly}K$, $\tau^{-1}>2R$, and

Mark a scale exceptional if some possible nonzero $\lambda$ or center difference $\gamma-\gamma'$ satisfies

Section 5 bounds the total number of exceptional scales by $\operatorname{poly}K$. At each such scale, apply the original convolution multiplier bound $CA_r\le Ce^{-m}$ directly. Do not first sum the exponentially many main coefficients absolutely. The total loss from exceptional scales is therefore $e^{-m}\operatorname{poly}(K,1+L)\|F_a\|_2\|F_b\|_2$.

6.1 Replacement errors at nonexceptional scales

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Enabled quadratic terms satisfy $|\lambda|r^2\le R<\tau^{-1}$, so at a nonexceptional scale they must satisfy $<\tau$. The linear centers form small clusters of diameter $<\tau/r$, separated by distances $>\tau^{-1}/r$. They can be defined as the connected components of the tiny-distance relation. Since intermediate distances are absent and $2\tau<\tau^{-1}$, this relation is itself transitive. The source’s alternative of choosing a smaller threshold and then adjusting for the catalog size also works.

Replace the quadratic phase by 1 and the centers in each cluster by one representative from the original catalog. The small-quantity sum in Section 5 bounds the total replacement error over all scales by

which meets the budget. We have not yet used the band estimate to be proved.

6.2 Why the combined coefficients are $O(e^{-m})$

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Let $c_C(r)$ be the combined coefficient of cluster C, with representative $\gamma_C$. Pair the original function $Q(r\cdot)$ with $\zeta(u)e(-r\gamma_Cu)$. This cluster contributes

The other clusters have normalized separation exceeding $\tau^{-1}$, so their total contribution has absolute value at most $C_NA\tau^N$. Truncation, quadratic replacement, and small-cluster replacement have already supplied controlled uniformly small errors. Since $\zeta\ge0$ and $\int\zeta>0$, and the original pairing is at most $A_r\le e^{-m}$, we obtain

This argument uses the Fourier supremum of the original Q. The main catalog’s absolute coefficient sum A alone does not imply the conclusion.

6.3 Joint Bessel counting over centers and scales

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For each scale and cluster representative, take $\nu=\pm\gamma_C/(b-a)$ and consider the rows from Section 4. First absorb the fixed bounded multiplier $\psi$ as a fixed multiplication operator on the input. Decompose by

At a fixed scale and center, the row-operator norms are still

After fixing any Fourier variable $\xi$, all nonzero $|\xi+\nu|$ are bounded-height rational evaluations of the same parameter vector

Thus Section 5 shows that, for fixed $\xi,\ell$, only $\operatorname{poly}K$ scales occur, independently of the number of catalog entries. At one scale, the centers are separated by more than $1/r$. A shell of length $O(2^\ell/r)$ contains at most $C(1+2^\ell)$ centers. The total Fourier overlap multiplicity is

Apply Plancherel, take the square root of the overlap multiplicity, and sum. This yields

This is the joint Bessel bound. Exact centers $\xi=-\nu$ form only finitely many null points and create no additional atomic term in L².

For projection sequences, apply Lemma 3.4 to this whole local Bessel row family; only then does the factor $\operatorname{poly}(\log(2D_h))$ enter. For energy sequences, use the separated-center Bessel bound on each individual I and sum the given energy bound, without compression. Cauchy–Schwarz between the two slots, $|c_C(r)|\le Ce^{-m}$, and summation of the tensor coefficients imply

The estimate for one Fourier-supremum band has now been proved from the preceding analytic steps, without using the conclusion of Lemma 5.1.

7. From the band estimate to a uniform finite list of triples

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We complete the error budget in the order in which the parameters are chosen.

1. Expand the fixed low-column pair as in Section 3. Shell T has coefficient mass $DT^{-100}$, and each term in that shell has complexity at most $(DT)^C$.
2. Apply the band estimate just proved to each term and sum in m. This yields its all-scale bound, with complexity loss only $\operatorname{poly}(\log(2H_s))$. The total loss from high separation shells is therefore

Choose $\log T_0\le\operatorname{poly}(B+\log(2H_{\rm low}))$ so that this is at most a fixed portion of the budget.

3. Only finitely many terms remain, with quasipolynomial bounds on their total coefficient mass and complexity. For these terms discard sufficiently small Fourier-supremum bands, namely m above some $M\le\operatorname{poly}(B+\log(2H_{\rm low}))$. Since $\sum_{m>M}e^{-m}\operatorname{poly}(m+\log H_s)$ decays exponentially, this error also meets the budget.
4. For each retained separated term, use $\delta=e^{-M-1}$ uniformly to obtain all admissible rational projections. Truncate using the fixed total labels of Section 2 and the absolute scale-sum tail estimate of Section 4.1. When choosing the tail mass, first divide by the known separation-coefficient budget, quadratic-size budget, and finite number of terms. Their logarithms are all polynomial, so the final Fourier truncation remains quasipolynomial.
5. Distribute the retained linear phase exactly as

and define

These are fixed bounded atoms. Unrestricted real linear modulations do not increase complexity. $\lambda$ also comes from a fixed label. At each scale only the enabled labels and coefficients change.

The finite term count, label count, coefficient bounds, and atom complexities all satisfy

The total absolute length-weighted error from the three discarded parts is at most

The whole catalog is independent of the tree depth, the particular inputs in slots a,b, and the selected descendant collection. It depends only on the fixed low columns, the kernel class, and B. The projection budget enters only through the compression constant in the error bound.

8. Conditions to retain throughout the proof

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8.1 Five points about all-scale summation

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• Finite pooling of projections and tail envelopes in Lemma 4.4 follows directly if repeated labels are retained.
• For the absolute scale sum of a fixed tail phase, first prove fixed-modulation Bessel and then use the quadratic geometric tail. The order is noncircular.
• To use height bins in Bessel, temporarily add the real parameter $\xi$ for each Fourier variable. This increases the dimension by only 1. Do not add a separate Bessel constant for every frequency.
• At arbitrarily fine depths, the zero quadratic phase does not supply a summable absolute-error factor. The linear part is instead controlled by mean-zero Bessel, so no depth cutoff is needed.
• Small divisors near resonance only shift the locations of scale intervals. They do not increase the number of occupied magnitude bins beyond a polynomial.

8.2 The order of the steps

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• The fixed-linear-frequency Bessel estimate needed on pp. 30–31 is proved first in Section 4 of this appendix. The tail estimate therefore does not use the multifrequency conclusion still to be proved.
• The cutoffs, periods, and lattice labels on pp. 19–20 are chosen uniformly under one height budget, and $\lambda,\gamma$ are unscaled parameters.
• The per-depth contraction estimates on p. 30 are applied separately to fixed inputs, projection inputs, and energy sequences. When energy parameters occur in both slots, an L² factor is allowed and absorbed into a fixed-degree polynomial.
• The fixed multiplier on p. 31 is applied by precomposing the Bessel row family. The bump after multiplication need not have mean zero.
• The truncations on p. 32 are ordered as follows: first separation shells, then small $A_r$ bands, and finally the pooled Fourier tail. The final accuracy cannot be fixed before allowing for the preceding coefficient budgets.

8.3 Two further analytic results

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• The full-depth spatial compression in Lemma 3.4 is proved in the spatial compression appendix. The row family here is local and jointly Bessel, and the columns are inherited, so its assumptions hold. Fixed inputs and energy sequences do not use this compression.
• Rational splitting comes from Lemma 4.3 and finite expansion on a fixed box from Lemma 4.4. The forms needed here are developed in the technical appendix to Chapter 10. Only these analytic results are used here, not active-cell counting. The required all-scale pooled tails are proved in this appendix.

8.4 Returning to the basic catalog construction

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This appendix supplies a fixed finite prediction catalog for each low-column pair and an absolute scale-sum error bound. Sections 7–8 of the main text use them to produce and count prediction columns. Sections 10–14 also require delayed replacement, high-slot detection, and successive accuracy choices to obtain the basic budget needed for Proposition 5.6. This appendix does not establish the actual later histories, the whole helper basis, or the main theorem of H.