Appendix: three-variable chart calculations

This appendix concerns Definition 4.1 and Lemmas 4.2–4.4 on pp. 15–20 of H, and proves only the forms needed for the fixed-triple scale estimate. We denote the number of spatial variables by $s\in\{1,2,3\}$ to distinguish it from the accuracy parameter $k$ in the main text.

1. Charts, translations, and quantitative dependence

H source, p. 15 · H source, p. 16 · This chapter: main text, Section 4

Write $\mathrm e(t)=\exp(2\pi it)$. Consider

where $\Lambda$ is symmetric, $V$ and $b$ are an arbitrary real matrix and vector, and $C$ is a rational matrix. Complexity $H\ge\mathrm e^2$ means

The fixed-order Gevrey constant $c_0$ does not change with any real frequency. At each point there are at most $(2H+3)^d$ nonzero lattice terms, so the absolute value of $Q$ has a quasipolynomial upper bound in $H$. No control of $V,\Lambda,b$ is needed.

Conjugation, products of a fixed number of factors, and affine substitutions preserve this form. Translation deserves an explicit calculation. Choose an integer $P$ such that $PC$ is an integer matrix; we may take $P\le H^{d^2}$. For a horizontal shift $a$, write

Relabel the lattice points and set $u=Vz,w=u-n$. Modulo integers,

The last term is constant. The two middle terms enter the unrestricted ordinary linear phase, and the small-shift term involving $w$ is absorbed into the smooth amplitude. The support and derivatives of the amplitude depend only on the bounded $a_0$, not on the possibly large $m_0$. Translating the ordinary quadratic phase changes only its linear and constant terms. Thus, after recentering at a fixed path point $y$, the complexity remains uniform, and all new real parameters are fixed along that path.

Below, $D_1,D_2,\ldots$ denote quasipolynomial quantities in $H$ and the reciprocal of the current threshold. Only a fixed number of such enlargements are made at each step.

2. A large Fourier value controls every small return

H source, p. 16 · H source, p. 17 · This chapter: main text, Section 6

Let $\vartheta$ be supported in a fixed box, with absolute value and Lipschitz norm at most $H$. Suppose

The Fourier transform uses the exponential $\mathrm e(-\xi\cdot z)$. We may choose a uniform $D_1$ such that

and such that $\widehat F$ is $D_1$-Lipschitz.

Write

The last definition applies to any square matrix $B$. For any return

translate the variable and relabel the lattice points by $n+m$. This yields

whenever the parenthesized quantity on the right is at most $1$.

No error depending on $V$ is hidden here. Set $u=Vz,w=u-n,d=d_h$. The nonconstant increment of the chart phase is

The first two terms equal

The last term contains only $w$, bounded on the amplitude support, and the small quantity $d$. The discarded $m^TCn$ is an integer. The ordinary quadratic phase contributes $2\Lambda h$, while the increment of the ordinary linear phase is constant. Errors from translating the amplitude and $\vartheta$ are controlled by the same right side.

Choose $\xi_0$ with $|\widehat F(\xi_0)|\ge3\gamma/4$, and set

Take the integer

where $C_s$ is sufficiently large to include the volume constants of unit balls in dimensions at most three. Then set

If the return satisfies $\|h\|+\|d_h\|\le a$, then for every $1\le j\le N$ we may apply the return identity to $(jh,jm)$. Consequently, the absolute values of the Fourier transform satisfy

If $\|\Omega(h,m)\|\ge1$, the balls of radius $\rho$ around these points are disjoint. The Lipschitz bound implies $|\widehat F|\ge\gamma/3$ on each ball. Plancherel then yields

a contradiction. Thus every sufficiently small return satisfies

The reciprocals of all the chosen small quantities remain quasipolynomial in $H,\gamma^{-1}$. We have not assumed that the Fourier supremum is attained.

3. Constructing a rational fast space from small returns

H source, p. 17 · H source, p. 18 · Minkowski: Henk, Theorem 1.3, p. 2 · This chapter: main text, Section 6

For two small returns, direct expansion yields

By the explicit choice of $a$ above,

The pairing on the left is an integer, so it is zero.

First suppose $d\ge1$. Define the centrally symmetric convex body

It contains the cube and has nonempty interior. Let $\lambda_i$ and $\mu_i$ be the successive minima of $\mathcal K$ and $\mathcal K^\circ$, respectively, with respect to the integer lattice. The required transference bound is

For the lower bound, two linearly independent families of integer vectors whose dimensions sum to $d+1$ must have a nonzero integer pairing between them. Apply the polar-body inequality.

The upper bound needs only the upper half of Minkowski's second theorem. Its precise statement appears in Martin Henk, Successive Minima and Lattice Points, Theorem 1.3, p. 2:

where $B$ is a centrally symmetric convex body with nonempty interior and $\Gamma$ is a full-rank lattice.

To apply it, choose from $\mathcal K$ the columns of a matrix $T$ with largest absolute determinant. Maximality and convexity yield the following inclusions:

Therefore

Multiplying the two Minkowski upper bounds yields

Every reverse-order product is already at least $1$, so each is at most their total product. This proves the transference bound.

Take

Among the $d+1$ half-open intervals

one contains no successive minimum. Denote its left endpoint by $b_0$. Set

Every generating integer vector $n$ can be written as

Then $m=Pn,d_h=-Pw$ defines a small return within the threshold $a$. The integer pairing above is zero, proving

The threshold also has an explicit lower bound:

Since $d\le\log H$, this reciprocal is still a quasipolynomial quantity of the required type.

Suppose $r_0=\dim U<d$. The empty interval ensures $\lambda_{r_0+1}>L_0b_0$. The transference bound provides $d-r_0$ independent integer vectors

Their pairings with the short integer vectors generating $U$ have absolute value less than $1$, so the pairings vanish. These vectors therefore span exactly $U^\perp$. The definition of the polar body also implies

Let $S$ be the full-row-rank integer matrix with these rows, and define

The Gram matrix $SS^T$ is an invertible integer matrix with determinant at least $1$. Cramer's rule and the bounded integer minors control both the rational height of $P_s$ and the size of

All bounds are quasipolynomial. If $r_0=d$, simply take $P_s=0$.

4. Controlling the effective quadratic matrix in each of three coordinate directions

H source, p. 17 · H source, p. 18 · This chapter: main text, Section 6

Write $V_f=P_fV$ and define

For each coordinate vector $e_j\in\mathbb R^s$, sample at equally spaced points in $[0,b_0]$ the values

Partition the horizontal torus into cubes of side length at most $b_0/4$. With more sample points than cubes, two points lie in the same cube. Their difference yields

Increase the absolute constant if needed to handle noninteger partition counts. This return still satisfies the threshold $a$. Its integer part satisfies $P^{-1}m\in b_0\mathcal K$, so $m\in U$.

Thus

and isotropy removes the fast component:

The already controlled quantities $V_s,A,d_h$ bound this expression quasipolynomially.

The matrix identity is

Substituting into the return increment yields

Every term on the right is controlled. Dividing by $t_j$, which has a quantitative lower bound, controls $\Lambda_*e_j$ column by column. There are at most three columns, so

When $d=0$, there is no horizontal matrix or chart lattice. Every sufficiently small $h$ is a return, and $\Omega=2\Lambda h$. Taking $h=ae_j/2$ directly controls $\Lambda$, without introducing a zero-dimensional convex body.

5. Obtaining frequencies of a fixed form after splitting

H source, p. 18 · H source, p. 19 · H source, p. 20 · This chapter: main text, Section 6

Now assume only that a rational splitting has been obtained, with its height and the sizes of $V_s$ and $\Lambda_*$ at most $B$. This section no longer uses a large Fourier value.

Set

The symmetric part satisfies

Isotropy means that the bilinear form defined by $C_{\mathrm{sk}}$ vanishes whenever both arguments lie in the fast space, so

The ordinary quadratic part in the first identity and the first term in the second, together with $z^T\Lambda z$, combine to form $z^T\Lambda_*z$.

Choose an even integer $P_0$ clearing the denominators of $C_{\mathrm{sy}}$. Then

depends only on $n\bmod P_0\mathbb Z^d$. Let $q_0$ be a common denominator of $P_s$. On the fixed $z$ box, $u_s=V_sz$ is bounded and $w$ lies in the amplitude support, so

lies in a bounded box of $q_0^{-1}\mathbb Z^d$. There are only quasipolynomially many possible $N$ and residue classes $a$.

Group terms by $P_sn=N$ and $n\equiv a\pmod{P_0\mathbb Z^d}$. Remove the known residue-class constant and the linear phase

To Fourier expand the remaining amplitude, introduce two independent variables $s_0,u\in\mathbb R^d$. All cutoffs below come from one fixed Gevrey cutoff by coordinate products, translations, and dilations at controlled scales, so they have uniform corresponding Gevrey derivative bounds. Choose $\eta(s_0)$ so that on all possible $V_sz$ its value is $1$, with support in $[-L_\eta,L_\eta]^d$. We may take $L_\eta=4(1+C_{\mathrm{box}}B)$, where the fixed constant $C_{\mathrm{box}}$ bounds the Euclidean norm on the $z$ box in use. Choose a smooth function $E$ such that

Define

At $s_0=P_su$, the input to the equality cutoff is $P_sn-N$. This is a lattice point with denominator $q_0$, so the cutoff keeps exactly the specified group, not an approximate group.

In $u$, this function has period $P_0$. Choose the controlled integer $P_1=\lceil4L_\eta+2\rceil$. The support of $\eta$ then lies strictly inside this period box, allowing periodization in the $s_0$ direction. The Fourier modes are

Substitute $s_0=P_su$ and restore the linear phase above to obtain

Thus we have the exact, absolutely convergent expansion

This explains the form $\omega_\nu=\beta_y+V_y^T\sigma_\nu$ in the main text. The remaining real horizontal values enter only through $Vz$; they do not become unbounded rational coefficients.

6. Truncation, different splittings, and a fixed rational evaluation family

H source, p. 19 · H source, p. 20 · This chapter: main text, Section 5 · This chapter: main text, Section 6 · This chapter: main text, Section 7

All cutoff scales, periods, numbers of lattice groups, amplitude supports, and bounded phase coefficients are controlled by a quasipolynomial quantity in $H,B$, denoted by $D_2$. After normalizing each periodic variable to period one, the product and chain rules yield

This bound does not involve the unbounded fast part of $V$. The differentiation variables are $s_0,u$; we have not differentiated the composition with $Vz$.

For a nonzero Fourier index $v=(p,\ell)\in\mathbb Z^{2d}$, choose its coordinate with largest absolute value and integrate by parts $m$ times. This yields

A shell $T\le1+\|v\|<2T$ contains at most $(CT)^{2d}$ indices. Taking $m$ to be a sufficiently large fixed multiple of $(d+1)^2$ bounds the absolute coefficient sum over all groups in this shell by

Since $d\le\log H$, the prefactor for this fixed high derivative order is still quasipolynomial. The total absolute coefficient sum is therefore at most $CD_3$, and the tail above $T_0$ is at most $CD_3T_0^{-2}$. Given $\varepsilon_0>0$, choose $T_0$ quasipolynomial in $H,B,\varepsilon_0^{-1}$. This yields the uniform sup-norm truncation required in the main text.

The entries of the rational projection matrices have height at most $B$. The number of such matrices is at most

We may first take the coefficient bounds above for every possible matrix and bounded group, then sum them. The cost is still quasipolynomial. Not every matrix needs to occur as a Weyl splitting at some scale. The actual splitting may vary within this fixed finite set.

Within the truncation range, all rational entries of $N,P_0,P_1,P_f,P_s,p,\ell$ have uniformly quasipolynomial height. Matrix multiplication, summation in fixed dimension, and clearing denominators also bound the height of $\sigma_\nu$ quasipolynomially. For the rescaled family

the unscaled entries of the effective quadratic matrix are rational linear combinations of the entries of $\Lambda_y$ and pairwise products of entries of $V_y$. The linear frequencies are rational linear combinations of the entries of $\beta_y$ and $V_y$.

We may therefore collect these finitely many unscaled real quantities into $\chi_y$. Its dimension is bounded by a fixed polynomial in the horizontal dimension, and all evaluation coefficients have height at most

This is all the uniformity needed for the fixed-triple scale count. No numerical upper bound on large real frequencies is needed, and the same rational fast space need not work at every scale.

7. Scope of the verification

H source, p. 15 · H source, p. 16 · H source, p. 17 · H source, p. 18 · H source, p. 19 · H source, p. 20 · This chapter: main text, Section 14

The derivation above reconstructs the Weyl output and finite expansion on pp. 16–20 used for the three-variable six-atom integral. The classical inputs are the explicitly stated upper bound in Minkowski's second theorem, together with Plancherel, integration by parts, and smooth periodic Fourier expansion in finite-dimensional Fourier analysis. The absolute convergence and quantitative tail of the last expansion have been developed in this appendix.

This appendix does not reconstruct the inverse theorem or detection lemma in Section 4, or supply the basic-catalog construction in Proposition 5.6. The main text develops the path budget and deletion error from the supplied basic data.