5 Layer L2: continuum estimates
Deterministic Fourier core, the torus Green’s function, singular convolutions, and the discretization that feeds the lattice machine.
Functions on \(\mathbb {T}^4\) (resp. \(\mathbb {R}^4\)) invariant under coordinate permutations and per-coordinate sign flips (hyperoctahedral \(B_4\)). The formalization uses this single-slot interpretation.
Characters \(e_k\), kernel coefficients (3.23) \(\widehat K(\alpha ,\beta ) = \langle e_{-\alpha }, K e_\beta \rangle \), Plancherel on the product torus, multiplier calculus, and the normalization ledger dictionary.
\(G := \int _0^\infty e^{-t}\, \Theta (t,\cdot )\, \mathrm{d}t\) with \(\Theta \) the periodized Gaussian (Bessel-potential representation; a Fourier tsum is ill-defined since \(\langle k\rangle ^{-2}\notin \ell ^1\cup \ell ^2\) and \(G \notin L^2\)). Lemmas: \(\widehat G(k) = \langle k\rangle ^{-2}\); \(G \in \mathcal{E}\); \(\lvert \nabla ^k G(z)\rvert \lesssim \lvert z\rvert ^{-2-k}\) (\(k \le 2\)); the improved near-diagonal bound for \(G - \frac{1}{4\pi ^2\lvert z\rvert ^2}\).
Chain convolution bounds for \(\lvert z\rvert ^{-2}\)-kernels on \(\mathbb {T}^4\); the estimates (5.3)–(5.4); \(\eta _\varepsilon \)-support handling.
Reduction of 26 to the lattice counting bound (5.5): choice of \(\mathcal{A}\), discretization \(y_j = \lfloor \varepsilon ^{-1}x_j\rfloor \), and the singular-integral bounds (5.2)–(5.4).
The bounds (4.3) and (4.4) on \(\mathcal{J}_{2n,\mathrm{prim}}\), with and without the \(\varepsilon ^2 + \max \lvert x_i - x_j\rvert ^2\) insertion, for \(1 \le n \lesssim \lvert \log \varepsilon \rvert \).