8 Layer L5: random parametrix
The chaos-integrated kernels, the parametrix identity, and the \(L^2\)-theory on the good event (paper §3.1–§3.3 random content and §3.4 Step 1).
\(\mathcal{I}_m = \sum _\kappa \mathcal{I}_{m,\kappa }\): the deterministic profile kernels integrated against chaos projections of \(\prod _{i\in S}\xi _\varepsilon (x_i)\).
Random \(\mathcal{RI}_{m,\kappa }\), \(\mathcal{P}_m\), \(\mathcal{P}_\varepsilon = \sum _{m\le A}\mathcal{P}_m\). Operator realization: \(K_\varepsilon := G \circ M_\varepsilon \) with \(M_\varepsilon \) multiplication by \(\lambda _\varepsilon \xi _\varepsilon - \mathcal{C}_\varepsilon \); invertibility predicate \(:=\) invertibility of \(1 - K_\varepsilon \) in \(\mathcal{B}(L^2)\); primary mode objects \(\mathrm{modeCoeffH}(\alpha ,\beta ) = \big\langle e_{-\alpha }, \big((1-K_\varepsilon )^{-1}G - G\big)e_\beta \big\rangle \) on the invertibility event and \(0\) off it; the conditioned HS-difference lemma \((1-K_\varepsilon )^{-1}G - G = (1-K_\varepsilon )^{-1}\, G M_\varepsilon G\) with the \((1+\log )\)-weighted Hilbert–Schmidt bound.
\(\lambda _\varepsilon \xi _\varepsilon \circ \mathcal{P}_{m-1} = (1-\Delta )\mathcal{P}_m + \sum _{1\le q\le m/2} \mathcal{C}_{2q}\mathcal{P}_{m-2q}\) and its mirror (3.17), as Fourier-multiplier identities of random kernels; proof by the three-case analysis of pairings of the new index.
Three-case analysis of the pairings of the new index at graded-kernel level, under an explicit Fubini-integrability ledger; the ledger is discharged almost everywhere from almost-sure continuity of the mollified noise, giving the almost-everywhere kernel and coefficient agreement forms.
\(\mathcal{L}_\varepsilon \mathcal{P}_\varepsilon = 1 + \mathcal{R}_\varepsilon \) in multiplier form, with the explicit error kernels \(\mathcal{R}_\varepsilon , \mathcal{R}'_\varepsilon \).
Telescoping the graded parametrix identity across orders \(m \le A\) on both sides, under the same Fubini ledger as 44 (discharged almost everywhere by the a.e. closure chain).
\(\mathbb {E}\lvert \widehat{\mathcal{P}_m}(\alpha ,\beta )\rvert ^2 \le \lambda _\varepsilon ^2\, C(C\lambda )^{2m-2}\min \big(1, \varepsilon ^{-8}\langle \alpha \rangle ^{-4}\langle \beta \rangle ^{-4} \langle \varepsilon ^2(\alpha +\beta )\rangle ^{-8}\big)\): the Wick reduction to 34.
\(\mathbb {E}\| \mathcal{P}_m\| ^2_{L^2\to L^2}\) via Plancherel double sums ((3.30)–(3.31)); \(\mathbb {E}\| \mathcal{P}_\varepsilon \| \le \varepsilon ^{-12}\), \(\mathbb {E}(\| \mathcal{R}_\varepsilon \| + \| \mathcal{R}'_\varepsilon \| ) \le \varepsilon ^{30}\); the Chebyshev good event \(Z_\varepsilon ^c\); Neumann inversion and \(\| \mathcal{L}_\varepsilon ^{-1} - \mathcal{P}_\varepsilon \| _{L^2\to L^2} \le \varepsilon ^{12}\) (3.33).
Kernels vs. \(\mathcal{B}(L^2(\mathbb {T}^4))\): operator norms from Plancherel double sums (no named Hilbert–Schmidt API assumed); Neumann-series inversion.
Kernel coefficients against mode vectors identified with operator inner products; the coefficient-level Neumann series agrees with the operator inverse below norm one.