Long time stability for cubic nonlinear Schrödinger equations on non-rectangular flat tori

The paper proves almost-global H^s-stability for cubic NLS on admissible (non-rectangular) flat tori via a finite-dimensional reduction and a parameter-dependent normal-form scheme; it establishes the quartic resonance identity Ω = 2 g(n1−n2, n1−n4) and a quantitative Diophantine lower bound on g(a,b), showing that nontrivial four-wave resonances are excluded and near-resonant quartic terms are integrable (Q(4)) under a cutoff κ, with M constrained by ε−μ_d r/s ≤ M ≤ ε−ν (Theorem 1.3) . Crucially, the stability result holds for a typical set Θε of initial data of almost full measure, not for the entire small ball; the paper explicitly constructs such a set and proves meas(Θε) ≥ (1 − ε^c) meas(Π_M B_s(ε)) (e.g. 1 − ε^{1/39} in Theorem 2.7) . By contrast, the model’s solution incorrectly sets Θε equal to the whole ball Π_M B_s(ε), thereby claiming stability for all small data and bypassing the parameter-dependent small-divisor exclusions that are essential in the paper’s argument. The model’s high-level steps (quartic factorization, normal-form/DBP iteration, M-scale choices) qualitatively mirror the paper, but it omits the internal-parameter normal form and Lipschitz tracking central to controlling small divisors and near-resonances, and it overclaims typicality. Hence the paper is correct, while the model’s solution is not.