This preprint proves two static sieve estimates arising from a largest-prime-factor dynamical system. The first is a two-step delta-square bad-set upper bound obtained by fixing two successive cofactors and applying a Selberg upper-bound sieve to a three-linear-form normal form. The second is a regular long-window singular-series partition estimate for content-free reduced families, with all cofactors restricted above a parameterized regular threshold. The paper does not claim tightness for random initial values or a full low-cofactor-layer theorem.
Hu et al. (2026) studied this question.