A recent note of Goda disproves the Kazashi--Sloan counting conjecture and proves, without that conjecture, the dimension-independent averaged squared worst-case-error estimate \(\eav^2(n)=O(n^{-1/2}\log n)\) for unshifted rank-one lattice rules in weighted unanchored Sobolev spaces of smoothness one, under the corresponding summability condition on the weights. The same note asks whether this rate can be improved. We give an affirmative answer. The key observation is the exact identity \(\sum_{\kappa=1}^{n-1}T_n(\kappa)=2H_{(n-1)/2}^2\), combined with the uniform Kazashi--Sloan estimate \(T_n(\kappa)\le \pi^2/6\). This replaces threshold counting by a first-moment-to-all-moments estimate. Consequently, for every odd prime \(n\) and every finite dimension \(d\), the averaged squared worst-case error is bounded by \(n^{-1}(\log n)^2\) times an explicit weighted subset sum. For product weights, this proves the existence of an unshifted rank-one lattice rule satisfying \(e(n,\bm z)=O_\gamma(n^{-1/2}\log n)\), uniformly in \(d\), whenever \(\sum_{j\ge1}\gamma_j<\infty\). Numerical experiments confirm the exact arithmetic identity, illustrate the resulting moment conversion, and compare the new bounds with the previous rate scale.

The first rigorous analysis of the compact genetic algorithm (cGA) on $\mathrm{LeadingOnes}$ established an $O(\mu n\log n)$ high-probability upper bound and left open an asymptotically matching lower bound. We show that the logarithmic factor is not intrinsic. For the standard cGA with borders $[1/n,1-1/n]$, a lattice-aligned update parameter $\mu$, and $\mu\ge Cn\log n$, the optimization time $\tau_{\mathrm{opt}}$ satisfies $\Pp(\tau_{\mathrm{opt}}>C'\mu n)\le\exp(-c\mu/n)$. For every fixed $\eta\in(0,1)$ and every such parameter with $\mu\le(2-\eta)^n/n$, a matching lower bound holds with probability $1-\exp(-\Omega_\eta(\mu/n))-\exp(-\Omega_\eta(n))$. Consequently, $\tau_{\mathrm{opt}}=\Theta(\mu n)$ with high probability throughout every polynomially bounded low-genetic-drift regime covered by the theorem. The proof derives the exact coordinate transition kernel, establishes a localized multiscale contraction theorem for adaptive reflected biased walks, and advances all marginals along diagonals in coordinate--accuracy space. A deterministic log-odds flow reveals the associated traveling wave. An independent aggregate-mass martingale argument yields the lower bound and explains why an exponential-order upper restriction on $\mu$ is unavoidable in any uniform high-probability $\Omega(\mu n)$ statement.

We study Wilf’s conjecture for a numerical semigroup S using only the first Kunz layer S ∩ (m, 2m). Let η = |S ∩ (m, 2m)| and write the conductor as c = qm − ρ with 0 ≤ ρ < m. We prove the staircase bound |L| ≥ q + (q − 2)η + ηρ, where L = S ∩ [0, c) and ηρ = |S ∩ (m, 2m − ρ)|. This yields a lower bound for Wilf’s number and the criterion e(η + 2) ≥ 2m, hence also (η + 1)(η + 2) ≥ 2m, implying Wilf’s conjecture. When m | c, we obtain the stronger condition e(3η + 4) ≥ 4m, and in particular 3η² + 7η + 4 ≥ 4m. We also derive an exact cumulative-layer formula for |L| and apply it to interval-generated semigroups, where the second cumulative layer gives a strictly stronger infinite family.