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.

Arrange the integers $1,2,\dots ,n$ on a circle and, for a fixed $k\ge 1$, let $s_i$ be the sum of the k consecutive entries starting at position i (indices taken modulo n). For a circular permutation $\pi$, define the range $R\left(\pi \right)={max}_i{s}_i-{min}_i{s}_i,$ and let $w(n,k)$ be the minimum value of $R\left(\pi \right)$ over all circular permutations of $\{1,\dots ,n\}$. We obtain three structural results. First, we prove the complement symmetry $w(n,k)=w(n,n-k)$. Second, we determine the first nontrivial arithmetic progression case $n=2k+1$ exactly: $w(2k+1,k)=2⌈{\displaystyle \frac{k}{2}}⌉.$ Third, we determine the structured regime $n=k^2+1$ exactly: $w(k^2+1,k)=k.$ The proofs combine averaging lower bounds on the progression $n\equiv 1\left(\mathrm{mod}k\right)$ with explicit constructions: a parity-sensitive two-block arrangement for $n=2k+1$ and a $k\times k$ array construction for $n=k^2+1$.

Bounded monomial sets are basic relaxable objects in deterministic global optimization. Belotti's 2025 Mathematical Programming article established the upper envelope for two-variable cones and the lower envelope in dimension two, while leaving the lower envelope open in dimensions greater than two. We close this gap by proving that, in every dimension of at least three, the lower convex envelope over the bounded monomial set intersected with a two-variable cone is flat: after convexifying the restricted epigraph, the only lower constraint on the graph variable is the imposed lower value bound. The formula is independent of the upper value bound and the total homogeneity degree. The proof is based on a level-set convexification identity: every point in the monomial superlevel set inside the cone is either on the target level or an explicit convex combination of two points on that level. A single coordinate outside the constrained pair supplies the required compensation, which also explains why the result fails sharply in dimension two. Combining this lower-envelope formula with Belotti's known upper envelope yields the complete convex hull of the bounded graph for all homogeneity regimes in dimensions of at least three. We also give constructive certificates, separation conditions, a notation summary in the back matter, and reproducible numerical diagnostics showing that the two-point certificates are recovered to roundoff-level reconstruction accuracy over representative regimes.

We give a constructive high-dimensional escape sequence for the equation (∆ − x ·∇)u = u associated with the symmetric Ornstein–Uhlenbeck operator in Gaussian space. Let (a_i)_i≥1 be a positive square summable sequence and let B_n = {x ∈ Rⁿ : ∑ⁿ_i=1 a_i² x_i² < 1}. We construct functions u_n that are continuous on Rⁿ, smooth on both sides of ∂B_n, solve the positive spectral equation away from ∂B_n, and have finite Gaussian H¹ energy. The construction uses a single real harmonic polynomial, Re(x1 +ix2)^m_n with m_n = ⌊n^1/8⌋, multiplied by the finite-energy Tricomi branch of the separated radial Ornstein–Uhlenbeck equation and then extended into B_n by the weighted Dirichlet principle. The exterior energy has a lower bound of order (2π)^n/2n^−1/2(2m_n/e)^m_n, whereas the interior minimizing energy is bounded by (2π)^n/2n^CC_a^m_n. Hence the ratio of total Gaussian H¹ energy to the energy inside B_n tends to infinity. The proof is written with all non-standard notation defined explicitly, and two examples, including an ℓ¹-small sequence with ∑_i a_i < 1, are included as checks of the hypotheses.