Wilf's Conjecture from the First Kunz Layer

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.