On the absence of uniform spectral gap for the linearized Boltzmann operator with positive collision kernel: Model B_n = δ_{cos(1/n)} ≥ 0 and absolute vs relative distinction

We study the uniform spectral gap problem for the linearized Boltzmann operator in the Maxwell molecules case. It is known that a kernel with sign-changing Legendre expansion, such as B = P2 < 0, yields inf_l μ_l = 0, but such a kernel violates the H-theorem. We introduce a family of physically admissible kernels B_n = δ_{cos(1/n)} satisfying: - B_n ≥ 0, Legendre coefficients P_l ≥ 0 - C^∞ regularization possible, bounded, compact support - Grad cutoff condition, ||B_n||_{L1} = 1 - H-theorem preserved, compatible with DiPerna-Lions theory Main results: Spectral gap Δ_n = inf_{l≥2} μ_l^{(n)} = 1.5 sin^2(1/n) ~ 1.5/n^2 → 0, with μ_n ~ 1/(4n) → 0 for angular scale θ_n = 1/n and dominant mode l_n = √n. If μ2 = 1 is imposed, the required total mass is n^2/1.5 → ∞ and μ_l/μ2 ≥ l(l+1)/6 ≥ 1 for large l, showing no relative closure within the class B ≥ 0. This clarifies the absolute vs relative distinction for the spectral gap problem without breaking physical admissibility. MSC: 82C40, 76P05, 35Q20, 35P15