Sharp Neumann eigenvalue means on triangles:an exact certificate proof and stability

We prove both inequalities in Laugesen and Siudeja’s Conjecture~6.34 for the first two positive Neumann eigenvalues of triangles. The equilateral triangle uniquely maximizes their harmonic mean at fixed perimeter and their geometric mean at fixed area. Two-dimensional trial spaces reduce the proof to polynomial positivity. Projected cubic corrections repair the second-order error near equality; elementary estimates and rational Bernstein certificates cover the remaining shapes. We provide exact data and a verifier, and derive an explicit quadratic stability estimate in terms of side-length asymmetry.