KCBS Contextuality Under Non-Depolarizing Noise: Analytic Thresholds for Dephasing and Amplitude Damping

We derive closed-form expressions for the KCBS witness K(p) under two noise channels not previously analyzed analytically for a spin-1 qutrit: dephasing and full-cascade amplitude damping. For the KCBS-optimal state |2><2|, dephasing leaves the witness invariant (K(p) = sqrt(5) for all p) because the optimal state carries no coherences. Amplitude damping yields K(p) = A*p*(2-p) + sqrt(5)*(1-p)^2, where A = 5/2 - sqrt(5)/2, with noise threshold p* = 1 - sqrt((sqrt(5)-1)/(3*sqrt(5)-5)) ≈ 0.149. This is 2.78x lower than the depolarizing threshold p* ≈ 0.415, showing that the geometry of the noise channel — not only its strength — determines how quickly contextuality is lost. All results are verified numerically to machine precision.