Exact Robustness Certificates for Five-Qubit Stabilizer Decoding: Decoder Switching and a Deterministic–Randomized Minimax Separation

We give an exact-arithmetic case study of maximum-likelihood (ML) stabilizer decoding under uncertain dephasing, separating channel-adaptive, fixed deterministic, and fixed randomized recovery decisions. For the five-qubit [[5, 1, 3]] code, we enumerate syndrome–stabilizer-coset masses and certify the homogeneous dephasing success polynomial 1 − 10t³ + 15t⁴ − 6t⁵. On the full homogeneous interval, adaptive ML has robust success 1/2, every fixed deterministic syndrome decoder has worst-case success zero, and fixed randomization attains 1/2. For a heterogeneous but conditionally independent family with site rates (t, t, 2t, 2t, 2t), 0 ≤ t ≤ 1/2, we obtain four algebraic switching thresholds and an exact five-piece adaptive ML curve. The adaptive and optimal fixed randomized worst-case success both equal 3/4, whereas the best fixed deterministic decoder has success min₀≤ₜ≤₁⁄₂ (16t⁵ − 28t⁴ + 10t³ + 3t² − 2t + 1) ≈ 0.749591899. A finite exhaustive reduction and rational witness checks certify strict separation. A CSS-code theorem and independent Steane-code calculation establish a second, strictly separated heterogeneous minimax instance with R_det < 0.4999 < R_rand = R_ML = 1/2. We retain biased-noise orientation sensitivity and two counterexamples to invalid vertex shortcuts, and distinguish these instance-level certificates from general decoding algorithms and established enumerator or robust-optimization methods.