Exact bit-flip noise thresholds of complete-graph Kochen–Specker sets

For odd N, a Kochen–Specker set of type K_N consists of N(N−1)/2 rays in C^(N−1), one for each edge of the complete graph K_N, such that the N−1 rays at each vertex form an orthonormal basis; such sets are known for N = 7, 9, 11, 13. We determine the exact threshold of independent bit-flip noise (every outcome bit flipped with probability η/2) at the maximally mixed state: the noisy statistics are noncontextual if and only if (1−η)^(N−1) ≤ (N−2)/N, so η*F = 1 − (1−2/N)^(1/(N−1)), and every other state has a threshold at least as large. More generally, for every family of context distributions that is the same in all contexts and invariant under permutations of the N−1 rays of a context, noncontextuality is equivalent to P(odd row weight) ≤ (N−1)/N. The lower side is the parity (handshake) inequality; the upper side is an explicit measure on two classes of near-regular graphs. For N = 7 this gives a uniform proof of the threshold 1 − (5/7)^(1/6) of the 21-ray set of Lisoněk, Badziąg, Portillo and Cabello. The same criterion gives the exact thresholds of every noise that acts on the bits of a row independently and identically, for instance asymmetric flips, dark counts and losses. For even N every symmetric family is noncontextual. We show that Lisoněk's construction from S-Hadamard matrices gives a realisation of K(n+1) exactly when the entrywise products of pairs of rows span pairwise distinct lines, a condition invariant under monomial equivalence, and that a product of Diţă type with suitably permuted columns preserves it. Hence K_N has a realisation over Q(ζ_3) for every N = 12·6^t + 1, t ≥ 0; this gives an infinite family of Kochen–Specker sets with pairwise distinct rays whose bit-flip threshold at the maximally mixed state is known exactly. We also give exact certificates for K_61 and K_73. The statement for model F at the maximally mixed state is formalised in Lean 4 for the abstract one-hot family and all odd N ≥ 3.