Exact Kochen–Specker joint-measurability noise thresholds of the KCBS n-cycle: an arc lemma for all n, a lattice-rounding sandwich, and the continuum limit

For the regular odd KCBS n-cycle of qutrit projectors under white noise, E ↦ (1−η)E + ηI/3, let η_KS(n) be the least noise at which the noisy effects admit a joint measurement whose outcomes are the Kochen–Specker (KS) value assignments of the cycle, i.e. its independent sets. The dihedral symmetry reduces the semidefinite program over the ≈ 1.6^n independent sets to a second-order cone program on the orbits of arcs (runs of physically consecutive rays), whose data are Dirichlet kernels; its value η_arc(n) is an upper bound. We prove the arc lemma η_KS(n) = η_arc(n) for every odd n ≥ 5. The proof combines a combinatorial lemma — for weights b0 + b1 cos φ + b2 cos²φ with b2 ≤ 0 a minimum-weight independent set of the cycle is an arc — proved on paper and, independently, in Lean 4, with a computer-assisted certificate, in ball arithmetic, of the sign of y2 − y1 for all n. Comparing the discrete problem with its relaxation in which the arc length is a continuous parameter (value η_s(n)) we prove the sandwich η_s(n) ≤ η_KS(n) ≤ η_s(n) + C t(1−t)/n², t = frac(n ρ1(n)), for all odd n ≥ 7, with C ≤ 3.55 for n ≥ 401 (direct construction; the Lagrangian bound gives 3.59). As a consequence η_KS(n) → η_∞ = 0.42473994622455799193…, the value of an explicit continuum problem, and the rounding constant tends to c_∞ = 3.47856584115611146…, with a rate (computer-assisted): |η_KS(n) − η_s(n) − c_∞ t(1−t)/n²| ≤ 1.97/n³ for all odd n ≥ 7, and with an explicit second term. The thresholds are not monotone (computer-assisted): η_KS(n+2) > η_KS(n) and η_KS(n+2) < η_KS(n) each for infinitely many n, with approximate period 4 and a slow beat. For 5 ≤ n ≤ 401 and five larger n the thresholds are enclosed to 10^−35 as roots of explicit algebraic systems. A parameter-free form of the rounding law, fixed by a recorded hash before the corresponding discrete values were read, predicted the optimal support for 60 of 60 values 83 ≤ n ≤ 201.