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.
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.
We study the all-state noise threshold η_b(C_m ⊗ C_n) of the product of two regular odd n-cycle (KCBS-type) scenarios in ℝ³, with every context encoded by two bits and local white noise of the same strength on each factor. The threshold is bounded below by the single-cycle thresholds (product states) and by the maximum M_face(m,n) of the threshold over the two-qubit faces v_j^⊥ ⊗ w_k^⊥. We prove that, for all odd m ≥ 9 and n ≥ 5, the face maximum strictly exceeds the single-cycle threshold of C_m (for m ≥ 13 via the published completeness theorem of Araújo et al. for the n-cycle inequalities). Hence entanglement makes C_m ⊗ C_n strictly more robust than both factors for all odd m, n ≥ 9, and the face maximum is at least of order c/m: liminf_m m·inf_n M_face(m,n) ≥ 3√6·π/8 = 2.8857371… > 2 = lim_m m·E(m); no upper bound of order 1/m is proved. The proof uses one Clauser–Horne witness whose violation reduces to a 2×2 form. On each face a six-bit marginal, the CHSH window, is a CHSH test with visibility (1 − η′)², η′ = η/(2 − η); after this reduction Landau's bound gives the closed lower bound Φ(g_m, g_n) = 2(1 − u)/(2 − u), u = (1 S)^(−1/4). For KCBS × KCBS the face threshold equals the window value exactly, M_face(5,5) = 0.26560777045727252964 (a root of an explicit octic), while for 13 pairs (m,n) the face maximum provably exceeds the window value through a second, Clauser–Horne branch whose value stays bounded away from zero as n → ∞. Pure face dominance fails for C₅ ⊗ C₁₁; we formulate a two-regime conjecture and give the numerical evidence for it, together with the known rigorous upper bounds for all states. For KCBS × KCBS we reduce the all-state upper bound to a positivity hypothesis at noise 1/4 on the non-window facets of the noncontextual polytope, verified exactly on all known facet orbits (flip images checked for the lists of 156 and 437 orbits); under it η_b(C₅ ⊗ C₅) = M_face(5,5), the window facets being settled exactly. The stronger hypothesis at the single-cycle threshold η_b(C₅) is false: at least 25 facet orbits outside the window class are violated beyond η_b(C₅), the largest violating noise, attained by two of them, being t_max ∈ [0.2485455, 0.2485456), the lower end being the noise level up to which a planar singlet violates them (numerically its threshold). All values refer to local bit noise and the regular realisation; several of them change under other noise models. Checkers, certificates and cone covers: https://doi.org/10.5281/zenodo.23204387. Lean library (release 1.4.0): https://doi.org/10.5281/zenodo.23204263
We compare noise thresholds of the products C5⊗C5 (KCBS × KCBS) and C5⊗C7 of odd-cycle contextuality scenarios under local bit noise, for quantum realisations and for the level-1+AB (almost-quantum) relaxation. For the window class, the noncontextuality inequalities supported on the four contexts through one vertex of each cycle, we prove that the level-1+AB threshold equals Φ(1) = (2·2^(1/4) − 2)/(2·2^(1/4) − 1) ≈ 0.2745287, the maximum of the window law; the upper bound is formalised in Lean, and both ends are confirmed by exact rational certificates. Free real representations of the cycles attain Φ(1); on five further near-tight orbits of C5⊗C5 they come numerically within 7.2·10⁻⁸ of it. On the window class, the gap to the regular-ray thresholds 0.2656078 and 0.2634363 is a gap between regular and free measurements, not between quantum and almost-quantum. Pure level 1 gives about 1/2 (violation at 1/2 − 10⁻⁶). For C5⊗C7 we settle 52 of the 63 support classes of facets exactly at one noise value; 11 remain open.
We study the orthogonality graphs Gn of the Kochen–Specker configurations Zn,n(kn, kn) recently obtained by Favacchio from grids on dual quadrics: for every odd n ≥ 3 they consist of 2n2 rays in R4 and n2 orthogonal bases, and G3 is the 18-ray set of Cabello, Estebaranz and García-Alcaine. We determine all orthogonal pairs, not only those inside the bases: Gn is (2n + 1)-regular, and the “only if” direction rests on the fact that 2 cos(2πt/n) is a unit in the ring of algebraic integers for odd n and t ̸≡ 0, while 2 is not. Consequently the independence number is α(Gn) = (n2 − 1)/2, with an explicit maximum independent set, whereas ϑ(Gn) = α∗(Gn) = n2/2 by the standard argument of Cabello, Severini and Winter (CSW). Through the CSW correspondence, every member thus gives a state-independent noncontextuality inequality with classical bound (n2−1)/2 and quantum value n2/2. The list of orthogonal pairs, α, α∗ and the value of Lovász’s max-characterization of ϑ are formalized in Lean 4 with Mathlib, for the rays given by explicit coordinates.
We place several known indicators of Kochen--Specker contextuality -- the KCBS correlator χ, the contextual fraction $\CF$, the Shannon-entropic n-cycle inequality of Chaves and Fritz, and the operational commutator witness D of Paper~I -- into a single projector-geometric framework organized around the overlap matrix $\Tcal_{ij} = d^{-1}\tr[(\hat P_i \hat Q_j)^2]$, where P^i and Q^j are the joint-eigenspace projectors of the two compatible observable pairs within a measurement context. The state-independent scalar content of $\Tcal$ is carried by two independent contractions: the mutual information energy $E = \sum_{ij}\Tcal_{ij}$ of Paper~I (equivalently, its logarithmic form S2=−log2E), and the Maassen--Uffink extremal overlap $c_\MU = \max_{i,j}|\langle a_i,b_i | c_j,b_j\rangle|$. We prove that S2 is non-increasing under coarse-graining, that $S_2(\Gcal) > 0$ is a necessary configuration-level condition for observable contextuality, and that the additive composition $S_2(\Gcal) = \sum_αS_2(\Gcal_α)$ is exact for the KCBS pentagon. We further show that in the spin-1 realization of the KCBS pentagon, a shared ms=0 eigenstate in each context forces $c_\MU = 1$, rendering every Maassen--Uffink-type bound trivial -- a structural mechanism that makes explicit why outcome-entropic uncertainty relations based on $c_\MU$ are silent on KCBS contextuality, while S2≈2.7266~bits throughout. Applied to KCBS and CHSH, the framework identifies regimes in which every state-dependent witness considered here is silent yet $S_2(\Gcal) > 0$ by an amount set by the projector geometry alone.
Within the program of identifying rigid Kochen-Specker (KS) sets (Xu-Saha-Bharti-Cabello, PRL 132, 140201; Trandafir-Cabello, PRA 111, 052204; Kernaghan, arXiv:2603.16988), this paper contributes exact-arithmetic certificates, one unification, and one correction. The two flexible islands of Kernaghan's landscape (Peres-33 and Z[sqrt(-2)]-33) are the two antipodes of a single exact circle of complex 33-ray KS sets, the closed form of the Gould-Aravind phase family. We certify flex=1 exactly at both islands, prove local completeness, and correct Kernaghan's Observation 28: the flex is finite, not merely infinitesimal. The circle contains no degenerations (proved exactly) and its Sum P = 11 I mechanism is resolved via hyperoctahedral orbit tight frames. Exact rigidity certificates across the full known census make Peres-Penrose-33 the unique flexible KS set. The Hermitian rigidity matrix block-diagonalizes at real points (flex_C = flex_R + flex_skew) with the graded cycle split proved, and a torsion layer is proved flex-invariant. Every claim carries an explicit evidence label (PROVED / EXACT / NUMERICAL / LITERATURE / OPEN). The attached verification suite (35+ self-checking scripts) reproduces every certificate; companion hardware notebooks (IBM Quantum, two devices) confirm the witness-invariance prediction Sum P = 11 to 0.45% and measure the loop holonomy eigenphase to ~2%, consistent with the exact value 2*pi*sqrt(1867)/33. Note on methods: large-language-model assistants (Anthropic Claude) were used substantially (drafting, verification-script development, literature search, proof proposals); all mathematical claims are machine-verified by the attached suite, and the author takes full responsibility. Code and data: https://github.com/manuflog/contextuality-obstructions
A Kochen-Specker (KS) set is a specific set of projectors and measurement contexts that prove the Bell-Kochen-Specker contextuality theorem. The simplest known KS sets in Hilbert space dimensions d=3,4,5,6,8 are reproduced, and several methods by which a new KS set can be constructed using one or more known KS sets in lower dimensions are reviewed and improved. These KS sets and improved methods enable the construction of explicitly critical new KS sets in all dimensions, where critical refers to the irreducibility of the set of contexts. The simplest known critical KS sets are derived in all even dimensions $d\ensuremath{\ge}10$ with at most nine contexts and 30 projectors, and in all odd dimensions $d\ensuremath{\ge}7$ with at most 13 contexts and 39 projectors. These results show that neither the number of contexts nor the number of projectors in a minimal KS set scales with dimension d.