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 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
Noisy Bell mixtures may carry entanglement, standard teleportation usefulness, projective-measurement steerability, and optimized Cavalcanti-Jones-Wiseman-Reid (CJWR) and Clauser-Horne-Shimony-Holt (CHSH) violations. If the noise already has an ability, the mixture may lose and later recover it at distinct boundary crossings as the Bell weight decreases. We characterize ability-absence intervals and definitive thresholds for mixtures of ∣Φ+⟩ with arbitrary complex two-qubit X noise. A closed-form singular-value flow of the Pauli correlation tensor gives the exact teleportation-useless, CJWR-satisfying, and CHSH-local intervals, while the positive partial transpose criterion gives the exact separable interval. At fixed populations and coherence magnitudes, these intervals widen as the relative phase between the Φ-block coherence of the noise and the Bell coherence increases from 0 to π. The definitive thresholds form a universal chain from entanglement through teleportation usefulness and the three-setting CJWR witness to the common two-setting CJWR witness and CHSH-nonlocal threshold. Replacing teleportation usefulness by steerability gives a second chain, although their thresholds are not mutually ordered. For arbitrary pure two-qubit noise, the steerability thresholds in both directions for projective measurements and arbitrary positive operator-valued measures equal the entanglement threshold. For arbitrary product noise, an effective X-state singular-value flow gives the teleportation usefulness, CJWR witness, and CHSH-nonlocal thresholds. If a local noise factor is pure, the entanglement threshold and all four steerability thresholds are zero. For generic full-rank mixed X noise, finite-setting semidefinite programs give upper bounds on the unknown directional projective-measurement steerability thresholds.
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.
Preprint. Third paper of the contextuality-obstructions program, alongside Paper A (the canonical class of the MASA context stack), Paper B (the obstruction spectrum of Weyl context families), Paper D (the observer-context category) and a synthesis note. The slogan "contextual holonomy encodes quantum contextuality" is attractive and, as usually stated, false. This paper gives it a sharp form. On the context stack of maximal abelian subalgebras the canonical degree-2 class has order exactly n, and its cocycle holonomy along closed context loops detects EXACTLY the all-versus-nothing sector of contextuality for closed-triple Weyl families at even local dimension. Both containments are strict, each by an explicit certificate: at d=3 the class is nontrivial of order 3 while the achievable state-independent AvN value is 0, and KCBS exhibits state-dependent contextuality where the class shadow vanishes. The Wigner-friend loop generated by H and S splits by level: its projective (Pancharatnam) holonomy is +1 while its determinant-section holonomy is exactly -i, so any interpretational claim about "the" holonomy of that loop must state which one it means. ATTRIBUTION. The classification of state-independent values for qudit Weyl families, the even/odd dichotomy and the fixed value d/2, on which the state-independent side of the equivalence rests, is DUE TO Abramsky, Cercelescu and Constantin (FSCD 2024). Attainment at general even d is the Raussendorf-Okay-Zurel-Feldmann construction. This paper uses both and rediscovers neither. A RETRACTION IS CARRIED IN THIS PAPER, NOT BURIED. An earlier draft presented the tier-two facets of the d=4 census on a (1/(2*sqrt2))Z grid with a single coefficient magnitude 3/(2*sqrt2), and read the resulting unit-coefficient bounds 4*sqrt2 and 14*sqrt2/3 as placing Tsirelson's sqrt2 inside the tier-two geometry. The whole presentation was a normalization artifact and is withdrawn (Retraction 1): the stored lift has coefficients in {0,+-1} with integer bounds 6 and 7, every facet functional in the census is rational, and Q(sqrt2) is not among the eighteen canonical arithmetic fields. Certificate: d4_arithmetic_profile.py. WHERE THE EVIDENCE IS THIN, STATED IN THE PAPER. Two sweeps quoted here drew no positive instances at all: the d=4 half of the 2,100-family equivalence test, and the 6,500-trial false-certificate sweep, which draws no contextual family at any of d=2,4,8 and therefore establishes the absence of false certificates only vacuously. Both are disclosed where they appear. Theorem 2 is labelled machine-validated rather than proved: the closed form is established end to end for the deletion families treated, while the general closed-triple d=2 statement in compressed coordinates remains open and is recorded as open in the verification ledger. HARDWARE CAVEAT, STATED UP FRONT. The quoted significances, about 56 sigma for the ray-level Wigner-friend holonomy and 35.4 sigma for the Peres-Mermin witness above the deterministic bound 4, are shot-noise statistical only, computed under the implemented witness model and conditional on compatibility and nondisturbance assumptions. They are not systematics-aware and neither measurement is loophole-free; a roughly 3.5 sigma preparation offset on one strand is itself evidence of unmodelled systematics. We also record that the pinned value S = 4.6125 cannot be recomputed from the stored counts (a naive recomputation gives 4.5662, a gap of about 2.7 standard errors); the archived value 4.558 +- 0.018 is the one an independent reader can reproduce from the deposited data. Read these as consistency checks on real hardware, not as precision measurements or certifications. STATUS AND SCOPE. This is a preprint, not a peer-reviewed article. It was held back from the first publication batch and released only after three adversarial review passes; what each pass found, and how each item was closed, is recorded in verification/INDEX.md rather than quietly fixed. One defect found there is NOT closed and belongs to the companion Paper B, which cites a false-certificate artifact that is not in fact in the repository; it is recorded rather than left for a reader to discover. COMPANIONS. Paper A: https://doi.org/10.5281/zenodo.21630431 . Paper B: https://doi.org/10.5281/zenodo.21630630 . Paper D: https://doi.org/10.5281/zenodo.21630697 . Note: https://doi.org/10.5281/zenodo.21630739 . Code archive: https://doi.org/10.5281/zenodo.21498498 AI-ASSISTANCE DISCLOSURE. Large-language-model assistants (Anthropic Claude) were used substantially in drafting, in developing the verification scripts, in literature search and in proposing proofs. Every load-bearing claim is machine-verified independently of that assistance by the scripts named in the text, and the author takes full responsibility for the results, including any errors. VERIFICATION CODE. https://github.com/manuflog/contextuality-obstructions
Abstract We formulate and prove the main properties of the generalized Gell–Mann representation for traceless qudit observables with eigenvalues in and analyze via this representation violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality by a general two-qudit state. For the maximal value of the CHSH expectation in a two-qudit state with an arbitrary qudit dimension , this allows us to find two new bounds, lower and upper , expressed via the spectral properties of the correlation matrix for a two-qudit state. We have not yet been able to specify a class of two-qudit states where the new upper bound improves the upper bound of Tsirelson. However, this is the case for all two-qubit states, where the new lower bound and the new upper bound coincide and reduce to the precise two-qubit CHSH result of Horodeckis, and also, for the Greenberger–Horne–Zeilinger (GHZ) state with an odd where the new upper bound is lower than the upper bound of Tsirelson. Moreover, we explicitly find the correlation matrix for the two-qudit GHZ state and prove that, for this state, the new upper bound is attained for each dimension and this specifies the following new result: for the two-qudit GHZ state, the maximum of the CHSH expectation over traceless qudit observables with eigenvalues in is equal to if is even and to if d > 2 is odd.
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.