Aravind, Burton, Nunez Ponasso and Richter (2026) proved that the 60-ray Kochen-Specker system of the 600-cell is not rigid and conjectured that the 300-ray/675-basis system of the 120-cell is rigid. This note shows it is not. The golden-ring lift phi from Z[alpha]^4 to Z^8 underlying their 600-cell construction preserves orthogonality for arbitrary golden-ring vectors, because the Z^8 inner product of two lifts equals the rational part of their R^4 inner product. Applied to the 300 rays of the 120-cell, the lift is an 8-dimensional rank-one realization satisfying all 4050 orthogonality relations while acquiring 7200 additional ones (exact integer census), so no unitary can map it into the original rays. An explicit integer witness pair is given, and the same computation reproduces the known 600-cell counts (450 orthogonal pairs, 360 extra). The rigidity of Gosset's polytope 4_21, and dimension-preserving rigidity of the 120-cell in R^4, remain open. Verification: run python3 verify_rigidity.py with rays300.txt in the same directory; it re-verifies the identity, the full census (4050 and 7200), and the witness in seconds. The bonus tool solve120.py is a self-contained long-run decider for the still-open exact 120-cell noncontextuality maximum (usage in its header).
Aravind, Burton, Nunez Ponasso and Richter (2026) proved that the 60-ray Kochen-Specker system of the 600-cell is not rigid and conjectured that the 300-ray/675-basis system of the 120-cell is rigid. This note shows it is not. The golden-ring lift phi from Z[alpha]^4 to Z^8 underlying their 600-cell construction preserves orthogonality for arbitrary golden-ring vectors, because the Z^8 inner product of two lifts equals the rational part of their R^4 inner product. Applied to the 300 rays of the 120-cell, the lift is an 8-dimensional rank-one realization satisfying all 4050 orthogonality relations while acquiring 7200 additional ones (exact integer census), so no unitary can map it into the original rays. An explicit integer witness pair is given, and the same computation reproduces the known 600-cell counts (450 orthogonal pairs, 360 extra). The rigidity of Gosset's polytope 4_21, and dimension-preserving rigidity of the 120-cell in R^4, remain open. Verification: run python3 verify_rigidity.py with rays300.txt in the same directory; it re-verifies the identity, the full census (4050 and 7200), and the witness in seconds. The bonus tool solve120.py is a self-contained long-run decider for the still-open exact 120-cell noncontextuality maximum (usage in its header).
The 60 rays and 75 orthonormal bases derived from the vertices of the 600-cell form the well-known Kochen-Specker system of Waegell and Aravind. This note determines exactly the largest number of the 75 bases that a single {0,1} ray assignment can satisfy (exactly one ray valued 1 per satisfied basis): the answer is 65, so at least ten bases must fail. This improves the lower bound of nine failures obtained by Waegell, Aravind, Megill and Pavicic (2011) from parity-proof coverings, answers the question they left open, and settles the corresponding open problem restated by Aravind, Burton, Nunez Ponasso and Richter (2026) for the 600-cell. The lower bound is an explicit 17-ray assignment verified in exact Q(sqrt5) arithmetic; the upper bound is certified both by two independent integer-programming solvers and by an explicit family of 150 parity proofs (ten orbits under the triacontagonal C15 symmetry) admitting no nine-element hitting set. The tight noncontextuality inequality is 65 versus the quantum value 75 (ratio 15/11 in the +/-1 normalization). A transfer proposition brackets the analogous 120-cell maximum in [600, 625]. Verification: run python3 verify.py with proofs150.txt in the same directory. It rebuilds the system from the two stated generators and re-verifies the construction anchors, the certificate count of 65, the ten failing bases, and the validity of all 150 parity proofs in seconds, without any solver.
The 300 rays and 675 orthonormal bases of the 120-cell form the Kochen-Specker system of Waegell and Aravind. This note determines exactly the largest number of the 675 bases satisfiable by a single {0,1} ray assignment (exactly one ray valued 1 per satisfied basis): the answer is 600, so at least seventy-five bases must fail. The proof avoids the intractable full-system optimization entirely: left and right multiplication by the binary icosahedral group partition the rays into five left and five right blocks, each an isometric copy of the 600-cell system; a single 60-ray integer program, proven optimal and transported to all ten blocks by the icosian symmetry, gives the per-block frontier 2*(block-only satisfied) + (doubly satisfied) <= 120; summing over each partition double-counts the doubly embedded bases and collapses the bound to 600. An explicit assignment attains 600. This improves the transfer bracket [600, 625] of DOI 10.5281/zenodo.21249805 to an exact value and closes the third member of the open family of Aravind, Burton, Nunez Ponasso and Richter. Whether parity proofs alone force the bound (as they provably do for the 600-cell and the two-qubit system) is posed as the sharpened open question: the argument gives hitting number h <= 75, and h = 75 remains to be decided. Verification: run python3 verify_max600.py with rays300.txt in the same directory. Structural checks are instant; the two 60-ray integer programs finish in minutes; no full-system optimization is used anywhere.
The 60 complex rays and 105 orthonormal bases arising from the two-qubit Pauli group form the Kochen-Specker system of Waegell and Aravind (2011), recently named by Aravind, Burton, Nunez Ponasso and Richter (2026) as a natural target for symmetry-based analysis. This note determines exactly the largest number of the 105 bases satisfiable by a single {0,1} ray assignment: the answer is 93, so at least twelve bases must fail. The optimum has clean structure: an attaining assignment selects exactly one state from each of the fifteen maximal abelian subgroups of the Pauli group, so all fifteen stabilizer bases are satisfied and all twelve failures are hybrid bases. The bound is certified by two independent integer-programming solvers (CBC proven optimum; HiGHS optimality and infeasibility of 94), and the attaining assignment is re-verified in exact Gaussian-integer arithmetic. The tight inequality is 93 versus the quantum value 105 (ratio 35/27 in the +/-1 normalization). This is the exact two-qubit analog of the value 65 of 75 determined by the author for the 600-cell system. Verification: run python3 verify2q.py with rays60c.txt in the same directory; it rebuilds the system from the fixed data and re-verifies the anchors (60 rays, 105 bases, seven per ray), the certificate count of 93, the transversal property, and the twelve failing bases in seconds, without any solver.
Fourth paper of the rigidity program (companions: https://doi.org/10.5281/zenodo.21488473, https://doi.org/10.5281/zenodo.21498342, https://doi.org/10.5281/zenodo.21498385). In 2010, Blanchfield asked how common parameterized generalisations are among Kochen-Specker sets, and found every d=4 set she examined fully determined by its orthogonalities; the only known continuous family lived in dimension 3 (the Gould-Aravind circle, whose uniqueness among two-symbol d=3 alphabets the companion papers prove). This paper answers her question in dimension four. Results: (1) an anchor theorem: every two-symbol d=4 mechanism-stable graph contains Peres-24 and is therefore KS-uncolorable, so the d=3 uniqueness argument structurally cannot run in d=4; (2) the M9 family: for the unimodular mechanism |x|^2=1, an 89-ray, 35-basis KS core with entries {0, +-1, +-e^(i theta)} whose 433 orthogonalities hold identically in theta, KS-uncolorable at every point of the circle, with exact flex = 1 - to our knowledge the first continuous fixed-graph KS family in dimension 4; (3) a complete triage of all 16 two-symbol d=4 mechanisms: exactly two carry families - the M9 circle (flex 1) and the M10 line Re(x)=0 (exact flex 2, 21 rays) - while the off-center-circle mechanisms are proved rigid by a Moebius-stability method; (4) the geometry of the M9 circle: its Wilczek-Zee connection is exactly constant, with eigenphase quartic irreducible over Q with Galois group S4 (so no closed form of the d=3 type 2*pi*sqrt(1867)/33 exists - an honest structural contrast), and det W = exp(2*pi*i*23/35), nontrivial where d=3's is trivial; (5) a structure card: all four conjugation-fixed points of the circle degenerate and are exactly rigid while generic points flex - the reverse of the d=3 pattern. Every claim carries an explicit evidence label (PROVED / EXACT / NUMERICAL / CONJECTURE / OPEN), each backed by a script in the attached verification suite (also public: https://github.com/manuflog/contextuality-obstructions/tree/main/rigidity). A final adversarial literature guard (including the Pavicic-Megill coordinatization program, the Waegell-Aravind catalogs, and the 2022 RMP review) found no prior continuous fixed-graph d=4 KS family; the claim is stated as first known. Methods note: large-language-model assistants (Anthropic Claude) were used substantially in drafting and in developing the verification suite; every mathematical claim is machine-verified independently of that assistance, and the author takes full responsibility for the results. v2 (2026-07-23): adds the refined reading of M10's flex (Remark 7.x): the second deformation is an exactly-integrable core-local twin-ray rotation, not portable to the full stable pool, so the mechanism-level flex of M10 is 1 = dim V; adds the conditional d=4 uniqueness results (decoupling certificates for all 8 rigid mechanisms; integer-pool sweep: 2082 exact flex computations, all critical {0,+-1}^4 cores rigid) and the guard-mandated citations (RMP 2022 review; Trandafir-Cabello rigid-sense disambiguation; Pavicic-Megill coordinatization distinguished; Waegell-Aravind catalogs). Verification suite updated with branch_d4unique.py, branch_intflex.py, branch_m10x.py. v3 (2026-07-25): structural pass on Sections 5-7. v4 (2026-07-27): the stale "SCAFFOLD DRAFT" banner and a stale Discussion paragraph removed; and, importantly, an attribution remark added -- the anchor property used here is NOT ours, it is due to Pavicic and co-authors (Pavicic/Waegell, Pavicic/Megill), and earlier versions did not say so. This deposit is document version v4, and the Zenodo version counter now reads v4 to match; do not read the two as independent.
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
Abstract Coxeter pointed out that a number of polytopes can be projected orthogonally into two dimensions in such a way that their vertices lie on a number of concentric regular triacontagons (or 30-gons). Among them are the 600-cell and 120-cell in four dimensions and Gosset’s polytope 4 21 in eight dimensions. We show how these projections can be modified into Kochen–Specker diagrams from which parity proofs of the Bell–Kochen–Specker theorem are easily extracted. Our construction trivially yields parity proofs of fifteen bases for all three polytopes and also allows many other proofs of the same type to be constructed for two of them. The defining feature of these proofs is that they have a fifteen-fold symmetry about the center of the Kochen–Specker diagram and thus involve both rays and bases that are multiples of fifteen. Any proof of this type can be written as a word made up of an odd number of distinct letters, each representing an orbit of fifteen bases. Knowing the word representing a proof makes it possible to infer all its characteristics without first having to recover its bases. A comparison is made with earlier approaches that have been used to obtain parity proofs in these polytopes, and some directions in which this work can be extended are discussed.