We present a computational survey of Kochen-Specker (KS) uncolorability in three-dimensional Hilbert space across two-symbol coordinate alphabets $\mathcal{A} = \{0, \pm 1, \pm x\}$ drawn from quadratic, cyclotomic, and golden-ratio number fields. In every tested alphabet, KS sets arise only when $x$ supports one of two cancellation mechanisms: modulus-2 cancellation (the generator satisfies $|x|^2 = 2$, as in $|\sqrt{2}|^2=2$, $|\sqrt{-2}|^2=2$, or $|α|^2=2$; the integer case $1+1=2$ is the degenerate additive instance) or phase cancellation (a vanishing sum of unit-modulus terms, as in $1+ω+ω^2=0$). Alphabets whose generators have $|x|^2 \geq 3$ and are not roots of unity produce orthogonal triples but not KS-uncolorability in our survey. This empirical pattern explains why constructions cluster into six discrete algebraic islands among the tested fields. Two yield potentially new KS graph types: the Heegner-7 ring $\mathbb{Z}[(1+\sqrt{-7})/2]$ (43 vectors) and the golden ratio field $\mathbb{Q}(\varphi)$ (52 vectors, revealed only by cross-product completion); $\mathbb{Z}[\sqrt{-2}]$ provides a new algebraic realization of a known Peres-type graph. Using SAT-based bipartite KS-uncolorability, we verify and extend the input counts of Trandafir and Cabello for bipartite perfect quantum strategies across all six islands. Whether the two-mechanism pattern extends to all number fields remains an open question.
We present a computational survey of Kochen-Specker (KS) uncolorability in three-dimensional Hilbert space across two-symbol coordinate alphabets A={0,±1,±x} drawn from quadratic, cyclotomic, and golden-ratio number fields. In every tested alphabet, KS sets arise only when x supports one of two cancellation mechanisms: modulus-2 cancellation (the generator satisfies ∣x∣2=2, as in ∣2∣2=2, ∣−2∣2=2, or ∣α∣2=2; the integer case 1+1=2 is the degenerate additive instance) or phase cancellation (a vanishing sum of unit-modulus terms, as in 1+ω+ω2=0). Alphabets whose generators have ∣x∣2≥3 and are not roots of unity produce orthogonal triples but not KS-uncolorability in our survey. This empirical pattern explains why constructions cluster into six discrete algebraic islands among the tested fields. Two yield potentially new KS graph types: the Heegner-7 ring Z[(1+−7)/2] (43 vectors) and the golden ratio field Q(φ) (52 vectors, revealed only by cross-product completion); Z[−2] provides a new algebraic realization of a known Peres-type graph. Using SAT-based bipartite KS-uncolorability, we verify and extend the input counts of Trandafir and Cabello for bipartite perfect quantum strategies across all six islands. Whether the two-mechanism pattern extends to all number fields remains an open question.
Companion paper to "Exact rigidity certificates for Kochen-Specker sets, the graded flex dictionary, and the Peres-Penrose circle" (https://doi.org/10.5281/zenodo.21488473). A complete algebraic theory of Kernaghan's two-symbol alphabet landscape (arXiv:2603.16988). Results: (1) the exhaustive 80-case d=3 classification: exactly FOUR independent cancellation mechanisms (rational, modulus-2, phase, golden), not two - a fifth candidate is proved identical to the phase mechanism under the alphabet's own x<->1/x gauge duality - and any generator avoiding all four provably yields a 3-colorable pool (structure theorem via Godsil-Zaks); (2) the d=4 extension: a term-alphabet lemma, 16 new mechanisms from the 280-case 4-term classification, and no colorable regime (Peres-24 anchors every pool); (3) the mechanism-moduli audit across the census, proving the forward flex bound in general; (4) the uniqueness theorem: a machine-certified KS-coloring of the full 145-ray M3-line-stable graph proves the Gould-Aravind circle is the unique positive-dimensional flexible family of two-symbol d=3 KS sets - including the honest retraction of a tempting arithmetic explanation that failed direct verification; (5) the M2 mirror: the maximal circle-stable graph is KS-uncolorable, compactness pins the M2 circle to exactly five integer-trace exact points, and the Gaussian points x=+-(1+i) carry a NEW 49-ray KS core (SAT-certified with a three-anchor-validated encoder, critical, and exactly rigid) distinct from all six census islands and from the survey's own Gaussian island (shown here to reduce to Peres-33). Flexibility tracks the circle's four conjugation-fixed points exactly (Hypothesis L), and a J-splitting construction makes the real/skew flex decomposition well-posed at conjugation-symmetric non-real points. 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). 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. Version 4 (July 2026) — status of the Observation 28/29 point. This paper extends and in one place corrects Part II of Kernaghan's survey (arXiv:2603.16988). The specific disputed point, the reported non-persistence of the Peres/Penrose flex, was conceded by its author in v8 of that preprint, in a labelled correction note that also publishes the cause. References are updated to v8 and the passage now describes the matter as settled. As elsewhere in this series, v8 is not treated as independent confirmation of finiteness: its continuation is double precision and its modular rank computation bounds flex only from above, as its author states. Version 5 (July 2026) — a proof step is withdrawn and replaced. The theorem is unchanged. Versions up to v4 justified the last step of the structure theorem (necessity of a mechanism for the raw pool) by invoking Godsil–Zaks, paraphrased as “rational-coordinate projective ray sets in R^n are always 3-colorable”, and concluded that the x=3 graph is 3-colorable. Both halves are false and are withdrawn. The paraphrase is false: the x=2 raw pool is a rational-coordinate ray set (49 rays, 138 orthogonal pairs) with chromatic number 4. The application is false at the pool it was applied to: the x=3 graph is not 3-colorable either, also chromatic number 4. Godsil–Zaks concerns rational points of the unit sphere, and a rational ray such as (1,1,1) does not normalize to one, so the theorem never applied to these pools at all. The theorem itself is unaffected. Graph 3-colorability was only ever a sufficient route to KS-colorability, and KS-colorability at x=3 is established directly by exhaustive certificate — 31488 distinct valid KS colorings, recomputed independently for this version. Both chromatic numbers were verified twice, by DSATUR backtracking and by SAT (Glucose3). An earlier section of the paper had already retracted this same invocation for the Re(x) = -1/2 line; that retraction should have been applied to the structure theorem at the same time and was not. It is now. Also corrected: earlier versions described cross-product completion as “the closure step that rescues the Golden and CK-31 islands from initial colorability”. That is wrong for CK-31, whose raw pool at x=2 is already KS-uncolorable (49 rays, 138 pairs, 26 triads, no valid coloring). Completion enlarges it but is not needed for uncolorability. Only the Golden island is genuinely rescued by completion. Version 6 (July 2026) — a theorem is scoped, its closed-pool reading withdrawn, and the resolution identified. The structure theorem (“no mechanism, no KS set”) was stated through v5 without an alphabet hypothesis. For Hermitian-closed complex pools it is FALSE, and the cause is a term the classification missed: with x* as an entry, (x*)*·x = x² appears, so the closed term set is {±1, ±x, ±x*, ±N, ±x², ±(x*)²}. Counterexample, verified twice in independent exact implementations: x = √3 + i avoids every locus M1–M5, yet its closed pool (145 rays / 390 pairs / 30 triads) is KS-uncolorable; unclosed it is colorable. The identity x² + (x*)² = N holds along the whole lines arg x ≡ ±30° (mod 180°), at every modulus — a positive-dimensional uncolorable locus outside the classification. The resolution, also in this version: the new lines carry no new mechanism at core level and no new flexible family. Along them the ratio x/x* is a primitive sixth root of unity at every modulus, and projective rescaling identifies the lines’ uncolorable content with the known Eisenstein island: the line-stable graph is constant in the modulus away from a finite exceptional set, all 23 of its minimal uncolorable cores (complete enumeration) consist of modulus-independent rays, every core has mechanism flex exactly 0, and all 43 rays of the certified core embed projectively into the closed ω-pool. The pool’s one genuine deformation (flex exactly 1) lives entirely on colorable decoration rays. The uniqueness theorem — the Gould–Aravind circle as the unique positive-dimensional flexible family, scoped to M1–M4 — is unaffected. A complete x²-extended classification of closed-pool identities is recorded as open. The theorem is restated with the unclosed-alphabet hypothesis (under which it is true, and true for all real x); the term-alphabet lemma, correct as literally stated, now carries its scope explicitly. Found by the program’s own adversarial verification gate; every number above is from exact arithmetic with SAT and backtracking engines agreeing. Version 7 (July 2026) — the open problem left by v6 is closed, and the two-mechanism dichotomy is restored. v6 corrected a scope error (with x* as an entry the product x² appears, so the published 80-case classification is incomplete for Hermitian-closed pools) and recorded the complete x²-extended classification as open. This version carries it out. The classification. The closed term set has six types, giving 224 three-term cases; 126 are impossible for every nonzero x, and the rest reduce to the published loci M1–M5 together with exactly four new positive-dimensional loci — x²+(x*)² = ±N (the lines arg x = ±30° and ±60°) and x²+(x*)² = ±1 (two rectangular hyperbolae) — plus eight isolated points, all of which lie on the 60° line. Restricted to the 80 published cases the same computation returns exactly the 54 impossible cases and exactly M1–M5, reproducing the original theorem. The corrected dichotomy, now a theorem rather than an open problem. The closed raw pool is KS-uncolorable if and only if (A) |x|² ∈ {2, 1/2} or x ∈ {±2, ±1/2}, or (B′) x/x* is a primitive third or sixth root of unity. Established by an audit of all 58 realizable locus-membership classes, using the fact that the index-labelled closed graph depends only on which loci x belongs to. Why this is deflationary, and the point of the version. A ratio lemma explains the new uncolorable loci rather than adding a mechanism: projective rescaling by 1/x* carries the degree-one sub-pool isomorphically onto the pool over {0, ±1, ±r} with r = x/x*, multiplying every inner product by 1/N and so preserving orthogonality exactly. Since |r| = 1, the uncolorable cases are precisely those where r is a primitive third or sixth root of unity, and there the sub-pool is the unclosed Eisenstein pool at every modulus. The two hyperbola loci are genuinely new identities but carry no uncolorable content at any point tested; the arg = 45° locus, where r = i has order four and the pool is colorable, shows that being a root of unity is necessary but not sufficient. Consequence for the two-mechanism reading. Hermitian closure does not break the dichotomy reported in arXiv:2603.16988; it relocates one half of it. The literal rule “|x|² ≥ 3 and x not a root of unity implies colorable” fails — x = √3 + i is a counterexample — but that x has x/x* = e^{iπ/3}, and the corrected statement is again exactly two mechanisms: modulus, acting on x; phase, acting on x/x*. We read this as identifying the correct invariant for the phase mechanism rather than as a counterexample to the pattern. Verification scripts x2class_derive.py, x2class_pools.py, x2class_scan.py, x2class_deflate.py and x2class_closure.py accompany this version in the public repository; every colorability verdict was decided by SAT and by an independent backtracker in agreement, in exact arithmetic in the appropriate number field.
What this paper does. It builds a tower of Kochen–Specker circles: continuous one-parameter families of KS sets in even dimensions 4, 6, 8, 10 and 12, all carried by the unimodular cancellation mechanism |x|2 = 1 over two-symbol coordinate alphabets. For each rung it computes the Wilczek–Zee holonomy of the KS loop and determines the Galois group of the holonomy's characteristic polynomial, which is the full symmetric group Sd at all five even rungs for cores with irreducible coupling. It also proves a trace law — |Tr S| = d/2 identically on the working stratum, hence det W(2π) = 1 — and assembles a p-cycle certificate route whose required prime exists at every even dimension. Attribution. The three-dimensional Peres/Penrose family originates with Gould and Aravind; this program's contribution there is the gauge reduction showing the moduli space is a circle rather than a 3-torus, which corrects their pairwise-inequivalence claim. The Kochen–Specker uncolorability of the integer {0,±1} ray pools is due to Pavičić and collaborators, not to this program; the paper records only an independent re-derivation and a mechanism-independence observation downstream of their result. The dimension-three two-symbol classification is Kernaghan's (arXiv:2603.16988). The Galois machinery is classical and is not claimed as new: Jordan's 1873 theorem, the block argument of Wielandt (Finite Permutation Groups, Thm. 13.9), the standard practice recorded by Isaacs and Zieschang, and Nagura's 1952 prime-interval theorem. Only the application of that machinery to these holonomy polynomials is this program's. Status and scope. This is a preprint, not a peer-reviewed article. Every claim in the paper carries an explicit evidence label (proved, exact, numerical, conjecture, literature, open), and the scope limits are stated rather than implied. In particular: the Sd law is established only at the rungs actually built, and the general even-dimensional statement is a conjecture; the hypothesis of irreducible coupling is necessary, and the paper exhibits the counterexamples that show it; existence of cores beyond d = 12 is not claimed constructively; and flex certificates have been run at d = 10 only, not at d = 8 or d = 12. All arithmetic is exact; no claim rests on floating-point evidence. Relation to arXiv:2603.16988. An earlier version of that survey reported the Peres/Penrose flex as infinitesimal only. In v8 its author retracts that statement in a labelled correction note and publishes the cause, a constraint-ordering error. The present paper describes the matter as settled, and does not treat v8 as independent confirmation of finiteness: v8's own continuation is double precision and its modular rank computation bounds flex only from above, as its author states. The finite-flexibility verdict rests on the exact certificates of the companion paper. Use of AI assistance. This work was produced with substantial assistance from AI language models, used for exact symbolic and combinatorial computation, for drafting, for literature search, and for adversarial review of the arguments. Every mathematical claim was independently recomputed, and an adversarial pre-publication audit was run against the manuscript. The author is responsible for the content. This disclosure is made unconditionally. Verification code. https://github.com/manuflog/contextuality-obstructions Version 2 (July 2026) — a theorem is corrected and a claim is withdrawn. Version 1 stated a “reachability law” asserting that the unimodular cancellation mechanism |x|²=1 is algebraically realizable in dimension d if and only if d is even and d ≥ 4, and concluded that a flexible KS circle of this kind can exist in every even dimension and in no odd dimension. The odd-exclusion is false and is withdrawn. The proof partitioned the terms of a vanishing sum by symbol type into four groups, {1}, {x}, {x*}, {N}. The alphabet also contains 0, and the zero terms were omitted from that partition; padding with zeros defeats the parity argument. Explicitly, in C&sup5; the rays u = (1, x, 0, 0, 0) and v = (1, −x, 0, 0, 0), both over {0, ±1, ±x}, have inner product 1 − |x|², which vanishes exactly when |x|² = 1 — the unimodular mechanism, in an odd dimension. What survives, and is now stated as the theorem, is the full-support form: a vanishing sum of m nonzero alphabet terms forcing |x|² = p/q independently of arg x exists iff m ≥ p+q and m ≡ p+q (mod 2), so the unimodular mechanism needs an even m ≥ 4. Since an overlap in C^d contributes one term per coordinate and zeros are permitted, the mechanism is available in every dimension d ≥ 4, of either parity. This is not a technicality about phrasing, and the correction is stated at length in the paper rather than absorbed. Every core built here from d = 6 upward is drawn from the support-four stratum, so its overlaps carry at most four nonzero terms whatever d is, and the identity actually in use is the four-term one available at all d ≥ 4. The parity law governs the full-support case only, which is not the case the tower is built from. Consequently the evenness of the rungs constructed here is no longer explained by this paper, and no obstruction to an odd rung is claimed. Why the constructed rungs are all even is now recorded as open. Everything else is unchanged: the cores at d = 4, 6, 8, 10, 12, the S_d Galois certificates, the trace law, and the anchor obstruction are unaffected, since none of them uses the withdrawn direction.
We look at generalisations of sets of vectors proving the Kochen-Specker theorem in 3 and 4 dimensions. It has been shown that two such sets, although unitarily inequivalent, are part of a larger 3-parameter family of vectors that share the same orthogonality graph. We find that these sets are unusual, in that the vectors in all other Kochen-Specker sets investigated here are fully determined by orthog-onality conditions and thus admit no free parameters. The Kochen-Specker (KS) theorem [1] is a statement about the fundamen-tal nature of quantum mechanics: no non-contextual hidden variable theory can reliably reproduce quantum mechanical predictions. The KS theorem states that in a Hilbert space with dimension n ≥ 3 it is impossible to assign definite values v ∈ {0, 1} to m projection operators Pm so that for every commuting set of projection operators satisfying Pm = I, the correspond-ing definite values obey v (Pm) = 1. The KS theorem is often visualised by means of a set of "uncolourable" vectors, where projection operators are represented by their constituent rays in Hilbert space and rays can be simplified to vectors as only directions are relevant here. The vectors are coloured according to their pre-assigned val-ues and the summation conditions above translate into colouring rules. This allows us to construct orthogonality graphs that, when uncolourable, pro-vide a contradiction within non-contextual hidden variable models and thus a proof of the KS theorem. Sets of vectors exemplifying the KS theorem have been discovered by nu-merous authors, with an aim to reduce the number of vectors required to reach a contradiction. Most notably, in a 3-dimensional Hilbert space, the original KS proof of 117 vectors has been replicated with sets using between 31 and 33 vectors [2, 3, 4, 5] and in a 4-dimensional Hilbert space, sets of between 24 and the lower limit of 18 vectors have been realised [2, 6, 7]. Additionally, an extensive computer search has recently revealed over one thousand of such KS sets in R4 [8]. In a publication by E. Gould and P. K. Aravind [10], two of these sets in 3 dimensions have been shown to share the same orthogonality graph. Specifically, the sets of 33 vectors found by Penrose [3] and Peres [2] have
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
Logical Kochen-Specker (KS) contextuality is widely regarded as an intrinsic property of specially constructed measurement configurations. We show instead that it can emerge from KS-colorable vector sets through a constructive procedure we call the Hamilton extension. Defined for four-dimensional vector sets, the Hamilton extension associates each real vector with a measurement context while inducing additional measurement contexts among Hamilton-extended children of distinct parent vectors. These emergent contexts fundamentally alter the compatibility structure, transforming KS-colorable configurations into KS-uncolorable ones and recovering logical contextuality lost under apex-vertex augmentation. We establish a sharp and optimal threshold -- the Hamilton extension of every five-vector parent set remains KS-colorable, whereas suitably chosen six-vector parent sets already generate logical KS contradictions. Thus, six vectors constitute the smallest parent set capable of generating KS contradiction through this mechanism. Our results reveal a new structural route to contextuality, provide a systematic framework for constructing compact KS sets, and have implications for contextuality-based quantum information protocols and graph-theoretic approaches to nonclassicality.