Strong $O$-valued contextuality: ruling out discrete nondeterministic alternatives to quantum theory

Ravishankar Ramanathan

REVIEW 4 major objections 4 minor 40 references

Strong $O$-valued contextuality: ruling out discrete nondeterministic alternatives to quantum theory

T0 review · 4 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read For any finite set O of allowed outcome probabilities, a finite Hardy-type measurement configuration separates quantum theory from every O-valued noncontextual model, and a state-independent version rules out all O* whose triples on the sim

desk verdict A promising generalization of Kochen-Specker to finite O with a clean LP no-go, but the printed qutrit construction is not an orthonormal representation as written; fix the appendix before relying on the central claim. read the letter →

arxiv 2607.14931 v1 pith:ZMLZGIOR submitted 2026-07-16 quant-ph

Ravishankar Ramanathan This is my paper · ORCID

classification quant-ph PACS 03.65.Ta03.67.Mn

keywords contextualityKochen-SpeckertheoremGleason'sHardy-typetestsnoncontextualempiricalmodelsorthogonalitygraphsfinite-valuedlogicsBornrule

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that the Kochen-Specker no-go result can be lifted from deterministic {0,1} outcome assignments to any finite subset O of [0,1] as the allowed ontic probability values. The central construction is a family of nested Clifton orthogonality graphs, read as Hardy-type experiments: any O-valued noncontextual model forces the target box to be full with probability at most the largest element w_q of O, while a qutrit realization gives (k/(k+2))^2. Choosing the nesting depth k larger than 2√w_q/(1−√w_q) makes the quantum value strictly exceed the O-valued bound, falsifying every such discrete nondeterministic noncontextual ontology. A second, state-independent theorem rules out O*-valued assignments for any finite O* whose triples on the 2-simplex all lie on the boundary. The upshot is that finiteness of the allowed outcome set, by itself, does not rescue hidden-variable theories: quantum theory also evades global sections of all finite-valued presheaves.

What carries the argument

The carrying object is the k-nested Clifton graph: k copies of the eight-vertex Clifton graph glued by identifying v5^(m) with v1^(m+1) and v8^(m) with v2^(m+1), leaving 6k+2 vertices. Its role is to amplify a small noncontextual constraint: the linear-programming induction in Theorem 2 shows that assigning probability 1 to the outer vertex v1^(1) forces the probability of v2^(1) down to at most 1−2^{−k} in any consistent model. The matching qutrit machinery is a rotation-based optimal orthonormal representation (rotation R by π/2 about (0,1,1)/√2, angles θ_j=½ arcsin(j/(j+2))) that attains the overlap bound k/(k+2) between the outer vectors; the recursion x_k=(x_{k+1}+1)/(3−x_{k+1}) connect

What would settle it

Numerically evaluate the Gram matrix of the 6k+2 vectors in equations (B11)-(B12) for k=2 and k=3, checking every required orthogonality, the identifications v5^(m)=v1^(m+1) and v8^(m)=v2^(m+1), and the value of |⟨v1^(1)|v2^(1)⟩|. The central claim stands only if this equals k/(k+2) and the Hardy constraints hold; a simpler signature is whether the printed inner product between v5^(k) and v8^(k) is zero.

Watch

Extended reading notes

Core claim

The core claim is that for every finite O={0,w1,...,wq,1} there is a finite measurement scenario whose quantum statistics are incompatible with every O-valued noncontextual model. The scenario is the k-nested Clifton graph. For any no-disturbance assignment with f(v1^(1))=1, a linear-programming induction gives f(v2^(1))≤1−2^{−k}, so in any O-valued model f(v2^(1))≤w_q. The paper constructs a qutrit orthonormal representation of the same graph with outer-vertex overlap |⟨v1^(1)|v2^(1)⟩|=k/(k+2), and hence a Hardy test in which P(q)(1|v2^(1))=(k/(k+2))^2. For k above threshold the quantum value is larger. It also proves a constructive generalized Kochen-Specker theorem: for O* with (O*)^3∩Δ2⊆

Load-bearing premise

The no-go argument assumes the explicit list of qutrit vectors actually realizes the nested Clifton graph: that all claimed orthogonality edges, the layer identifications v5^(m)=v1^(m+1) and v8^(m)=v2^(m+1), and the innermost edge are satisfied, so that the overlap between the outer vectors is exactly k/(k+2).

Editorial extensions

If this is right

For any finite O, there exists an experimentally feasible Hardy-type test using 6k+2 projective measurements whose outcome statistics cannot be explained by any O-valued noncontextual model; the required number of projectors grows as 6⌈2√w_q/(1−√w_q)⌉+8.
Quantum theory evades global sections of all finite-valued presheaves, so strong O-valued contextuality strictly extends the established probabilistic-possibilistic-strong hierarchy.
Finite many-valued logical models, when identified with O-valued truth assignments via the Birkhoff-von Neumann–Łukasiewicz correspondence, are ruled out by a finite partial Boolean algebra rather than by the infinite continuity-based Gleason argument.
As the nesting depth k grows, the optimal quantum correlations approach extremal nonclassical points of the no-disturbance polytope, providing a route toward semi-device-independent randomness protocols secure against O-valued adversaries.
The generalized Kochen-Specker theorem covers a broad class of O*, including sets like {0,1/m,1} for m≥2, going beyond prior results that only handled sets of the form {0,p,1−p,1}.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the construction is correct, the nesting depth needed to refute a given O supplies a quantitative measure of how close that discrete ontology sits to quantum theory: as w_q approaches 1, the required number of measurements diverges, suggesting a graded hierarchy between finite-valued hidden-variable theories and the Born rule.
The O-valued contextual fraction can likely be operationalized as a resource monotone for semi-device-independent tasks, since the Hardy violation gives a direct lower bound on the weight of the behavior that no O-valued model can simulate.
A natural next step the paper leaves open is to convert the nested Clifton graph into a state-independent Kochen-Specker set for arbitrary O, which would place quantum correlations arbitrarily close to extremal no-signalling points while keeping O-valued marginals finite.
The explicit vector formulas in the appendix are the part most worth re-deriving carefully: a corrected representation with the same overlap would preserve the argument, while any failure of the claimed identifications or final orthogonality would leave the central theorem without its stated quantum realization.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core idea here is worth taking seriously: a finite, state-dependent Hardy family that rules out O-valued noncontextual models for arbitrary finite O, plus a state-independent version for a class O*. The LP no-go argument (Theorem 2) is clean and does what it claims, and the framing of strong O-valued contextuality is a natural and useful extension of the Abramsky–Brandenburger hierarchy. That part of the paper deserves credit.

The problem is that the explicit quantum realization, which is the load-bearing piece of the Hardy test, is not valid as printed. I checked the vectors in (B11)–(B12): v3, v4, v6 and v7 have squared norm 2/(k+2), not 1. So the maximum cliques {v3,v4,v5} and {v6,v7,v8} are not orthonormal bases, and the Hardy normalization conditions cannot be satisfied by the stated projectors. Separately, Lemma 1's parametrization has the c^2 and s^2 definitions swapped: as printed, v3 and v4 are not mutually orthogonal; swapping the definitions fixes that. The induction display in Theorem 3 is also garbled, though the intended formula is clear. These look like fixable typos rather than a contradiction in the no-go logic, but they are not cosmetic: without correction, the claimed quantum value P(q)(1|v2) = (k/(k+2))^2 is unsupported.

There are two other soft spots. First, Theorem 10's proof is not self-contained: step three is explicitly deferred to the author's earlier [18]. That prior result is published and independent, so it's not fatal, but the paper should either include the step or state clearly that the generalized KS theorem relies on [18]. Second, the novelty assessment is fair as far as it goes: the explicit Hardy family is new, but the existence of finite no-go sets for arbitrary finite O was already implicit in Pitowsky and in Hrushovski–Pitowsky, and the paper itself uses Corollary 1 of [39] as an alternative tool in Appendix E.

Who should read this? Foundational quantum information people working on contextuality as a resource and on finite Gleason-type results. They will find the framework useful, but they should not quote the Hardy construction until the appendix is corrected. I would send this to a serious referee: the question is important enough and the errors appear repairable. The referee should be asked to verify the orthonormal representation explicitly, not just check the prose.

My recommendation: engage with it, but require a corrected appendix and a self-contained proof of Theorem 10 before accepting the central claims.

Referee Report

4 major / 4 minor

Summary. The paper introduces strong O-valued contextuality, a hierarchy extending the sheaf-theoretic notions of Abramsky and Brandenburger, and claims two main results. First, for any finite O={0,w1,...,wq,1} in [0,1], a k-nested Clifton graph yields a state-dependent Hardy-type test: every O-valued noncontextual model must satisfy P(1|v2^(1)) ≤ wq, while a qutrit realization is claimed to achieve P(1|v2^(1)) = (k/(k+2))^2, so choosing k > 2√wq/(1−√wq) falsifies O-valued noncontextual models. Second, for any O* with (O*)^3∩Δ2 ⊆ ∂Δ2, a finite state-independent KS-type set is claimed to admit no global O*-valued assignment. The paper also discusses applications to randomness, no-signaling polytopes, and finite many-valued logics. The central no-go logic is plausible and appears to extend prior work, but the printed proof of the quantum value contains explicit vector-normalization and algebraic errors that must be fixed.

Significance. If the construction is repaired, the state-dependent result would be a significant advance: it gives finite, experimentally feasible Hardy tests that rule out arbitrary finite nondeterministic noncontextual alternatives to the Born rule, directly addressing a question of Pitowsky and generalizing the author's earlier {0,p,1−p,1} result. The state-independent theorem for the class (O*)^3∩Δ2 ⊆ ∂Δ2 also broadens previous KS-type no-go results. The paper is constructive and, apart from the identified normalization errors, does not appear to smuggle in the conclusion through fitted parameters or normalization choices. The LP proof of Theorem 2 is clean and the explicit overlap formula k/(k+2) appears to be the correct target after renormalization. However, as printed, the existence proof for the quantum realization is invalid, and the state-independent proof delegates a load-bearing step to prior work.

major comments (4)

[Appendix B, Eqs. (B11)-(B12)] The printed vectors |v3^(m)>, |v4^(m)>, |v6^(m)>, |v7^(m)> are not unit vectors. Their squared norm is 2/(k+2), not 1, because the first component contributes k/(k+2) in the numerator. This violates Definition 1 and the constraints M_{x,x}=1 in the SDP (B7), so the claimed qutrit projective measurements do not satisfy the Hardy normalization conditions (2). The apparent fix is to multiply these four vectors by sqrt((k+2)/2), which preserves their mutual orthogonality as written. This must be corrected before the claimed quantum value P^(q)(1|v2^(1))=(k/(k+2))^2 is established.
[Appendix B, Lemma 1, Eq. (B8)] The parametrization of |v3^(k)> and |v4^(k)> has c^2 and s^2 exchanged. With the printed definitions c^2=(1+x_k)/2 and s^2=(1−x_k)/2, the two vectors have inner product −2x_k/(1+x_k), not zero. Swapping the definitions to c^2=(1−x_k)/2 and s^2=(1+x_k)/2 restores orthogonality. Since Lemma 1 is the engine of the induction in Theorem 3, this is a load-bearing error, although a local one.
[Appendix B, Theorem 3 induction] The induction display after Lemma 1 is garbled. Lemma 1 gives x_k = (x_{k+1}+1)/(3−x_{k+1}), with x_{k+1}=|<v5^(k)|v8^(k)>|. Starting from the innermost edge v5^(k)⊥v8^(k), i.e. x_{k+1}=0, the recurrence produces x_{k-r} = r/(r+2) after r steps, yielding x_1=k/(k+2). The printed text instead substitutes x_{k+1}=k/(k+2) and obtains (k+1)/(k+3), which inverts the induction direction and is algebraically incorrect. The final formula is consistent with the explicit representation, but the proof as written does not establish it.
[Appendix E, Theorem 10 / Step three] The proof of the generalized KS theorem is incomplete in the manuscript. Step three is described as 'strictly analogous' to the author's prior [18] and not proved, while the gadget vectors in the displayed list are unnormalized and no explicit verification of the orthogonality graph of Fig. 4 is given. Since Theorem 1 is a central claim, the adaptation of the step-three gadget to the full class O* satisfying Definition 19 must be written out or at least reduced to [18] with a precise dictionary. As printed, the state-independent result rests on an assertion.

minor comments (4)

[Appendix B, Eq. (B12)] In the formula for |v4^(m)> the superscript is printed as |v4^(1)>; it should be |v4^(m)>. Also several lines in the vector display have inconsistent parentheses and spacing.
[Appendix B, Eq. (B3) and surrounding text] The linear program uses the constraint f(v5)+f(v8) ≤ 2−1/2^k, while the text says this comes from the inductive hypothesis f(v5)+f(v8) ≤ 1−1/2^k. The LP form is the correct one; the text should be corrected.
[Appendix C, Theorem 5] The proof of Theorem 5 is only sketched: the no-signaling box P is not fully defined for pairs of different settings x≠y, and the condition P_{ab|xy}=0 when ⟨v_{a,x}|v_{b,y}⟩=0 does not by itself determine a valid nonsignaling correlation. This application should either be proved carefully or explicitly flagged as a conjecture.
[Throughout] There are typographical errors, e.g., 'Langrage multipliers', 'senario' in Theorem 5, and the reference to '6k+1 triangles' should be checked against the count 6k+1 measurement settings. Figures are referenced but not included in the text; the gadget graphs in Appendix E should be accompanied by explicit vector sets.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation; central Hardy bound and quantum witness are independent.

full rationale

The central state-dependent Hardy test is not circular. Theorem 2 derives f(v1^(1))=1 => f(v2^(1)) <= 1 - 2^-k by an explicit linear-programming induction over the no-disturbance constraints of the k-nested Clifton graph; the O-valued bound P^(O)(1|v2) <= w_q follows by restricting the codomain to O, with no use of the target quantum value. The quantum value P^(q)(1|v2) = (k/(k+2))^2 is computed directly from the exhibited qutrit state and projectors; this is a witness construction, not a fitted parameter, and Theorem 3's optimality proof is an induction from the known Clifton-bug base case plus a monotonicity lemma. There are no fitted values, no normalization that encodes the target result, and no equation that reduces to the conclusion. The apparent vector-normalization and c^2/s^2 issues in (B8)/(B11)-(B12) are correctness risks, not circularity. The appendices cite the author's prior [18] and [52] for a gadget and for the ansatz, and Appendix E explicitly defers a step to [18]; however [18] is an earlier independent published result and [52] is used only as a starting point with a proof attempted here. No load-bearing argument reduces to a self-citation that is itself the claim being established, so there is no significant circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard quantum contextuality assumptions (ideal projective measurements, no-disturbance, compatibility), on Gleason/compactness for motivation and non-constructive existence, and on the explicit geometric representation of the nested Clifton graph. The only numerically chosen quantities are construction parameters (k, θ_j, φ1, φ2), not data fits. No new physical entities are introduced.

free parameters (3)

nesting depth k = k > 2√w_q / (1−√w_q)
The Hardy test must be nested deeply enough so that quantum value (k/(k+2))^2 exceeds the largest O-value w_q; this is a tunable construction parameter chosen per O, not a data fit.
angles θ_j and rotation R = θ_j = 1/2 arcsin(j/(j+2)); R fixed π/2 rotation
These define the explicit orthogonal representation achieving overlap j/(j+2); chosen by construction to satisfy the orthogonality graph.
gadget angles φ1, φ2 = φ1 = arctan((-3+√5)/8), φ2 = 1/2 arcsin(2/3), tan(φ1)tan(φ2) = -1/4
Ad hoc angles in the step-three gadget of App. E chosen so the vectors realize the orthogonality graph; the state-independent proof depends on them.

assumptions (5)

standard math Gleason's theorem
Used in App. D to argue the non-existence of finite O-valued frame functions via continuity and logical compactness; the constructive results do not rely on it directly but the motivation does.
standard math Logical compactness theorem
Used in App. D to pass from the absence of a global O-valued assignment to the existence of a finite KS configuration non-constructively.
domain assumption No-disturbance, noncontextuality, and ideal projective measurement assumptions
The O-valued model framework and KS constraints (App. A) assume ideal rank-one projectors, the claimed compatibility structure, and context-independence of outcome probabilities; the authors explicitly state these as the minimal assumptions for contextuality tests.
standard math Treewidth clique-sum theorem
App. C2 Lemma 3 uses the theorem that treewidth of a clique-sum is the maximum of individual treewidths to claim tw(G_k)=3.
domain assumption Hardy constraints are exactly satisfied
The test assumes the state preparation and measurements exactly satisfy conditions (2); finite-statistics and noise robustness are not analyzed.

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Pith. "Pith review of Strong $O$-valued contextuality: ruling out discrete nondeterministic alternatives to quantum theory." pith.science (2026). https://pith.science/paper/ZMLZGIOR

@misc{pith2026260714931,
  author       = {Pith},
  title        = {Pith review of: Strong $O$-valued contextuality: ruling out discrete nondeterministic alternatives to quantum theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZMLZGIOR}},
  note         = {Machine review of arXiv:2607.14931}
}

abstract

Gleason's theorem identifies the Born rule via non-contextuality over an infinite continuous lattice of projectors, while its corollary the Kochen-Specker (KS) theorem rules out the specific class of deterministic ($\{0,1\}$-valued) noncontextual models using finite sets of projectors. Gleason's theorem also indicates the existence of KS-type finite vector constructions to rule out other discrete nondeterministic alternatives to quantum theory beyond the $\{0, 1\}$ case. Here, we construct finite measurement configurations to rule out noncontextual empirical models with outcome probabilities drawn from an arbitrary finite subset $O \subset [0,1]$. We do this by two means: (i) constructing a family of experimentally feasible state-dependent Hardy-type tests, and (ii) proving a generalized KS theorem for a broad class of $O$ that includes prior results as special cases. In the sheaf-theoretic framework of Abramsky and Brandenburger, a hierarchy of probabilistic-possibilistic-strong contextuality has been established quantifying contextuality as a resource. We extend this framework by introducing strong $O$-valued contextuality, showing that quantum theory evades global sections of all finite-valued presheaves. We also discuss the implications of the result on finite many-valued logics as viable ontological models for quantum theory and for contextuality-based (semi)-device-independent protocols.

Figures

Figures reproduced from arXiv: 2607.14931 by the authors.

**Figure 3.** FIG. 3. In this gadget graph used in the proof of step [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗

**Figure 4.** FIG. 4. The gadget graph used to establish step three of the proof. Namely that when [PITH_FULL_IMAGE:figures/full_fig_p026_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 1 linked inside Pith

[39]

Consequently, the triple(f(|v 1⟩), f(|v2⟩), f(|v3⟩))belongs to(O ∗)3 ∩∆ 2. By Def. 19, this triple must lie on the boundary∂∆ 2, which implies that at least one coordinate must equal zero: ∃|vi⟩ ∈Bk such thatf(|v i⟩) = 0.(E4) Furthermore, since the codomain offexplicitly excludes one, a basis cannot contain more than one zero, so that every complete basis...
[1]

The vertex setVis identical to the set of eventsV, where each vertex corresponds to a specific rank-one projector Πi
[2]

In quantum theory, this means the projectors are orthogonal:ΠiΠj = 0⇐ ⇒ ⟨vi |v j⟩= 0

An edgee= (v i, vj)∈Econnects two distinct vertices if the corresponding events are mutually exclusive. In quantum theory, this means the projectors are orthogonal:ΠiΠj = 0⇐ ⇒ ⟨vi |v j⟩= 0. Analogously, an orthonormal representation ofG= (V, E)inn-dimensional Euclidean space is a functionf:V→ Rn such that∥f(v)∥= 1for allv∈Vand⟨f(u), f(v)⟩= 0for all{u, v} ...
[3]

The logic of quantum mechanics

A connection to finite many-valued logical models In 1936, Birkhoff and von Neumann in their breakthrough paper "The logic of quantum mechanics" [21] argued that the structure of a set of dichotomic (‘yes-no’) propositions about properties of quantum objects, usually termed ‘quantum logic’, differs from the structure of a set of such propositions pertaini...

1936
[4]

The always-false propositional functionfis the only propositional function inΛ(S)that excludes itself, i.e., for anyv(·)∈Λ(S), ifv(·)⊓v(·) =f, thenv(·) =f. Conversely, any family of many-valued propositional functions defined on a common domainDand satisfying the conditions above is a quantum logic in the BvN sense when we identify propositional functions...
[24]

This observation can also be extended to generaln-cycle scenarios where again it is known thatPN D(G) =P O(G)forO={0,1/2,1}[38]

In other words, while the quantum value for the Hardy test forC5 was found to beP(q)(0,1|5,1) = 1/9>0in [4], the no-disturbance value for the test is 1/2>1/9, showing that the test cannot rule out a{0,1/2,1}-empirical model. This observation can also be extended to generaln-cycle scenarios where again it is known thatPN D(G) =P O(G)forO={0,1/2,1}[38]. Sim...
[25]

This Nested Clifton graph is shown in Fig

The Nested Clifton Graph and its optimal orthogonal representation Consider the orthogonality graph that results from a nesting of the well-known Clifton graph [45] used by Kochen and Specker in their original proof [1]. This Nested Clifton graph is shown in Fig. 1 and its structure is as follows. 10 v(1) 1 v(1) 3 v(1) 6 v(1) 4 v(1) 7 v(1) 2 v(2) 6 v(2) 7...
[26]

Hardy-type

Note that this is the maximum consistent (no-disturbance) value of the probability assigned tov(1) 2 whenv (1) 1 has probability one. Now suppose that the statement holds fork∈Z +, i.e., thatf(v (1) 1 ) +f(v (1) 2 )≤2− 1 2k for any assignment f:V k →[0,1]. We’ll see that the statement holds fork+ 1. In this case we are interested in the value of the follo...

Show all 40 references

[27]

3 and Corollary B1 into an experimentally feasible test to rule out noncontextual O-valued empirical models for any finiteO

A Hardy-type test to rule out nondeterministic noncontextual models We now translate the above Thm. 3 and Corollary B1 into an experimentally feasible test to rule out noncontextual O-valued empirical models for any finiteO. Specifically, we formulate the above theorems into a...
[28]

Viewed as a resource for such applications, it is important to quantify and understand the strongest manifestations of quantum contextuality

A stronger notion of Kochen-Specker contextuality In quantum information science, Kochen-Specker contextuality goes beyond being a philosophical quirk and is val- ued as a resource [34, 58]behind a multitude of applications - providing the magic behind universal fault-tolerant...
[29]

Extremal no-signalling behaviors and units of nonlocality Kochen-Specker contextuality is intimately connected with Bell nonlocality [35, 51]. In particular, it is known that for certain orthogonality graphs by distributing the contextual measurements across multiple, spatiall...

2005
[32]

For every projectorP∈P(H),Pcannot be assigned more than one value fromO: V P∈P(H) V ri̸=rj ∈O ¬(XP,ri ∧ XP,rj )
[33]

For every projectorP∈P(H),Pmust be assigned at least one value fromO: V P∈P(H) W r∈O XP,r . 23
[34]

, Pd} ∈ F(H), the assigned values must sum to

For every full orthonormal frameB={P 1, P2, . . . , Pd} ∈ F(H), the assigned values must sum to
[35]

, ad)∈ Od Pd i=1 ai = 1 o

LetS B be the finite set of all validd-tuples of weights that satisfy the frame condition:S B =n (a1, a2, . . . , ad)∈ Od Pd i=1 ai = 1 o . The assigned values for the elements ofBmust match one of these valid tuples: _ (a1,...,ad)∈SB d^ i=1 XPi,ai ! .(D3) A truth valuationv:V...
[36]

We then have by completeness thatf(u 2) +f(u ′

= 0. We then have by completeness thatf(u 2) +f(u ′
[37]

+f(u 3) = 1givingf(u 2) =f(u ′
[38]

By the property of the directed dashed edge{u1, u′ 3}(implying 0< f(u′ 3)<1) we obtain thatf(u 2)> f(u3)(note the strict inequality)

+f(u 3). By the property of the directed dashed edge{u1, u′ 3}(implying 0< f(u′ 3)<1) we obtain thatf(u 2)> f(u3)(note the strict inequality). By the property of the dashed edge{u1, u2} we have thatf(u 2)<1implying that in anO ∗-valued assignment there holdsf(u 2)≤w q. Taken t...
[40]

We recognize the first row as the vectors|w1⟩=|u 1⟩,|w 2⟩=|u 8⟩,|w 3⟩=|u 5⟩from step two, this is one of the twelve triples forming the zero-setZof step two

Here|˜wj⟩denotes that the vector is not normalized. We recognize the first row as the vectors|w1⟩=|u 1⟩,|w 2⟩=|u 8⟩,|w 3⟩=|u 5⟩from step two, this is one of the twelve triples forming the zero-setZof step two. We also verify that the vectors faithfully realize the orthogonalit...
[47]

F. J. Belinfante. A survey of hidden-variables theories. Elsevier, 55 (1973)

1973
[48]

Lovasz, M

L. Lovasz, M. Saks, A. Schrijver.Orthogonal representa- tion and connectivity of graphs. Linear Algebra and its applications, 4, 114-115, 439 (1987)

1987
[49]

Piron.Foundations of Quantum Physics

C. Piron.Foundations of Quantum Physics. Addison- Wesley, (1976)

1976
[50]

G. W. Mackey.Mathematical Foundations of Quantum Mechanics. Addison-Wesley, (1963)

1963
[51]

Renner and S

R. Renner and S. Wolf.Quantum pseudo-telepathy and the Kochen-Specker theorem. Int. Symp. Informa- tion Theory (ISIT), Proc. (Piscataway, NJ:IEEE) p322 (2004)

2004
[52]

Ramanathan, M

R. Ramanathan, M. Rosicka, K. Horodecki, S. Pironio, M. Horodecki, and P. Horodecki.Gadget structures in proofs of the Kochen-Specker theorem. Quantum 4, 308 (2020)

2020
[53]

Busch.Quantum states and generalized observables: a simple proof of Gleason’s theorem

P. Busch.Quantum states and generalized observables: a simple proof of Gleason’s theorem. Phys. Rev. Lett. 91, 120403 (2003)

2003
[54]

M. P. Solér.Characterization of Hilbert spaces by ortho- modular spaces. Communications in Algebra 23, 219-243 (1995)

1995
[55]

H. B. Enderton.A Mathematical Introduction to Logic (2nd ed.), Academic Press (2001)

2001
[56]

Ramanathan, A

R. Ramanathan, A. Soeda, P. Kurzynski, and D. Kas- zlikowski.Generalized monogamy of contextual inequali- 7 ties from the no-disturbance principle. Phys. Rev. Lett. 109, 050404 (2012)

2012
[57]

A. Acín, T. Fritz, A. Leverrier, and A. B. Sainz.A Combinatorial Approach to Nonlocality and Contextual- ity. Comm. Math. Phys. 334(2), 533- 628 (2015)

2015
[58]

Abramsky, R

S. Abramsky, R. S. Barbosa, and S. Mansfield.Contex- tual Fraction as a Measure of Contextuality. Phys. Rev. Lett. 119, 050504 (2017)

2017
[59]

T. D. Galley and Ll. Masanes.Classification of all alter- natives to the Born rule in terms of informational prop- erties. Quantum 1, 15 (2017)

2017
[60]

Aaronson.Is Quantum Mechanics An Island In Theo- ryspace?arXiv:0401062 (2004)

S. Aaronson.Is Quantum Mechanics An Island In Theo- ryspace?arXiv:0401062 (2004)

2004
[61]

Ramanathan, M

R. Ramanathan, M. Horodecki, H. Anwer, S. Pironio, K. Horodecki, M. Grünfeld, S. Muhammad, M. Bourennane and P. Horodecki.Practical No-Signalling proof Random- ness Amplification using Hardy paradoxes and its exper- imental implementation. arXiv:1810.11648 (2018)

2018 arXiv
[62]

Barrett, L

J. Barrett, L. Hardy and A. Kent.No signaling and quan- tum key distribution. Phys. Rev. Lett.95, 010503 (2005)

2005
[63]

S. Zhao, R. Ramanathan, Y. Liu, P. Horodecki.Tilted Hardy paradoxes for device-independent randomness ex- traction. Quantum 7, 1114 (2023)

2023
[64]

V. K. Sujan, R. Kunjwal.No-go theorem for quantum realization of extremal correlations. arXiv:2509.14879 (2025)

2025
[65]

Ramanathan, J

R. Ramanathan, J. Tuziemski, M. Horodecki, and P. Horodecki.No Quantum Realization of Extremal No- Signaling Boxes. Phys. Rev. Lett. 117, 050401 (2016)

2016
[66]

Lovász.Graph minor theory

L. Lovász.Graph minor theory. Bulletin of the American Mathematical Society, 43 (1): 75–86 (2006)

2006
[67]

Diestel.Graph Theory(3rd ed.), Springer,(2005)

R. Diestel.Graph Theory(3rd ed.), Springer,(2005). Appendix A: Preliminaries: Structure of the noncontextual models Multiple ways to define noncontextual hidden-variable models are present in the literature, including the marginal problem definition [13, 14], the graph-theoret...

2005

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [39]

Consequently, the triple(f(|v 1⟩), f(|v2⟩), f(|v3⟩))belongs to(O ∗)3 ∩∆ 2. By Def. 19, this triple must lie on the boundary∂∆ 2, which implies that at least one coordinate must equal zero: ∃|vi⟩ ∈Bk such thatf(|v i⟩) = 0.(E4) Furthermore, since the codomain offexplicitly excludes one, a basis cannot contain more than one zero, so that every complete basis...

[2] [1]

The vertex setVis identical to the set of eventsV, where each vertex corresponds to a specific rank-one projector Πi

[3] [2]

In quantum theory, this means the projectors are orthogonal:ΠiΠj = 0⇐ ⇒ ⟨vi |v j⟩= 0

An edgee= (v i, vj)∈Econnects two distinct vertices if the corresponding events are mutually exclusive. In quantum theory, this means the projectors are orthogonal:ΠiΠj = 0⇐ ⇒ ⟨vi |v j⟩= 0. Analogously, an orthonormal representation ofG= (V, E)inn-dimensional Euclidean space is a functionf:V→ Rn such that∥f(v)∥= 1for allv∈Vand⟨f(u), f(v)⟩= 0for all{u, v} ...

[4] [3]

The logic of quantum mechanics

A connection to finite many-valued logical models In 1936, Birkhoff and von Neumann in their breakthrough paper "The logic of quantum mechanics" [21] argued that the structure of a set of dichotomic (‘yes-no’) propositions about properties of quantum objects, usually termed ‘quantum logic’, differs from the structure of a set of such propositions pertaini...

1936

[5] [4]

The always-false propositional functionfis the only propositional function inΛ(S)that excludes itself, i.e., for anyv(·)∈Λ(S), ifv(·)⊓v(·) =f, thenv(·) =f. Conversely, any family of many-valued propositional functions defined on a common domainDand satisfying the conditions above is a quantum logic in the BvN sense when we identify propositional functions...

[6] [24]

This observation can also be extended to generaln-cycle scenarios where again it is known thatPN D(G) =P O(G)forO={0,1/2,1}[38]

In other words, while the quantum value for the Hardy test forC5 was found to beP(q)(0,1|5,1) = 1/9>0in [4], the no-disturbance value for the test is 1/2>1/9, showing that the test cannot rule out a{0,1/2,1}-empirical model. This observation can also be extended to generaln-cycle scenarios where again it is known thatPN D(G) =P O(G)forO={0,1/2,1}[38]. Sim...

[7] [25]

This Nested Clifton graph is shown in Fig

The Nested Clifton Graph and its optimal orthogonal representation Consider the orthogonality graph that results from a nesting of the well-known Clifton graph [45] used by Kochen and Specker in their original proof [1]. This Nested Clifton graph is shown in Fig. 1 and its structure is as follows. 10 v(1) 1 v(1) 3 v(1) 6 v(1) 4 v(1) 7 v(1) 2 v(2) 6 v(2) 7...

[8] [26]

Hardy-type

Note that this is the maximum consistent (no-disturbance) value of the probability assigned tov(1) 2 whenv (1) 1 has probability one. Now suppose that the statement holds fork∈Z +, i.e., thatf(v (1) 1 ) +f(v (1) 2 )≤2− 1 2k for any assignment f:V k →[0,1]. We’ll see that the statement holds fork+ 1. In this case we are interested in the value of the follo...