Contextuality and the Kochen–Specker Theorem in Biquaternionic Form

Introduction

A context is a set of observables that can be measured together — a maximal set of mutually commuting Hermitian elements. Contextuality is the failure of the assumption that the outcome assigned to an observable is independent of which context it is measured in. The Kochen–Specker theorem is the sharpest statement of that failure: in dimension three and higher there is no assignment of definite outcomes to the projectors that is consistent in every context. This article develops contexts, value assignments, and the Kochen–Specker obstruction in the biquaternion language.

The framework's algebra makes the ingredients explicit. Observables are Hermitian elements of $\mathbb{M}_+$ (or of $\mathbb{M}_+^{\otimes n}$ for $n$ qubits). A context is a maximal commutative subalgebra of Hermitian elements, generated by a complete set of commuting involutions; its spectral resolution is a complete family of orthogonal idempotents, which in the two-qubit case is an idempotent basis such as the Bell basis. A noncontextual value assignment is a map from the Hermitian elements to the reals that restricts to an algebra homomorphism on each context, $v(\tilde{A}\tilde{B}) = v(\tilde{A})v(\tilde{B})$ and $v(\tilde{A}+\tilde{B}) = v(\tilde{A})+v(\tilde{B})$ whenever $\tilde{A}$ and $\tilde{B}$ commute, with $v(e_0)=1$ and $v(\tilde\Pi)\in\{0,1\}$ on idempotents. The Kochen–Specker theorem says that for a system whose defining module has dimension at least three, no such assignment exists.

The framework's native module is two-dimensional. In dimension two the obstruction is empty: every context contains at most two orthogonal rank-one idempotents, and a consistent assignment can be constructed. The smallest arena in which the obstruction appears within the framework is therefore the two-qubit algebra $\mathbb{B}\otimes\mathbb{B}\cong M_4(\mathbb{C})$, whose module has dimension four, and there the obstruction has a particularly compact form: the Mermin–Peres magic square, nine commuting observables arranged so that the product of every row is $+e_0\otimes e_0$ and the product of two columns is $+e_0\otimes e_0$ while the product of the third is $-e_0\otimes e_0$. A noncontextual $\pm1$ assignment would have to make the product of all nine entries both $+1$ and $-1$. This article constructs that square in biquaternion form, verifies its product relations, and relates it to the state-dependent contextuality of the CHSH inequality treated elsewhere in the subcategory.

Two boundaries are respected. The physics of spin and of measurement is not re-derived; the article is about the information-theoretic structure of contexts and their obstruction. And the results of the algebra that are purely structural — the classification of commutative subalgebras, the idempotent bases — are taken from the companion articles and read here for their contextual content.

Contexts and Value Assignments

Observables and contexts

An observable is a Hermitian element $\tilde{A}\in\mathbb{M}_+$, with spectral decomposition $\tilde{A} = \sum_a a\,\tilde\Pi_a$ into orthogonal idempotents $\tilde\Pi_a$ and real eigenvalues $a$. Two observables are compatible when they commute, $[\tilde{A},\tilde{B}] = 0$; then they possess a common spectral resolution and can be measured together.

A context is a maximal set of mutually commuting observables. Equivalently, it is a maximal commutative $^\dagger$-subalgebra of $\mathbb{B}$ consisting of Hermitian elements, together with its joint spectral resolution. In the two-qubit algebra, a context generated by two commuting involutions has four joint idempotents; the Bell basis article in this series shows that the stabilizer pair $S_1 = -e_1\otimes e_1$ and $S_3 = -e_3\otimes e_3$ generates the Cartan subalgebra whose four joint spectral projectors are the Bell idempotents $P_\epsilon$. That is the framework's canonical two-qubit context, and it is a complete commuting set of observables.

Three features of the framework's contexts matter here.

A context is commutative, so its elements can be simultaneously diagonalized; within a context the algebra behaves as a classical algebra of functions on the joint spectrum.
The joint spectrum is realized by a complete family of orthogonal idempotents summing to the identity, $\sum_a\tilde\Pi_a = e_0$ with $\tilde\Pi_a\tilde\Pi_b = \delta_{ab}\tilde\Pi_a$.
A given idempotent belongs to many contexts: any maximal commutative subalgebra containing it. Contextuality is precisely the statement that the value assigned to that idempotent cannot be chosen independently of which of these subalgebras is used.

Noncontextual value assignments

A noncontextual value assignment is a function $v$ on the observables satisfying

$$ v(e_0) = 1, \qquad v(\tilde{A}+\tilde{B}) = v(\tilde{A})+v(\tilde{B}), \qquad v(\tilde{A}\tilde{B}) = v(\tilde{A})\,v(\tilde{B}) \quad\text{whenever }[\tilde{A},\tilde{B}]=0, $$

and, for observables with spectrum in $\{0,1\}$, taking values in $\{0,1\}$. The first two conditions make $v$ linear on each context; the third makes it multiplicative, hence an algebra homomorphism on each context. Restricted to the idempotents of a context, $v$ is a $\{0,1\}$-valued function with

$$ v(\tilde\Pi_a)\in\{0,1\}, \qquad \sum_a v(\tilde\Pi_a) = v\!\left(\sum_a\tilde\Pi_a\right) = v(e_0) = 1 , $$

so exactly one idempotent of each context is assigned the value one. Noncontextuality is the requirement that $v(\tilde\Pi)$ be the same whichever context containing $\tilde\Pi$ is used to compute it. A hidden-variable model of the Kochen–Specker type is precisely such a noncontextual assignment, possibly averaged over a hidden variable; the theorem below shows that none exists in dimension at least three.

The single qubit: no obstruction

For the native qubit the module has complex dimension two, and every context contains at most two orthogonal rank-one idempotents, $\tilde\Pi_+(\hat{n})$ and $\tilde\Pi_-(\hat{n})$. Two orthogonal rank-one idempotents of a qubit are the only possible pair, and a $\{0,1\}$ assignment to them consistent with $\tilde\Pi_++\tilde\Pi_- = e_0$ is immediate. Extending to all observables, one may assign to each direction $\hat{n}$ the value

$$ v_\lambda\bigl(\tilde\Pi_\pm(\hat{n})\bigr) = \begin{cases}1 & \pm\,\hat{n}\cdot\hat{\lambda} > 0,\\ 0 & \text{otherwise,}\end{cases} $$

for a fixed hidden direction $\hat{\lambda}$, with a convention on the measure-zero equator. Every context of the qubit is a pair of antipodal projectors, and the assignment is consistent within each; there is no constraint linking two different directions, because two non-collinear rank-one projectors of a qubit are not orthogonal and therefore do not lie in a common context. Hence no Kochen–Specker obstruction exists for a single qubit, which is the familiar statement that the theorem requires dimension at least three. The single-qubit noncontextual model is due to Bell (1966).

The conclusion for the framework is structural: the obstruction cannot be exhibited by the native unit alone. It requires a composite, or a system of dimension at least three, and the smallest composite the framework supplies is the two-qubit algebra $\mathbb{B}\otimes\mathbb{B}$, whose module has dimension four.

Why the single-qubit model cannot be extended. The reason is the scarcity of contexts. Two rank-one projectors of a qubit lie in a common context only if they are orthogonal, and for a qubit orthogonality of $\tilde\Pi(\hat{n})$ and $\tilde\Pi(\hat{m})$ means the directions are antipodal. Consequently every context of the qubit is the pair $\{\tilde\Pi_+(\hat{n}),\tilde\Pi_-(\hat{n})\}$ for a single direction, and the value assignment of Bell is consistent on each such context by construction. Noncontextuality is then trivially satisfied, because no observable belongs to two distinct contexts: an observable's ray determines its context uniquely. In dimension $d\geq3$ this fails — a rank-one projector belongs to a continuum of orthogonal resolutions, and those resolutions impose incompatible constraints — and it is the sharing of observables among many contexts that drives the obstruction. In the two-qubit algebra the sharing is realized by the magic square: each of the nine entries belongs to one row context and one column context simultaneously, and it is precisely that double membership — for instance $e_0\otimes\tilde{Z}$ sitting in row $2$ and column $1$ at once — that the parity argument exploits.

The Kochen–Specker Theorem

Statement

Kochen–Specker theorem. Let $\mathcal{H}$ be a Hilbert space of dimension $d\geq3$. There is no function $v$ from the rank-one projectors of $\mathcal{H}$ to $\{0,1\}$ that is noncontextual and consistent on every orthogonal resolution of the identity:

$$ v(P)\in\{0,1\}, \qquad \sum_{i}P_i = I \ \Longrightarrow\ \sum_i v(P_i) = 1 . $$

Equivalently, no noncontextual deterministic hidden-variable model reproduces the predictions of quantum mechanics for a system of dimension at least three. The theorem is state-independent: it concerns the structure of observables, not any particular state.

The hypothesis $d\geq3$ is necessary. For $d=2$ the constraints are satisfiable, as the previous subsection showed. The original proof of Kochen and Specker used a finite set of vectors in $\mathbb{R}^3$; simpler proofs use the Peres–Mermin configuration in $d=4$, which is the two-qubit case relevant to the framework.

The dimension-four arena

The two-qubit algebra $\mathbb{B}\otimes\mathbb{B}$ is isomorphic to $M_4(\mathbb{C})$, and its defining module has complex dimension four. This is the smallest dimension in which the Kochen–Specker obstruction has a compact, easily verified form: the Mermin–Peres square, a $3\times3$ array of two-qubit observables each squaring to the identity, with every row and column a commuting set, and with product relations that are globally inconsistent. The next section constructs it in the framework's notation and verifies the relations.

The Mermin–Peres Magic Square in Biquaternion Form

The observables

On a single qubit introduce the three Hermitian involutions

$$ \tilde{Q} = ie_1, \qquad \tilde{Y} = ie_2, \qquad \tilde{Z} = ie_3, $$

each of which satisfies $\tilde{Q}^2 = \tilde{Y}^2 = \tilde{Z}^2 = e_0$ and which multiply as the Pauli matrices do up to central phases, for example

$$ \tilde{Q}\tilde{Z} = (ie_1)(ie_3) = -e_1e_3 = e_2 = -i\tilde{Y}, \qquad \tilde{Z}\tilde{Q} = -e_2 = i\tilde{Y}. $$

On the two-qubit algebra $\mathbb{B}\otimes\mathbb{B}$, tensor these with $e_0$ and with one another to form the nine elements

$\tilde{Q}\otimes e_0$	$e_0\otimes\tilde{Q}$	$\tilde{Q}\otimes\tilde{Q}$
$e_0\otimes\tilde{Z}$	$\tilde{Z}\otimes e_0$	$\tilde{Z}\otimes\tilde{Z}$
$\tilde{Q}\otimes\tilde{Z}$	$\tilde{Z}\otimes\tilde{Q}$	$\tilde{Y}\otimes\tilde{Y}$

Each entry is Hermitian and squares to $e_0\otimes e_0$, so each has spectrum $\{\pm1\}$; this is why the array is called a magic square. The entries are the framework's transcription of the standard Mermin–Peres observables built from $\sigma_1,\sigma_2,\sigma_3$ and the identity, using the identification $\sigma_k\leftrightarrow ie_k$.

The product relations

Within each row and each column the three entries commute, so their product is unambiguous. Direct computation in the biquaternion algebra — using $(x\otimes y)(x'\otimes y') = (xx')\otimes(yy')$ and the products of the units — gives

	$\text{column }1$	$\text{column }2$	$\text{column }3$	$\text{row product}$
row 1	$\tilde{Q}\otimes e_0$	$e_0\otimes\tilde{Q}$	$\tilde{Q}\otimes\tilde{Q}$	$\prod = e_0\otimes e_0$
row 2	$e_0\otimes\tilde{Z}$	$\tilde{Z}\otimes e_0$	$\tilde{Z}\otimes\tilde{Z}$	$\prod = e_0\otimes e_0$
row 3	$\tilde{Q}\otimes\tilde{Z}$	$\tilde{Z}\otimes\tilde{Q}$	$\tilde{Y}\otimes\tilde{Y}$	$\prod = e_0\otimes e_0$
column product	$\prod = e_0\otimes e_0$	$\prod = e_0\otimes e_0$	$\prod = -e_0\otimes e_0$

That is: the product of every row is $+e_0\otimes e_0$, the products of the first two columns are $+e_0\otimes e_0$, and the product of the third column is $-e_0\otimes e_0$. Each of these statements was verified by explicit multiplication in the algebra of the sixteen basis elements $e_a\otimes e_b$, and each row and column was verified to be a commuting set.

Why multiplicativity holds on a context

The value assignments of the parity argument use multiplicativity within each row and column. This is not an assumption about hidden variables; it is a theorem about commuting Hermitian elements, and it is worth deriving in the framework's terms.

Lemma. Let $\tilde{A}_1,\dots,\tilde{A}_m$ be mutually commuting Hermitian elements of $\mathbb{M}_+^{\otimes n}$ with common idempotent basis $\{\tilde{Q}_\alpha\}$, $\sum_\alpha \tilde{Q}_\alpha = e_0^{\otimes n}$, on which $\tilde{A}_i\tilde{Q}_\alpha = a_i^{(\alpha)}\tilde{Q}_\alpha$. Then the product $\tilde{A}_1\cdots\tilde{A}_m$ has eigenvalues $a^{(\alpha)}_1\cdots a^{(\alpha)}_m$ on the same idempotents.

Proof. Multiplying the eigenvalue equations gives $\tilde{A}_1\cdots\tilde{A}_m\tilde{Q}_\alpha = a^{(\alpha)}_1\cdots a^{(\alpha)}_m\tilde{Q}_\alpha$.

For the magic square each generator has spectrum $\{\pm1\}$, so on each joint eigen-idempotent the product relation reads as a product of signs. Applying the lemma to row $1$, whose joint idempotents are built from $\tilde\Pi_\pm(\hat{e}_1)$ on both factors, the identity $\tilde{Q}\otimes\tilde{Q} = (\tilde{Q}\otimes e_0)(e_0\otimes\tilde{Q})$ forces the third entry's value to be the product of the first two on every joint eigen-idempotent, so the row product relation holds automatically. The same argument applies to each row and column. The parity contradiction then needs only the two evaluations of the product of all nine values, one from the rows and one from the columns.

The parity contradiction

Suppose a noncontextual value assignment $v$ exists, with $v(e_0\otimes e_0) = 1$ and $v(\cdot)$ taking values in $\{\pm1\}$ on the nine entries, since each squares to the identity. Multiplicativity on commuting sets then forces, for each row,

$$ v(\text{row }i\text{ product}) = \prod_{\text{entries in row }i} v(\cdot) = v(e_0\otimes e_0) = +1 , $$

and likewise for the first two columns, while the third column requires

$$ \prod_{\text{entries in column }3} v(\cdot) = v(-e_0\otimes e_0) = -1 . $$

Multiplying the three row equations gives

$$ \prod_{\text{all nine entries}} v(\cdot) = (+1)(+1)(+1) = +1 , $$

while multiplying the three column equations gives the same product equal to

$$ (+1)(+1)(-1) = -1 . $$

The two evaluations of one and the same product disagree. Hence no noncontextual $\{\pm1\}$ assignment exists for the two-qubit observables: the Kochen–Specker obstruction is present in the two-qubit algebra $\mathbb{B}\otimes\mathbb{B}$. The contradiction uses only the multiplicativity of the value assignment on each commuting context and the noncontextuality that lets a single value be used for each entry in both its row and its column.

Relation to the idempotent contexts

The magic square can also be read through idempotents. Each entry squares to the identity, so each entry defines a pair of orthogonal idempotents, $\tilde{A} = \tilde\Pi_+^A - \tilde\Pi_-^A$. A context — a row or a column — is a set of three commuting observables whose joint spectral resolution on the four-dimensional module is a complete family of four orthogonal rank-one idempotents; the product relation is a relation among the signs of the joint eigenvalues. The parity contradiction is a statement about the joint spectra of the rows and columns, and it is the operator form of the fact that the six commuting idempotent bases attached to the three rows and three columns cannot be consistently oriented. In the framework's vocabulary, the context is a maximal commutative subalgebra with an idempotent basis, and the theorem says those bases cannot be glued along shared entries.

Value Assignments as Characters

The contradiction can be recast in a way that exhibits its algebraic character and shows exactly where the framework's commutative subalgebras enter.

A context is a finite abelian group

A context — a maximal set of mutually commuting Hermitian involutions — generates a finite abelian group $S$, of order $2^m$ when it is generated by $m$ independent elements. Its joint spectral resolution is by the central idempotents of the group algebra,

$$ e_\chi = \frac{1}{|S|}\sum_{\tilde{g}\in S}\chi(\tilde{g})^{-1}\tilde{g}, \qquad \chi : S \longrightarrow \{\pm1\}, $$

one for each character $\chi$ of $S$; on the idempotent $e_\chi$ every element $\tilde{g}\in S$ has eigenvalue $\chi(\tilde{g})$. On a module of dimension $d$ only the characters that occur in the joint spectrum contribute a nonzero idempotent, the remaining group-algebra idempotents vanishing as operators — in the magic square this happens in the third column, whose entries generate a group of order eight while the module has dimension four. A value assignment on the context is precisely the choice of one character: assigning $v(\tilde{g}) = \chi(\tilde{g})$ for a fixed $\chi$ is the only way to give every element of the context a value in $\{\pm1\}$ that is multiplicative on the context and sums to one on each resolution of the identity. In the qubit's language, a context generated by one involution has two characters and the assignment is a choice of one of the two eigenspaces; a context generated by $m$ commuting involutions has $2^m$ characters and the assignment is a choice of one of its joint eigenspaces.

Noncontextuality is a global section

A noncontextual hidden-variable model is therefore a rule that selects one character on each context, with the requirement that the value assigned to an observable be the same in every context containing it:

$$ \tilde{g}\in S\cap S' \ \Longrightarrow\ \chi_S(\tilde{g}) = \chi_{S'}(\tilde{g}) . $$

The Kochen–Specker theorem says that for a module of dimension at least three no such globally consistent selection exists. This is the statement that the assignment of characters to the commuting subalgebras of $M_d(\mathbb{C})$ — a finite diagram of finite abelian groups, glued along their intersections — admits no global character; the obstruction is topological in form and algebraic in content, and the framework's contribution is to identify the contexts with the maximal commutative subalgebras of $\mathbb{M}_+^{\otimes2}$ and the local sections with their characters.

The magic square in this language

Each row and column of the magic square is a context: its entries generate a finite abelian group of commuting involutions — of order four for the rows and the first two columns, where the third entry is the product of the first two, and of order eight for the third column, where the sign in the product relation makes the three entries independent — and the joint spectral resolution on the four-dimensional module has four idempotents, one for each character that occurs. A noncontextual model would select one character per row and per column, agreeing on the entries that a row and a column share. The parity argument shows that the product of the selected values over the nine entries is forced to be $+1$ by the rows and $-1$ by the columns. Equivalently: no selection of characters is simultaneously consistent with the row and column group structures. The contradiction is entirely about the characters, and it uses neither the biquaternion norm nor the state cone.

State-Dependent and State-Independent Contextuality

The Kochen–Specker theorem is state-independent: it obstructs a noncontextual value assignment for all states at once, and the magic-square contradiction involves no state. There is a second, weaker form of contextuality, state-dependent contextuality, in which a noncontextual hidden-variable model is possible for some states but not others; the CHSH scenario and its Bell inequalities are its standard representatives.

The two forms sit differently in the framework.

State-independent contextuality is a property of the algebra of observables alone, here $\mathbb{B}\otimes\mathbb{B}$; the magic square is an algebraic identity and requires no state, no biquaternion norm, and no cone. This is the sense in which the Kochen–Specker theorem is a statement about the structure of $\mathbb{M}_+^{\otimes2}$.
State-dependent contextuality is a property of a state together with a set of observables. The companion article Exercise: The CHSH Inequality and Tsirelson's Bound computes the correlation function $E(\hat{a},\hat{b}) = -\hat{a}\cdot\hat{b}$ for the singlet and the Tsirelson bound $2\sqrt2$, and the companion article Entangled Subsystems in the Biquaternion Framework shows that the correlation function is the trace pairing between the state idempotent and a tensor-product observable. Violation of a Bell inequality is state-dependent contextuality, and the framework locates it in the pairing of a particular state with a particular context.

The two are related but not identical. The magic square is the stronger statement, because it needs no state; the Bell inequalities are the more operational statement, because they are tested by counting coincidences. Both are consequences of the same algebraic structure: the noncommutativity of the observables and the noncommutativity of the idempotent bases.

What the Framework Does and Does Not Add

What it does.

It exhibits a context as a maximal commutative subalgebra with an idempotent basis, and a noncontextual value assignment as an algebra homomorphism on each context with agreement across contexts.
It shows that the native qubit admits a noncontextual model and that the obstruction first appears in the two-qubit algebra, where the module has dimension four.
It constructs the Mermin–Peres magic square in the algebra's own elements and verifies the product relations that produce the parity contradiction.
It distinguishes state-independent from state-dependent contextuality structurally: the first is a property of $\mathbb{M}_+^{\otimes2}$, the second of a state paired with a context.

What it does not.

It does not prove the Kochen–Specker theorem in its general dimension-$\geq3$ form; only the dimension-four magic-square instance is constructed and verified here. The general theorem, and the original Kochen–Specker and Peres proofs, are imported as standard.
It does not resolve the interpretation of contextuality, nor does it supply a mechanism for the context-dependence; the algebra exhibits the obstruction as a fact about commuting subalgebras.
It does not go beyond the two-qubit arena; the framework's native composite is $\mathbb{B}\otimes\mathbb{B}$, and larger systems inherit the same tensor-product question raised in the entanglement articles.
It does not make an empirical prediction that quantum mechanics does not already make; the Kochen–Specker theorem is a theorem of the standard formalism, and the biquaternion version is its transcription.

Open Questions

1. Contexts and the two sectors. The contexts used here are commutative subalgebras of Hermitian elements. Is there an information-theoretic reading, in the framework's division between $\mathbb{M}_+$ and $\mathbb{M}_-$, of why the maximal commutative subalgebras are the natural contexts, and of the role of the idempotent bases?

2. The minimal obstruction in the framework. The smallest Hilbert-space dimension for a Kochen–Specker proof is three. The framework's native module is two-dimensional and the smallest composite is four-dimensional. Is there an obstruction intrinsic to a three-dimensional module that the framework can express without leaving the qubit's two factors, for instance via a three-outcome measurement?

3. Contextuality and the biquaternion norm. The magic square uses only the algebra and its involutions; the biquaternion norm and the positive cone play no role. Is there a version of the obstruction that is sensitive to the state cone, and does it distinguish pure from mixed states in a way the state-independent proof cannot?

4. Many-qubit contextuality. For $n>2$ qubits the Mermin–Ardehali–Belinskii–Klyshko inequalities generalize the magic square. What is their biquaternion form, and does the tensor-product structure of $\mathbb{B}^{\otimes n}$ organize them?

5. Empirical content. Contextuality is already tested; the framework reproduces the standard predictions and adds no observable.

Summary

A context is a maximal set of mutually commuting Hermitian elements of $\mathbb{M}_+$, realized by a complete family of orthogonal idempotents. A noncontextual value assignment is an algebra homomorphism on each context, with a value for each observable independent of which context contains it. The Kochen–Specker theorem states that for a module of complex dimension at least three no such assignment exists; the hypothesis is necessary, and the native qubit, of dimension two, admits a noncontextual model.

The smallest arena in which the obstruction appears in the framework is the two-qubit algebra $\mathbb{B}\otimes\mathbb{B}\cong M_4(\mathbb{C})$, whose module has dimension four. There the Mermin–Peres magic square gives the obstruction explicitly: nine Hermitian involutions $\tilde{Q}=ie_1,\tilde{Y}=ie_2,\tilde{Z}=ie_3$ and $e_0$ arranged in a $3\times3$ array, every row and column a commuting set, with every row product equal to $+e_0\otimes e_0$, the first two column products equal to $+e_0\otimes e_0$, and the third column product equal to $-e_0\otimes e_0$. Multiplying the row equations and the column equations evaluates the product of all nine entries as $+1$ and as $-1$ respectively, so no noncontextual $\{\pm1\}$ assignment exists.

State-independent contextuality (the Kochen–Specker theorem and the magic square) is a property of the algebra $\mathbb{M}_+^{\otimes2}$ alone; state-dependent contextuality (the Bell and CHSH inequalities) is a property of a state paired with a context and is expressed in the framework by the trace pairing. The framework transcribes both; it does not change their predictions.

Summary of Notation

Symbol	Meaning
$\mathbb{B}\otimes\mathbb{B}$	Two-qubit algebra, $\cong M_4(\mathbb{C})$
$\mathbb{M}_+$	Hermitian subspace (observables)
$e_0 = 1, e_1, e_2, e_3$	Quaternion units, $e_k^2 = -e_0$
$i$	Central scalar imaginary
$[\tilde{A},\tilde{B}]=0$	Compatibility (common measurement)
$\tilde\Pi_a$, $\sum_a\tilde\Pi_a = e_0$	Idempotent basis of a context
$v(\cdot)$	Noncontextual value assignment
$v(\tilde{A}\tilde{B}) = v(\tilde{A})v(\tilde{B})$ on a context	Multiplicativity
$\tilde{Q}=ie_1,\ \tilde{Y}=ie_2,\ \tilde{Z}=ie_3$	Hermitian involutions ($\sigma_1,\sigma_2,\sigma_3$)
$S_1=-e_1\otimes e_1,\ S_3=-e_3\otimes e_3$	Context generators (Bell stabilizers)
$P_\epsilon$	Bell idempotents (joint spectral basis)
Magic square	$3\times3$ array of commuting involutions
Row products, first two column products $= +e_0\otimes e_0$	Magic-square relations
Third column product $= -e_0\otimes e_0$	Parity obstruction