Degrees, Levels, and Profiles of Contextuality

Ehtibar N. Dzhafarov; Victor H. Cervantes

quant-phMar 16, 2026

Degrees, Levels, and Profiles of Contextuality

Authors: Ehtibar N. Dzhafarov, Victor H. Cervantes

Organizations: Purdue University, USA · University of Illinois at Urbana-Champaign, USA

Abstract

We introduce a new notion, that of a contextuality profile of a system of random variables. Rather than characterizing a system's contextuality by a single number, its overall degree of contextuality, we show how it can be characterized by a curve relating degree of contextuality to level at which the system is considered. A system is represented at level n if one only considers the joint distributions with no more than n variables, ignoring higher-order joint distributions. We show that the level-wise contextuality analysis can be used in conjunction with any well-constructed measure of contextuality. We present a method of concatenated systems to explore contextuality profiles systematically, and we apply it to the contextuality profiles for three major measures of contextuality proposed in the literature.

Figures & tables

Figure 1: Four possible contextuality profiles with the same final degree of contextuality at level 5.

Figure 2: A two-dimensional projection of a vector $\mathbf{v}^{*}$ and a noncontextuality polytope $\mathbb{V}$ , with the $L^{1}$ -distance between them.

Figure 3: Contextuality profiles for the method of concatenated systems, $n=2$ . The boxes represent the systems being concatenated, with the numbers in them indicating their final level of contextuality. Symbols attached to the curves indicate contextuality values.

Figure 4: Four possible types of the contextuality profiles for concatenated systems ( $n=2$ ): superadditive (top left panel), additive (top right), subadditive (bottom left), and, as the extreme case of subadditivity, plateau (bottom right).

Figure 5: Contextuality profiles for a selection of undisturbed concatenated systems. Symbols

\mathcal{A}

and

\mathcal{B}

with indices refer to

\mathcal{A}

- and

\mathcal{B}

-subsystems, respectively (as specified in Appendix A). The dashed lines attached to each profile show the increment from

d_{2}

d_{2}+\Delta_{3}

: if it is above the corresponding segment of the profile we have subadditivity, and when the dashed line is not seen (coincides with the segment) we have additivity.

Figure 6: The same as in Figure 5 , for another selection of the

\mathcal{A}

-subsystems.

Figure 7: Contextuality profiles for a selection of disturbed

\mathcal{A}

-subsystems concatenated with a disturbed subsystem

\mathcal{B}_{3}

. The rest is the same as in Figure 5 .

Figure 8: The same as in Figure 7 but for another selection of

\mathcal{A}

-subsystems concatenated with a disturbed subsystem

\mathcal{B}_{3}

Figure 9:

\mathsf{CNTF}

profiles for a selection of concatenated systems (

\mathcal{A}_{1}\ast\mathcal{B}_{2}

left and

\mathcal{A}_{3}\ast\mathcal{B}_{2}

right). The dashed lines represent the

\mathsf{CNTF}

profiles for the systems’

\mathcal{B}

-parts. The dotted lines represent the

\mathsf{CNTF}

profiles for the system’s

\mathcal{A}

-parts (invisible if it coincides with a system’s profile).

Figure 10: The same as in Figure 9 but for

\mathsf{CNT_{3}}

	$\mathsf{CNT_{2}}$	$\mathcal{A}_{0}$	$\mathcal{A}_{1}$	$\mathcal{A}_{2}$	$\mathcal{A}_{3}$	$\mathcal{A}_{4}$	$\mathcal{A}_{5}$
	$\mathsf{CNT_{2}}$	0	$\nicefrac{{1}}{{10}}$	$\nicefrac{{1}}{{5}}$	$\nicefrac{{3}}{{10}}$	$\nicefrac{{2}}{{5}}$	$\nicefrac{{1}}{{2}}$
$\mathcal{B}_{1}$	$\nicefrac{{1}}{{24}}$	$\nicefrac{{1}}{{24}}$	$\nicefrac{{17}}{{120}}$	$\nicefrac{{29}}{{120}}$	$\nicefrac{{41}}{{120}}$	$\nicefrac{{53}}{{120}}$	$\nicefrac{{13}}{{24}}$
$\mathcal{B}_{2}$	$\nicefrac{{1}}{{8}}$	$\nicefrac{{1}}{{8}}$	$\nicefrac{{9}}{{40}}$	$\nicefrac{{13}}{{40}}$	$\nicefrac{{17}}{{40}}$	$\nicefrac{{21}}{{40}}$	$\nicefrac{{5}}{{8}}$
	$\mathsf{CNT_{3}}$	$\mathcal{A}_{0}$	$\mathcal{A}_{1}$	$\mathcal{A}_{2}$	$\mathcal{A}_{3}$	$\mathcal{A}_{4}$	$\mathcal{A}_{5}$
	$\mathsf{CNT_{3}}$	0	$\nicefrac{{1}}{{10}}$	$\nicefrac{{1}}{{5}}$	$\nicefrac{{3}}{{10}}$	$\nicefrac{{2}}{{5}}$	$\nicefrac{{1}}{{2}}$
$\mathcal{B}_{1}$	$\nicefrac{{1}}{{18}}$	$\nicefrac{{1}}{{18}}$	$\nicefrac{{1}}{{10}}$	$\nicefrac{{1}}{{5}}$	$\nicefrac{{3}}{{10}}$	$\nicefrac{{2}}{{5}}$	$\nicefrac{{1}}{{2}}$

Table 1: Contextuality value

d_{3}

of undisturbed concatenated systems for the measures

\mathsf{CNT_{2}}

\mathsf{CNT_{3}}

, and

\mathsf{CNTF}.

The value of

d_{3}

for

\mathcal{A}_{i}*\mathcal{B}_{j}

is in the intersection of column

\mathcal{A}_{i}

and row

\mathcal{B}_{j}

. The corresponding values of

d_{2}

for

\mathcal{A}_{i}

and

\Delta_{3}

for

\mathcal{B}_{j}

are shown, respectively, in the row and the column containing the measure’s name. Observe that for

\mathsf{CNT_{2}}

d_{3}

is the sum of the corresponding values of

d_{2}

and

\Delta_{3}

; and for both

\mathsf{CNT_{3}}

and

\mathsf{CNTF}

d_{3}

is the larger of the corresponding values of

d_{2}

and

\Delta_{3}

. (Note that for the

\mathcal{A}

-subsystems,

\mathsf{CNT_{2}=\mathsf{CNT_{3}}=\mathsf{\frac{1}{2}CNTF}}

, as it was previously established for all cyclic systems [ 5 ] ).

	$\mathsf{CNT_{2}}$	$\mathcal{A}^{\prime}_{0}$	$\mathcal{A}^{\prime}_{1}$	$\mathcal{A}^{\prime}_{2}$	$\mathcal{A}^{\prime}_{3}$	$\mathcal{A}^{\prime}_{4}$	$\mathcal{A}^{\prime}_{5}$	$\mathcal{A}^{\prime}_{6}$
	$\mathsf{CNT_{2}}$	0	$\nicefrac{{1}}{{18}}$	$\nicefrac{{1}}{{9}}$	$\nicefrac{{1}}{{6}}$	$\nicefrac{{2}}{{9}}$	$\nicefrac{{5}}{{18}}$	$\nicefrac{{3}}{{9}}$
$\mathcal{B}^{\prime}_{1}$	$\nicefrac{{1}}{{100}}$	$\nicefrac{{1}}{{100}}$	$\nicefrac{{59}}{{900}}$	$\nicefrac{{109}}{{900}}$	$\nicefrac{{159}}{{900}}$	$\nicefrac{{209}}{{900}}$	$\nicefrac{{259}}{{900}}$	$\nicefrac{{309}}{{900}}$
$\mathcal{B}^{\prime}_{2}$	$\nicefrac{{5}}{{100}}$	$\nicefrac{{5}}{{100}}$	$\nicefrac{{19}}{{180}}$	$\nicefrac{{29}}{{180}}$	$\nicefrac{{13}}{{60}}$	$\nicefrac{{49}}{{180}}$	$\nicefrac{{59}}{{180}}$	$\nicefrac{{23}}{{180}}$
$\mathcal{B}^{\prime}_{3}$	$\nicefrac{{9}}{{100}}$	$\nicefrac{{9}}{{100}}$	$\nicefrac{{131}}{{900}}$	$\nicefrac{{181}}{{900}}$	$\nicefrac{{77}}{{300}}$	$\nicefrac{{281}}{{900}}$	$\nicefrac{{331}}{{900}}$	$\nicefrac{{127}}{{300}}$
	$\mathsf{CNT_{3}}$	$\mathcal{A}^{\prime}_{0}$	$\mathcal{A}^{\prime}_{1}$	$\mathcal{A}^{\prime}_{2}$	$\mathcal{A}^{\prime}_{3}$	$\mathcal{A}^{\prime}_{4}$	$\mathcal{A}^{\prime}_{5}$	$\mathcal{A}^{\prime}_{6}$
	$\mathsf{CNT_{3}}$	0	$\nicefrac{{1}}{{18}}$	$\nicefrac{{1}}{{9}}$	$\nicefrac{{1}}{{6}}$	$\nicefrac{{2}}{{9}}$	$\nicefrac{{5}}{{18}}$	$\nicefrac{{3}}{{9}}$

Table 2: Contextuality value

d_{3}

of disturbed concatenated systems for the measures

\mathsf{CNT_{2}}

\mathsf{CNT_{3}}

, and

\mathsf{CNTF}.

The rest as in Table 1 .

Figure 11: Contextuality profiles for a selection of undisturbed hypercyclic systems of order 3 and rank 4.

Figure 12: Contextuality profiles for a selection of disturbed hypercyclic systems of order 3 and rank 4.

Figure 13: Relationship between the overall contextuality degrees generated by two contextuality measures applied to our selection of undisturbed hypercyclic systems. The horizontal lines show that the abscissa measure cannot be a function of the ordinate one; the vertical lines show that the ordinate measure cannot be a function of the abscissa one.

Figure 14: Relationship between the segments of contextuality profiles (between levels 2 and 3, as indicated by arrows) generated by two measures applied to the same disturbed hypercyclic system. The horizontal lines show that the abscissa profile cannot be a function of the ordinate one; the vertical lines show that the ordinate profile cannot be a function of the abscissa one.

Figure 15: Contextuality profiles for a triple-concatenation of systems. The boxes represent the systems being concatenated, with the numbers in them indicating their final level of contextuality.

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Mar 20, 2026J/K move · Enter open · S save

The production of meaning in the processing of natural language

Christopher J. Agostino, Quan Le Thien, Nayan D'Souza, Louis van der Elst

NPC Worldwide, Bloomington, IN, USA · Department of Physics, Indiana University, Bloomington, IN, USA · Department of Linguistics, Indiana University, Bloomington, IN, USA+1

LLM Interpretability Language Modeling

Open paper AskPDF arXiv

Understanding the fundamental mechanisms governing the production of meaning in the processing of natural language is critical for designing safe, thoughtful, engaging, and empowering human-agent interactions. If meaning is constituted rather than retrieved, then the search for context-independent features or circuits in the pursuit of mechanistic interpretability may be fundamentally limited. Experiments in cognitive science and social psychology have demonstrated that human semantic processing exhibits contextuality more consistent with quantum logical mechanisms than classical Boolean theories, and recent works have found similar results in large language models---in particular, clear violations of the Bell inequality in experiments of contextuality during interpretation of ambiguous expressions. In this work, we explore the CHSH

|S|

parameter---the metric associated with the inequality---across the inference parameter space of models spanning four orders of magnitude in scale and cross-reference our findings with MMLU, hallucination rate, and nonsense detection benchmarks. We find that the interquartile range of the

|S|

distribution is completely orthogonal to all external benchmarks, while overall violation rate shows weak anticorrelation with all three benchmarks. We investigate how

|S|

varies with sampling parameters and word order, and discuss the information-theoretic constraints that genuine contextuality imposes on prompt injection defenses and its human analogue, whereby careful construction and maintenance of social contextuality can be carried out at scale, shaping the space of possible interpretations before any particular one is reached. We consider the implications for mechanistic interpretability and how genuine contextuality sets an information-theoretic bound on the decomposability of semantic processing.

Abstract

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