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arxiv: 2603.26692 · v4 · submitted 2026-03-16 · 🪐 quant-ph · cs.AI· math.PR

Degrees, Levels, and Profiles of Contextuality

Ehtibar N. Dzhafarov , Victor H. Cervantes This is my paper

Pith reviewed 2026-05-15 10:37 UTC · model grok-4.3

classification 🪐 quant-ph cs.AImath.PR
keywords contextualitycontextuality measuresrandom variablesquantum foundationscontextuality profileslevels of analysisconcatenated systems
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The pith

Contextuality of a system of random variables is characterized by a curve relating its degree to the level of joint distributions considered rather than by a single overall number.

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

The paper introduces contextuality profiles to describe how the degree of contextuality in a system of random variables changes as one moves from lower to higher levels of analysis. A system at level n is represented only by its joint distributions involving at most n variables, with higher-order joints ignored. This level-wise approach works with any established measure of contextuality and is explored systematically through a concatenated-systems construction that builds larger systems from smaller ones. The result is a curve rather than a scalar that shows whether contextuality appears, strengthens, or changes character at successive levels.

Core claim

A system of random variables has a contextuality profile given by the function that assigns to each level n the degree of contextuality computed from the joints involving at most n variables. This profile is obtained for any well-constructed contextuality measure by using the method of concatenated systems, which systematically assembles families of systems whose profiles can be compared across the three major measures examined in the paper.

What carries the argument

The contextuality profile: a curve that plots the numerical degree of contextuality (from any chosen measure) against the level n, where level n restricts attention to joint distributions of at most n variables.

If this is right

  • Contextuality may be absent at low levels and appear only at higher levels for some systems.
  • Different contextuality measures can produce qualitatively different profile shapes for the same underlying system.
  • The concatenated-systems method generates infinite families of systems with prescribed profile features for systematic study.
  • Any existing measure of contextuality immediately yields a corresponding family of profiles without modification.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Profiles could distinguish systems that share the same overall degree but differ in when their contextuality emerges across levels.
  • The approach may extend to other notions of non-classicality such as incompatibility or non-locality by replacing the contextuality measure.
  • Experimental tests could focus on measuring only low-order joints to see whether observed contextuality matches the predicted low-level part of the profile.

Load-bearing premise

That level-wise restriction to joints of size at most n can be meaningfully combined with any existing contextuality measure without introducing artifacts in the resulting curve.

What would settle it

Compute the contextuality profile for the CHSH or Kochen-Specker system using the concatenated-systems method and three different measures; if the curves are not reproducible or change shape inconsistently when the same systems are reconstructed at different concatenations, the profile notion collapses.

Figures

Figures reproduced from arXiv: 2603.26692 by Ehtibar N. Dzhafarov, Victor H. Cervantes.

Figure 1
Figure 1. Figure 1: Four possible contextuality profiles with the same final degree of contextu [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A two-dimensional projection of a vector [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Contextuality profiles for the method of concatenated systems, [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Four possible types of the contextuality profiles for concatenated systems [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Contextuality profiles for a selection of undisturbed concatenated systems. [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The same as in Figure 5, for another selection of the [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Contextuality profiles for a selection of disturbed [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The same as in Figure 7 but for another selection of [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: CNTF profiles for a selection of concatenated systems (A1 ∗ B2 left and A3 ∗ B2 right). The dashed lines represent the CNTF profiles for the systems’ B-parts. The dotted lines represent the CNTF profiles for the system’s A-parts (invisible if it coincides with a system’s profile). d2 Δ3 0 1/9 1/6 1 2 3 Level CNT 3 d2 Δ3 0 1/6 3/10 1 2 3 Level CNT 3 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The same as in Figure 9 but for [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Contextuality profiles for a selection of undisturbed hypercyclic systems of [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Contextuality profiles for a selection of disturbed hypercyclic systems of [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Relationship between the overall contextuality degrees generated by two [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Relationship between the segments of contextuality profiles (between levels [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Contextuality profiles for a triple-concatenation of systems. The boxes [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces the notion of a contextuality profile for a system of random variables, characterizing contextuality via a curve of degree versus level n (where level n restricts attention to joint distributions involving at most n variables). It claims that any well-constructed contextuality measure can be applied level-wise and presents a concatenated-systems construction to systematically generate and explore such profiles, with applications to three major measures in the literature.

Significance. If the concatenated-systems method is free of artifacts and the level-wise application is shown to be measure-independent in the required sense, the profile concept would offer a finer-grained alternative to scalar contextuality degrees, potentially revealing structural features of contextuality in both quantum and classical settings that single-number summaries obscure.

major comments (2)
  1. [Concatenated-systems method] The concatenated-systems construction (described after the level definition) is asserted to explore profiles systematically without introducing spurious correlations, yet no argument or verification is given that concatenation commutes with the marginalization implicit in the level-n restriction; higher-order dependencies created by concatenation may persist under projection to level n and change the computed degree relative to the original system.
  2. [Application to three major measures] The central claim that level-wise analysis works with any well-constructed measure rests on the assumption that the profile curve is independent of the particular measure chosen, but the manuscript supplies no explicit check or counter-example showing that different measures produce qualitatively similar profiles on the same concatenated family.
minor comments (1)
  1. [Abstract] The abstract states the conceptual framework but contains no explicit definitions of 'level' or 'profile', no worked numerical example, and no derivation of the concatenated construction; moving at least one concrete example into the abstract or introduction would improve accessibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which have helped us identify points where the manuscript can be clarified and strengthened. We address each major comment below and will incorporate the necessary revisions.

read point-by-point responses

  1. Referee: [Concatenated-systems method] The concatenated-systems construction (described after the level definition) is asserted to explore profiles systematically without introducing spurious correlations, yet no argument or verification is given that concatenation commutes with the marginalization implicit in the level-n restriction; higher-order dependencies created by concatenation may persist under projection to level n and change the computed degree relative to the original system.

    Authors: We acknowledge that the original manuscript asserts the utility of the concatenated-systems construction without supplying a formal argument that it commutes with level-n marginalization or a verification that higher-order dependencies do not alter the computed degrees. The construction is intended to control the level-n joints explicitly, but this property was not demonstrated rigorously. In the revised version we will insert a dedicated subsection that (i) defines the concatenation operation formally, (ii) proves that the level-n marginals of the concatenated system coincide with those of the component systems, and (iii) includes a small explicit example in which the contextuality degree at each level is recomputed both before and after concatenation to confirm invariance. This will eliminate the possibility of spurious correlations at the levels under consideration. revision: yes

  2. Referee: [Application to three major measures] The central claim that level-wise analysis works with any well-constructed measure rests on the assumption that the profile curve is independent of the particular measure chosen, but the manuscript supplies no explicit check or counter-example showing that different measures produce qualitatively similar profiles on the same concatenated family.

    Authors: The manuscript states that the level-wise framework is compatible with any well-constructed contextuality measure; it does not claim that the resulting profiles must be identical across measures. Different measures can and do emphasize distinct aspects of contextuality, so qualitative differences are expected. To address the referee’s request for an explicit check, we will add a short comparative subsection that computes the profiles of one representative concatenated family under all three measures discussed in the paper. The comparison will illustrate both the common qualitative features (e.g., monotonicity or the location of transitions) and any measure-specific differences, thereby clarifying the scope of the claim without asserting measure-independence of the profiles themselves. revision: yes

Circularity Check

0 steps flagged

No significant circularity in contextuality profile derivation

full rationale

The paper defines the contextuality profile directly from standard joint-distribution marginals at level n (considering only joints of size ≤ n) and applies any independent well-constructed measure to those levels. The concatenated-systems construction is presented as an exploratory method that generates the resulting curves without the profile or degree values reducing to a fit, self-definition, or self-citation chain. No load-bearing steps equate outputs to inputs by construction; the central claim remains an extension applied to external measures from the literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on redefining contextuality analysis via levels of joint distributions and introducing the profile curve; no free parameters are mentioned, and the only invented entity is the conceptual profile itself.

axioms (1)
  • standard math Joint distributions of random variables exist and can be restricted to subsets of size at most n
    The level-n definition directly invokes this standard probability concept.
invented entities (1)
  • contextuality profile no independent evidence
    purpose: To represent contextuality as a curve over levels rather than a single scalar degree
    Newly introduced conceptual object with no independent evidence supplied in the abstract.

pith-pipeline@v0.9.0 · 5422 in / 1221 out tokens · 57420 ms · 2026-05-15T10:37:28.009493+00:00 · methodology

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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