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REVIEW 5 major objections 6 minor 39 references

Quantum-Like Contextuality in Large Language Models

T0 review · 5 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Quantum-like contextuality occurs in natural language at scale: out of 51,966,480 pronoun-reference instances built from Simple English Wikipedia and BERT, 77,118 are sheaf-contextual and 36,938,948 are Contextuality-by-Default contextual.

desk verdict A serious large-scale study of contextuality in LLM probability tables, but the headline counts need null baselines and the semantic-similarity claim is weaker than presented. read the letter →

arxiv 2412.16806 v1 pith:FTSWJDHF submitted 2024-12-21 cs.CL quant-ph

Kin Ian Lo , Mehrnoosh Sadrzadeh , Shane Mansfield This is my paper · ORCID
classification cs.CLquant-ph
keywords quantumcontextualitysheaftheorylargelanguagemodelsBERTanaphoraresolutioncontextuality-by-defaultsemanticsimilaritynatural
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 seeks to establish that quantum-style contextuality—the failure of a set of local observations to admit one global joint explanation—occurs in ordinary natural language, and at a scale far beyond hand-built examples. The authors build a pronoun-reference schema whose measurement structure is the minimal contextual quantum scenario, instantiate it automatically from Simple English Wikipedia, and treat BERT's masked-word predictions as probability distributions over the two candidate referents. Of 51,966,480 generated instances, 77,118 meet the signalling-corrected sheaf criterion and 36,938,948 meet the Contextuality-by-Default criterion. If the claim is right, a statistical structure known to be a resource for quantum advantage is present in a core language task, which would make quantum-inspired methods a plausible target for coreference resolution and related language engines.

What carries the argument

The load-bearing object is the PR-anaphora schema: a three-context measurement scenario whose observables are anaphoric modifiers and whose outcomes are two candidate noun referents, instantiated as sentences such as 'There is an apple and a strawberry. One of them is red and the very same one is round.' Its support is the PR prism, the strongly contextual model of the minimal 3-cyclic scenario, so contextuality is decided by the specialised inequalities $SF < 1/6$ (signalling-corrected sheaf theory) and $\Delta < 2$ (Contextuality-by-Default). The analytical engine is Proposition 7.1, which connects BERT's masked-token logits to the empirical table: $\varepsilon = \tanh\bigl(\tfrac12 (p\cdot \Delta x + \Delta b)\bigr)$. Under the paper's isotropy assumption, this identity turns contextuality into geometry: the farther apart the two noun embeddings are, the more extreme BERT's probabilities become and the less contextual the instance, which is why Euclidean distance, not entropy or bias difference, emerges as the dominant statistical predictor.

What would settle it

Re-probe the same 77,118 sheaf-contextual templates with human plausibility judgments or with a different masked language model; if the contextual instances largely disappear, or if randomly permuting the adjectives across contexts preserves the same fraction of contextual instances, the effect is an artifact of the probe rather than a property of natural language.

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Extended reading notes

Core claim

The central discovery is that a systematically constructed linguistic scenario—the PR-anaphora schema, a variant of coreference resolution with two candidate noun phrases and three adjectival modifiers—produces contextual empirical models when the probabilities come from a large language model. An instance is sheaf-contextual when its signalling fraction satisfies $SF < 1/6$ and CbD-contextual when its direct influence satisfies $\Delta < 2$; the first condition is the signalling-corrected sheaf criterion specialised to three contexts, and the second is the Contextuality-by-Default criterion for cyclic systems. Of the 51,966,480 instances built from 866,108 noun pairs, 77,118 are sheaf-contextual and 36,938,948 are CbD-contextual, and restricting to the 1% most semantically similar noun pairs raises these fractions to 0.50% and 81.83%. The paper also proves that the probability imbalance in each context is governed by the identity $\varepsilon = \tanh\bigl(\tfrac12 (p\cdot \Delta x + \Delta b)\bigr)$, which ties contextuality to the geometry of BERT's embedding space through the Euclidean distance between the two candidate nouns; regression over several candidate features confirms that Euclidean distance is the best statistical predictor of contextuality.

Load-bearing premise

The load-bearing premise is that BERT's normalized [MASK] probabilities for the two candidate nouns are faithful empirical probability distributions over referents; if those numbers largely reflect word co-occurrence or template artifacts, the contextuality belongs to BERT's softmax geometry rather than to natural language.

Editorial extensions

If this is right

  • Coreference resolution becomes a candidate domain for quantum-inspired algorithms, since its ambiguity structure can host contextual empirical models at scale.
  • The PR-anaphora schema offers an automatically instantiable generalisation of the Winograd Schema Challenge, moving beyond hand-crafted pronoun puzzles.
  • Because Euclidean distance between embeddings predicts contextuality, corpus searches can be seeded by semantic similarity rather than by enumerating all adjective triples.
  • The earlier anecdotal evidence from eleven hand-picked noun pairs is replaced by corpus-level evidence from 866,108 noun pairs, making the phenomenon reproducible and measurable.
  • If quantum-like contextuality behaves like the quantum resource, contextual instances mark language tasks where quantum methods could in principle outperform classical ones.

Reading between the lines

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

  • Editorial extension: collecting human plausibility judgments for the same PR-anaphora templates would test whether the inequalities survive without BERT in the loop; survival would move the phenomenon from language-model geometry to natural language itself.
  • Editorial extension: the same 51 million instances could be probed with a decoder-only model prompted to fill the anaphor, checking whether contextuality is specific to bidirectional masked language modelling or generic to neural language-model probability geometry.
  • Editorial extension: because the sheaf-contextual region is a strict subset of the CbD-contextual region, the huge gap between 0.148% and 71.1% means claims about 'contextuality in language' must specify which framework's resource is meant; the two frameworks are not interchangeable measures.
  • Editorial extension: a control with the adjectives randomly permuted across the three contexts would show what fraction of contextual instances is carried by lexical co-occurrence statistics rather than by the semantic relations the schema is designed to probe.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. The paper constructs a PR-prism-inspired anaphora schema, instantiates it with 51,966,480 instances from Simple English Wikipedia, and uses BERT's [MASK] predictions (normalized to the two candidate nouns) to define empirical models. It reports 77,118 sheaf-contextual instances (SF < 1/6) and 36,938,948 CbD-contextual instances (Delta < 2), derives an algebraic relation between BERT logit differences and the PR-like epsilon parameter (Prop. 7.1), and uses correlation/regression analyses to argue that Euclidean distance between BERT noun embeddings is the best statistical predictor of contextuality. The paper also provides analytic formulas for the contextual and signalling fractions of PR-like models.

Significance. If the headline counts were properly benchmarked, this would be the first large-scale evidence for quantum-like contextuality in natural language, with potential implications for using contextuality as a resource in language models. The paper has genuine strengths: the analytic derivation of CF and SF for PR-like models (Props. 3.3 and 3.4), the large automatically constructed dataset, the clean algebraic identity in Prop. 7.1, and the release of code and data in a public repository. However, the significance is currently conditional because the CbD count is close to a tautology of the template, the sheaf count lacks a null baseline, the similarity claim rests on an unverified isotropy assumption, and the reported correlations are very weak.

major comments (5)
  1. [Section 6(b)] The claim that 36,938,948 CbD-contextual instances (71.1%) constitute evidence of natural-language contextuality is not supported, because for the PR-prism template the CbD criterion Delta < 2 is satisfied by construction for almost any non-extreme BERT predictions: when the three epsilon_i values are equal to a common non-zero value, Delta = 2|epsilon| < 2, and since BERT's softmax outputs are almost never exactly 1 or 0, the majority of instances automatically fall in this region. A null model that randomizes the noun pairs or replaces BERT predictions with a plausible random distribution should be used to show that the observed rate exceeds chance; without this, the count is a property of the template and softmax geometry, not of natural language.
  2. [Section 6(b)] The sheaf-contextual rate of 0.148% (77,118 instances) is the main non-trivial evidence, but it is not compared to any null distribution. The criterion SF < 1/6 with three contexts and normalized two-outcome probabilities can be satisfied when all three contexts produce near-uniform predictions (epsilon_i all near 0); the observed 0.148% may simply be the base rate at which BERT is within 1/6 of uniform on three arbitrary contexts. A permutation or randomized-pair baseline is needed to establish that this rate is significantly elevated.
  3. [Section 7(a), Prop. 7.1] Equation (7.3) is an algebraic identity that follows from the softmax definition (7.2) and the PR-like parametrization; it is correct but does not by itself provide empirical evidence about language. The subsequent conclusion that contextual instances come from semantically similar words relies on the 'isotropic distribution of prediction vectors' assumption, which is stated in Section 7(a) without verification, and the regression results (Tables 4 and 6, R^2 approximately 0.006-0.009 full dataset, approximately 0.08 subset) show that Euclidean distance explains only a small fraction of variance. The strength of the 'best predictor' and 'came from' claims should be scaled to these effect sizes, or the isotropy assumption should be tested directly.
  4. [Section 6(c)] The similar-noun subset is constructed by selecting noun pairs with the highest cosine similarity after seeing the full data, and the contextual rates are then recomputed on this selected subset. Without a matched control (e.g., random subsets of equal size or a prespecified hypothesis), the observed increase from 0.148% to 0.50% sheaf-contextual and from 71.1% to 81.83% CbD-contextual could be a selection effect; the claim that similarity causes contextuality is not established by this procedure.
  5. [Section 5(a)] The interpretation of normalized BERT [MASK] probabilities as empirical probability distributions over referents is assumed but not validated. If these probabilities reflect word co-occurrence, template artifacts, or BERT's training biases rather than referent likelihood, then the reported 'contextuality' may be a property of BERT's softmax geometry rather than of natural language. The paper should either validate this interpretation or explicitly frame the results as properties of BERT's predictions.
minor comments (6)
  1. [Section 3(a)] The word 'quantity' should be 'quantify' in the sentence 'One can try to define a signalling fraction (SF), in the same way CF is defined, to quantity the degree of signalling.'
  2. [Section 7(b)] There are several typos: 'contexutaliy' should be 'contextuality' in the text above Table 3, 'descirbing' should be 'describing' in Section 7(a), and 'polynomail' should be 'polynomial' in the caption of Figure 10.
  3. [Section 6] Two different tables are both labelled 'Table 2' (the random sample in Section 6(a) and the similar-noun sample in Section 6(c)), which makes cross-referencing confusing.
  4. [Section 6(b), Figure 5] The caption says the contextual models are 'Highlighted', but the sheaf-contextual bars are too small to be visible in the histogram; consider using a log-scale inset or a different visualization.
  5. [Eq. (4.2)] The definition of s_odd uses sigma dot x, but the notation 'p(sigma) = product_i sigma_i' combined with the maximization over sigma is not fully explained; please spell out the parity condition explicitly.
  6. [Tables 4 and 6] The R^2 values are reported to four decimal places, but the caption and text inconsistently use 'R2'; please standardize the notation as R² throughout.

Circularity Check

1 steps flagged · score 3.0 of 10

One claimed 'proof' (Eq. 7.3) is an algebraic identity from the softmax and epsilon definitions; the headline contextuality counts are empirical and not circular.

  1. self definitional [Section 7(a), Proposition 7.1, Eq. (7.3)]
    "Using equations 7.1 and 7.2, below in Prop. 7.1 we prove a result which connects the BERT logit scores to the empirical table of the PR-like model descirbing the PR-anaphora schema: ... Proposition 7.1. The logit scores of the masked token given by BERT relates to the ϵ parametrisation of the PR-like model as follows: ϵ = tanh(1/2 (p · ∆x + ∆b)). (7.3)"

    The parameter ϵ is defined by the PR-like table entries (1±ϵ)/2, and those entries are taken to be the renormalized BERT softmax probabilities of Eq. (7.2). Substituting P_O1 = e^{l_O1}/(e^{l_O1}+e^{l_O2}) and P_O2 = e^{l_O2}/(e^{l_O1}+e^{l_O2}) into P_O1=(1+ϵ)/2 immediately gives log((1+ϵ)/(1-ϵ)) = l_O1-l_O2, i.e. Eq. (7.3). The proposition is therefore an algebraic identity that unpacks definitions, not an independent first-principles result. The abstract's claim that this 'proved that the contextual instances came from semantically similar words' goes beyond the identity: it additionally requires the untested isotropic-distribution assumption on p and treats the tautological link as empirical support.

full rationale

The two headline counts are empirical measurements: 51,966,480 empirical tables are constructed from BERT's renormalized [MASK] probabilities and evaluated with the external sheaf-theoretic criterion SF < 1/6 and the CbD criterion Delta < 2. These counts do not reduce by construction to a fitted parameter, and no parameter is fitted to a subset and then renamed as a prediction. The sheaf criterion is imported from ref. [9], which includes an author of the present paper, but it is a published external inequality and is not derived from the present target result, so it is not load-bearing circularity. The CbD count's closeness to a tautology for the PR-prism template (for all-equal epsilon, Delta = 2|epsilon|) is a real null-baseline and interpretation concern, but it is a validity issue rather than a circular derivation. The regression analysis is empirical, and the 'Euclidean distance best predictor' claim is not fully forced by Eq. (7.3) because the logit difference also contains the bias difference and the isotropy assumption is additional. The main circular element is Proposition 7.1: it is presented as a proof connecting contextuality to Euclidean distance, but it is an analytic identity following from the definitions of epsilon and the BERT softmax. This warrants a score of 3 rather than 0, while the overall empirical contribution remains substantially independent.

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

The main free parameters are experimental design choices, not fitted physical constants. The load-bearing domain assumptions concern the interpretation of BERT probabilities as referent probabilities and the application of quantum contextuality criteria to language-model output. The 'isotropic distribution' assumption is ad hoc and unverified, and it is needed for the Euclidean-distance interpretation.

free parameters (3)
  • top_common_adjectives_count = 5
    Chosen in Section 6(a) to balance overlap and dataset size; determines which adjectives populate the schema and therefore which instances are counted.
  • similar_noun_percentile = 1%
    Threshold for the 'similar noun subset' in Section 6(c); post hoc choice that raises contextuality percentages and correlations.
  • polynomial_degree_cutoff = 3 (cubic)
    The regression analysis in Section 7(b) chooses cubic degree as the cutoff for comparing linear versus polynomial fits.
assumptions (5)
  • domain assumption BERT's [MASK] probabilities, renormalized to the two candidate nouns, represent empirical probability distributions over the referents.
    Introduced in Section 5(a) when constructing probabilistic PR-anaphora models.
  • domain assumption The PR-anaphora schema's linguistic constraints (same one/other one) force the empirical table to have PR-prism support.
    Definition 5.1 and Section 5(a); the zero entries in the empirical table come from these constraints.
  • domain assumption The signalling-corrected criterion CF > 2|M| SF from Ref [9] applies to natural-language empirical models.
    Used in Section 3(a), Eq. (3.4), to decide sheaf contextuality for signalling models.
  • ad hoc to paper BERT prediction vectors are isotropically distributed, so Delta_l is proportional to ||Delta x||.
    Stated in Section 7(a); unverified assumption used to interpret the derived equation as a Euclidean-distance law.
  • standard math Standard linear algebra and convex decomposition properties of empirical models.
    Used in Propositions 3.3 and 3.4 for CF and SF calculations.

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Pith. "Pith review of Quantum-Like Contextuality in Large Language Models." pith.science (2026). https://pith.science/paper/FTSWJDHF

@misc{pith2026241216806,
  author       = {Pith},
  title        = {Pith review of: Quantum-Like Contextuality in Large Language Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FTSWJDHF}},
  note         = {Machine review of arXiv:2412.16806}
}
read the original abstract

Contextuality is a distinguishing feature of quantum mechanics and there is growing evidence that it is a necessary condition for quantum advantage. In order to make use of it, researchers have been asking whether similar phenomena arise in other domains. The answer has been yes, e.g. in behavioural sciences. However, one has to move to frameworks that take some degree of signalling into account. Two such frameworks exist: (1) a signalling-corrected sheaf theoretic model, and (2) the Contextuality-by-Default (CbD) framework. This paper provides the first large scale experimental evidence for a yes answer in natural language. We construct a linguistic schema modelled over a contextual quantum scenario, instantiate it in the Simple English Wikipedia and extract probability distributions for the instances using the large language model BERT. This led to the discovery of 77,118 sheaf-contextual and 36,938,948 CbD contextual instances. We proved that the contextual instances came from semantically similar words, by deriving an equation between degrees of contextuality and Euclidean distances of BERT's embedding vectors. A regression model further reveals that Euclidean distance is indeed the best statistical predictor of contextuality. Our linguistic schema is a variant of the co-reference resolution challenge. These results are an indication that quantum methods may be advantageous in language tasks.

Figures

Figures reproduced from arXiv: 2412.16806 by the authors.

Figure 1
Figure 1. Empirical tables of measurement scenarios: Bell-CHSH (left), possibilistic Bell-CHSH [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Bundle diagrams of possibilistic empirical models: (left) the Bell-CHSH test; (middle) a [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Example of the PR-anaphora schema with adjectival modifiers. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Examples of the PR-anaphora schema with verbs (left) and preposition modifiers (right) [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Histograms of (a) the signalling fraction and (b) the direct influence of the 51,966,480 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: The distribution of the instances in the space of direct influence and signalling fraction, [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Histograms of (a) the signalling fraction and (b) the direct influence of the 519,660 [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: A flow chart illustrating how the embedding vectors are transformed into the output [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: A 2-dimensional sketch of a geometric interpretation of the mask predictions from BERT [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: The R 2 scores of the polynomial regression models at different polynomail degrees predicting (left) the signalling fraction and (right) the direct influence. shown in [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: The fraction of sheaf-contextual (left) and CbD-contextual (right) instances at different [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.