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arxiv: 2605.13154 · v1 · submitted 2026-05-13 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Three ways to find comfort with the Bell proof and the results of the Bell experiments

Bart Jongejan, Inge S. Helland, Richard D Gill

Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:51 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Bell's theoremCHSH inequalitycounterfactual definitenessloophole-free Bell testshidden variable modelsquantum foundationscausal inferenceaccessible variables
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The pith

Bell experiment results can be accommodated without counterfactual definiteness or conspiratorial setting dependence.

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

The paper explores three distinct ways to reconcile the observed violations of Bell inequalities in recent loophole-free experiments with the rejection of two key classical assumptions: counterfactual definiteness and conspiratorial correlations between measurement settings and hidden states. Using causal graphs to formalize the classical part of Bell's theorem, the authors summarize the experimental confirmations and then present their individual reconstructions. Gill accepts non-local irreducible quantum randomness, viewing the locality-realism dichotomy as false. Helland builds the quantum formalism from a theory of accessible variables, concluding that observers face inherent limitations. Jongejan develops a geometric hidden-variable model where the CHSH violation strength depends on the dimension of space. These approaches matter because they provide coherent alternatives that fit the data without invoking conspiracy or abandoning statistical independence.

Core claim

Bell's theorem shows that no local realistic theory without conspiracy can match quantum predictions, and experiments confirm the violation. The authors each propose a coherent worldview that drops counterfactual definiteness and conspiratorial independence violation: one by accepting irreducible non-local randomness, one by limiting observers via accessible variables, and one by dimension-dependent geometric hidden variables that achieve Tsirelson's bound in three dimensions.

What carries the argument

Rejection of counterfactual definiteness paired with either accessible variables theory, geometric dimension dependence, or acceptance of non-local randomness to explain the CHSH violation.

Load-bearing premise

The recent loophole-free Bell experiments have closed all relevant loopholes and the proposed reconstructions can be made consistent with quantum predictions.

What would settle it

Discovery of a remaining loophole in the experiments that allows a local realistic theory with conspiracy, or a quantum prediction that none of the three reconstructions can match.

Figures

Figures reproduced from arXiv: 2605.13154 by Bart Jongejan, Inge S. Helland, Richard D Gill.

Figure 1
Figure 1. Figure 1: Spatio-temporal disposition of one trial of a Bell experiment. Distance (left to right) is so large [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Graphical model of one trial of a Bell experiment [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The essential hardware for Carol’s experiment [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Distribution of Eab(XY) for |K × L | = 30 × 30 settings. The distribution of Ea10b(XY) is in red. The distribution of Eab15 (XY) is in blue. The expectation value Ea10b15 (XY) is in black and its position and height are marked with black triangles. All other 841 expectation values are in grey. There is a mathematical model that explains these results. The model functions at two levels. At the high level, t… view at source ↗
Figure 5
Figure 5. Figure 5: Cumulative distribution functions nH (γ) for several values of n. 2H (γ) = 1−cos(γ) (40) 3H (γ) = 2 π (γ −sin(γ) cos(γ)) (41) 4H (γ) = 1−cos(γ)− 1 2 sin2 (γ) cos(γ) (42) Let us now associate four nodes labelled ap,bq,ar ,bs of the bipartite graph with four points on the unit sphere. Set w [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Alice’s apparatus, cover over the fluid level gauge removed [PITH_FULL_IMAGE:figures/full_fig_p032_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The values of λA and λB for the first hundred trials in a run of Carol’s experiment that has been simulated on a computer, for key choices (a10,b15). The horizontal and vertical green lines represent Alice’s and Bob’s green fluid columns observed during a single trial. Red dots (lower left and upper right quadrant) are trials where Alice and Bob would have made the same colour observations if the covers st… view at source ↗
Figure 8
Figure 8. Figure 8: Observations by Alice and Bob as explained by a shared hidden parameter. A 4-loop has a [PITH_FULL_IMAGE:figures/full_fig_p034_8.png] view at source ↗
read the original abstract

Bell's theorem states that no description of a Bell experiment can be simultaneously local, realistic in the sense of counterfactual definiteness, and free of conspiracy between settings and hidden state. The recent generation of experiments has confirmed the predicted violation of the CHSH inequality, so one of the assumptions must be abandoned. Which one, and how one reconstructs a coherent worldview after doing so, is a question on which many authors disagree. This paper is written by three such authors. All three reject both counterfactual definiteness and conspiratorial violation of statistical independence of setting choices and state. After a joint exposition of the classical half of Bell's theorem in the language of Pearl-style causal graphs, a joint summary of the loophole-free experiments, and a joint survey of the recent literature, each author states where they have presently arrived. Gill accepts irreducible and non-local quantum randomness and finds the choice between locality and realism a false dichotomy. In his later works, Bell derives counterfactual definiteness from classical local causality, and that is what has to go. The metaphysical concepts "realism", "locality", "causality" need to be reconsidered. Helland reconstructs the Hilbert-space formalism from a theory of accessible variables, and from this theory he concludes that every observer must be limited in a specific sense. Jongejan proposes a geometric hidden-variable construction in which the degree of violation of the CHSH inequality depends on the number of dimensions of space, Tsirelson's bound corresponding to three dimensions. The authors conclude with a discussion.

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

3 major / 2 minor

Summary. The manuscript presents three personal reconstructions by Gill, Helland, and Jongejan for reconciling Bell's theorem with the results of loophole-free experiments, all rejecting both counterfactual definiteness and conspiratorial violation of statistical independence. After a joint causal-graph exposition of the classical assumptions, a summary of the experimental results, and a literature survey, each author outlines their view: Gill via irreducible non-local randomness, Helland via a theory of accessible variables that reconstructs the Hilbert-space formalism, and Jongejan via a geometric hidden-variable model in which the CHSH violation depends on the dimension of space (with Tsirelson's bound at three dimensions).

Significance. If the reconstructions can be shown to reproduce the full set of quantum predictions without additional constraints, the paper would supply concrete examples of coherent worldviews that accommodate the experimental data while dropping the usual assumptions, thereby clarifying options in the foundations debate. The joint causal-graph and experimental sections are standard and accurate; the individual sections remain at a conceptual level.

major comments (3)
  1. [Jongejan's section] Jongejan's geometric construction: the statement that Tsirelson's bound corresponds to three dimensions requires an explicit derivation showing that the model reproduces all two-qubit marginals, the full set of quantum correlations, and higher-order inequalities without imposing extra empirical restrictions on the accessible variables or the embedding.
  2. [Helland's section] Helland's accessible-variables derivation: the reconstruction of the Hilbert-space formalism must demonstrate that the observer limitation does not introduce effective setting dependence or restrict the allowed observables in a manner that would violate the no-conspiracy assumption while still matching all quantum predictions.
  3. [Gill's section] Gill's account of irreducible non-local randomness: the view needs to be shown to remain consistent with the joint causal-graph framework without reintroducing locality or realism issues that would undermine the explicit rejection of counterfactual definiteness.
minor comments (2)
  1. [Literature survey] The literature survey would benefit from explicit citations to quantitative comparisons of the three reconstructions with standard quantum predictions beyond CHSH.
  2. [Joint exposition] Notation for the causal graphs and accessible variables could be unified across the joint and individual sections for clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. The comments correctly note that the individual sections are presented at a conceptual level. We address each major comment below, clarifying the intended scope of the paper while agreeing to add explicit statements about limitations and references to fuller derivations where they exist in the authors' prior work.

read point-by-point responses

  1. Referee: [Jongejan's section] Jongejan's geometric construction: the statement that Tsirelson's bound corresponds to three dimensions requires an explicit derivation showing that the model reproduces all two-qubit marginals, the full set of quantum correlations, and higher-order inequalities without imposing extra empirical restrictions on the accessible variables or the embedding.

    Authors: The Jongejan section offers a geometric hidden-variable proposal in which CHSH violation strength is tied to the dimension of the embedding space, with Tsirelson's bound emerging at three dimensions. This is presented as an illustrative construction rather than a complete proof that all quantum marginals and higher-order inequalities are reproduced without further restrictions. We agree that an explicit derivation would be valuable and will revise the text to state clearly that the section outlines the geometric idea and refers to separate ongoing work for the full verification against the complete set of two-qubit predictions. revision: partial

  2. Referee: [Helland's section] Helland's accessible-variables derivation: the reconstruction of the Hilbert-space formalism must demonstrate that the observer limitation does not introduce effective setting dependence or restrict the allowed observables in a manner that would violate the no-conspiracy assumption while still matching all quantum predictions.

    Authors: Helland's reconstruction starts from a theory of accessible variables and derives the Hilbert-space formalism together with a specific observer limitation. The limitation is formulated to be independent of the choice of measurement settings and is therefore compatible with the no-conspiracy assumption used in the joint causal-graph section. The section summarizes the main steps; the full technical development, including verification that all quantum predictions are recovered without additional empirical constraints, appears in Helland's earlier papers. We will add a short clarifying paragraph that points to those references and reiterates the independence from setting choice. revision: partial

  3. Referee: [Gill's section] Gill's account of irreducible non-local randomness: the view needs to be shown to remain consistent with the joint causal-graph framework without reintroducing locality or realism issues that would undermine the explicit rejection of counterfactual definiteness.

    Authors: Gill's contribution accepts irreducible non-local randomness while rejecting both counterfactual definiteness and conspiratorial dependence, exactly as required by the joint causal-graph exposition earlier in the manuscript. The causal-graph framework already encodes the absence of local causality and the rejection of counterfactual definiteness; the non-local randomness is introduced as the remaining degree of freedom that is consistent with those graphs. We will revise the section to include an explicit cross-reference to the causal-graph diagrams, making the compatibility more transparent without altering the underlying position. revision: partial

Circularity Check

0 steps flagged

No significant circularity; arguments rest on standard QM predictions and causal modeling

full rationale

The paper is expository and philosophical rather than predictive. It begins from the established CHSH violation in loophole-free experiments and Pearl-style causal graphs, then presents three independent reconstructions (Gill on non-local randomness, Helland on accessible variables yielding Hilbert space, Jongejan on dimension-dependent geometry). No step fits a parameter to a subset of data and renames the fit as a prediction, nor does any central claim reduce by definition or self-citation chain to its own inputs. Self-references to prior technical work serve only as background; the rejection of counterfactual definiteness and conspiratorial dependence follows directly from the experimental facts and the authors' stated metaphysical choices without circular closure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard quantum mechanics, Pearl-style causal graphs, and the reported experimental outcomes without introducing new free parameters or postulated entities.

axioms (2)
  • domain assumption Quantum mechanics makes correct predictions for Bell experiments
    Invoked throughout the joint summary of experiments and theorem.
  • standard math Causal graphs correctly represent locality and statistical independence
    Used in the classical half of the exposition.

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