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Reply to Gill, Helland and Jongejan: A Fourth Way — The Nilpotent Universe

Anton Lorenz Vrba

Published Jul 18, 2026• 0 citations
Publisher

Abstract

I propose a fourth response to the Bell--CHSH violation that lies outside the three positions surveyed by Gill, Helland, and Jongejan. Rather than relinquishing realism, locality, or counterfactual definiteness, I challenge the assumption that the universe can be correctly decomposed into conditionally independent local parts. In a nilpotent universe, the global conservation laws associated with Noether's theorem are treated as primitive ontological constraints rather than merely as emergent consequences of autonomous local interactions. The framework is developed as a corpuscle--pilot-wave test theory: a circularly structured pilot wave supplies local susceptibility to reorientation, while an explicit global steering postulate supplies the compensating relation between partners of a common-origin pair. The postulate yields the singlet correlation $E(a,b)=-\cos 2(a-b)$ and hence $S_{\mathrm{CHSH}}=2\sqrt{2}$ in a configuration for which standard quantum mechanics predicts no violation: a single-pass Type~I SPDC source producing $\ket{HH}$ pairs, converted to $\ket{RL}$ by quarter-wave plates and analysed by rotatable two-channel polarisers. Standard quantum mechanics treats the plates as local unitaries acting on a separable state and predicts $S_{\mathrm{CHSH}}\leq2$. The proposed arrangement therefore provides a direct experimental test of Hilbert-space quantum mechanics, under any interpretation, against the nilpotent steering theory defined here.

DOI
10.5281/zenodo.21426049

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Anton Lorenz Vrba

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I propose a fourth response to the Bell--CHSH violation that lies outside the three positions surveyed by Gill, Helland, and Jongejan. Rather than relinquishing realism, locality, or counterfactual definiteness, I challenge the assumption that the universe can be correctly decomposed into conditionally independent local parts. In a nilpotent universe, the global conservation laws associated with Noether's theorem are treated as primitive ontological constraints rather than merely as emergent consequences of autonomous local interactions. The framework is developed as a corpuscle--pilot-wave test theory: a circularly structured pilot wave supplies local susceptibility to reorientation, while an explicit global steering postulate supplies the compensating relation between partners of a common-origin pair. The postulate yields the singlet correlation E(a,b)=−cos⁡2(a−b)E(a,b)=-\cos 2(a-b)E(a,b)=−cos2(a−b) and hence SCHSH=22S_{\mathrm{CHSH}}=2\sqrt{2}SCHSH​=22​ in a configuration for which standard quantum mechanics predicts no violation: a single-pass Type~I SPDC source producing ∣HH⟩\ket{HH}∣HH⟩ pairs, converted to ∣RL⟩\ket{RL}∣RL⟩ by quarter-wave plates and analysed by rotatable two-channel polarisers. Standard quantum mechanics treats the plates as local unitaries acting on a separable state and predicts SCHSH≤2S_{\mathrm{CHSH}}\leq2SCHSH​≤2. The proposed arrangement therefore provides a direct experimental test of Hilbert-space quantum mechanics, under any interpretation, against the nilpotent steering theory defined here.
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.
Whether Nature is deterministic or probabilistic has stood outside the reach of experiment since 1926: Bohmian mechanics reproduces the predictions of quantum mechanics without residue, so the choice between chance and determinism has been one of interpretation, not of measurement. This paper constructs a nilpotent steering theory --- definite corpuscles guided by physically real waves, under a global conservation constraint --- that is empirically equivalent to quantum mechanics save in one experimental arrangement, stated here. The theory treats the conservation laws as primitive ontological constraints --- a zero-totality principle called the nilpotent condition --- under which every photon pair is entangled at its creation event, carrying its conserved content as a single ledger entry. What the celebrated source arrangements of the Bell literature manufacture is not entanglement but the condition for its expression: a circularly structured pilot wave, supplied natively by the atomic-cascade source of the first Bell test and constructed after creation by the beam-splitter, crossed-crystal, and cone-intersection arrangements of the down-conversion era. This reading admits a configuration the record leaves untested: a single-pass Type-I source whose pair, separable on the orthodox account, is converted to circular guidance by a quarter-wave plate in each arm. There quantum mechanics, under any interpretation, predicts a vanishing correlation and no Bell--CHSH violation, EQM(a,b)=0E_{\mathrm{QM}}(a,b)=0EQM​(a,b)=0 and S=0S=0S=0, while the nilpotent steering postulate predicts the maximal violation, E=±cos⁡2(a−b)E=\pm\cos2(a-b)E=±cos2(a−b) and S=22S=2\sqrt2S=22​. A violation would point to globally constrained dynamics (interpreted here as deterministic); a null result rejects the theory proposed here while leaving orthodox de~Broglie--Bohm mechanics untouched. The ledger accounting offered here is bookkeeping, not yet dynamics; its completion --- a formalism in which the bookkeeping becomes the evolution of physical fields --- is stated as the theory's principal debt. Either outcome of the experiment is informative, and the enabling conditions are stated in advance.
In this paper, I respond to a critique of one of my papers previously published in this journal, entitled ‘Dr. Bertlmann’s socks in a quaternionic world of ambidextral reality.’ The geometrical framework presented in my paper is based on a quaternionic 3-sphere, orS3S^{3}S3, taken as a model of the physical space in which we are inescapably confined to perform all our experiments. The framework intrinsically circumvents Bell’s theorem by reproducing the singlet correlations local-realistically, without resorting to backward causation, superdeterminism, or any other conspiracy loophole. In this response, I demonstrate point by point that, contrary to its claims, the critique has not found any mistakes in my paper, either in the analytical model of the singlet correlations or in its event-by-event numerical simulation based on Geometric Algebra.
Whether Nature is deterministic or probabilistic has stood outside the reach of experiment since 1926 --- a gap this paper proposes to close. Bohmian mechanics reproduces the predictions of quantum mechanics without residue, so the choice between chance and determinism has so far been one of interpretation, not of measurement; we identify a regime in which the two can be told apart experimentally, using a modified version of the Kwiat et al.\ (1995) configuration, with a quarter-wave plate (QWP) in each arm, as in modern photonic state tomography. The paper's premise is that Bohmian mechanics requires a global constraint --- a zero-totality principle called the nilpotent condition --- under which every photon pair is entangled at its creation event, so that Bohmian guidance is supplemented by a postulate we call Universal Nilpotency (UNP). What the celebrated source arrangements of the Bell literature manufacture is not entanglement but the condition for its expression: we identify this condition as a circular state, supplied natively by the atomic-cascade source of the first Bell test and constructed after creation by the beam-splitter, crossed-crystal, and cone-intersection arrangements of the down-conversion era. This reading admits a configuration the record leaves untested: a single-pass Type-I source whose pair, separable on the orthodox account, is converted to circular guidance by a quarter-wave plate in each arm. There quantum mechanics predicts a vanishing correlation, EQM(α,β)=0E_{\mathrm{QM}}(\alpha,\beta)=0EQM​(α,β)=0 and S=0S=0S=0, while the UNP guidance postulate of Bohmian mechanics predicts a structured correlation, EUNP(α,β)=12[±cos⁡2(α−β)∓cos⁡2(α+β)]E_{\mathrm{UNP}}(\alpha,\beta)=\tfrac12\bigl[\pm\cos2(\alpha -\beta)\mp\cos2(\alpha +\beta)\bigr]EUNP​(α,β)=21​[±cos2(α−β)∓cos2(α+β)], and a calculated S=2S=\sqrt2S=2​. The test is not on a CHSH violation but on the contrast between the vanishing EQME_{\mathrm{QM}}EQM​ and the structured EUNPE_{\mathrm{UNP}}EUNP​.
We propose an optical experiment to distinguish orthodox quantum mechanics from a deterministic Bohmian framework, using standard, laboratory-proven photonics. In a single-pass Type-I parametric down-conversion source with a quarter-wave plate in each arm --- an arrangement standard to modern photonic state tomography --- circularizing the photon pair at π⁄4 leaves it separable on the orthodox account, predicting a vanishing correlation, EQM(α,β) and S=0, at the fixed CHSH angles used throughout. Under the hypothesis of a global constraint on Bohmian guidance --- Universal Nilpotency (UNP), a zero-totality condition under which a photon pair is entangled at its creation event rather than by the subsequent optical arrangement --- the same configuration is instead predicted to yield a structured correlation, EUNP(α,β)= 1/2 ±cos2(α−β)∓cos2(α+β), and S = √2.. The test is not designed to search for a CHSH violation or to corroborate the UNP hypothesis, but to provide a direct contrast, at fixed measurement settings, between a vanishing and a structured correlation. Because Bohmian mechanics otherwise reproduces every orthodox prediction, this previously untested configuration bears directly on whether the statistical character of quantum mechanics is fundamental or describes an underlying deterministic dynamics.