Superdeterministic Theories

• Superdeterministic theories are physical models that explicitly violate the assumption of statistical independence by linking hidden variables with measurement settings.
• They employ methodologies such as fine-tuned initial conditions, statistical flukes, and nomic exclusion to reproduce quantum correlations while upholding local causality.
• These models raise profound implications for experimental design, free will, and the interpretation of quantum probabilities in foundational physics.

Superdeterministic theories constitute a class of physical models in which all events—including the choices of measurement settings and outcomes—are fully determined by prior conditions, and, critically, in which the hidden variables describing physical systems are not statistically independent of those measurement settings. The defining feature is a systematic violation of the Statistical Independence (SI) assumption, which posits that the probability distribution over ontic (hidden) variables is uncorrelated with the settings of measurement devices. This SI violation allows superdeterministic theories to evade the conclusions of Bell’s theorem regarding the incompatibility of local realism and quantum predictions, not by invoking nonlocality or indeterminism, but by correlating the “random choices” of experimenters with the microscopic state of the system under paper. Superdeterminism thereby offers a pathway to retaining local causality in hidden-variable completions of quantum mechanics, though at the cost of introducing deep challenges to the standard methodology of science and to notions of free will, contingency, and the autonomy of experimental choices.

1. Definition, Mathematical Structure, and Historical Context

Superdeterminism is defined by the failure of Statistical Independence: if $\lambda$ denotes the set of hidden variables and $Z$ a label encoding the selection of measurement settings (e.g., in Bell-type or interference experiments), superdeterministic models satisfy

$\rho(\lambda | Z) \neq \rho(\lambda)$

for at least some choices of $Z$, in contrast to the independence posited by Bell and most of his successors (Waegell et al., 27 Sep 2025). This ties the ontic state of the system to the settings selected, whether those settings are assigned by human researchers, cosmic light, or random number generators.

Historical context traces debate to Einstein’s search for an “epistemic” interpretation of the wavefunction, in which quantum states merely reflect ensembles of possible classical configurations—or our knowledge thereof. It has been argued that if one rejects quantum nonlocality but demands conservation laws at the level of individual events (e.g., “one photon, one count”/no double-detection), then any epistemic model for interference experiments necessarily implies a preordained, superdeterministic structure in which both settings and outcomes must be fixed by the underlying ontic variables (Suarez, 2012).

A general mathematical structure for superdeterministic hidden-variable models can be summarized by:

$$p(k | \psi, M) = \int d\lambda \, p(k | \psi, M, \lambda) \, \rho(\lambda | \psi, M)$$

where $M$ are measurement settings, $\psi$ is the prepared state, $k$ is the outcome, and crucially the distribution $\rho(\lambda | \psi, M)$ depends on $M$ (settings dependence) (Sen et al., 2020).

2. Types and Constructions of Superdeterministic Theories

Superdeterministic theories can be categorized according to the type of SI violation and the mechanism by which they enforce it (Waegell et al., 27 Sep 2025):

Category	Mechanism of SI Violation	Features/Examples
Deterministic, Fine-tuned Initial Conditions	Special, atypical initial state fixes all	Homogeneous internal state (e.g., LQM)
Fluke Theories	Atypical but possible sample sequence	Statistical flukes, Everettian branches
Nomic Exclusion Theories	Dynamics forbids certain events by law	“Goblin” toy models, forbidden assignments

Deterministic, Fine-tuned Initial Conditions: The universe’s initial condition is so fine-tuned that the hidden variables predetermine both settings and outcomes, yielding only those outcomes compatible with quantum predictions. Models such as Leibnizian Quantum Mechanics (LQM) posit a collection of spatially distributed internal “worlds” sharing identical initial conditions (resulting in perfect concordance across hidden variables and experimental choices), formalized through a statistical dependence such as $\rho(\lambda|Z)$ sharply peaked on only the quantum-allowed assignments (Waegell et al., 27 Sep 2025).

Fluke Theories: Represent scenarios where SI is violated not as a matter of law, but by rare statistical accidents—a sequence of outcomes that, while possible under the standard theory, are not typical. This is analogous to a sequence of fair coin flips giving all heads in a million trials—a non-representative sample not anticipated by the global probability law but permitted in principle (Waegell et al., 27 Sep 2025).

Nomic Exclusion Theories: These models, illustrated by various “goblin” metaphors, invoke constraints that explicitly prevent some combinations of hidden variables and experimental settings from ever arising. Access to different settings is dynamically restricted by constraints built into the structure of the theory, rather than achieved through improbable initial conditions (“nomic” here refers to “by law”) (Waegell et al., 27 Sep 2025).

Mathematically, nomic exclusion manifests as certain $(\lambda, Z)$ pairs having probability zero: $P(\lambda | Z') = 0$ for forbidden settings $Z'$.

3. Mechanisms, Motivations, and Theoretical Frameworks

Superdeterministic approaches to quantum theory are often motivated by two central desiderata:

1. Restoration of Local Causality: By correlating settings and ontic variables, superdeterministic models reproduce the quantum violations of Bell inequalities without invoking nonlocal influences or outcome dependence (Hossenfelder et al., 2019).
2. A Deterministic Foundation for Probability: Viewing probability as a shadow of deterministic evolution (cf. hypothesis HYP-1: “Any probability emerges from underlying deterministic processes”), superdeterminism offers explanations for the origin of quantum probabilities, the prevalence of normal (Gaussian) statistics via the Central Limit Theorem, and the contextuality problem in probabilistic dependence (Vervoort, 2018, Nikolaev et al., 2022).

Specific frameworks include:

• Discretised Hilbert Space: Only a finely discretized subset of complex Hilbert space (with rational amplitudes/phases) is physically realized. Violations of SI arise because the rationality (number-theoretic) constraints prevent the existence of counterfactual assignments required for Bell inequalities; e.g., in the “Impossible Triangle Corollary,” most counterfactual settings are assigned zero probability (Palmer, 2022, Palmer, 2023).
• Invariant Set Theory: States belong to a dynamically invariant, measure-zero fractal (often with p-adic metric structure) in state space. Only histories on the invariant set are ontologically real, causing SI violation to be an emergent geometric property that is not conspiratorial, but rather a consequence of global consistency conditions (e.g., consistent histories) (Palmer, 2016).
• Pilot-wave and Local Pilot-wave Models: Some local pilot-wave models evade Bell’s theorem not by nonlocality, but by settings-dependent initial conditions—a measure-zero, homogeneous internal state that imprints the correlations needed for SI violation (Ciepielewski et al., 2020).

4. Empirical Tests, Fine-tuning, and the Problem of Conspiracy

A recurring criticism is that superdeterministic models are ‘conspiratorial’: they require that nature arranges correlations between settings and hidden variables with such precision that measurement statistics depend only on the settings, but never on the mechanism by which those settings were chosen (Sen et al., 2020). This necessitates enormous fine-tuning, often quantified by the “fine-tuning parameter” $F$ (which approaches unity as the configuration space grows), or by a dramatic entropy drop at the hidden-variable level as settings and sample sub-ensembles are matched up precisely.

Potential avenues for empirical testability include proposals for repeated measurements on single systems to detect persistent time auto-correlation in hidden-variable-dominated sequences, in contrast to the exponential decoherence expected in standard quantum theory (Hossenfelder, 2014). If SI violation mediated by hidden variables could be observed as a statistical deviation from Born rule predictions, this would provide evidence supportive of superdeterminism—but such empirical signatures have not been decisively detected.

Some approaches argue that superdeterministic correlations, though present, may be undetectable for all practical purposes due to the complexity of causal chains and experimental limitations (e.g., in “billiard table” models, correlations are quickly washed out through chaotic dynamics and environmental factors) (Nikolaev et al., 2022).

5. Foundational, Philosophical, and Methodological Implications

Superdeterministic theories have profound implications for science:

• Statistical Methodology: A robust SI assumption underpins experimental inference and the scientific method. If hidden-variable/sample correlations can exist in principle, then any statistical sample may not be representative. While in practice this may be negligible, at a foundational level it calls into question induction and experimental reproducibility (Waegell et al., 27 Sep 2025, Baas et al., 2020).
• Status of Scientific Laws: Superdeterminism, especially in the form of fine-tuned initial conditions, makes the observed quantum laws contingent on a highly atypical, cosmologically determined initial state, supporting a neo-Humean “mosaic” view where laws simply supervene on particular facts rather than representing independently necessary constraints (Baas et al., 2020).
• Free Will and Compatibilism: SI violation inherently restricts the space of possible alternative measurement settings, posing a challenge to free will (even of the compatibilist or dispositional variant), as there is never genuine freedom to carry out counterfactual interventions (Waegell et al., 27 Sep 2025).
• Retrocausality and Related Models: While some retrocausal and invariant-set theories are presented as SI-violating, closer examination shows that under certain causal-ordering conventions, they need not be superdeterministic in the strict sense. In retrocausal frameworks, ontic states are built up after settings are chosen, allowing random sampling and statistical independence of outcomes up to boundary conditions fixed by the experiment (Waegell et al., 27 Sep 2025).
• Conspiratorial Character: Superdeterministic models are characterized by correlations that systematically “deceive” experimenters about the representativeness of their samples. The theory space encompasses both highly fine-tuned and more dynamically enforced SI violation, but all share the feature that freedom to vary settings independently is fundamentally illusory (Sen et al., 2020, Waegell et al., 27 Sep 2025).

6. Observer Scope, Axiomatic Formulations, and Holism

An axiomatic reformulation of superdeterminism suggests that only the deterministic evolution of the observer scope—the collection of states relevant to all observers—is necessary to account for quantum measurement correlations, rather than a cosmic-level, all-encompassing determinism (Shackell, 2023). This transition reduces the demand for universal determinism and renders superdeterminism more plausible, as all observed events and “choices” are captured in the evolution of observer states:

$$\Psi(t, X) = F(t, \Psi(0, X)) \qquad \text{and} \qquad \Psi(t) = G(X)$$

where $\Psi$ denotes observer (or universe) states, $X$ initial conditions, and $F$, $G$ deterministic mappings. This observer focus suggests greater potential for integration with disciplines such as evolutionary biology, neurology, and complexity theory, where deterministic processes govern observer evolution and behavior.

A broader implication from the discretized Hilbert space and invariant set approaches is a shift toward holistic frameworks—state-space geometry, p-adic metrics, and top-down global constraints determined by gravitational and cosmological boundary conditions (e.g., the “all-at-once” fractal attractor). In these models, the fine-grained details (e.g., rationality constraints on measurement orientations) enforce SI violation non-conspiratorially, while quantum mechanics emerges in the singular limit of these more fundamental theories (Palmer, 2016, Palmer, 2022, Palmer, 2023).

7. Outlook and Ongoing Debates

Debate continues regarding the scope, consistency, and experimental viability of superdeterministic theories. Key open questions include:

• Is SI violation a feature unique to quantum experiments, or could it undermine experimental methodology more broadly?
• Can retrocausal or invariant set theories truly avoid the core superdeterministic pathology by arranging the causal order differently?
• Do superdeterministic theories offer practical pathways to new physics, or are they relegated to the “pre-scientific” stage due to the lack of distinguishing predictions and the challenge of specifying their scope and dynamics? (Waegell et al., 27 Sep 2025)

A notable perspective is that while superdeterminism is unambiguously conspiratorial (in the sense of systematic SI violation), it need not be unscientific—it may offer greater explanatory power regarding the unification of probability, causality, and quantum correlations, provided its implications are properly understood and its scope carefully defined (Nikolaev et al., 2022, Vervoort, 2018).

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In summary, superdeterministic theories represent a rigorously defined, systematically SI-violating approach to quantum foundations in which local causality and determinism can be preserved at the cost of deeply entwined correlations between hidden variables and experimental settings. The resulting frameworks challenge traditional notions of scientific inference, free will, and lawhood, and prompt novel questions at the intersection of physics, mathematics, and philosophy. Ongoing research aims to clarify the theoretical structure, experimental consequences, and broader significance of these proposals in the landscape of quantum theory and beyond.