Superdeterministic Action Principle

Updated 15 October 2025

Superdeterministic Action Principle is a deterministic framework in quantum mechanics that asserts all measurement outcomes and settings are predetermined by initial conditions, defying the assumption of statistical independence.
It employs extended action functionals, such as complex and discretized formulations, to mathematically structure the selection of physical histories and reproduce quantum correlations.
The principle has profound implications for free will and scientific inference, challenging conventional statistical methods and the interpretation of quantum experiments.

The Superdeterministic Action Principle is a theoretical construct in the foundations of quantum mechanics and statistical physics that asserts the evolution of physical systems—including measurement outcomes and choices of experimental settings—is fully determined by initial conditions, thereby violating the standard assumption of statistical independence central to Bell’s theorem. This principle provides a deterministic framework where quantum correlations, violations of Bell inequalities, and apparent randomness all stem from underlying causal structure and, in some models, nontrivial selection criteria on the universe’s history. Across various formulations, it serves as the cornerstone for superdeterministic models of quantum theory, linking deep issues of causality, measurement, statistical inference, and the status of physical law.

1. Formal Structure: Action Principles and Superdeterminism

At the heart of the superdeterministic action principle is an extension or generalization of conventional action-based formulations of dynamics. In the complex action model (Nielsen et al., 2010), the action functional takes a complex form: $S[\text{path}] = S_R[\text{path}] + i S_I[\text{path}],$ with the realized history not only satisfying the classical equation of motion (δS_R = 0) but being uniquely selected by minimizing the imaginary component: $S_I[\text{sol}] = \min\{ S_I[\text{path}] \}.$ This selection mechanism renders all evolution, including initial conditions and future events, calculable in principle.

The principle recurs in discretized or computational models as well. In action-based cellular automaton frameworks (&&&1&&&), a fully deterministic, integer-valued dynamical system is governed by a discrete action whose stationary points under integer-valued variations produce deterministic finite-difference equations. In both continuous and discrete settings, the action principle is superdeterministic because future measurement settings and outcomes are fixed by, or part of, the causal chain initiated at the origin (e.g., the Big Bang or “initial bang” in model analogies (Nikolaev et al., 2022)).

The mathematical architecture often employs LaTeX-expressible formulae such as: $S[\text{history}] = \int_{-\infty}^{+\infty} L_I(\text{history}(t))\, dt$ for the imaginary Lagrangian’s contribution to the action (Nielsen et al., 2010), and discrete analogs for cellular automata: $S = \sum_n \left[(P_n + P_{n-1})A x^2 + (T_n + T_{n-1})^T T_n - A_n\right]$ (Elze, 2013).

In axiomatic formulations (Shackell, 2023), the state at time $t$ is a deterministic function of an initial state $X$ : $\Psi(t, X) = F(t, \Psi(0, X)),\qquad \forall t>0.$

2. Violation of Statistical Independence and Bell’s Theorem

The superdeterministic action principle systematically breaks the statistical independence (SI) assumption required for the derivation of Bell inequalities. In standard Bell-local models, hidden variables $\lambda$ and measurement settings (often denoted $a, b$ , or $x, y$ ) are assumed uncorrelated: $\rho(\lambda | a, b) = \rho(\lambda).$ Superdeterministic action principles fundamentally reject this, positing $\,\rho(\lambda | a, b) \neq \rho(\lambda)$ , so that measurement settings and system states possess common causal ancestry or are otherwise entangled via selection rules, e.g., through minimization of $S_I$ or deterministic initial conditions (Nielsen et al., 2010, Hossenfelder, 2020, Vervoort, 2018).

This explicit correlation blocks the application of Bell's inequalities: since $\lambda$ and the choice of measurement basis are entwined, statistical averages over $\lambda$ reflect this dependency, and quantum correlations can be reproduced within a locally causal framework—provided one accepts that measurement “choices” are not genuinely free.

A more technical treatment (Waegell et al., 27 Sep 2025) invokes a frequency interpretation of SI: $\rho(\lambda) \approx \rho(\lambda|Z),$ where $Z$ labels sub-ensembles defined by measurement choices. Superdeterminism corresponds to systematic SI violation such that $\rho(\lambda|Z)$ diverges from the global ensemble $\rho(\lambda)$ .

3. Classes, Implementations, and Model Variants

Superdeterministic theories instantiate the action principle in several structurally distinct ways:

Class	Selection Mechanism	Typical Model Features
Deterministic with fine-tuned initial cond.	Atypical/deterministic selection of global state	Global constraint on $\lambda$ and settings (Ciepielewski et al., 2020, Waegell et al., 27 Sep 2025)
Statistical fluke	Branches/sample flukes yield SI violation	Coincidence or branch selection (Waegell et al., 27 Sep 2025)
Nomic exclusion	Laws prohibit certain $\lambda$ -setting combos	“Goblin” toy models or invariant set restrictions (Waegell et al., 27 Sep 2025, Palmer, 2023)

In cellular automaton models (Elze, 2013), an action principle—constrained to quadratic form—enforces deterministic updates that, when mapped to the continuum via Shannon sampling, recapitulate linear quantum evolution and conservation laws. Here, linearity and the superposition principle emerge from an underlying discrete, deterministic, and thus superdeterministic, substrate.

In discretized Hilbert space models (Palmer, 2022, Palmer, 2023), the set of allowed quantum states and settings is restricted by arithmetic constraints (e.g., rational squared amplitudes and rational phases), enforced by hidden variable structure and “invariant set” fractal geometry. Counterfactual configurations central to Bell’s theorem are not defined for a fixed $\lambda$ due to number-theoretic obstructions (Niven’s theorem/Impossible Triangle Corollary), hence the SI violation is “structural” rather than conspiratorial.

4. Fine-Tuning, Conspiracy, and Critique

A central critique of the superdeterministic action principle is its reliance on apparent fine-tuning and the specter of conspiracy. In multi-device Bell scenarios (Sen et al., 2020), mathematical analysis demonstrates that to ensure measurement statistics depend only on settings (not on “which mechanism” provided those settings), the hidden variable distribution must be severely constrained. This is quantified by an “overhead fine-tuning parameter”: $F = 1 - \frac{N_f}{V(\Lambda, L)^{\Omega}},$ where the volume of consistent hidden variable distributions $N_f$ becomes negligible as the number of mechanisms $N$ grows.

Alternatively, the mutual information required to align device choice and hidden variable correlation per run grows as $2\log_2 N$ , resulting in a divergence in the infinite device limit—a signature of “extreme conspiracy.” In contrast, nonlocal and retrocausal models avoid this fine-tuning as the dependency is absorbed naturally within the nonlocal or retarded-causal framework.

However, in some invariant set and discretized models (Palmer, 2022, Palmer, 2023), fine-tuning is argued to be avoided altogether: the fractal geometry and p-adic metric punish small Euclidean perturbations, so that even “close” counterfactuals are maximally distant off the invariant set. Thus, SI violation is a necessary, intrinsic geometric selection, not a dynamical or conspiratorial adjustment.

5. Philosophical and Methodological Consequences

The superdeterministic action principle forces reevaluation of key concepts:

Free will: SI violation entails that, under many compatibilist accounts, the agent “could not have done otherwise”—choices are implicit in the initial state (or observer scope (Shackell, 2023)).
Nature of laws: Laws become contingent on (possibly atypical) initial conditions. This motivates a neo-Humean account, where regularities are patterns emergent from the initial data rather than external governing necessities (Baas et al., 2020).
Scientific method: Since SI is foundational in experimental design and statistical inference, its systematic violation in principle undermines conventional approaches to statistical analysis and theory confirmation. However, some proposals (Nikolaev et al., 2022, Hossenfelder, 2020) argue that correlations responsible for SI violation are so diluted in practice (due to “washing out” over many collisions or chaotic amplification) that effective independence arises at macroscales, thus preserving operational science.
Interdisciplinary analogies: Restricting determinism to observer substrate (the "observer scope" formulation (Shackell, 2023)) rather than all degrees of freedom aligns superdeterminism with deterministic models in evolutionary biology, neurology, and cultural studies, strengthening the plausibility of observer-based determinism without overcommitting universal scope.

6. Prospects, Open Problems, and Empirical Tests

Despite theoretical coherence, the superdeterministic action principle faces several open issues and prospects for further development:

Model completion: Fully specified, generally applicable, and non-linear dynamical laws that yield Born-type statistics and accommodate future-input dependence are not yet constructed for generic experiments (Hossenfelder, 2020).
Discriminating tests: Standard Bell-type experiments cannot distinguish superdeterministic models from quantum theory if the distribution over $\lambda$ and settings is appropriately arranged; only memory effects, autocorrelation in repeated measurement outcomes, or deviations from quantum statistics in controlled repeated-experiment protocols (as discussed in (Hossenfelder, 2014)) could serve as diagnostics.
Continuum limit and unification: Discrete and fractal-based action principles reproduce quantum predictions in the large- $p$ (dense discretization) or continuum limits, introducing a possible “singular emergence” paradigm for quantum mechanics from deterministic, superdeterministic underpinnings (Palmer, 2022, Palmer, 2023).
Integration with gauge theory and auxiliary variables: Recent work (Struyve, 2023) discusses the formulation of Bohmian mechanics via an action principle using auxiliary (gauge) variables, extending the metaphysical landscape for superdeterministic interpretations.

7. Comparative Perspectives and Theoretical Landscape

Superdeterministic action principles define a distinct region of theories attempting to account for quantum nonlocal correlations while upholding locality. Comparative analysis clarifies:

Fine-tuned determinism: Relies on universal or observer-based initial state selection, with all subsequent correlations fixed (Ciepielewski et al., 2020, Shackell, 2023).
Statistical fluke: Involves rare but possible statistical anomalies producing SI violation (Waegell et al., 27 Sep 2025).
Nomic exclusion/invariant set approaches: Forbids certain configurations, structurally precluding some counterfactuals central to Bell’s theorem (Palmer, 2022, Palmer, 2023).
Nonlocal and retrocausal models: Do not require SI violation or superdeterminism in the strict sense; correlations are accounted for via nonlocal mechanisms or backward-in-time causation (Sen et al., 2020, Waegell et al., 27 Sep 2025).

A crucial insight is that—contrary to some prior treatments—retrocausal and some invariant set models can, with causal-order analysis, avoid SI violation, making them distinct from core superdeterministic action principles (Waegell et al., 27 Sep 2025).

The superdeterministic action principle establishes a deterministic foundation for quantum and statistical phenomena, centered on the breakdown of statistical independence between hidden variables and measurement settings. It provides constructive mathematical frameworks but also faces stringent requirements regarding fine-tuning, philosophical implications for free will and scientific inference, and significant challenges in differentiating its predictions from quantum mechanics under laboratory conditions. Its diverse manifestations—from complex action models and cellular automata to discretized Hilbert space and observer-scope deterministic dynamics—underscore its central role in ongoing efforts to reconcile quantum theory, causality, and realism.