The Physics of Causation

Abstract: Assembly theory (AT) introduces a concept of causation as a material property, constitutive of a metrology of evolution and selection. The physical scale for causation is quantified with the assembly index, defined as the minimum number of steps necessary for a distinguishable object to exist, where steps are assembled recursively. Observing countable copies of high assembly index objects indicates that a mechanism to produce them is persistent, such that the object's environment builds a memory that traps causation within a contingent chain. Copy number and assembly index underlie the standardized metrology for detecting causation (assembly index), and evidence of contingency (copy number). Together, these allow the precise definition of a selective threshold in assembly space, understood as the set of all causal possibilities. This threshold demarcates life (and its derivative agential, intelligent and technological forms) as structures with persistent copies beyond the threshold. In introducing a fundamental concept of material causation to explain and measure life, AT represents a departure from prior theories of causation, such as interventional ones, which have so far proven incompatible with fundamental physics. We discuss how AT's concept of causation provides the foundation for a theory of physics where novelty, contingency and the potential for open-endedness are fundamental, and determinism is emergent along assembled lineages.

• The paper introduces Assembly Theory, a framework that quantifies causation by calculating assembly indices as the minimum causal steps required for an object's formation.
• It demonstrates that a high assembly index signals selective construction, establishing a measurable threshold to differentiate spontaneous formations from those arising from persistent causal mechanisms.
• Empirical studies on molecular systems validate the theory, linking object copy number and assembly thresholds to the emergence of complexity, open-ended evolution, and potential biosignatures.

The Physics of Causation: A Formalization via Assembly Theory

Introduction

"The Physics of Causation" (2601.00515) proposes a radical re-conceptualization of causation as a material, physically quantifiable property, rather than an interventionist or counterfactual notion. Assembly Theory (AT), central to this work, introduces a substrate-specific topological space—assembly space—where the assembly index (AI) quantifies the minimum number of causal steps required for a physical object to exist. By formalizing the physical limits for spontaneous object formation, the work underpins a rigorous, quantitative framework with significant implications for the physics of emergence, life detection, and the characterization of open-ended evolution.

Assembly Theory: Foundations and Metrology

Assembly Theory reconceives objects as the terminus of recursively assembled causal chains, where causation is embodied in the joining operations that create objects from units. The assembly space encapsulates all possible constructions, with each object assigned an assembly index, $a_i$, representing its minimal causal depth. The assembly threshold $a_M$ demarcates the AI above which objects cannot form spontaneously in a given physical system, bounded by system size ($N_T$), measurement resolution ($M$), and the combinatorial substrate branching factor ($b$):

$a_M = 1 + \frac{\ln M - \ln((1-B)N_T + B)}{\ln b}$

where $B = 1/b$ in the absence of environmental selection. The notion of "copy number" ($n_i$)—the countable abundance of an object—provides empirical evidence for persistent, contingent causal mechanisms, i.e., constructors.

To operationalize causation as a measurable physical attribute, the assembly index is shown to be invariant across distinct experimental modalities (mass spectrometry, IR, NMR), reflecting its observer-independence and compatibility with quantum mechanical descriptions. This enables the definition and application of a standardized, physically intrinsic scale for constructed complexity.

The Spontaneous-Selected Threshold and Detection of Life

By quantifying the exponential expansion of combinatorial possibilities in assembly space, AT predicts a universal, albeit substrate-dependent, abiotic upper bound on observable AI. Objects with $a_i < a_1$ (ontological threshold) may arise spontaneously, but $a_i > a_1$ necessitate selective construction via persistent causal lineages—an essential physical signature of life and intelligence. Empirical studies on molecular systems (e.g., Taxol, ribosome, ATP) corroborate the theory: no abiotically-derived molecule with $a_i > 15$ has been detected in molar-scale experiments, reflecting steep combinatorial suppression in the absence of selection.

Importantly, AT enables extrapolation from laboratory, to planetary, to cosmological scales; for instance, a molecule with $a_i > 58$ cannot arise abiotically even utilizing the observable universe's atom count. This provides a rigorous, system-size-dependent criterion for the detection of biosignature objects irrespective of molecular identity.

Materiality of Causation and the Assembly Space Ontology

Unlike counterfactual or interventionist accounts (e.g., Pearl's do-calculus), AT posits that causation is inherent in the material construction of objects, encoded in their AI and lineage. The assembly space, being both combinatorial and recursive, defines the universe of causal possibility without presupposing the fixed existence of all objects as in block-universe models.

Objects exist as endpoints of causal chains, and their repeated occurrence (high copy number) is direct evidentiary support for persistent contingency—selection that is quantifiable independent of environment. The concept of "virtual objects" formalizes the partially realized structures in the assembly space, while "assemblage" ($A$) aggregates the total density of causal possibility (weighted by AI and copy number) in a system.

Contingency, Open-endedness, and Transition to Life

AT permits a rigorous account of contingency: an object's existence depends on the reproducibility of its causal ancestors, encoded in the AI and validated by copy number. The existence of lineages with cumulative AI and persistent copy number marks a phase transition: the system becomes "alive" when it supports a nontrivial burden of high-AI objects above the abiotic threshold.

Open-ended evolution is explicated by the expansion of assembly space through the stabilization of novel object types—novelty emerges from rare, persistent interactions in the under-resolved environment that cross the discretization threshold. Deterministic dynamics and predictability are emergent, not fundamental; the state-space expands noncomputably over time as new objects and operations are defined only retrospectively, invalidating precomputable causal closure.

Entropy, Path-dependence, and Comparison to Existing Paradigms

Assembly index is fundamentally distinct from entropy—AI is a non-extensive, algorithmic invariant of an individual object, whereas entropy is an ensemble property descriptive of state multiplicities. The work highlights the critical role of path-dependence in non-equilibrium systems and asserts that AT discards the unphysical assumption of ergodic sampling over unbounded ensembles. Instead, AT founds non-equilibrium path-dependency on precise, measurable causation and contingency, reconciling physical description with the empirical uncomputability of possibilities in complex systems.

Further, AT challenges the interpretation of possibility versus probability: the existence of an object is not governed by ensemble likelihood but by the presence of a causal mechanism—probability statements pertain to systems, not objects, reinforcing the substrate-centric, material notion of causation.

Selection, Phase Transitions, and Universality

The assembly, $A$, condenses the cumulative selection and contingent causation in a system, providing a physically grounded parameter that marks the transition from abiotic matter to life (and further to intelligence and technology). The formalism admits precise, quantitative phase transitions, where a growing density of high-AI, high-copy objects signals the emergence (or extinction) of life-like behavior.

Prebiotic chemistry, often confounded with truly abiotic conditions, is shown to operate fundamentally within this selective regime, as human (or otherwise constructed) experimenter interventions effectively instantiate selective lineages.

Implications and Future Directions

By declaring the physical system-size and substrate as fundamental determinants of causal possibility, Assembly Theory yields a highly testable, resource-bound, and quantifiable framework. The formalization of material causation—distinct from existing interventionist accounts—enables the detection of living, technological, or potentially intelligent systems in principle via physical measurement alone.

The emergence of complexity, persistent selection, and dynamical "possibility expansion" are rendered consequences of real, measurable causation, not artifacts of fine-tuning or block-universe initial conditions. Future AI and astrobiology research may leverage AT in experimental life detection, understanding major evolutionary transitions, analyzing technological lineages, and elucidating the origins—and boundaries—of open-ended evolution.

Conclusion

"The Physics of Causation" establishes a rigorous, physically intrinsic account of causation, fundamentally distinct from extant formalizations. Assembly Theory introduces substrate-specific, quantifiable thresholds demarcating spontaneous from selected (and hence, living) objects based on assembly index and copy number. This formalism reconstitutes causation as a material property, underpinning a unified approach to the emergence of complexity, life, and intelligence, with broad implications for physical science, AI, and theories of evolution. The framework redefines the boundaries of what is possible and observable, offering a new lens for experimental and theoretical exploration.