Contribution
A map of where the many notions of “closure” in theories of autonomy came from, and a classification of them (Table 3) by what is closed, what kind of circularity, what causal regime, whether it is material, whether it is computable, where semantics sits, and what it explains. On that map the authors rest an interpretive thesis: closure, properly so called, is a circular organization of efficient causes, and it is not captured by any state-determined syntactic description. They apply the thesis to current AI and conclude that its loops are “engineered approximations of closure rather than instances of organizational closure itself”.
Key insight
The same word names different things that close. Ashby closed information, Maturana and Varela operations, Rosen efficient causes, Montévil and Mossio constraints, Kauffman catalysts. In the authors’ words, “each explicative account apparently identifies a different “thing” that closes upon itself”, while “every closure theory attempts to solve the same problem: How can a living system become the cause of the conditions required for its own persistence?” A loop that only feeds a signal back, or a pattern that persists only while a gradient is maintained from outside, answers a different question.
Assumptions
- Genealogy as conceptual analysis. Not “merely as a historical reconstruction of facts”, but a mapping of origins to show how research programmes “progressively converged”. The transition from logic to biology “is not deductive”; logic supplies precursors, not a theory.
- A stipulated sense of “non-computable”. It “does not refer simply to practical computational limitations, computational complexity, or the impossibility of simulation”. It means that closure “cannot be exhaustively characterized as a state-determined syntactic procedure in the sense inherited from the Church–Turing thesis”.
- A reading of the Church–Turing thesis. “Within the interpretation adopted here”, effective procedures “characterize physical phenomena insofar as they are non-self-referential and formally specifiable”. The thesis is thereby read as a claim about natural systems, which is not the usual reading.
- Rosen’s relational biology taken as the strongest causal account. The paper says it “does not seek to adjudicate these controversies or to provide a definitive assessment of Rosen’s formal claims”, but its conclusions lean on them.
Key results
- Closure is not feedback (§3). First-order cybernetics recast circular causality as a feedback mechanism taken from the governor: “Circular causality was thus equated with feedback loops”. The BCL turn was to read circular causality as self-reference instead. Von Foerster called first-order cybernetics “rather unfascinating”.
- Order is not organization (§4). Reaction–diffusion, Turing patterns and dissipative structures produce physical order. Organization is “the relations between components which define a system as a unity” (Maturana). Open systems are “necessary but insufficient” for Maturana and Piaget; Piaget’s “cyclic closure” is the complement.
- Regulation belongs to the observer, for Maturana and Varela (note 17). “for Maturana and Varela, regulation (and self-regulation in particular—a central piece of cybernetic terminology at the time) depends entirely on the observer and does not belong to the intrinsic organization of the living.”
- Self-reproducing automata are heteronomous (note 4). “Their efficient causal structures execute preprogramed instructions and thus remain heteronomous”. Rosen (1959) argued the von Neumann automaton is an inconsistent model of life, conflating efficient with material causes. Varela’s 1974 autopoiesis model also had catalysts supplied from outside.
- Rosen’s (M, R)-systems (§6). Metabolism M, repair R, and a replication map produced inside the system close the regress of catalysts. Rosen’s theorem: “a material system is an organism if, and only if, it is closed to efficient causation”. The authors add that the four Aristotelian causes are interchangeable within the structure. Protein folding is offered as a paradigm and as “essentially non-syntactic (non-algorithmic)”; AlphaFold’s results are called statistical approximations.
- A halting-problem argument (§6). Mossio, Longo & Stewart (2009, not held; cited in the paper as “Mossio et al. (2009)”, which is not LIT-570) gave a computable expression of closure to efficient causation. The authors reply that “if we enforce the requirement that every computation must terminate in a finite number of steps, the Mossio et al. (2009) model of closure-to-efficient causation collapses into uncomputability”. Their own conclusion is conditional: “Rosen’s claim that (M, R)-systems are non-computable holds true if and only if computability is defined by halting algorithms”, and “these conclusions remain provisional”.
- Three senses of non-computability (Table 1). Not Turing-computable; not formally axiomatizable; not adequately modelled by classical computational cognitive science. Rosen’s claims are said to concern the first two, enactivism mainly the third.
- Convergence and its limits (§7). Catalytic closure is “not the organizational closure found in autopoietic or (M, R)-systems”, because catalysts are not produced by a closed network of efficient causes. Viability theory, with its kernel, “complements rather than replaces autopoiesis, closure-to-efficient causation, or closure of constraints”: it asks under what conditions the system can continue, not how its causes are produced.
- The AI verdict (§7). In-context learning, agentic loops, persistent memory and RLHF “introduce forms of operational circularity and adaptive regulation”. “Their feedback loops do not regenerate the constraints that make those loops possible, nor do they fabricate the conditions of their own continued existence. What recurs are information flows, policy updates, and state transitions, not the mutual production of the efficient causes that constitute the system as an autonomous unity.” LLMs remain “computationally mediated heteronomy”.
- Detection (§7 and note 35). “A robust understanding of closure—and the methods used to model it—must ultimately rely on the ability to detect when closure occurs” (after Chemero and Turvey). Note 35: “a degree of heteronomy is typically embedded within the design process itself”, and “questions whether true organizational closure has been achieved frequently re-emerge when the analysis is shifted to descriptive levels other than those initially specified by the model.”
- Conclusion (§7 a–d). Closure is “the circular entailment of efficient causes that fabricate and regenerate one another while maintaining their own boundary conditions”. Such systems “actively conserve and re-establish the constraints that make their own continuation possible”. “It provides a proposed necessary and sufficient criterion for distinguishing living from non-living systems.”
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | The notions of closure derive from distinct research programmes and differ in what they take to be closed | moderate | historical reconstruction from published sources and recollections; Table 3 |
| C2 | Closure is not feedback, self-organization, physical order or a catalytic cycle | moderate | conceptual distinctions drawn from the sources (Maturana, Rosen, Chemero and Turvey) |
| C3 | Closure is best read as a circular organization of efficient causes that produces the conditions of the system’s own existence | weak to moderate | stated as an “interpretative thesis”, supported by the convergence C1 traces |
| C4 | Closure cannot be exhaustively characterized as a state-determined syntactic procedure | weak | Rosen’s arguments, reported and not assessed; a halting-problem argument the authors call provisional |
| C5 | Current AI, agentic LLM systems included, approximates closure without instantiating it | weak | asserted from C3: loops not regenerating their constraints; two neurolinguistic citations on grounding |
| C6 | Closure is a necessary and sufficient criterion of life | weak | stated in the conclusion, called “proposed” |
Concepts
- Organizational closure. As the paper concludes, the circular entailment of efficient causes that produce and regenerate one another while maintaining their own boundary conditions; open to matter and energy, with the causal roles needed for persistence produced inside.
- Closure to information (Ashby). Systems “open to energy but closed to information and control”; a formalism over Shannon information, with no causal thesis.
- Operational closure (Maturana and Varela). A closed domain of relations of material production; the authors note the 1973 book does not use the word “closure”.
- Closure to efficient causation (Rosen). Every efficient cause in the system is produced inside it; here the replication map.
- Closure of constraints (Montévil and Mossio). Mutual dependence among constraints, each produced by and productive of another.
- Catalytic closure (Kauffman). Every molecule is food or is made by reactions the set catalyses; not closure of efficient causes.
- Heteronomy. Organization specified from outside: by design, objectives, training or supplied catalysts.
Connections
The paper reconstructs Ashby, von Foerster, Piaget, Maturana and Varela, Rosen, Pattee, Kauffman, Letelier and Montévil and Mossio. In the record, Montévil & Mossio (2015) is LIT-582 and the organizational account of function is LIT-570; Moreno & Mossio’s book is LIT-536, unavailable. Weber & Varela (2002, LIT-1018) appears in note 29 as Varela’s move to “immanent teleology”, lacking “a complete formalization of its organizational basis”. It cites Di Paolo et al. (2022) on structural coupling and treats the free energy principle as a formalism for the relational domain of autopoiesis, not for its constitution. Aubin’s viability theory, which the record holds through THEORY-134, is named as a complement to closure accounts.
Bearing on the record
On CLAIM-284 and CLAIM-288. The paper gives the thesis an adversary sharper than the counter the record filed. CLAIM-288 says a simulation whose operators maximize objectives the modeller fixed has borrowed its norms. The paper says the same of AI architectures, which “remain fundamentally dependent on externally specified objectives, training procedures, evaluation criteria, computational infrastructures, and energy flows”. It then goes further. Even a system that revises its own policy is, on this account, not closed, because what recurs is “information flows, policy updates, and state transitions, not the mutual production of the efficient causes”. So the route CLAIM-288 leaves open, operators whose objectives the organization produces and revises, would not satisfy these authors either, unless the organization also produced the substrate those processes run on. That is support for the counter and a reason to think it cannot be answered inside a simulation. Whether CLAIM-284 needs closure in this material sense is the record’s question, not the paper’s. Its “self” is organizational, and it does not say that the processes must produce their own efficient causes.
On CLAIM-116. The paper’s distinction between feedback and closure, and note 17’s report that for Maturana and Varela regulation “depends entirely on the observer”, agree with the disclaimer that regulation does not make an agent. But the paper goes the other way at the top. It follows Maturana in equating life with mind and calls closure the basis of “the continuity between life, mind, and organized agency”. CLAIM-116 lists organizationally closed systems among those that regulate without warranting a will. The two are not in contradiction, since mind and cognition in Maturana’s sense are not a will, but they are not saying the same thing.
On QUESTION-086 and CASE-076. The paper offers no intervention that tells a closed system from a matched heteronomous one. It says detection is the condition of a robust understanding and leaves it there. Note 35 is the case’s complication stated by these authors: whether closure has been achieved changes “when the analysis is shifted to descriptive levels other than those initially specified by the model”.
On THEORY-072. That account, following LIT-570, grants a candle flame closure and draws the line at differentiation. This paper’s efficient-cause closure would deny the flame closure in the first place: “Unlike transient self-organizing processes that persist only while external gradients are maintained, organizationally closed systems actively conserve and re-establish the constraints”. It does not discuss differentiation or analyse a flame. The two place the same line by different criteria, and neither has applied its criterion to both sides.
On THEORY-071 and CLAIM-194. Consistent with both: closure is a matter of production, not of coupling. It adds nothing that tells a produced boundary from an imposed one by observation.
Not ML practice. The AI section is a verdict about what current systems are, not an instruction for building them.
Limitations
- The non-computability thesis rests on Rosen, whose arguments the paper reports and does not assess. The halting-problem argument is weak as given. That a non-terminating process has no halting output is a consequence of the definition chosen, and the undecidability of halting for arbitrary programs does not show that a particular model’s closure cannot be computed. The authors make the conclusion conditional on the definition and call it provisional.
- The reading of the Church–Turing thesis as a thesis about natural systems is adopted, not argued.
- The protein-folding claim, that a folded shape “cannot be deduced or predicted” from sequence or landscape, is asserted. The paper’s own evidence is that predictors rely on crystallographic data.
- The AI verdict is drawn from the definition, not from an analysis of any system. No architecture is examined against the criteria of Table 3.
- The “necessary and sufficient criterion” of life is stated in the conclusion without argument beyond the genealogy.
- Much of the history rests on participants’ recollections.
Open questions
- Can closure of efficient causes be detected in a system whose substrate is supplied, as every computational system’s is? The paper’s criterion seems to say no by definition. A criterion that admits degrees of closure relative to a level would make it an empirical question.
- Does the computational-heteronomy verdict survive a system whose objectives, evaluation criteria and constraints are all produced by its own processes, with only energy and hardware supplied? Organisms also have their energy supplied.
- Which notion of closure does the record need for CLAIM-284: efficient causes, constraints, or operations? Table 3’s criteria are a way to state the choice.