1.3.1 Metabolic and Replicative Closure
1.3.1.1 Let’s turn our alphabet soup into an organic soup. Instead of As and Bs, we now have C, H, N, O, P, and S—carbon, hydrogen, nitrogen, oxygen, phosphorus, and sulfur—each with its own bonding capacity (valences). Instead of strings, we have combinations that form three-dimensional shapes.
We run the system forward. If each pulse is a second, every million iterations takes just under 12 days. Symmetry breaks. Chains form. Structures diversify. We’re doing chemistry. As we advance through time, complexity increases.
Consider protein interactions as one example. Beyond the basic three dimensions, proteins can interact across six or more dimensions—including electric charge, dipole moment, and hydrophobicity (Kauffman 2000, 61). We now have not just descriptive complexity but interactional complexity (Wimsatt 2007, 185–186). Our combinatorial space explodes with possibilities.
Shapes floating in our soup now influence which future shapes are created. A curly one might bring two strings close together, making a combination more likely. Another shape might shield a segment from negation. A loop might bind and hold other strings in place. The shapes become tiny causal mechanisms written in the medium of our soup, increasing the likelihood of specific outcomes.
1.3.1.2 The leap from chemistry to life seems vast, but even simple chemistries can bootstrap into an autocatalytic set—a network of reactions where the catalysts needed to sustain the network are themselves produced by it. The set doesn’t create matter. Instead, it pulls surrounding feedstock (“food,” or in our analogy, bits) through a closed web of catalyzed transformations (Lloyd 2007, 203–204).
A key idea we’ll return to repeatedly is closure. Here, closure marks the threshold where a network stops being a collection of one-off reactions and becomes a self-sustaining organization (Kauffman 2000, 61–62).
Kauffman identifies this as central to life’s emergence: “The radical new view of life that I adhere to is that life is based on collectively autocatalytic sets of molecules, not on template replication per se” (Kauffman 2000, 32).
Autocatalytic closure gives you something metabolic—a self-maintaining organization that persists by pulling feedstock through a closed web of transformations. Call this metabolic closure.
“Any life must eat before it can multiply” (Gleizer 2023, 153).
1.3.1.3 But persistence alone isn’t hereditary. Negation explains why most configurations fail—it doesn’t explain how constraints become inheritable. To ratchet complexity up, the constraints that stabilize organization must also become copyable with enough fidelity, as we see in DNA.
Claiming that replication can emerge from our soup of shapes again seems far-fetched, but we’ve already seen how chemical processes bootstrap themselves (metabolic closure). Something remarkably similar happens when we simulate a comparable system computationally.
Consider a simple programming language called Brainfuck, generally referred to as BFF, that consists of minimal commands—think of our little causal shapes. Create an environment (a combinatorial space) that allows combinations and negations of commands (i.e., bits) to happen. Pulse the clock forward millions of times. Replicators emerge: programs that can generate their own code (Agüera y Arcas et al. 2024, 5–9). Earlier experiments with Lisp—a language that can self-reference—showed similar phenomena (Kauffman 2000, 121–122). In the BFF experiments, there’s a long period where nothing happens—and then boom: the soup fills with replicators. The kettle whistles. The ice freezes. We’ve undergone a phase transition. We’ve obtained a form of closure again—we’ll call this replicative closure.
Since things that can replicate crowd out those that cannot, replicators create dynamic stability (Agüera y Arcas and Manyika 2025, 57). As we’d expect, once this happens, the tape is full of regularities and can be compressed to a fraction of its original size (Agüera y Arcas 2025, 85).
“Life is the universe making a memory” (Walker 2024, 105).
Lucretius already had the intuition: “To atoms through the fathomless void but, rather, they are driven by sundry restless motions. After colliding, some will leap” (Lucretius 2007, 39).
Life and complexity ratchet up.
1.3.1.4 These experiments tie directly back to John von Neumann’s ideas of a universal constructor and self-replication. Neumann demonstrated that it was possible to generate something as complex as yourself—essentially bootstrapping yourself into existence—with a universal constructor implemented on a Turing machine (Von Neumann 1966). And DNA works more or less as Neumann suggested.
It’s not hard to imagine our system as DNA. The bits become A, T, C, G, and As can only combine with Ts, and Cs can only combine with Gs—this gives us our ladder rungs. These combine and twist to form the staircase that life walks up—DNA (Watson and Crick 1953).
1.3.1.5 Let’s now tighten our definition of closure. Closure occurs when a set of bits becomes self-stabilizing: through feedback and mutual reinforcement, they preserve their organization against noise and persist as a unit over time—a constraint loop that stabilizes a pattern. It’s how sets of bits become a bit—a many-to-one stabilization where parallel interactions collapse into a new composable bit.
Nucleotides become the bits for DNA; amino acids become the bits for proteins.
Closure is not merely the appearance of a macro-pattern. Emergence names the fact that higher-level patterns arise from lower-level interactions; closure names the condition under which such a pattern becomes robust, persistent, and composable—a structure that survives long enough, and coheres tightly enough, to function as a bit itself.
As a guiding principle, closure requires an envelope to localize these dynamics and feedback in the form of negation—which restricts certain trajectories—so that most variations fail while some persist and become self-reinforcing (able to re-enter the future as the same pattern). Closure occurs when stabilization outruns noise: the pattern converges faster than noise can dissolve it.
Nagel gestures at this kind of non-teleological constraint when he writes that “it would have to be restrictive in what it makes likely, but without depending on intentions or motives” (Nagel 2012, 93).
One way to test for closure is to look for a high number of compressible bits (i.e., high copy number and regularity within their tapes).
Life requires at least two types of closures—metabolic (the creation of an envelope in its own regard) and replicative (the propagation of that envelope through time with inheritable characteristics).
The challenge is that once we pass through these phase transitions and obtain these closures, many of the shapes that got us there may no longer exist, or may exist in very small quantities. The ladder gets thrown away after the shapes have climbed up, and we’re left looking for prints in the dust.
1.3.2 Computation and Closure
1.3.2.1 There’s a tendency to see everything as a computer and to assume this explains complexity. Seth Lloyd even frames chemistry as a form of computation: “Chemical reactions can readily produce AND, NOT, and COPY operations. By adding more chemicals to the set, such logic operations combine to produce a set of reactions corresponding to any desired logic circuit. Thus, chemical reactions are computationally universal” (Lloyd 2007, 203).
But computation, or at least Turing completeness, isn’t the whole story.
As an example, simulations with programming languages like BFF and Lisp produce replicators, but other languages, like SUBLEQ, provide a counterexample: replicators are possible in principle, yet the transition is never observed, despite the language also being Turing complete (Agüera y Arcas et al. 2024, 16–17).
Recall our earlier discussion: searching some combinatorial spaces is impossible, even in the lifetime of the universe. Achieving closure in finite time requires not only a supply of bits, but bits that can combine in a particular way.
BFF likely obtains closure while SUBLEQ cannot—it finds the stable combinations needed to create replicators within the search threshold. The BFF paper supports this idea: replicators arise through composition of partial motifs (symbiosis) rather than through a single lucky global mutation: “self-replicators arise mostly due to self-modification and interaction among different programs and are not simply due to random initialization and random mutations” (Agüera y Arcas et al. 2024, 19).
Similarly, if the origin of life depended on something like a specific protein forming through random combination, it would almost certainly never happen (1.2.1.5).
1.3.2.2 BFF has more bits than SUBLEQ as we’ve defined it, and thus a larger combinatorial space to explore—yet it achieves closure. Notably, SUBLEQ has only one arithmetic operator (subtraction), whereas BFF has two (add one, subtract one). This suggests a fundamental difference: the paired add-and-subtract operators versus the single subtract operator. BFF finds composable parts early and combines them to reach replication. Does the presence of certain operators lead to closure? If so, which ones, and why?
More generally, it would seem that closure emerges only in systems whose representational primitives make the search for mutually reinforcing structures tractable. It’s striking that addition and subtraction work—they mirror our earlier pattern of variation through combination and negation to create constraints. Modern computation rests on similar operators (AND and NOT), but universality isn’t what matters here. What matters is how these operators enable reusable building blocks.
Universality guarantees encodability, not discoverability. The primitives help reshape the local neighborhood of mutations and therefore the probability of bootstrapping motifs.
This implies that computation is part of, but not the entire story. Both languages are Turing complete, yet closure and replicators emerge in one and not the other. Turing completeness tells us that a system can, in principle, realize arbitrary computation. But that is a very different claim from saying it will discover stable, reusable structures early enough for complexity to build.
So whether a system is Turing complete does not tell us whether it will reach closure, nor whether closure will be reachable within any realistic search process. Turing completeness guarantees universality of computation, not the emergence of complexity; the real question is when closure becomes reachable.
1.3.3 Harmonics and Hierarchies
1.3.3.1 This discussion reveals another feature of complex systems: the emergence of hierarchical structure.
Imagine two watchmakers, Tick and Tock. Each builds a watch in 100 steps. If Tick takes a break at any point, his watch falls apart. Tock, however, can pause every 10 steps without losing her progress. In a chaotic world that constantly interrupts them, who makes more watches? Tock, obviously. She builds her watch from intermediate subassemblies that survive interruption. In a noisy world, hierarchy itself becomes a form of robustness (Simon 1962, 470–71).
This directly mirrors our idea of closure. Tock has found a way to create subassemblies—she obtains closure—and can then assemble the subassemblies into higher-order structures. Tick’s watch always falls apart. Tock races ahead while Tick is stuck searching. Tock rhymes with BFF; Tick rhymes with SUBLEQ.
Small, persistent collections of shapes that have reached closure can combine into larger, more complex shapes. Simon showed that in some cases complexity can arise only through this reduction of search space—the combinatorial space is simply too large for the things we observe to have arisen by chance (Simon 1962, 496).
At each closure, a phase change occurs. Collections of shapes become bits themselves, setting the stage for the process to start over. Small molecules yield nucleotides and amino acids, nucleotides become the bits of nucleic acids, and amino acids become the bits of proteins. And up and down we go from there—systems composed of interrelated subsystems, each hierarchical in structure, until we reach some lowest or uppermost level (Simon 1962, 496).
Similarly, Walker and Cronin have shown that only life produces molecules above an assembly index of 15 steps (molecules made by random processes cannot get beyond this threshold). Above this threshold, the physics of evolution must take over to explain the formation of more complex molecules (Walker and Cronin 2023).
1.3.3.2 We can think of these levels as complexity harmonics: each new collection of pieces lies within the search threshold and can be explored and stabilized. By composing these pieces, the system reaches regions of combinatorial space that would be astronomically improbable to find in a single leap.
What we call harmonics, William Wimsatt calls levels with part–whole relations, “in which wholes at one level function as parts at the next (and at all higher levels)” (Wimsatt 2007, 201–205).
Harmonics are what closure looks like when it repeats. They’re the accumulation of survivable subassemblies in a world that won’t let you finish in one uninterrupted run.
What we’ve been calling phase changes that create new harmonics can be thought of as what biologists characterize as major evolutionary transitions. These transitions share three features: previously independent replicators become parts of a larger whole; there is division of labor, differentiation, or specialization; and they involve new ways of storing and transmitting information (Szathmáry and Smith 1995, 227).
1.3.3.3 Biologists often describe scaling laws—the relationships between these harmonics. As size increases, vibrations slow, time scales lengthen, and movement slows (Wimsatt 2007, 216–217). Lower harmonics are noisy; upper ones are quieter.
Freeman Dyson captures this dynamic well: “For life to originate spontaneously, it was essential to have an ordered state with a high error rate, but when life was once established, the whole course of evolution was toward more specialized structures with lower tolerance for errors” (Dyson 1988, 85–86).
This process is again tied to noise. Murray Gell-Mann explains: “Noise gives the system a chance to escape from a shallow depression and seek out a deeper one nearby, and to perform that operation over and over, until the bottom of a really deep basin is reached. The noise must be such that the amplitude of the chance excursions are not too great, however. Otherwise, there would be too much interference with the process of descent, and the system would not remain in a deep basin even after finding one” (Gell-Mann 1994, 267).
We can also consider at which level selection occurs (i.e., our combinations and negations). Dawkins argued for selection at the gene level (Dawkins 2016), Darwin at the phenotype level (Darwin 2012), and E. O. Wilson at the group or social level (Wilson 2012).
1.3.4 Life and Complexity
1.3.4.1 We might now define life as a complex system in which sets of shapes obtain both metabolic and replicative closure, enabling phase transitions into reusable parts that create the harmonic on which the next level is constructed. Life oscillates upward, establishing new harmonics along the way.
Defined this way, life emerges from our original complex system. To be alive is to be complex, but complexity doesn’t guarantee life. And while we’ve focused mainly on biology, this definition isn’t necessarily substrate-dependent. We’ve also checked off robustness and nested structure on our complexity features list (1.1.4.1).
This brings us close to some non-standard definitions of life that others have proposed. Walker writes: “Life is the high-dimensional combinatory space of what is possible for our universe to build that can be selected to exist as finite, distinguishable physical objects” (Walker 2024, 141). Blaise Agüera y Arcas offers: “Life is a self-modifying computronium arising from selection for dynamic stability; it evolves through the symbolic composition of similar dynamically stable entities” (Agüera y Arcas 2025, 89).
1.3.4.2 A living system is a collection of bits (molecules, nucleotides, amino acids) driven through a vast combinatorial space by noise but held together by an envelope (a boundary that keeps reactions from diffusing into the world). It stays far from equilibrium by tapping free energy (a gradient it can metabolize), and it earns persistence by repeated negation: configurations that don’t contribute to self-maintenance collapse, while those that reinforce the system recur.
Life is an envelope that persists against noise.
“Ever tried. Ever failed. No matter. Try again. Fail again. Fail better” (Beckett 1996, 77).
1.3.4.3 This framing aligns well with theories on the origin of life. Hot ocean vents or salty tide pools are noisy envelopes with free energy that can make metabolic closure reachable within the search threshold. Supporting this idea of life arising from a noisy envelope, recent research has shown that increases in complexity of subsurface microbiomes are related to seismic-induced generation of chemical disequilibria (Boyd et al. 2025)—add noise and you get more complex microbes.
RNA-world scenarios fit our earlier discussion of replication (replicative closure). And the importance of combination as a source of complexity, like we saw in the BFF example—rather than random search alone—appears in the creation of eukaryotic cells from two prokaryotes through symbiosis, for example (Sagan 1967).
1.3.4.4 Over time robustness develops in response to the environment, and in that sense we might call our sets of shapes adaptive. Each new shape is an adaptive prediction, and through this process they accumulate what Daniel Dennett called “competence without comprehension” (Dennett 2009, 51).
Think of this competence as the accumulation of resilient information—that is, information that is capable of keeping itself instantiated in physical systems, thereby allowing adaptations to persist (Marletto 2022, 13).
But evolution pays a brutal sampling cost: early on, each prediction is a life, and most negations are fatal. Thus, variation is slow because it must accumulate across lifetimes.
“Life is that which predicts itself into continued existence” (Agüera y Arcas and Manyika 2025, 61).
1.3.4.5 At this point, I feel obligated to apologize to the biologists—I’m sure there’s a much more elegant way to say all this: fitness landscapes, functional information, and all that. But we’ve got places to go, and we haven’t even started talking (language). And after all, if there were an easier way to explain this, we’d have an answer to the origin of life.
The important takeaway is that any system that evolves complexity—languages, proteins, brains—must structure search through levels that reuse patterns rather than enumerate possibilities. Those levels arise when sets of shapes achieve closure and become parts in themselves: new bits for the next harmonic. Thus brains have regions, books have chapters, and biology has cells—carving out a niche of improbability one level at a time by breaking the problem into smaller pieces.