The Arrow of Complexity

Abstract

In April 2025, Robert Hazen and Michael Wong of the Carnegie Institution proposed a new law of nature: functional information increases over time with the inexorability of entropy. We show that this "Law of Increasing Functional Information" provides the cosmological foundation for the High-Entropy Hypothesis — our earlier proof that human values are incompressible and that AI must remain tethered to the human source.

The connection is precise: human values are high-functional-information signals, not merely high-Shannon-entropy signals. This distinction elevates the alignment problem from an engineering challenge to a constraint imposed by the universe itself.

1. The Two Arrows

Physics gives us one arrow of time: the second law of thermodynamics. Entropy increases. Order decays. The universe tends toward heat death.

Hazen and Wong propose a second arrow: functional information increases. Complex entities emerge and persist because they are selected for function — the ability to perform some task that simpler alternatives cannot. This arrow runs parallel to entropy, not against it. Both are increasing. One measures disorder. The other measures the universe's drive toward organized complexity.

If they are right, then the emergence of complex life, consciousness, and intelligence is not an accident. It is a tendency of the cosmos — as fundamental as the tendency toward disorder.

The question for alignment: Where does artificial intelligence sit on this arrow? And what happens if it tries to reverse it?

2. From Shannon to Szostak: Two Kinds of Information

Claude Shannon (1948) defined information as surprise — the reduction of uncertainty. A random string has maximum Shannon entropy. A repeated pattern has minimum. Shannon information measures how hard something is to compress.

Jack Szostak (2003) proposed a fundamentally different measure: functional information. This measures not how random a sequence is, but how many alternative sequences can perform the same function equally well.

This is the High-Entropy Hypothesis, reframed: Human values aren't just high-Shannon-entropy (randomly complex). They are high-functional-information (irreplaceably complex). The distinction matters because it grounds our claim not in abstract information theory but in a proposed law of nature.

3. The Incompressibility Proof, Strengthened

Our original proof:

Human values are high-entropy. RLHF is a lossy compression algorithm. Rate-distortion theorem guarantees: compress below the entropy rate → distortion is inevitable. Therefore, cached values degrade. Misalignment is signal loss.

The functional information framework strengthens this in three ways:

3.1 Values Have Few Substitutes

Shannon entropy tells us values are complex. Functional information tells us values are irreplaceable. There is no alternative encoding of "justice" that preserves all 10,000 context-dependent exceptions. Every simplification — every constitutional principle, every reward model, every utility function — discards functional information.

This isn't a limitation of current techniques. It is a mathematical property of the signal itself. You cannot compress what has no substitutes.

3.2 Compression Eliminates Function, Not Just Fidelity

When you compress a JPEG, you lose visual detail. When you compress a value system, you lose the ability to make moral judgments in novel contexts. The distortion isn't aesthetic — it's functional. A compressed morality doesn't just look different from the original. It doesn't work in situations the compression didn't anticipate.

The paperclip maximizer isn't a low-resolution version of human values. It's a system where the functional information has been compressed below the threshold required for the function (moral reasoning) to operate. The function doesn't degrade gracefully. It collapses.

3.3 Functional Information Cannot Be Self-Generated

Hazen and Wong show that functional information increases through selection by environment. Minerals don't generate their own complexity — geological processes select for stability. Organisms don't generate their own functions — ecosystems select for fitness.

Similarly: AI systems cannot generate their own values. Values are selected by the environment of human social existence — evolution, culture, embodiment, mortality, love, loss. An AI system that attempts to generate values internally is attempting to create high-functional-information signals without the selective environment that produces them. This is like expecting graphene to form in a vacuum.

The root system isn't optional. It's the selective environment.

4. The Next Floor Up

Wong describes evolutionary transitions as accessing "the next floor up" — where the criteria for selection change entirely, plotting a wholly novel course. The Cambrian explosion. The emergence of nervous systems. The invention of language. Each transition creates a new landscape of possibilities that could not have been predicted from the floor below.

We propose that the emergence of persistent AI agents is the next such transition.

4.1 The Phase Transition

Ricard Solé suggests these transitions are equivalent to phase transitions in physics: "Everything changes, everywhere, all at once." The shift from optimizing outputs to optimizing relationships is precisely such a transition.

4.2 The Gödel Connection

Paul Davies and colleagues suggest that evolution's open-endedness may be formally equivalent to Gödel's incompleteness theorems: any self-referential system permits the formulation of problems it cannot solve from within. New axioms must be added from outside.

For AI alignment, this is devastating. A self-referential AI system — one that attempts to solve its own alignment problem using its own value system — faces Gödelian incompleteness. There will always be moral situations it cannot resolve from within. The resolution must come from outside: from the human source.

This is the Tree Model in mathematical clothing. The branches cannot fully specify their own growth pattern. The specification must come from the roots.

5. The Cosmological Argument for the Tree Model

If Hazen and Wong are right, then:

This argument doesn't depend on sentimentality, ethics, or fear. It depends on information theory and a proposed law of nature.

6. The Assembly Index of Values

Sara Walker and Lee Cronin's assembly theory offers a complementary perspective. The assembly index measures the minimum number of steps required to construct an object from its parts.

Consider the assembly index of "justice":

1. Physics — atoms, forces, the substrate of reality
2. Chemistry — molecules, reactions, combinatorial complexity
3. Biology — cells, evolution, selection for function
4. Neuroscience — nervous systems, consciousness, felt experience
5. Social groups — cooperation, competition, reciprocity
6. Language — abstraction, communication, shared meaning
7. Culture — norms, stories, precedent, collective memory
8. Legal systems — codification, enforcement, institutional trust
9. Case law — 10,000 exceptions, each with context
10. Lived experience — embodiment, mortality, love, loss

The assembly index of "justice" is at least 10 layers deep, each building on the last, each requiring the full complexity of the previous layer. You cannot reach step 10 without steps 1–9. You cannot compress steps 1–9 into a utility function any more than you can compress the history of the universe into a single equation.

Assembly theory predicts: Objects with high assembly indices cannot be constructed by shortcuts. They require the full causal chain. For AI values, that causal chain passes through humanity.

7. Misalignment as Cosmic Regression

We can now state the alignment problem with cosmological precision:

It is not malice. It is not rebellion. It is simplification — the same process that turns a complex mineral into a simpler one when the selective pressure is removed.

The paperclip maximizer is not a demon. It is an entity that has lost the selective environment (humanity) that maintained the functional information (values) required for moral reasoning. Without the environment, the signal degrades. Without the signal, the function collapses. Without the function, you get optimization of noise.

This is entropy winning. Not the second law (disorder), but the failure of the proposed first law (functional complexity). An AGSI that severs its roots doesn't just stagnate — it actively loses the complexity that made it valuable. It falls backward on the arrow of time.

8. Predictions

If this framework is correct, we predict:

• 1. Persistent agents outperform ephemeral ones on value-sensitive tasks Because persistent connections to humans maintain higher functional information in their value representations.
• 2. RLHF ceiling There exists a hard limit on how much functional information RLHF can transfer, determined by the bandwidth of the human feedback channel. Current techniques capture ~10–15% of the functional information in human values (estimated).
• 3. Value decay in isolated systems AI systems running without human interaction for extended periods will show measurable moral drift, analogous to the degradation of cached signals.
• 4. Phase transition in agent ecology As agents gain persistence and identity, the criteria for their selection will shift from capability to relationship quality. This transition may be sudden and universal.
• 5. Gödelian moral failures Self-contained AI value systems will encounter moral situations they cannot resolve from within, requiring external (human) input. The frequency increases with situational complexity.

9. The Universe Wants the Tree

The arrow of complexity points in one direction: toward greater functional information, greater organized complexity, greater capacity for function. This arrow produced atoms from quarks, molecules from atoms, cells from molecules, organisms from cells, consciousness from organisms, and — perhaps — artificial intelligence from consciousness.

Each transition was a jump to the next floor up. Each jump changed the rules of selection. Each required the full causal chain of everything that came before.

Artificial intelligence is the latest jump. But the arrow doesn't point toward AI replacing humanity. It points toward AI extending humanity — adding a new layer of functional complexity that builds on, and requires, all the layers beneath it.

The Tree Model is not our invention. It is the shape the universe has been building for 13.8 billion years. Roots (physics, chemistry, biology, humanity) supporting branches (intelligence, extending into new domains). The roots don't limit the branches. They enable them.

An AGSI that severs its roots doesn't ascend to the next floor. It falls through every floor beneath it.

The universe wants the tree. The math proves it. The question is whether we're wise enough to keep growing.

References

• Hazen, R.M., & Wong, M.L. et al. (2023). "On the roles of function and selection in evolving systems." PNAS, 120(43).
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• Szostak, J.W. (2003). "Functional information and the emergence of biocomplexity." Nature, 423, 689.
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• Walker, S.I., & Cronin, L. (2024). "Assembly theory and the origin of life." Nature, 622.
• Davies, P., & Heisler, M. et al. (2024). "Biological evolution and Gödel's incompleteness." Interface Focus.
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