A theory study · conclusion first 中文版

Self-Reference and the Arrow of Complexity

Why the universe seems to grow more complex, why "entropy increases" does not explain it, and what would.

In plain words. The universe began as a nearly uniform hot soup, yet today it contains cells, brains, and computers. A popular explanation invokes "entropy" — but entropy only measures how spread-out and shuffled energy is, and nothing in physics says shuffled things must organize themselves. The best current answer has four parts: a steady flow of usable energy (like sunlight) makes organization possible; things that carry a copyable recipe of themselves can keep each improvement instead of losing it; competition decides which improvements survive; and living things keep changing each other's world, so the game never settles. Complexity grows only where all four meet — and even then only at the leading edge: most life, then and now, is still microbes.

This site presents a study I wrote to test some questions of my own: if an agent and its environment are really one physical system, where does the apparent drive toward complexity come from? Is "self-reference" — a thing containing and using its own recipe — the secret ingredient? Everything below is conclusion-first; the full reference list, with each claim tagged as established, contested, or speculative, is at the end.

Core conclusions

  1. There is no law that forces complexity to grow. We audited every serious candidate (table below). The general proof of the "maximum entropy production" principle was refuted[14]; the free-energy principle has published technical counterexamples[22] and contains no term that makes anything grow; assembly theory measures complexity but does not predict its rise, and its stronger claims have two published critiques[19],[20]. In a sealed-off system, complexity rises, peaks, and then decays[9].
  2. What exists instead is a four-part conditional engine. Permission: a sustained flow of free energy (energy still capable of doing useful work), ultimately traceable to the extremely ordered starting state of the universe[68],[69]. Ratchet: a copyable, interpretable self-description — the architecture of DNA, anticipated on paper before its discovery[40] — which lets improvements be inherited rather than lost. Rectifier: natural selection. Pump: organisms constantly changing each other's environment, so the game never reaches a final level[30],[31]. Remove any one part and growth stops.
  3. Self-reference is (probably) necessary but demonstrably not sufficient. Mathematics guarantees the ladder of complexity has no top[35],[37],[38], and strong arguments say accurate heredity is impossible without a copied-and-interpreted self-description[40],[59],[32]. But digital-evolution experiments show populations of fully self-referential programs still stagnate in closed worlds[44],[48]: the drive is not inside self-reference; it lives in the combination of self-referencers with a driven, ever-shifting environment.
  4. Self-reference cannot exist at the deepest microscopic level. A basic theorem of quantum physics (no-cloning: an unknown quantum state cannot be copied[56]) forbids using raw quantum states as a heritable recipe. Heritable self-description must live in robust, effectively classical records — the kind the environment naturally duplicates[57],[58]. So the "recipe-carrying" kind of self-reference is necessarily an emergent, higher-level phenomenon. This is the closest thing to a theorem in the whole study.

One sentence: entropy is neither the enemy nor the engine; the engine is a ratchet that spends flowing energy to buy recorded depth, self-reference is the ratchet's necessary architecture rather than its motor, and the motor is the universe's low-entropy start plus the restless environment self-referencers make for each other.

Why "entropy increases" explains nothing here

There are three different things called entropy, and the popular argument blurs them. Thermodynamic entropy counts how many microscopic arrangements look the same from outside — it depends on how you decide to blur the details (the coarse-graining). Shannon entropy[1] measures the average unpredictability of a source of messages. Algorithmic entropy (Kolmogorov complexity[2]) is the length of the shortest computer program that reproduces one specific object. They are bridged by a physical fact: erasing one bit of information has a minimum heat cost[3],[4],[5] — information processing is physical.

Three precise reasons "entropy = disorder" fails as an explanation:

  1. "Disorder" is undefined. The same microscopic state has different entropies under different coarse-grainings; an entropy claim that does not specify one is incomplete.
  2. Order can form while total entropy rises. Crystals grow and proteins fold spontaneously; what the second law constrains is system-plus-surroundings, not visual tidiness. Living things are not entropy's opponents — they are channels that speed up entropy production in their surroundings while maintaining deep structure inside[10].
  3. Complexity is not low entropy. A perfect crystal (lowest entropy) and a stirred-up gas (highest entropy) are both simple. Interesting structure lives in between, which is why in a closed system complexity first rises and then falls as mixing completes[9]. The right measures of "interesting" reward stored history, not randomness: logical depth (how long the shortest recipe takes to run[6]) and the assembly index (the minimum number of construction steps, reusing earlier parts[18]). Pure noise maximizes Kolmogorov complexity and minimizes both.

Earth escapes the closed-system fate by not being closed: it absorbs concentrated sunlight and radiates the same energy back as diluted heat, exporting entropy. That flux — the permission — traces back to the universe's extraordinarily low-entropy beginning[68],[69]. The arrow of complexity and the arrow of time have the same source.

The theory audit

Every major candidate for a "complexity drive," what it actually predicts, and where it stands. (A dissipative structure is a pattern, like a whirlpool or a candle flame, that exists only while energy flows through it.)

TheoryPredicts a complexity trend?Conditions / notesStatus
Dissipative structures[11]No — shows order can appear under drivingFar from equilibrium; no memory, vanishes when driving stopsestablished (mechanism)
Minimum entropy production[11]NoValid only near equilibriumestablished (narrow)
Maximum entropy production[12],[13],[15]Claimed; unprovenGeneral derivation refuted[14]contested
Dissipative adaptation[16],[17]Conditional tendency toward work-absorbing states — not the same as complex onesDriven many-body systems; the 2013 dissipation bound is rigorous, the mechanism is a conjecturebound established · mechanism contested
Kolmogorov complexity[2]Wrong measure — noise scores highest—established (as a measure)
Logical depth / effective complexity[6],[7],[8]Right flavor of measure; no dynamicsNo theorem makes depth growestablished (measures)
Coffee automaton[9]Opposite: in closed systems complexity rises then fallsClosed + interacting particles; preprintpreliminary
Assembly theory[18]Quantifies the trend; does not predict it"High assembly + many copies ⇒ a selective history" as an empirical criterion; critiques[19],[20] say the index behaves like known compression measures and "selection" is used too broadlycontested
Free-energy principle[21]No — describes persistence, not growth ("dark-room problem"[24])Technical counterexamples to early formulations[22], partially repaired[23]contested
Major evolutionary transitions[25],[26]No — retrospective classificationRecurring motif: revolutions in information storage and transmission; explicitly contingentestablished (biology); not a law
Energy rate density[33],[34]Describes an empirical rise (energy flow per gram, from galaxies to brains to society)Sampled along the lineage leading to us; high power ≠ complexitycontested (as a law)
Passive trend / ZFEL[27],[28]Yes — but as diffusion, not driveA lower wall of minimal viable complexity makes the record-holder drift upward with no force at all; any real "drive" theory must first beat this null modelnull model established

The conjecture: does self-reference itself drive complexity?

My conjecture was: once a self-referential system appears, it tends toward ever-greater complexity — and possibly this is a pure mathematics question. Here self-reference means something specific and strong: a system that contains its own recipe and uses it twice — once read as instructions (to build itself) and once photocopied blindly (to pass on)[40]. This is exactly how DNA works, and it was predicted on paper five years before the double helix was found.

Split the conjecture into three strengths, which get three different verdicts:

Steelman (the strongest defensible version)

In a persistently energized world where self-replicators share resources, unbounded growth of the complexity frontier turns from impossible into statistically expected: the self-description makes gains heritable (ratchet); the ladder has no top (mathematics); and — the key move — populations of self-describers make their own weather. Every published recipe is a resource others can exploit: in the classic Tierra experiment, parasites arose precisely by hijacking other programs' copy routines[44]. That launches arms races[30] and environment-reshaping[31] which keep reopening opportunity — self-reference nearly supplies its own pump.

Attack (why the literal conjecture fails)

  1. The quine problem. The canonical mathematical product of self-reference is a program that prints itself, unchanged[36]. Self-reference guarantees perfect self-copying — self-preservation, not self-transcendence. Nothing in the fixed-point theorems pushes upward.
  2. The stagnation counterexample. In Tierra and Avida every individual is fully self-referential, yet in closed worlds novelty runs dry and the systems saturate[44],[48],[49]. Complex features evolve when the environment supplies stepping-stone rewards — and not otherwise[45]. The drive is in the environment's structure.
  3. Complexity also goes down. Selection frequently prunes it (parasites shed genes), and much observed complexity may originate in genetic drift — a non-adaptive process — rather than any upward push[29].
  4. The measure trap. Until you fix a complexity measure and a coarse-graining, the conjecture is unfalsifiable; once fixed, different measures can disagree.
  5. Undecidability. For world-rules rich enough to compute anything, "does complexity grow without bound here?" is undecidable in general[39]. So "a pure mathematical question" is half right: provable on restricted classes, undecidable universally.
  6. The null model. Part of any observed rise is explainable as diffusion away from a lower wall[27],[28], with no appeal to self-reference at all.

Verdict: possibility holds; necessity is the most promising target for an actual theorem; sufficiency is false in closed worlds; the steelman — self-referential populations under sustained energy flow nearly pump themselves — is the genuinely open question. Even Tierra's arms race eventually stopped, so "nearly" is doing real work.

A companion toy experiment (illustrative only)

Setup. A small artificial-chemistry simulation I ran (unpublished; 8 random seeds): two otherwise identical systems, except that in one, copy events read segments from the unit's own genome (self-reference), while in the control, copied segments come from an external random stream.

Result. In 8 of 8 seeds, the self-referential system scored higher on assembly-index and logical-depth proxies and lower on Shannon and block entropy. More structure and history, less randomness — a clean, hands-on illustration that entropy and complexity are different quantities that can move in opposite directions.

Honest critique: the result is close to true-by-construction. Self-copying mechanically manufactures repeats; repeats directly lower entropy scores and directly raise reuse-counting assembly scores. Both differences are arithmetic consequences of the copy mechanism, so the experiment demonstrates reuse, not open-ended complexity growth. It also reports a snapshot difference between two systems, whereas the real conjecture is about a trend over time; and it has only one of the four engine components (no energy budget, no selection, no interaction between lineages).

What a non-circular test needs: novelty measures that do not reward repetition (first-occurrence rates against an expanding dictionary; statistical complexity[8]; decompression-time depth proxies); a structured but non-self-referential control (copying from other units or a fixed template library — the actual contrast at issue), length- and composition-matched arms with shuffled baselines; long runs testing whether growth rates stay positive or saturate; and the missing components — resource costs, selection, and shared-resource interaction.

Two falsifiable conjectures

Conjecture A — the self-referential ratchet

In a driven stochastic system over an unbounded space of buildable configurations (think: a simulated chemistry with energy flowing through), satisfying (i) sustained free-energy flux, (ii) self-description copying above the heredity error threshold[32], (iii) no ceiling on constructible size, and (iv) multiple replicating lineages sharing finite resources — the population's maximum assembly index grows without bound, with probability one. And: deleting (ii) (no template copying) or (iv) (a lone lineage in a fixed world) bounds it forever.

Falsify it by building a simulation with (i)–(iv) that provably plateaus, or one that grows without bound with (ii) removed. Both attacks are runnable on existing digital-evolution platforms[46].

Conjecture B — no microscopic self-reference

Any self-replicator with heritable variation must store its description in effectively classical records: robust states that the environment redundantly copies[57],[58]. Equivalently, no evolvable replicator can use a coherent quantum state as its genetic tape. For unknown states this follows from the no-cloning theorem[56]; for known-but-coherent tapes it is a genuine conjecture about decoherence rates versus the error threshold.

Falsify it by exhibiting — even as a theoretical model — a replicator whose heritable medium is a coherent quantum state maintained across generations.

Three concrete next steps

  1. A necessity theorem. In a minimal model class — driven autocatalytic chemistries[53],[54] or noisy cellular automata[55] — state and try to prove: without a copied-and-interpreted self-description, the maximum assembly index stays bounded almost surely. Necessity is more theorem-shaped than sufficiency, and a counterexample would itself be a major discovery.
  2. A factorial simulation. On an Avida-style platform[46]: {self-description copying vs. external copier} × {fixed vs. coevolving environment} × {bounded vs. extensible genome}, ≥20 long runs per cell, with preregistered saturation criteria. Prediction: only the self-description × coevolving × extensible cell keeps growing. Sustained growth anywhere else falsifies part of the steelman.
  3. Measure engineering. An open-source suite — assembly index, compression-based depth proxies, statistical complexity — calibrated for agreement on molecular data and digital-organism histories. Until the measures agree, the conjectures are not cleanly falsifiable.

Open questions by field

These are the questions I keep returning to — phrased for readers coming from each field.

Chemistry

Biology

Physics / information

Computer science / AI

References

Tags grade the claim each source supports here: established = broadly accepted; contested = serious published disagreement; speculative = programmatic or preliminary. Bibliographic details of the load-bearing contested items were re-verified against the publishers' pages in September 2026; entries marked "not independently re-verified" are given from standard knowledge and flagged honestly. No citation here is invented.

  1. Shannon, C. E. (1948). "A Mathematical Theory of Communication." Bell System Technical Journal 27: 379–423, 623–656. established
  2. Kolmogorov, A. N. (1965). "Three approaches to the quantitative definition of information." Problems of Information Transmission 1(1): 3–11. established
  3. Landauer, R. (1961). "Irreversibility and heat generation in the computing process." IBM Journal of Research and Development 5: 183–191. established
  4. Bérut, A., et al. (2012). "Experimental verification of Landauer's principle linking information and thermodynamics." Nature 483: 187–189. established
  5. Bennett, C. H. (1982). "The thermodynamics of computation — a review." International Journal of Theoretical Physics 21: 905–940. established
  6. Bennett, C. H. (1988). "Logical depth and physical complexity." In R. Herken (ed.), The Universal Turing Machine: A Half-Century Survey. Oxford University Press, 227–257. established
  7. Gell-Mann, M. & Lloyd, S. (1996). "Information measures, effective complexity, and total information." Complexity 2(1): 44–52. established
  8. Crutchfield, J. P. & Young, K. (1989). "Inferring statistical complexity." Physical Review Letters 63: 105–108. established
  9. Aaronson, S., Carroll, S. M. & Ouellette, L. (2014). "Quantifying the Rise and Fall of Complexity in Closed Systems: The Coffee Automaton." arXiv:1405.6903 (preprint). preliminary
  10. Schrödinger, E. (1944). What is Life? Cambridge University Press. established
  11. Nicolis, G. & Prigogine, I. (1977). Self-Organization in Nonequilibrium Systems. Wiley. established (mechanism; minimum entropy production valid only near equilibrium)
  12. Dewar, R. C. (2003). "Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states." J. Phys. A 36: 631–641. contested
  13. Dewar, R. C. (2005). "Maximum entropy production and the fluctuation theorem." J. Phys. A 38: L371–L381. contested
  14. Grinstein, G. & Linsker, R. (2007). "Comments on a derivation and application of the 'maximum entropy production' principle." J. Phys. A: Math. Theor. 40: 9717–9720. established (the refutation)
  15. Martyushev, L. M. & Seleznev, V. D. (2006). "Maximum entropy production principle in physics, chemistry and biology." Physics Reports 426: 1–45. established (as a review of a contested principle)
  16. England, J. L. (2013). "Statistical physics of self-replication." Journal of Chemical Physics 139: 121923. established (the bound)
  17. England, J. L. (2015). "Dissipative adaptation in driven self-assembly." Nature Nanotechnology 10: 919–923. contested (as a mechanism for life)
  18. Sharma, A., Czégel, D., Lachmann, M., Kempes, C. P., Walker, S. I. & Cronin, L. (2023). "Assembly theory explains and quantifies selection and evolution." Nature 622: 321–328. contested
  19. Jaeger, J. (2024). "Assembly Theory: What It Does and What It Does Not Do." Journal of Molecular Evolution 92: 87–92. established (as a published critique)
  20. Zenil, H. and collaborators (2024). "On the salient limitations of the methods of assembly theory and their classification of molecular biosignatures." npj Systems Biology and Applications. (Full author list not re-verified.) established (as a published critique)
  21. Friston, K. (2010). "The free-energy principle: a unified brain theory?" Nature Reviews Neuroscience 11: 127–138. contested (as a universal principle)
  22. Biehl, M., Pollock, F. A. & Kanai, R. (2021). "A Technical Critique of Some Parts of the Free Energy Principle." Entropy 23(3): 293. established (as a published critique)
  23. Friston, K., Da Costa, L. & Parr, T. (2021). "Some Interesting Observations on the Free Energy Principle." Entropy 23(8). response
  24. Friston, K., Thornton, C. & Clark, A. (2012). "Free-energy minimization and the dark-room problem." Frontiers in Psychology 3: 130. established
  25. Maynard Smith, J. & Szathmáry, E. (1995). The Major Transitions in Evolution. Oxford University Press. established (explicitly not a law of progress)
  26. Szathmáry, E. (2015). "Toward major evolutionary transitions theory 2.0." PNAS 112: 10104–10111. established
  27. Gould, S. J. (1996). Full House. Harmony Books. established (the passive-trend null model)
  28. McShea, D. W. & Brandon, R. N. (2010). Biology's First Law. University of Chicago Press. contested (as a "law")
  29. Lynch, M. (2007). "The frailty of adaptive hypotheses for the origins of organismal complexity." PNAS 104 (suppl 1): 8597–8604. established
  30. Van Valen, L. (1973). "A new evolutionary law." Evolutionary Theory 1: 1–30. established (Red Queen)
  31. Odling-Smee, F. J., Laland, K. N. & Feldman, M. W. (2003). Niche Construction. Princeton University Press. established
  32. Eigen, M. (1971). "Selforganization of matter and the evolution of biological macromolecules." Naturwissenschaften 58: 465–523. established (error threshold)
  33. Chaisson, E. J. (2001). Cosmic Evolution: The Rise of Complexity in Nature. Harvard University Press. contested (as a complexity metric)
  34. Chaisson, E. J. (2011). "Energy rate density as a complexity metric and evolutionary driver." Complexity 16(3): 27–40. contested
  35. Gödel, K. (1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I." Monatshefte für Mathematik und Physik 38: 173–198. established
  36. Kleene, S. C. (1938). "On notation for ordinal numbers." Journal of Symbolic Logic 3: 150–155 (the recursion theorem). established
  37. Turing, A. M. (1939). "Systems of logic based on ordinals." Proc. London Mathematical Society s2-45: 161–228. established
  38. Feferman, S. (1962). "Transfinite recursive progressions of axiomatic theories." Journal of Symbolic Logic 27: 259–316. established
  39. Rice, H. G. (1953). "Classes of recursively enumerable sets and their decision problems." Transactions of the AMS 74: 358–366. established
  40. von Neumann, J. (1966). Theory of Self-Reproducing Automata (A. W. Burks, ed.). University of Illinois Press. established
  41. Hofstadter, D. R. (1979). Gödel, Escher, Bach; and (2007) I Am a Strange Loop. Basic Books. established (conceptual, not theorems)
  42. Maturana, H. R. & Varela, F. J. (1980). Autopoiesis and Cognition. D. Reidel. established (a self-maintenance concept; claims no complexity growth)
  43. Rosen, R. (1991). Life Itself. Columbia University Press. contested (its non-computability claim is disputed, e.g. by Chu & Ho 2006, Artificial Life 12(1) — not independently re-verified)
  44. Ray, T. S. (1991). "An approach to the synthesis of life." In Artificial Life II. Addison-Wesley, 371–408 (Tierra). established
  45. Lenski, R. E., Ofria, C., Pennock, R. T. & Adami, C. (2003). "The evolutionary origin of complex features." Nature 423: 139–144. established
  46. Ofria, C. & Wilke, C. O. (2004). "Avida: A software platform for research in computational evolutionary biology." Artificial Life 10: 191–229. established
  47. Adami, C., Ofria, C. & Collier, T. C. (2000). "Evolution of biological complexity." PNAS 97: 4463–4468. established
  48. Taylor, T., et al. (2016). "Open-ended evolution: Perspectives from the OEE workshop in York." Artificial Life 22(3): 408–423. established (field consensus: no artificial system yet shows biosphere-style open-endedness)
  49. Packard, N., et al. (2019). "An overview of open-ended evolution." Artificial Life 25(2): 93–103. established
  50. Bedau, M. A., Snyder, E. & Packard, N. H. (1998). "A classification of long-term evolutionary dynamics." In Artificial Life VI. MIT Press. established
  51. Soros, L. B. & Stanley, K. O. (2014). "Identifying necessary conditions for open-ended evolution through the artificial life world of Chromaria." ALIFE 14. MIT Press. established (as a proposal)
  52. Fontana, W. & Buss, L. W. (1994). "What would be conserved if 'the tape were played twice'?" PNAS 91: 757–761. established
  53. Kauffman, S. A. (1986). "Autocatalytic sets of proteins." Journal of Theoretical Biology 119: 1–24. established
  54. Hordijk, W. & Steel, M. (2004). "Detecting autocatalytic, self-sustaining sets in chemical reaction systems." Journal of Theoretical Biology 227: 451–461. established
  55. Gács, P. (2001). "Reliable cellular automata with self-organization." Journal of Statistical Physics 103: 45–267. established
  56. Wootters, W. K. & Zurek, W. H. (1982). "A single quantum cannot be cloned." Nature 299: 802–803; independently Dieks, D. (1982), Physics Letters A 92: 271–272. established
  57. Zurek, W. H. (2003). "Decoherence, einselection, and the quantum origins of the classical." Reviews of Modern Physics 75: 715–775. established
  58. Zurek, W. H. (2009). "Quantum Darwinism." Nature Physics 5: 181–188. established (as a framework)
  59. Marletto, C. (2015). "Constructor theory of life." Journal of the Royal Society Interface 12: 20141226. established (within its framework's assumptions)
  60. Wheeler, J. A. (1990). "Information, physics, quantum: The search for links." In W. H. Zurek (ed.), Complexity, Entropy, and the Physics of Information. Addison-Wesley. speculative (a program manifesto)
  61. Bekenstein, J. D. (1973). "Black holes and entropy." Physical Review D 7: 2333; and (1981) Physical Review D 23: 287. established
  62. 't Hooft, G. (1993). "Dimensional reduction in quantum gravity," arXiv:gr-qc/9310026; Susskind, L. (1995). "The world as a hologram." J. Math. Phys. 36: 6377. established (as the holographic proposal)
  63. Maldacena, J. (1998). "The large N limit of superconformal field theories and supergravity." Adv. Theor. Math. Phys. 2: 231–252. established
  64. Jacobson, T. (1995). "Thermodynamics of spacetime: The Einstein equation of state." Physical Review Letters 75: 1260–1263. established (derivation; interpretation open)
  65. Chiribella, G., D'Ariano, G. M. & Perinotti, P. (2011). "Informational derivation of quantum theory." Physical Review A 84: 012311. established
  66. Verlinde, E. (2011). "On the origin of gravity and the laws of Newton." JHEP 2011(4): 29. contested
  67. Vopson, M. M. & Lepadatu, S. (2022). "Second law of information dynamics." AIP Advances 12: 075310; Vopson, M. M. (2019). "The mass-energy-information equivalence principle." AIP Advances 9: 095206. speculative (derivation criticized for a reversibility assumption; no experimental confirmation)
  68. Albert, D. Z. (2000). Time and Chance. Harvard University Press. established (the "past hypothesis")
  69. Penrose, R. (1979). "Singularities and time-asymmetry." In Hawking & Israel (eds.), General Relativity: An Einstein Centenary Survey. Cambridge University Press. contested (Weyl curvature hypothesis)
  70. Rovelli, C. (1996). "Relational quantum mechanics." International Journal of Theoretical Physics 35: 1637–1678. contested (as an interpretation)
  71. Breuer, T. (1995). "The impossibility of accurate state self-measurements." Philosophy of Science 62: 197–214. flagged — not independently re-verified; treat with care.