Pith. sign in

REVIEW 5 major objections 6 minor 131 references

A non-ergodic framework for understanding emergent capabilities in Large Language Models

T0 review · 5 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper argues that large language models are non-ergodic systems whose capabilities emerge through discrete phase transitions, formalized as a resource-bounded growth equation over token combinations.

desk verdict A suggestive analogy with a promising non-ergodicity angle, but the central isomorphism is asserted and the experiments never measure the quantities in the central equation. read the letter →

arxiv 2501.01638 v2 pith:YCVWRGO2 submitted 2025-01-03 cs.CL cs.AIcs.LG

Javier Marín This is my paper · ORCID
classification cs.CLcs.AIcs.LG
keywords largelanguagemodelsnon-ergodicsystemstheoryoftheadjacentpossibleemergentcapabilitiesphasetransitionsconstraintinteractionspathdependenceresource-boundedTAPequation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper sets out to prove that large language models are non-ergodic systems: their behavior at any moment depends on the path of context and training that led there, so the usual assumption that time averages equal ensemble averages does not hold. It then argues that capabilities do not grow smoothly, but through discrete phase transitions in semantic space, and offers a mathematical framework for this growth based on the theory of the adjacent possible (TAP), adapted to computational constraints. The central object is a resource-bounded TAP equation in which the growth of the accessible state space is a constrained combinatorial expansion over the active vocabulary, with a single constraint function combining architectural, training, and contextual limits. If this is right, emergent capabilities are not mysterious accidents of scale; they are phase transitions in a constrained possibility space, with thresholds shaped by how constraints interact.

What carries the argument

The central object is the resource-bounded TAP equation, adapted from the theory of the adjacent possible, a framework describing how a system expands its space of possibilities by combining currently available elements under constraints. The load-bearing identity is the claimed isomorphism between softmax attention and TAP's combinatorial sum, $$\text{Attention}(Q,K,V) \approx \sum_i \alpha_i \binom{|M_t|}{i}$$, which lets the paper carry the combinatorial expansion from the original TAP equation into transformer architectures. The equation says that the next state of the model's accessible semantic space is built by taking the current state and adding hierarchical transformations of all constrained token combinations, with the whole increment capped by a resource bound $R(C_t)$. It is this isomorphism, rather than any empirical regularity, that carries the derivation from the original TAP equation to Eq. 36.

What would settle it

One concrete check: take a fixed context, compute the actual attention distribution, and compare the model's next-token predictions against the predictions of the TAP combination sum with any choice of constraint coefficients; if there exist contexts where attention is sharply concentrated but the combinatorially predicted spread is wide, the isomorphism fails. A second check would be to measure capability curves on a controlled benchmark and see whether jumps occur at thresholds predicted by Eq. 36 or whether the curves are continuous and scale-smooth.

Watch

Extended reading notes

Core claim

The paper's central claim is that language models are non-ergodic and that their capability emergence is governed by the resource-bounded TAP equation $$A_{t+1} = A_t + \sum_{l=1}^{L} g_l\left( \sum_{i=1}^{|V_t|} \$\alpha$(i,t) \binom{|V_t|}{i} \right)$$ where $A_t$ is the accessible semantic state space, $|V_t|$ the active vocabulary, $\binom{|V_t|}{i}$ counts token combinations, $\alpha(i,t)$ fuses architectural, training, and contextual constraints multiplicatively, and $g_l$ are hierarchical functions that map token combinations into semantic space while bounded by computational resources. The paper claims to prove that this equation follows by mapping the token-prediction probability space onto the TAP combinatorial structure, treating the softmax attention mechanism as an isomorphism to combinatorial selection. It reports experiments on three open models that show phase-transition-like jumps in performance, multiplicative constraint effects with distinct thresholds, and path-dependent problem-solving behavior, which it reads as empirical support for the framework.

Load-bearing premise

The load-bearing premise is that softmax attention really is a combinatorial selection process, so an attention computation can be rewritten as a constrained sum over token combinations; if that isomorphism fails, Equations 33 through 36 do not follow.

Editorial extensions

If this is right

  • Capability emergence should appear as discrete jumps at critical points rather than smooth improvements, so continuous scaling curves would indicate that a constraint threshold has not been reached.
  • Relaxing any single constraint (architecture, training data, or context) will produce only limited gains while the other constraints are binding; improvements need balanced relaxation of all three.
  • Because the system is path-dependent, the same prompt presented in different orders can lead to measurably different solution trajectories, so evaluation and alignment work must track trajectories rather than averages.
  • Architecture design can target phase-transition thresholds, since the hierarchical functions $g_l$ and the resource bound $R(C_t)$ set the limits of capability growth and can make emergence more predictable.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: if Eq. 36 is right, transition thresholds could in principle be estimated before training from the architecture's resource bounds and the vocabulary size, making at least some emergent capabilities predictable at design time.
  • Editorial inference: the attention-to-TAP isomorphism suggests that changing attention variants (sparse, linear, sliding-window) changes the effective constraint profile $\alpha(i,t)$, which predicts that different attention designs will shift phase-transition points; this is a testable extension the paper does not run.
  • Editorial inference: the framework implies a new scaling variable, namely the growth rate of the accessible state space under constraints, rather than parameter count alone; comparing models on effective-dimensionality growth could be a sharper predictor of benchmarks than raw scale.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 6 minor

Summary. The paper proposes a theoretical framework, based on Stuart Kauffman's theory of the adjacent possible (TAP), for explaining emergent capabilities in large language models. It claims to prove that LLMs are non-ergodic systems and to derive a 'resource-bounded TAP equation' (Eq. 36) that governs the growth of an accessible state space through constrained combinatorial expansion. The empirical section reports analyses of three open-source models (gpt2-xl, opt-1.3B, pythia-1.4B) on the MMLU high-school mathematics subset, measuring accuracy, attention entropy, PCA-based effective dimensionality, constraint-like quantities, and path-dependence metrics, and claims support for three hypotheses about phase transitions, multiplicative constraint interactions, and path dependence.

Significance. The ambition is substantial: a unifying quantitative theory of emergent LLM capabilities would be a major contribution, and the paper is to be credited for making the non-ergodicity intuition precise enough to state specific hypotheses and for evaluating them on multiple open models with a public benchmark. However, the central theoretical object (Eq. 36) is never rigorously derived, the asserted attention-to-TAP isomorphism is not established, and the experiments do not measure the quantities appearing in Eq. 36. The paper also contains explicit self-acknowledged limitations, notably in §10.2 ('we need additional independent evidence to validate whether this emphasis accurately represents the real dynamics of language models') and §9.2 (90 questions per difficulty level, a simple difficulty split). Consequently, although the direction is thought-provoking, the current manuscript does not support its central claims as stated.

major comments (5)
  1. [§8.2.1, Lemma 2, Eq. (18)] The asserted identification Attention(Q,K,V) ~= sum_i alpha_i (|M_t| choose i) is not an isomorphism or a derivation. The left-hand side is a d-dimensional vector (or a matrix for multiple heads), while the right-hand side is a scalar combinatorial sum; no dimension-matching, bijection, or preservation of operations is supplied. The justifications given (softmax outputs lie in [0,1]; attention patterns reflect combinatorial selection; the 1/sqrt(d) scale is a natural limit) are analogies, not mathematical conditions. Because Eqs. (33)-(36) rely on this step to replace the TAP combinatorial sum with attention, the central equation of the paper is not derived from transformer mechanics. Lemma 4 in §8.6 restates the same claim in category-theoretic language but again provides no construction or proof.
  2. [§8.5.3, Eqs. (33)-(36)] The derivation of the 'resource-bounded TAP equation' consists of asserted substitutions: M_t is replaced by |V_t|, the constraint pair (beta_i, gamma_i delta_i) is replaced by alpha(i,t) = min(beta_i, gamma_i delta_i, R(C_t)), and hierarchical functions g_l are inserted with a norm bound but without specifying their domain or codomain consistently. In particular, g_l is said to map P(V) to R^n, but its argument in Eq. (36) is a scalar sum; the equality A_{t+1} = A_t + sum_l g_l(...) therefore mixes a scalar input with a vector-valued output. The initial condition A_0 = |V_0| also conflates the accessible state space with the vocabulary size. No theorem states that this equation follows from transformer architecture, attention, or training; it is a proposed model, not a derived one.
  3. [§9-§10, experimental validation of Eq. (36)] The experiments measure accuracy, attention entropy, PCA effective dimensionality, and path-difference metrics, but never measure A_t or alpha(i,t) as defined in Eq. (36). The 'phase transitions' reported in Figures 1-3 and 8 are transitions in attention entropy, PCA dimensionality, and a heuristic combined constraint product; no evidence connects these to the combinatorial phase transitions of the TAP equation. The paper itself acknowledges this gap in §10.2, stating that additional independent evidence is needed to validate whether the framework's emphasis accurately represents the real dynamics of language models. As a result, the empirical sections do not constitute a test of the paper's central equation.
  4. [§9.2, §9.5, Fig. 5, Tables 4-5] The statistical evidence is too weak for the strength of the claims. Each difficulty level contains only 90 questions, no error bars or repeated-seed variability are reported, and §9.5 uses only 30 questions. Figure 5 reports R^2 values of -0.866 and 0.500 with p-values computed on what appear to be three data points; negative R^2 and values such as R^2=1.0 with p=0.000 indicate that the reported quantity is not the standard coefficient of determination but a signed correlation coefficient, and the text does not clarify the estimator. The thresholds in Figure 8 are drawn by inspection without uncertainty quantification. These issues undermine the specific claims about 'precise positive correlation' and 'perfect negative correlation' that are used to support the multiplicative-constraint hypothesis.
  5. [Abstract; §7.3; §8.4.5] The abstract and Section 7.3 claim that the paper 'prove[s] that language models are actually non-ergodic systems,' but no formal proof is provided. The argument consists of asserting path-dependent transition probabilities (Eq. 3, Eq. 27) and training-induced state-space restrictions, then equating path dependence with non-ergodicity. A rigorous proof would need a precise dynamical system or stochastic process model of LLM inference, a definition of the relevant invariant measure, and a demonstration that the Birkhoff ergodic theorem or its equivalent fails in that setting. The current text offers a plausible hypothesis and empirical illustrations, not a proof. This matters because the paper's title and abstract make the proof claim central.
minor comments (6)
  1. [§10.3 heading] The heading 'Constraint interactions shape capability boundaries (H3)' appears to be a copy-paste error; the section content tests Hypothesis 3, which is about path dependence, not constraint interactions.
  2. [§9.3, Eq. (51)] The formula for effective dimensionality, d_eff = min{k : sum_{i=1}^k lambda_i / sum_{i=1}^k lambda_i >= 0.9}, has the same index in numerator and denominator, so the ratio is identically 1; the denominator should presumably be the total variance sum_{i=1}^n lambda_i.
  3. [§9.4] The cross-references to equations are inconsistent: the text says architectural constraints are measured 'as in Equation 47' and later refers to 'Equation 46' and 'Equation 50', but the displayed formulas are Eqs. (52) and (53); the numbering appears to have shifted during revision.
  4. [§8.4.4, Eqs. (5)-(6)] Equations (5) and (6) are displayed with identical notation, although the text says Eq. (5) uses a constant alpha and Eq. (6) uses a sequence alpha_i; the distinction is not visible, making the discussion of Kauffman's reformulation confusing.
  5. [Throughout] There are frequent typographical and grammatical issues, including 'op-1.3B' vs 'opt-1.3B' in Table 1, 'sof tmax' in Eq. (18), 'Kaufman' vs 'Kauffman', and inconsistent citation formats (e.g., '[28,65,76,86]' vs '[87]'); a thorough editing pass is needed.
  6. [§9.2] The experimental setup does not state whether any code, prompts, or preprocessing scripts will be released, and it reports no hardware details, random seeds, or number of runs; a reproducibility statement would strengthen the empirical contribution.

Circularity Check

1 steps flagged · score 4.0 of 10

The H2 validation is partly self-referential: the 'training constraint' γ is measured as accuracy, and the same accuracy values are repackaged as the performance that constraint interactions are said to predict. The central TAP equation is proposed via an asserted attention–TAP isomorphism, which is unsupported rather than circular; no load-bearing self-citation appears.

  1. fitted input called prediction [Section 9.4 (Eq. 53) and Section 10.2 (Table 2, Figure 8)]
    "Training constraints: These constraints are computed using Equation 46 ... γ = 1/N Σ_{i=1}^N 1(y_i = by1)c_i ... Equation 46 evaluates the accuracy of the raw performance, while Equation 50 quantifies the limitations set by the system's training patterns. ... The performance column represents a weighted combination of architectural, training and contextual constraints effects (30% β, 40% γ, and 30% δ)."

    The 'training constraint' γ is, by Eq. 53, the model's own multiple-choice accuracy with a confidence term. Table 2 then defines its 'Performance' column as a weighted sum in which γ contributes 40%, and Figure 8 plots model performance against the product β×γ×δ, which therefore already contains the accuracy being predicted. Claiming that the data reveal multiplicative constraint interactions and phase transitions is thus partly a restatement of the defining equation: the outcome is included in the constraint product by construction, not discovered from independent measurements.

full rationale

The only step that reduces to its own inputs by construction is the Hypothesis 2 validation: the training constraint is operationalized as accuracy (Eq. 53), and the same accuracy is then embedded in the 'Performance' column and in the β×γ×δ product used to demonstrate constraint-driven phase transitions. This makes part of the experimental support self-referential. The non-ergodicity argument rests on ordinary, independently observable context/path dependence in token prediction and is not circular. The central resource-bounded TAP equation (Eq. 36) is not actually tested against A_t or α(i,t); the experiments measure proxies. That is a validity gap rather than a circularity: Eq. 36 is introduced as a proposed extension of Kauffman's equation, and the asserted attention–TAP isomorphism in Eq. 18 is an unsupported analogy, not a conclusion already assumed elsewhere in the paper. No load-bearing self-citations occur. The localized self-reference in H2 justifies a modest score of 4, not a higher score, because the main non-ergodicity claim and the qualitative phase-transition observations have independent empirical content.

Assumptions & free parameters 4 free parameters · 5 assumptions · 2 invented entities

The framework rests on several unproven assumptions: that LLMs are CAS, that softmax attention is isomorphic to TAP combination, that constraints multiply, and that hierarchical functions with resource bounds exist. The only fitted numbers appear in the experimental section (weights, thresholds, variance cutoff); the central equation (36) is never instantiated with real parameters.

free parameters (4)
  • Performance weights in Table 2 = 30% beta, 40% gamma, 30% delta plus 10% interaction
    Hand-chosen weights used to compute the 'Performance' column that is then interpreted as evidence for constraint interactions; no justification is given.
  • Critical thresholds in Figure 8 = gpt2-xl: 0.10; pythia-1.4B: 0.08; opt-1.3B: 0.07
    Thresholds are read off the same experimental data and then used to claim phase transitions; they are not predicted by the TAP equation.
  • Effective dimensionality variance cutoff = 90%
    Threshold for PCA components in deff is chosen ad hoc; changing the cutoff would change the reported semantic dimensionality.
  • Model architecture constant kappa in Lemma 3 = unspecified
    The bound f(x) = kappa * x * log(x) depends on an unspecified constant, so the resource bound is not quantitative.
assumptions (5)
  • domain assumption Language models are complex adaptive systems (CAS) with self-organization and phase transitions.
    Section 6.2 asserts this analogy without formal justification and then imports CAS machinery.
  • ad hoc to paper Softmax attention is isomorphic to the TAP combinatorial sum with alpha_i as softmax weights.
    Equation 18 in Section 8.2: softmax outputs in [0,1] are taken to map to alpha_i and attention selection to binomial combinations; this is an asserted analogy, not a proven isomorphism.
  • ad hoc to paper Constraints multiply: alpha(i,t) = min(beta_i, gamma_i * delta_i, R(Ct)).
    Section 8.5.2; the multiplicative interaction is justified by biological examples, not derived from language model mechanics.
  • ad hoc to paper The hierarchical functions g_l satisfy ||g_l(x)|| <= K*R(Ct) and map probability vectors to R^n.
    Section 8.5.3; existence and bound are assumed, and no concrete construction or example is provided.
  • domain assumption A context-dependent stochastic process is non-ergodic.
    Section 5.1; path dependence alone does not establish non-ergodicity because the process can be redefined on an expanded state space (context history) where ergodicity may hold.
invented entities (2)
  • Semantic state space St and its dimension dim(St)
    purpose: Quantifies the model's possibility space and its phase transitions; used in Equations 7, 22, and 29.
    Defined through attention embeddings and PCA within this paper; no external observable predicts its values.
  • Constraint thresholds in Figure 8
    purpose: Marked critical points where models transition between performance regimes.
    Read off the experimental curves post hoc; no independent prediction is made.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A non-ergodic framework for understanding emergent capabilities in Large Language Models." pith.science (2026). https://pith.science/paper/YCVWRGO2

@misc{pith2026250101638,
  author       = {Pith},
  title        = {Pith review of: A non-ergodic framework for understanding emergent capabilities in Large Language Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YCVWRGO2}},
  note         = {Machine review of arXiv:2501.01638}
}
read the original abstract

Large language models have emergent capabilities that come unexpectedly at scale, but we need a theoretical framework to explain why and how they emerge. We prove that language models are actually non-ergodic systems while providing a mathematical framework based on Stuart Kauffman's theory of the adjacent possible (TAP) to explain capability emergence. Our resource-constrained TAP equation demonstrates how architectural, training, and contextual constraints interact to shape model capabilities through phase transitions in semantic space. We prove through experiments with three different language models that capacities emerge through discrete transitions guided by constraint interactions and path-dependent exploration. This framework provides a theoretical basis for understanding emergence in language models and guides the development of architectures that can guide capability emergence.

Figures

Figures reproduced from arXiv: 2501.01638 by the authors.

Figure 1
Figure 1. Model performance and task difficulty [PITH_FULL_IMAGE:figures/full_fig_p028_1.png] view at source ↗
Figure 2
Figure 2. Semantic space analysis. In [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗
Figure 3
Figure 3. Correlation between performance and entropy. [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Training vs architectural constraints effects. [PITH_FULL_IMAGE:figures/full_fig_p033_4.png]
Figure 5
Figure 5. Figure 5: Training vs contextual constraints effects. [PITH_FULL_IMAGE:figures/full_fig_p034_5.png]
Figure 6
Figure 6. Figure 6: Architectural vs contextual constraints effects. [PITH_FULL_IMAGE:figures/full_fig_p034_6.png]
Figure 7
Figure 7. Figure 7: Constraints distribution for different models. [PITH_FULL_IMAGE:figures/full_fig_p035_7.png]
Figure 8
Figure 8. Figure 8: Phase transitions for combined constraints across different models. [PITH_FULL_IMAGE:figures/full_fig_p036_8.png]
Figure 9
Figure 9. Figure 9: Path differences across models [PITH_FULL_IMAGE:figures/full_fig_p037_9.png]
Figure 10
Figure 10. Figure 10: Step length analysis across models [PITH_FULL_IMAGE:figures/full_fig_p038_10.png]
Figure 11
Figure 11. Figure 11: Path differences across models 38 [PITH_FULL_IMAGE:figures/full_fig_p038_11.png]
Figure 12
Figure 12. Figure 12: Path differences across models. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_12.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

131 extracted references · 52 canonical work pages

  1. [1]

    Achille, A., & Soatto, S. (2018). Emergence of invariance and disentanglement in deep representations. Journal of Machine Learning Research , 19(50), 1–34

    2018

  2. [2]

    Ainslie, J., Lee-Thorp, J., de Jong, M., Zemlyanskiy, Y., Lebr´ on, F., & Sanghai, S. (2023). Gqa: Training generalized multi-query transformer models from multi-head checkpoints. ArXiv Preprint arXiv:2305.13245 . 44

    arXiv 2023

  3. [3]

    M., Zhai, X., Kolesnikov, A., & Beyer, L

    Alabdulmohsin, I. M., Zhai, X., Kolesnikov, A., & Beyer, L. (2024). Getting vit in shape: Scaling laws for compute-optimal model design. Advances in Neural Information Processing Systems, 36

    2024

  4. [4]

    Alon, U. (2019). An introduction to systems biology: design principles of biological circuits. Chapman and Hall/CRC

    2019

  5. [5]

    Amaral, L. A. N., Buldyrev, S. V., Havlin, S., Salinger, M. A., & Stanley, H. E. (1998). Power Law Scaling for a System of Interacting Units with Complex Internal Structure. Physical Review Letters, 80(7), 1385. https://doi.org/10.1103/PhysRevLett.80.1385

    work page doi:10.1103/physrevlett.80.1385 1998

  6. [6]

    Anderson, P. W. (1972). More Is Different: Broken symmetry and the nature of the hierarchical structure of science. Science, 177(4047), 393–396

    1972

  7. [7]

    Arnold, J., Holtorf, F., Sch¨ afer, F., & L¨ orch, N. (2024). Phase Transitions in the Output Distribution of Large Language Models. ArXiv Preprint arXiv:2405.17088

    arXiv 2024

  8. [8]

    Arora, S., & Goyal, A. (2023). A theory for emergence of complex skills in language models. ArXiv Preprint arXiv:2307.15936

    arXiv 2023

Show all 131 references
  1. [9]

    B., & Dol´ eans-Dade, C

    Ash, R. B., & Dol´ eans-Dade, C. A. (2000).Probability and measure theory. Academic press

    2000

  2. [10]

    Askell, A., Bai, Y., Chen, A., Drain, D., Ganguli, D., Henighan, T., Jones, A., Joseph, N., Mann, B., & DasSarma, N. (2021). A general language assistant as a laboratory for alignment. ArXiv Preprint arXiv:2112.00861

    2021 arXiv

  3. [11]

    Assran, M., Duval, Q., Misra, I., Bojanowski, P., Vincent, P., Rabbat, M., LeCun, Y., & Ballas, N. (2023). Self-supervised learning from images with a joint-embedding predictive architecture. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 15...

    2023

  4. [12]

    Attanasio, G., Nozza, D., Hovy, D., & Baralis, E. (2022). Entropy-based attention regu- larization frees unintended bias mitigation from lists. ArXiv Preprint arXiv:2203.09192

    2022 arXiv

  5. [13]

    Bak, P. (2013). How nature works: the science of self-organized criticality . Springer Science & Business Media

    2013

  6. [14]

    Basu, S., Choraria, M., & Varshney, L. R. (2023). Transformers are Universal Predictors. ArXiv Preprint arXiv:2307.07843

    2023 arXiv

  7. [15]

    Baxter, J. (2000). A model of inductive bias learning. Journal of Artificial Intelligence Research, 12, 149–198

    2000

  8. [16]

    Bereska, L., & Gavves, E. (2024). Mechanistic Interpretability for AI Safety-A Review. ArXiv Preprint arXiv:2404.14082 . 45

    2024 arXiv

  9. [17]

    G., Bradley, H., O’Brien, K., Hallahan, E., Khan, M

    Biderman, S., Schoelkopf, H., Anthony, Q. G., Bradley, H., O’Brien, K., Hallahan, E., Khan, M. A., Purohit, S., Prashanth, U. S., & Raff, E. (2023). Pythia: A suite for analyzing large language models across training and scaling. International Conference on Machine Learning , ...

    2023

  10. [18]

    Birkhoff, G. D. (1931). Proof of the ergodic theorem. Proceedings of the National Academy of Sciences, 17(12), 656–660

    1931

  11. [19]

    D., Collins, T

    Bootman, M. D., Collins, T. J., Peppiatt, C. M., Prothero, L. S., MacKenzie, L., De Smet, P., Travers, M., Tovey, S. C., Seo, J. T., & Berridge, M. J. (2001). Calcium signalling—an overview. Seminars in Cell & Developmental Biology , 12(1), 3–10

    2001

  12. [20]

    M., & Mauro, J

    Bossen, A. M., & Mauro, J. C. (2024). Frozen in time: A review of non-ergodic physical systems. Journal of the American Ceramic Society , 107(12), 7939–7950

    2024

  13. [21]

    Brown, T., Mann, B., Ryder, N., Subbiah, M., Kaplan, J. D., Dhariwal, P., Neelakantan, A., Shyam, P., Sastry, G., Askell, A., Agarwal, S., Herbert-Voss, A., Krueger, G., Henighan, T., Child, R., Ramesh, A., Ziegler, D., Wu, J., Winter, C., ... Amodei, D. (2020). Language Model...

    2020

  14. [22]

    Chen, H., Yang, X., Zhu, J., & Wang, W. (2024). Quantifying Emergence in Large Language Models. ArXiv Preprint arXiv:2405.12617

    2024 arXiv

  15. [23]

    Chomsky, N. (2014). Aspects of the Theory of Syntax (Issue 11). MIT press

    2014

  16. [24]

    W., Sutton, C., & Gehrmann, S

    Chowdhery, A., Narang, S., Devlin, J., Bosma, M., Mishra, G., Roberts, A., Barham, P., Chung, H. W., Sutton, C., & Gehrmann, S. (2023). Palm: Scaling language modeling with pathways. Journal of Machine Learning Research , 24(240), 1–113

    2023

  17. [25]

    Church, K., & Mercer, R. L. (1993). Introduction to the special issue on computational linguistics using large corpora. Computational Linguistics , 19(1), 1–24

    1993

  18. [26]

    Cilliers, P. (2002). Complexity and postmodernism: Understanding complex systems . routledge

    2002

  19. [27]

    Conmy, A., Mavor-Parker, A., Lynch, A., Heimersheim, S., & Garriga-Alonso, A. (2023). Towards automated circuit discovery for mechanistic interpretability. Advances in Neural Information Processing Systems, 36, 16318–16352

    2023

  20. [28]

    P., Fomin, S

    Cornfeld, I. P., Fomin, S. V., & Sinai, Y. G. (2012). Ergodic theory (Vol. 245). Springer Science & Business Media

    2012

  21. [29]

    A., Liddle, A

    Cortˆ es, M., Kauffman, S. A., Liddle, A. R., & Smolin, L. (2022). The TAP equation: evaluating combinatorial innovation in biocosmology. ArXiv Preprint arXiv:2204.14115

    2022

  22. [30]

    Crawford, J. D. (1991). Introduction to bifurcation theory. Reviews of Modern Physics , 63(4), 991. 46

    1991

  23. [31]

    C., & Hohenberg, P

    Cross, M. C., & Hohenberg, P. C. (1993). Pattern formation outside of equilibrium. Reviews of Modern Physics , 65(3), 851

    1993

  24. [32]

    Cross, M., & Greenside, H. (2009). Pattern formation and dynamics in nonequilibrium systems. Cambridge University Press

    2009

  25. [33]

    Crutchfield, J. P. (2012). Between order and chaos. Nature Physics, 8(1), 17–24

    2012

  26. [34]

    Dao, T., Fu, D., Ermon, S., Rudra, A., & R´ e, C. (2022). Flashattention: Fast and memory-efficient exact attention with io-awareness. Advances in Neural Information Processing Systems, 35, 16344–16359

    2022

  27. [35]

    Doshi-Velez, F., & Kim, B. (2017). Towards a rigorous science of interpretable machine learning. ArXiv Preprint arXiv:1702.08608

    2017 arXiv

  28. [36]

    Elhage, N., Nanda, N., Olsson, C., Henighan, T., Joseph, N., Mann, B., Askell, A., Bai, Y., Chen, A., & Conerly, T. (2021). A mathematical framework for transformer circuits. Transformer Circuits Thread, 1(1), 12

    2021

  29. [37]

    Elhage, N., Hume, T., Olsson, C., Nanda, N., Henighan, T., Johnston, S., ElShowk, S., Joseph, N., DasSarma, N., Mann, B., Hernandez, D., Askell, A., Ndousse, K., Jones, A., Drain, D., Chen, A., Bai, Y., Ganguli, D., Lovitt, L., ... Olah, C. (2022). Softmax Linear Units. Transf...

    2022

  30. [38]

    England, J. L. (2015). Dissipative adaptation in driven self-assembly. Nature Nanotech- nology, 10(11), 919–923

    2015

  31. [39]

    Fan, A., Lewis, M., & Dauphin, Y. (2018). Hierarchical neural story generation. ArXiv Preprint arXiv:1805.04833

    2018 arXiv

  32. [40]

    Gal, Y., & Ghahramani, Z. (2016). Dropout as a bayesian approximation: Representing model uncertainty in deep learning. International Conference on Machine Learning , 1050–1059

    2016

  33. [41]

    Ganguli, D., Hernandez, D., Lovitt, L., Askell, A., Bai, Y., Chen, A., Conerly, T., Dassarma, N., Drain, D., & Elhage, N. (2022). Predictability and surprise in large generative models. Proceedings of the 2022 ACM Conference on Fairness, Accountability, and Transparency, 1747–1764

    2022

  34. [42]

    Gell-Mann, M. (1995). The Quark and the Jaguar: Adventures in the Simple and the Complex. Macmillan

    1995

  35. [43]

    Glattfelder, J. B. (2019). The Semantics of Symmetry, Invariance, and Structure. Fron- tiers Collection , Part F1071, 65–92

    2019

  36. [44]

    Goldenfeld, N., & Woese, C. (2011). Life is physics: evolution as a collective phenomenon far from equilibrium. Annu. Rev. Condens. Matter Phys. , 2(1), 375–399

    2011

  37. [45]

    Goodfellow, I. (2016). Deep learning. MIT press. 47

    2016

  38. [46]

    Gould, S. J. (1989). Wonderful Life: The Burgess Shale and the Nature of History . WW Norton & Company

    1989

  39. [47]

    Guckenheimer, J., & Holmes, P. (2013). Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (Vol. 42). Springer Science & Business Media

    2013

  40. [48]

    Ha, D., & Schmidhuber, J. (2018). World models. ArXiv Preprint arXiv:1803.10122

    2018 arXiv

  41. [49]

    A., & Pascanu, R

    Hadsell, R., Rao, D., Rusu, A. A., & Pascanu, R. (2020). Embracing change: Continual learning in deep neural networks. Trends in Cognitive Sciences, 24(12), 1028–1040

    2020

  42. [50]

    Haken, H. (1973). Introduction to synergetics. Synergetics: Cooperative Phenomena in Multi-Component Systems, 9–19

    1973

  43. [51]

    Haken, H. (1989). Synergetics: an overview. Reports on Progress in Physics , 52(5), 515

    1989

  44. [52]

    Haken, H. (1993). Synergetics: From Pattern Formation to Pattern Recognition. Some Basic Mathematical Results. Dynamical Systems: Theory And Applications , 127

    1993

  45. [53]

    Hassabis, D., Kumaran, D., Summerfield, C., & Botvinick, M. (2017). Neuroscience- inspired artificial intelligence. Neuron, 95(2), 245–258

    2017

  46. [54]

    Haussler, D., & Warmuth, M. (2018). The probably approximately correct (PAC) and other learning models. The Mathematics of Generalization , 17–36

    2018

  47. [55]

    Heinrich, R., & Schuster, S. (2012). The regulation of cellular systems . Springer Science & Business Media

    2012

  48. [56]

    Hendrycks, D., Burns, C., Basart, S., Zou, A., Mazeika, M., Song, D., & Stein- hardt, J. (2020). Measuring massive multitask language understanding. ArXiv Preprint arXiv:2009.03300

    2020 arXiv

  49. [57]

    Holland, J. H. (1992). Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence . MIT press

    1992

  50. [58]

    Holland, J. H. (2006). Studying complex adaptive systems. Journal of Systems Science and Complexity, 19, 1–8

    2006

  51. [59]

    (2020, April 28)

    Holtzman, A., Buys, J., Forbes, M., & Choi, Y. (2020, April 28). The curious case of neural text degeneration. ICLR

    2020

  52. [60]

    Huang, C.-Y., & Ferrell Jr, J. E. (1996). Ultrasensitivity in the mitogen-activated protein kinase cascade. Proceedings of the National Academy of Sciences , 93(19), 10078–10083

    1996

  53. [61]

    Jelinek, F. (1980). Interpolated estimation of Markov source parameters from sparse data. Proc. Workshop on Pattern Recognition in Practice, 1980

    1980

  54. [62]

    Jolliffe, I. T. (2002). Principal component analysis for special types of data . Springer. 48

    2002

  55. [63]

    B., Chess, B., Child, R., Gray, S., Radford, A., Wu, J., & Amodei, D

    Kaplan, J., McCandlish, S., Henighan, T., Brown, T. B., Chess, B., Child, R., Gray, S., Radford, A., Wu, J., & Amodei, D. (2020). Scaling laws for neural language models. ArXiv Preprint arXiv:2001.08361

    2020 arXiv

  56. [64]

    Katok, A. (1995). Introduction to the Modern Theory of Dynamical Systems. Encyclo- pedia of Mathematics and Its Applications , 54

    1995

  57. [65]

    Kauffman, S. (2022). Is There a Fourth Law for Non-Ergodic Systems That Do Work to Construct Their Expanding Phase Space? Entropy 2022, Vol. 24, Page 1383 , 24(10),

    2022

  58. [66]

    Kauffman, S. A. (1993). The origins of order: Self-organization and selection in evolution . Oxford University Press

    1993

  59. [67]

    Kauffman, S. A. (2000). Investigations. Oxford University Press

    2000

  60. [68]

    Kauffman, S. A. (2019). A world beyond physics: the emergence and evolution of life . Oxford University Press

    2019

  61. [69]

    Kauffman, S., & Levin, S. (1987). Towards a general theory of adaptive walks on rugged landscapes. Journal of Theoretical Biology, 128(1), 11–45. https://doi.org/10.1016/S0022- 5193(87)80029-2

    1987 doi

  62. [70]

    (2006).Bifurcation theory: An introduction with applications to PDEs (Vol

    Kielh¨ ofer, H. (2006).Bifurcation theory: An introduction with applications to PDEs (Vol. 156). Springer Science & Business Media

    2006

  63. [71]

    Klipp, E., Liebermeister, W., Wierling, C., & Kowald, A. (2016). Systems biology: a textbook. John Wiley & Sons

    2016

  64. [72]

    Koppl, R., Devereaux, A., Herriot, J., & Kauffman, S. (2018). A simple combinatorial model of world economic history. ArXiv Preprint arXiv:1811.04502

    2018 arXiv

  65. [73]

    J., Zhang, C., & Slivkins, A

    Krishnamurthy, A., Harris, K., Foster, D. J., Zhang, C., & Slivkins, A. (2024). Can large language models explore in-context? ArXiv Preprint arXiv:2403.15371

    2024 arXiv

  66. [74]

    Kullback, S. (1997). Information theory and statistics . Courier Corporation

    1997

  67. [75]

    Lansing, J. S. (2003). Complex adaptive systems. Annual Review of Anthropology, 32(1), 183–204

    2003

  68. [76]

    L., & Penrose, O

    Lebowitz, J. L., & Penrose, O. (1973). Modern ergodic theory. Physics Today, 26(2), 23–29. https://doi.org/10.1063/1.3127948

    1973 doi

  69. [77]

    LeCun, Y. (2022). A path towards autonomous machine intelligence (version 09). Open Review, 62(1), 1–62

    2022

  70. [78]

    Levine, Y., Dalmedigos, I., Ram, O., Zeldes, Y., Jannai, D., Muhlgay, D., Osin, Y., Lieber, O., Lenz, B., & Shalev-Shwartz, S. (2022). Standing on the shoulders of giant frozen language models. ArXiv Preprint arXiv:2204.10019 . 49

    2022 arXiv

  71. [79]

    Liu, Z., Gan, E., & Tegmark, M. (2023). Seeing is believing: Brain-inspired modular training for mechanistic interpretability. Entropy, 26(1), 41

    2023

  72. [80]

    Longo, G., Mont´ evil, M., & Kauffman, S. (2012). No entailing laws, but enablement in the evolution of the biosphere. Proceedings of the 14th Annual Conference Companion on Genetic and Evolutionary Computation , 1379–1392

    2012

  73. [81]

    M., & Novick, M

    Lord, F. M., & Novick, M. R. (2008). Statistical theories of mental test scores . IAP

    2008

  74. [82]

    Manning, C., & Schutze, H. (1999). Foundations of statistical natural language processing. MIT press

    1999

  75. [83]

    Markovi´ c, D., & Gros, C. (2014). Power laws and self-organized criticality in theory and nature. Physics Reports, 536(2), 41–74

    2014

  76. [84]

    Marshall, W. F. (2011). Origins of cellular geometry. BMC Biology, 9, 1–9

    2011

  77. [85]

    Matsuo, Y., LeCun, Y., Sahani, M., Precup, D., Silver, D., Sugiyama, M., Uchibe, E., & Morimoto, J. (2022). Deep learning, reinforcement learning, and world models. Neural Networks, 152, 267–275

    2022

  78. [86]

    Mazur, P. (1969). Non-ergodicity of phase functions in certain systems. Physica, 43(4), 533–545. https://doi.org/10.1016/0031-8914(69)90185-2

    1969 doi

  79. [87]

    Meister, C., & Cotterell, R. (2021). Language model evaluation beyond perplexity. ArXiv Preprint arXiv:2106.00085

    2021 arXiv

  80. [88]

    Mitchell, M. (2009). Complexity: A guided tour . Oxford University Press

    2009

  81. [89]

    Nakaishi, K., Nishikawa, Y., & Hukushima, K. (2024). Critical Phase Transition in a Large Language Model. ArXiv Preprint arXiv:2406.05335

    2024 arXiv

  82. [90]

    Nanda, N., Chan, L., Lieberum, T., Smith, J., & Steinhardt, J. (2023). Progress measures for grokking via mechanistic interpretability. ArXiv Preprint arXiv:2301.05217

    2023 arXiv

  83. [91]

    Neumann, J. v. (1932). Proof of the quasi-ergodic hypothesis. Proceedings of the National Academy of Sciences, 18(1), 70–82

    1932

  84. [92]

    Pang, Z., Xie, Z., Man, Y., & Wang, Y.-X. (2023). Frozen transformers in language models are effective visual encoder layers. ArXiv Preprint arXiv:2310.12973

    2023 arXiv

  85. [93]

    Papadimitriou, C. H. (2003). Computational complexity. In Encyclopedia of computer science (pp. 260–265)

    2003

  86. [94]

    I., Kemker, R., Part, J

    Parisi, G. I., Kemker, R., Part, J. L., Kanan, C., & Wermter, S. (2019). Continual lifelong learning with neural networks: A review. Neural Networks, 113, 54–71

    2019

  87. [95]

    Peters, O. (2019). The ergodicity problem in economics. Nature Physics, 15(12), 1216– 1221. 50

    2019

  88. [96]

    A., & Lewis, M

    Press, O., Smith, N. A., & Lewis, M. (2021). Train short, test long: Attention with linear biases enables input length extrapolation. ArXiv Preprint arXiv:2108.12409

    2021 arXiv

  89. [97]

    Prigogine, I., & Stengers, I. (2018). Order out of chaos: Man ’s new dialogue with nature . Verso Books

    2018

  90. [98]

    Raaijmakers, J. G. W. (1987). Statistical Analysis of the Michaelis-Menten Equation. Biometrics, 43(4), 793. https://doi.org/10.2307/2531533

    1987 doi

  91. [99]

    Radford, A., Wu, J., Child, R., Luan, D., Amodei, D., & Sutskever, I. (2019). Language models are unsupervised multitask learners. OpenAI Blog, 1(8), 9

    2019

  92. [100]

    Raffel, C., Shazeer, N., Roberts, A., Lee, K., Narang, S., Matena, M., Zhou, Y., Li, W., & Liu, P. J. (2020). Exploring the limits of transfer learning with a unified text-to-text transformer. Journal of Machine Learning Research , 21(140), 1–67

    2020

  93. [101]

    Raghu, M., Gilmer, J., Yosinski, J., & Sohl-Dickstein, J. (2017). Svcca: Singular vector canonical correlation analysis for deep learning dynamics and interpretability. Advances in Neural Information Processing Systems , 30

    2017

  94. [102]

    Rai, D., Zhou, Y., Feng, S., Saparov, A., & Yao, Z. (2024). A practical review of mechanistic interpretability for transformer-based language models. ArXiv Preprint arXiv:2407.02646

    2024

  95. [103]

    Raman, M., Chen, W., & Cobb, M. H. (2007). Differential regulation and properties of MAPKs. Oncogene, 26(22), 3100–3112

    2007

  96. [104]

    Ramscar, M., Hendrix, P., Shaoul, C., Milin, P., & Baayen, H. (2014). The Myth of Cognitive Decline: Non-Linear Dynamics of Lifelong Learning. Topics in Cognitive Science, 6(1), 5–42. https://doi.org/10.1111/TOPS.12078

    2014 doi

  97. [105]

    Reed, M., & Simon, B. (1980). Methods of modern mathematical physics: Functional analysis (Vol. 1). Gulf Professional Publishing

    1980

  98. [106]

    A., Lillicrap, T

    Richards, B. A., Lillicrap, T. P., Beaudoin, P., Bengio, Y., Bogacz, R., Christensen, A., Clopath, C., Costa, R. P., de Berker, A., & Ganguli, S. (2019). A deep learning framework for neuroscience. Nature Neuroscience, 22(11), 1761–1770

    2019

  99. [107]

    Riehl, E. (2017). Category theory in context (Courier Dover Publications, Ed.)

    2017

  100. [108]

    Ruelle, D. (2004). Thermodynamic formalism: the mathematical structure of equilibrium statistical mechanics. Cambridge University Press

    2004

  101. [109]

    Schaeffer, R., Miranda, B., & Koyejo, S. (2024). Are emergent abilities of large language models a mirage? Advances in Neural Information Processing Systems , 36

    2024

  102. [110]

    Schlag, I., Irie, K., & Schmidhuber, J. (2021). Linear transformers are secretly fast weight programmers. International Conference on Machine Learning , 9355–9366. 51

    2021

  103. [111]

    Shannon, C. E. (1948). A mathematical theory of communication. The Bell System Technical Journal, 27(3), 379–423

    1948

  104. [112]

    Simon, H. A. (2012). The architecture of complexity. In The Roots of Logistics (pp. 335–361). Springer

    2012

  105. [113]

    Sornette, D. (2006). Critical phenomena in natural sciences: chaos, fractals, selforgani- zation and disorder: concepts and tools . Springer Science & Business Media

    2006

  106. [114]

    Srivastava, A., Rastogi, A., Rao, A., Shoeb, A. A. M., Abid, A., Fisch, A., Brown, A. R., Santoro, A., Gupta, A., & Garriga-Alonso, A. (2022). Beyond the imitation game: Quantifying and extrapolating the capabilities of language models. ArXiv Preprint arXiv:2206.04615

    2022 arXiv

  107. [115]

    Stanley, H. E. (1971). Phase transitions and critical phenomena . Clarendon Press, Oxford

    1971

  108. [116]

    Steel, M., Hordijk, W., & Kauffman, S. A. (2020). Dynamics of a birth–death process based on combinatorial innovation. Journal of Theoretical Biology , 491, 110187

    2020

  109. [117]

    Tishby, N., & Zaslavsky, N. (2015). Deep learning and the information bottleneck principle. 2015 Ieee Information Theory Workshop (Itw) , 1–5

    2015

  110. [118]

    Troger, H., & Steindl, A. (2012). Nonlinear stability and bifurcation theory: an intro- duction for engineers and applied scientists . Springer Science & Business Media

    2012

  111. [119]

    M., & Tolias, A

    Van de Ven, G. M., & Tolias, A. S. (2019). Three scenarios for continual learning. ArXiv Preprint arXiv:1904.07734

    2019 arXiv

  112. [120]

    Vapnik, V. N. (1999). An overview of statistical learning theory. IEEE Transactions on Neural Networks, 10(5), 988–999

    1999

  113. [121]

    Vaswani, A. (2017). Attention is all you need.Advances in Neural Information Processing Systems

    2017

  114. [122]

    Villani, C. (2009). Optimal transport: old and new (Vol. 338). Springer

    2009

  115. [123]

    Walters, P. (2000). An introduction to ergodic theory (Vol. 79). Springer Science & Business Media

    2000

  116. [124]

    Wang, B., Min, S., Deng, X., Shen, J., Wu, Y., Zettlemoyer, L., & Sun, H. (2022). Towards understanding chain-of-thought prompting: An empirical study of what matters. ArXiv Preprint arXiv:2212.10001

    2022 arXiv

  117. [125]

    Wang, L., Zhang, X., Su, H., & Zhu, J. (2024). A comprehensive survey of continual learning: theory, method and application. IEEE Transactions on Pattern Analysis and Machine Intelligence . 52

    2024

  118. [126]

    Wei, J., Tay, Y., Bommasani, R., Raffel, C., Zoph, B., Borgeaud, S., Yogatama, D., Bosma, M., Zhou, D., & Metzler, D. (2022). Emergent abilities of large language models. ArXiv Preprint arXiv:2206.07682

    2022 arXiv

  119. [127]

    Zador, A. M. (2019). A critique of pure learning and what artificial neural networks can learn from animal brains. Nature Communications, 10(1), 3770

    2019

  120. [128]

    Zhang, Y. (2019). Dialogpt: Large-Scale generative pre-training for conversational response generation. ArXiv Preprint arXiv:1911.00536

    2019 arXiv

  121. [129]

    Zhang, S., Roller, S., Goyal, N., Artetxe, M., Chen, M., Chen, S., Dewan, C., Diab, M., Li, X., & Lin, X. V. (2022). Opt: Open pre-trained transformer language models. ArXiv Preprint arXiv:2205.01068

    2022 arXiv

  122. [130]

    Ziemann, I., Matni, N., & Pappas, G. J. (2024). State space models, emergence, and ergodicity: How many parameters are needed for stable predictions? ArXiv Preprint arXiv:2409.13421. 53

    2024 arXiv

  123. [1383]

    https://doi.org/10.3390/E24101383

    doi

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.