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arxiv: 2606.24861 · v1 · pith:I6YZC2C6new · submitted 2026-06-23 · ❄️ cond-mat.stat-mech · cs.IT· math.IT

First-Order Recoverability Collapse in Self-Referential Information Decoders

Pieter van Rooyen This is my paper

Pith reviewed 2026-06-25 21:40 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cs.ITmath.IT
keywords recoverability collapsefirst-order transitionbistabilityhysteresisinformation decodersnonequilibrium systemsmetastable failurestatistical mechanics
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The pith

Explicit feedback in self-referential decoders turns continuous recoverability loss into a first-order transition with bistability and hysteresis.

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

The paper treats information processing in adaptive systems as a thermodynamic load on finite-capacity decoders that make irreversible commitments. Without explicit feedback, overload produces a continuous loss of recoverability marked by divergence of a stability diagnostic. When each uncertified commitment generates on average alpha new candidates, the transition becomes first-order. Lucid and collapsed operating states then coexist inside a cusp-organized bistable region bounded by closed-form spinodals. Collapse occurs before the continuous divergence would have, recovery becomes hysteretic, and for alpha at or above one, simply lowering the load cannot restore function.

Core claim

When the self-referential feedback is made explicit by having each uncertified commitment spawn on average alpha new candidates, the continuous transition in recoverability turns first-order. Lucid and collapsed states coexist in a cusp-organized bistable region with closed-form spinodals. Collapse pre-empts the continuous divergence at a finite stability ratio. Recovery is hysteretic, and for alpha greater than or equal to one load reduction alone cannot restore operation. Cascade sizes remain bounded by the grounded fraction of input through a genealogy-times-congestion factorization, with the mean-field exponent tau equal to 3/2 recovered away from the boundary and each cascade carrying a

What carries the argument

The explicit feedback parameter alpha, the average number of new candidates spawned by each uncertified commitment, which organizes the bistable region and converts the transition from continuous to first-order.

If this is right

  • Cascade sizes obey a cutoff that increases as the grounded fraction of input decreases, recovering the mean-field exponent tau of 3/2 far from the boundary.
  • Each cascade produces a Landauer-priced burst of synthetic entropy.
  • Event-driven simulations reproduce both the cutoff law and the overall phase structure.
  • Recoverable dissipation becomes a necessary condition for decoder stability under high flux.
  • For alpha at or above one, metastable failure cannot be reversed by load reduction alone.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • High-throughput AI systems may accelerate non-recoverable loss when capacity is increased without added certification or gating.
  • Introducing explicit gating to keep alpha below one could eliminate the bistable region and restore continuous behavior.
  • Similar first-order collapses may appear in other overloaded self-referential systems such as distributed databases or biological regulatory networks.
  • Direct measurement of the predicted spinodal points in controlled decoder simulations would test the closed-form expressions for the bistable boundaries.

Load-bearing premise

Information processing acts as a thermodynamic load on finite-capacity decoders whose irreversible commitments eliminate options, and recoverability is fixed by a feasibility margin and stability diagnostic that do not depend on optimization goals, control rules, or hardware.

What would settle it

An experiment or simulation that records simultaneous lucid and collapsed decoder states at identical load values, together with a discontinuous jump in the stability diagnostic at a predicted spinodal, would confirm the first-order character.

Figures

Figures reproduced from arXiv: 2606.24861 by Pieter van Rooyen.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
read the original abstract

We study adaptive systems coupling inference to irreversible action under sustained nonequilibrium driving. Treating information processing as a thermodynamic load, we model them as finite-capacity decoders whose irreversible commitments eliminate counterfactual options, and characterize recoverable operation by a feasibility margin and a stability diagnostic fixing when irreversible action is admissible. Under sustained overload -- induced flux exceeding effective integrative capacity -- loss of recoverability and divergence of the diagnostic arise as structural consequences of capacity saturation, independent of optimization objective, control policy, or substrate. Added capacity alone does not restore recoverability: absent certification or gating, higher throughput accelerates non-recoverable loss, with high-throughput AI a concrete application. Making the feedback explicit -- each uncertified commitment spawning on average alpha new candidates -- turns the continuous transition first-order: lucid and collapsed states coexist in a cusp-organized bistable region with closed-form spinodals, collapse pre-empts the continuous divergence at finite stability ratio, recovery is hysteretic, and for alpha >= 1 load reduction alone cannot restore operation. Cascade sizes are bounded by the grounded fraction of input: a genealogy-times-congestion factorization sets a cutoff that grows as grounding shrinks, with the mean-field exponent tau = 3/2 recovered away from the boundary and each cascade carrying a Landauer-priced burst of synthetic entropy; event-driven simulations confirm the cutoff law and phase structure. This supplies the statistical mechanics of the metastable failures seen in distributed systems. The analysis is constraint-based and substrate-agnostic, establishing recoverable dissipation as a necessary criterion for decoder stability in high-flux regimes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 0 minor

Summary. The manuscript models adaptive information-processing systems as finite-capacity decoders subject to irreversible commitments under sustained overload. It claims that explicit feedback—each uncertified commitment spawning on average α new candidates—converts a continuous loss-of-recoverability transition into a first-order transition featuring a cusp-organized bistable region with closed-form spinodals, hysteretic recovery, and the property that load reduction alone cannot restore operation for α ≥ 1. Cascade-size distributions recover the mean-field exponent au = 3/2 away from the boundary, a genealogy-times-congestion cutoff is derived, and event-driven simulations are said to confirm the phase structure and cutoff law. The analysis is presented as constraint-based and independent of optimization objective, control policy, or substrate.

Significance. If the derivations and independence claims hold, the work supplies a statistical-mechanical account of metastable failures in distributed systems and high-throughput AI, linking capacity saturation and self-referential feedback to first-order recoverability collapse and providing falsifiable predictions (closed-form spinodals, au = 3/2, cutoff law) that could be tested across substrates.

major comments (3)
  1. [Abstract] Abstract: the central claim that the feasibility margin and stability diagnostic remain independent of control policy after the α-feedback term is introduced is load-bearing for the substrate-agnostic conclusion, yet the definition of 'uncertified' and the rate at which new candidates are generated are likely to depend on the specific commitment rule; inserting such dependence into the rate equations would generically shift the spinodals and the α ≥ 1 non-recovery condition.
  2. [Abstract] Abstract: the mean-field exponent au = 3/2 is recovered 'in the standard way' once α is introduced; this raises the possibility that the reported first-order structure and cusp bistability are shaped by the choice of the feedback parameter rather than emerging as a purely structural consequence of capacity saturation.
  3. [Abstract] Abstract: closed-form spinodals, the cusp-organized bistable region, and the hysteretic recovery condition are asserted without any displayed rate equations, fixed-point analysis, or stability calculation, preventing verification that the first-order character is not an artifact of the mean-field closure.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting points that require clarification. We respond to each major comment below, focusing on the structural and mean-field aspects of the model.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the feasibility margin and stability diagnostic remain independent of control policy after the α-feedback term is introduced is load-bearing for the substrate-agnostic conclusion, yet the definition of 'uncertified' and the rate at which new candidates are generated are likely to depend on the specific commitment rule; inserting such dependence into the rate equations would generically shift the spinodals and the α ≥ 1 non-recovery condition.

    Authors: The model is formulated strictly in terms of capacity saturation and the mean feedback factor α arising from uncertified commitments. 'Uncertified' is defined as any commitment that exceeds the decoder's integrative capacity in the absence of external gating or certification; the rate equations close at the level of this average α. The resulting spinodals and the α ≥ 1 non-recovery threshold are therefore properties of the feedback structure under capacity constraint, independent of the microscopic commitment rule provided the mean α is preserved. Policy-specific effects would appear only in higher-order correlations outside the mean-field description, which is consistent with the constraint-based, substrate-agnostic framing. We will insert a clarifying sentence in the revised abstract. revision: partial

  2. Referee: [Abstract] Abstract: the mean-field exponent τ = 3/2 is recovered 'in the standard way' once α is introduced; this raises the possibility that the reported first-order structure and cusp bistability are shaped by the choice of the feedback parameter rather than emerging as a purely structural consequence of capacity saturation.

    Authors: Without the α-feedback term the loss-of-recoverability transition is a continuous supercritical pitchfork fixed by capacity saturation alone. Insertion of the linear feedback term α converts the bifurcation into a cusp catastrophe whose bistable region and spinodals are derived directly from the modified rate equations. The exponent τ = 3/2 is the universal mean-field branching-process value recovered in the interior of the collapsed phase, away from the spinodal boundary; its appearance does not determine the first-order character, which is controlled by the feedback-induced change in the fixed-point structure. revision: no

  3. Referee: [Abstract] Abstract: closed-form spinodals, the cusp-organized bistable region, and the hysteretic recovery condition are asserted without any displayed rate equations, fixed-point analysis, or stability calculation, preventing verification that the first-order character is not an artifact of the mean-field closure.

    Authors: The abstract summarizes results whose derivations appear in the body: the rate equations are stated in Section II, the fixed-point analysis and linear stability calculation are performed in Section III, and the closed-form spinodals together with the cusp condition follow from the resulting cubic equation. The mean-field closure is the standard large-N limit for the branching process; its validity is cross-checked by the event-driven simulations reported in Section IV. To aid readers we will add a parenthetical reference to the relevant sections in a revised abstract. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation remains self-contained under stated modeling assumptions

full rationale

The provided abstract and context introduce alpha explicitly as a tunable feedback parameter controlling the order of the transition, with the first-order character and spinodals derived from the resulting rate equations. The independence of feasibility margin and stability diagnostic from policy/substrate is stated as a premise of the constraint-based model rather than derived from the equations themselves. No self-citation chains, self-definitional loops, or fitted parameters renamed as predictions appear; the mean-field tau=3/2 is noted as recovered in the standard way. The central claims therefore do not reduce to their inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 2 invented entities

The central claim rests on modeling choices that treat decoder behavior as substrate-agnostic and on the explicit introduction of the feedback parameter alpha; these are not derived from more fundamental principles within the abstract.

free parameters (1)
  • alpha
    Average number of new candidates spawned per uncertified commitment; controls the order of the transition and the condition for irreversible collapse.
axioms (2)
  • domain assumption Information processing can be treated as a thermodynamic load on finite-capacity decoders whose irreversible commitments eliminate counterfactual options.
    Core modeling premise stated at the opening of the abstract.
  • domain assumption Loss of recoverability arises as a structural consequence of capacity saturation independent of optimization objective, control policy, or substrate.
    Independence claim that allows the result to be presented as general.
invented entities (2)
  • feasibility margin no independent evidence
    purpose: Characterizes recoverable operation of the decoder.
    Introduced as part of the recoverability definition; no independent evidence supplied in abstract.
  • stability diagnostic no independent evidence
    purpose: Fixes when irreversible action is admissible.
    Introduced as part of the recoverability definition; no independent evidence supplied in abstract.

pith-pipeline@v0.9.1-grok · 5810 in / 1717 out tokens · 31145 ms · 2026-06-25T21:40:56.410131+00:00 · methodology

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Works this paper leans on

56 extracted references · 4 linked inside Pith

  1. [1]

    Entropy Production Under Capacity Constraints a. Variational admissibility condition.For a de- coder with effective integrative capacityC n, consider the informational entropy production rateσ n(t) required to preserve recoverabilitywhile dissipating an induced flux Rself(t) — the cost of full certification, distinct from the synthetic production ˙Ssyn of...

  2. [2]

    Introducing a structural resolution parameterη∈(0,1), it is equivalently the sustained violationR self(t)> η C n over the decision horizon

    Marginal-Cost Crossover Between Decoder Classes The admissibility-elevation inequality (4) is the crossover at which the marginal dissipationcostof the current substrate exceeds that of an alternative operat- ing with margin to spare, so recoverable operation can no longer be sustained within the current capacity class. Introducing a structural resolution...

  3. [3]

    Falsifiability The framework is empirically falsifiable. It would be invalidated by a sustained regime in which the induced flux satisfiesR self ≳C self over extended intervals, ir- reversible actions continue, and yet both the feasibil- ity marginM(t) and the stability ratioSR(t) remain bounded without explicit gating or capacity increase — showing that ...

  4. [4]

    (9) solveF(x) =ℓ 0, and under the natural relaxation dy- namics ˙x∝ℓ 0 −F(x) a fixed point is stable iffF ′(x)>0, whereF ′(x) = 1−αe −(1−x)θ(1 +θx)

    Fixed points, fold, and cusp WithF(x)≜x−αxe −(1−x)θ, steady states of Eq. (9) solveF(x) =ℓ 0, and under the natural relaxation dy- namics ˙x∝ℓ 0 −F(x) a fixed point is stable iffF ′(x)>0, whereF ′(x) = 1−αe −(1−x)θ(1 +θx). Forα(1 +θ)< 1,Fis monotone and a unique lucid state exists for ℓ0 < F(1) = 1−α— the continuous boundary. For α(1 +θ)>1,Fhas an interio...

  5. [5]

    Branching identities Each commitment is uncertified with probabilityP u and then spawns meanαoffspring, so the genealogical ratio isb=αP u(x∗). From Eq. (9),αx ∗Pu =x ∗ −ℓ 0, henceb= 1−ℓ 0/x∗ and⟨s⟩= 1/(1−b) =x ∗/ℓ0 (cascade size = load amplification; a parameter-free internal check of the simulation, satisfied to∼1%). Subcritical Galton– Watson (Otter) s...

  6. [6]

    Simulation methods Event-driven, seeded simulation of the closed loop: ex- ogenous Poisson arrivals at rateλ 0 (aggregateN λ 0 for the fleet);NFCFS servers (defaultN= 1) with Exp(µ) service; a job whose sojourn exceeds ∆tfires uncertified at its deadlineand spawns Poisson(α) offspring (routed uniformly across servers in the fleet), with an optional ex- po...

  7. [7]

    Prigogine and I

    I. Prigogine and I. Stengers,Order Out of Chaos: Man ’s New Dialogue with Nature(Bantam Books, New York, 1984)

    1984

  8. [8]

    Nicolis and I

    G. Nicolis and I. Prigogine,Self-Organization in Nonequi- librium Systems(Wiley, New York, 1977)

    1977

  9. [9]

    J. L. England, Statistical physics of self-replication, J. Chem. Phys.139, 121923 (2013)

    2013

  10. [10]

    C. E. Shannon, A mathematical theory of communica- tion, The Bell System Technical Journal27, 379 (1948)

    1948

  11. [11]

    T. M. Cover and J. A. Thomas,Elements of Information Theory, 2nd ed. (Wiley, Hoboken, NJ, 2006)

    2006

  12. [12]

    Landauer, Irreversibility and heat generation in the computing process, IBM Journal of Research and Devel- opment5, 183 (1961)

    R. Landauer, Irreversibility and heat generation in the computing process, IBM Journal of Research and Devel- opment5, 183 (1961)

    1961

  13. [13]

    C. H. Bennett, The thermodynamics of computation—a review, International Journal of Theoretical Physics21, 905 (1982)

    1982

  14. [14]

    S. R. de Groot and P. Mazur,Non-Equilibrium Thermo- dynamics(Dover Publications, New York, 1984)

    1984

  15. [15]

    H. B. Callen,Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (Wiley, New York, 1985)

    1985

  16. [16]

    S. B. Laughlin, Energy as a constraint on the coding and processing of sensory information, Current Opinion in Neurobiology11, 475 (2001)

    2001

  17. [17]

    H. B. Barlow, Possible principles underlying the trans- formations of sensory messages, Sensory Communication , 217 (1961)

    1961

  18. [18]

    H. A. Simon,Models of Man: Social and Rational(Wiley, New York, 1957)

    1957

  19. [19]

    Tsallis, Introduction to nonextensive statistical me- chanics, Springer (2009), book-length monograph

    C. Tsallis, Introduction to nonextensive statistical me- chanics, Springer (2009), book-length monograph

    2009

  20. [20]

    van Rooyen, Entropy, annealing, and the continuity of agency in Human–AI systems, Preprints202601, 0688 (2026)

    P. van Rooyen, Entropy, annealing, and the continuity of agency in Human–AI systems, Preprints202601, 0688 (2026)

    2026

  21. [21]

    P. van Rooyen, Entropy, annealing, and the continuity of agency in Human–AI systems, inEntropy 2026: Explor- ing Complexity and Information in Science(Barcelona, Spain, 2026) peer-reviewed abstract accepted; Informa- tion Theory, Data Science and Artificial Intelligence ses- sion

    2026

  22. [22]

    Entropy 2026: Exploring Complexity and Information in Science

    P. van Rooyen, Entropy, capacity, and the continuity of agency in Human–AI systems: A recoverable self- coding account,Entropy(MDPI), Special Issue “Entropy 2026: Exploring Complexity and Information in Science” (2026), forthcoming; the open-loop companion to the present work

    2026

  23. [23]

    P. van Rooyen, Entropy, capacity, and the continuity of agency in Human–AI systems: A recoverable self-coding account, inPhysical Sciences Forum (Proceedings of En- tropy 2026)(2026) in press

    2026

  24. [24]

    Wald,Sequential Analysis(Wiley, New York, 1947)

    A. Wald,Sequential Analysis(Wiley, New York, 1947)

    1947

  25. [25]

    Bogacz, E

    R. Bogacz, E. Brown, J. Moehlis, P. Holmes, and J. D. Cohen, The physics of optimal decision mak- ing: A formal analysis of models of performance in two-alternative forced-choice tasks, Psychological Review 113, 700 (2006)

    2006

  26. [26]

    J. I. Gold and M. N. Shadlen, The neural basis of decision making, Annual Review of Neuroscience30, 535 (2007)

    2007

  27. [27]

    Bronson, A

    N. Bronson, A. Aghayev, A. Charapko, and T. Zhu, Metastable failures in distributed systems, inProceed- ings of the Workshop on Hot Topics in Operating Systems (HotOS ’21)(2021) pp. 221–227

    2021

  28. [28]

    Lloyd, Ultimate physical limits to computation, Na- ture406, 1047 (2000)

    S. Lloyd, Ultimate physical limits to computation, Na- ture406, 1047 (2000)

    2000

  29. [29]

    Lloyd, Computational capacity of the universe, Phys

    S. Lloyd, Computational capacity of the universe, Phys. Rev. Lett.88, 237901 (2002)

    2002

  30. [30]

    E. T. Jaynes, Information theory and statistical mechan- ics, Physical Review106, 620 (1957)

    1957

  31. [31]

    Kleinrock,Queueing Systems, Volume 1: Theory(Wi- ley, New York, 1975)

    L. Kleinrock,Queueing Systems, Volume 1: Theory(Wi- ley, New York, 1975)

    1975

  32. [32]

    J. F. C. Kingman, The single server queue in heavy traf- fic, Mathematical Proceedings of the Cambridge Philo- sophical Society57, 902 (1961)

    1961

  33. [33]

    Whitt,Stochastic-Process Limits(Springer, New York, 2002)

    W. Whitt,Stochastic-Process Limits(Springer, New York, 2002)

    2002

  34. [34]

    F. P. Kelly,Reversibility and Stochastic Networks(Wiley, Chichester, 1979)

    1979

  35. [35]

    Hu, W.-X

    M.-B. Hu, W.-X. Wang, R. Jiang, Q.-S. Wu, and Y.-H. Wu, Phase transition and hysteresis in scale-free network traffic, Physical Review E75, 036102 (2007)

    2007

  36. [36]

    Jamshidi, A

    S. Jamshidi, A. Moradi Dakhel, K. W. Nafi, and F. Khomh, Hallucination cascade: Analyzing error prop- agation in multi-agent LLM systems, arXiv preprint arXiv:2606.07937 (2026)

    Pith/arXiv arXiv 2026

  37. [37]

    J. R. Artalejo and A. G´ omez-Corral,Retrial Queueing Systems: A Computational Approach(Springer, Berlin, 2008)

    2008

  38. [38]

    Phung-Duc, Retrial queueing models: A survey on theory and applications, inStochastic Operations Re- search in Business and Industry(World Scientific, 2017)

    T. Phung-Duc, Retrial queueing models: A survey on theory and applications, inStochastic Operations Re- search in Business and Industry(World Scientific, 2017)

    2017

  39. [39]

    Touchette, The large deviation approach to statistical mechanics, Physics Reports478, 1 (2009)

    H. Touchette, The large deviation approach to statistical mechanics, Physics Reports478, 1 (2009)

    2009

  40. [40]

    Scheffer, J

    M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin, S. R. Carpenter, V. Dakos, H. Held, E. H. van Nes, M. Ri- etkerk, and G. Sugihara, Early-warning signals for critical transitions, Nature461, 53 (2009)

    2009

  41. [41]

    Sagawa and M

    T. Sagawa and M. Ueda, Second law of thermodynamics with discrete quantum feedback control, Physical Review Letters100, 080403 (2008)

    2008

  42. [42]

    Shumailov, Z

    I. Shumailov, Z. Shumaylov, Y. Zhao, N. Papernot, R. Anderson, and Y. Gal, AI models collapse when trained on recursively generated data, Nature631, 755 (2024)

    2024

  43. [43]

    Alemohammad, J

    S. Alemohammad, J. Casco-Rodriguez, L. Luzi, A. I. Hu- mayun, H. Babaei, D. LeJeune, A. Siahkoohi, and R. G. Baraniuk, Self-consuming generative models go MAD, inInternational Conference on Learning Representations (ICLR)(2024)

    2024

  44. [44]

    T. B. Brown, B. Mann, N. Ryder, M. Subbiah, J. Ka- plan,et al., Language models are few-shot learners, Ad- 19 vances in Neural Information Processing Systems33, 1877 (2020)

    2020

  45. [45]

    Bommasani, D

    R. Bommasani, D. A. Hudson, E. Adeli, R. Altman, S. Arora,et al., On the opportunities and risks of foun- dation models, arXiv preprint arXiv:2108.07258 (2021)

    Pith/arXiv arXiv 2021

  46. [46]

    Vaswani, N

    A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, At- tention is all you need, Advances in Neural Information Processing Systems30(2017)

    2017

  47. [47]

    Tishby, F

    N. Tishby, F. C. Pereira, and W. Bialek, The informa- tion bottleneck method, arXiv preprint physics/0004057 (2000)

    Pith/arXiv arXiv 2000

  48. [48]

    Shwartz-Ziv and N

    R. Shwartz-Ziv and N. Tishby, Opening the black box of deep neural networks via information, arXiv preprint arXiv:1703.00810 (2017)

    Pith/arXiv arXiv 2017

  49. [49]

    Achille and S

    A. Achille and S. Soatto, Information dropout: Learn- ing optimal representations through noisy computation, IEEE Transactions on Pattern Analysis and Machine In- telligence40, 2897 (2018)

    2018

  50. [50]

    Behrmann, W

    J. Behrmann, W. Grathwohl, R. T. Q. Chen, D. Duve- naud, and J.-H. Jacobsen, Invertible residual networks, Proceedings of the 36th International Conference on Ma- chine Learning (2019)

    2019

  51. [51]

    Geiger, H

    A. Geiger, H. Lu, T. Icard, and C. Potts, Causal abstrac- tions of neural networks, Advances in Neural Information Processing Systems34(2021)

    2021

  52. [52]

    E. J. Chaisson,Cosmic Evolution: The Rise of Com- plexity in Nature(Harvard University Press, Cambridge, MA, 2001)

    2001

  53. [53]

    R. J. Glauber, Time-dependent statistics of the ising model, Journal of Mathematical Physics4, 294 (1963)

    1963

  54. [54]

    Seifert, Stochastic thermodynamics, fluctuation the- orems and molecular machines, Reports on Progress in Physics75, 126001 (2012)

    U. Seifert, Stochastic thermodynamics, fluctuation the- orems and molecular machines, Reports on Progress in Physics75, 126001 (2012)

    2012

  55. [55]

    Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Re- views of Modern Physics48, 571 (1976)

    J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Re- views of Modern Physics48, 571 (1976)

    1976

  56. [56]

    van Rooyen, cascade-collapse: simulation code,https: //github.com/Pietervr/cascade-collapse(2026)

    P. van Rooyen, cascade-collapse: simulation code,https: //github.com/Pietervr/cascade-collapse(2026)

    2026