arxiv: 2602.17150 · v2 · submitted 2026-02-19 · ⚛️ physics.hist-ph

Recognition: 2 theorem links

· Lean Theorem

The Causal Second Law

Balazs Gyenis

Authors on Pith no claims yet

Pith reviewed 2026-05-15 21:24 UTC · model grok-4.3

classification ⚛️ physics.hist-ph

keywords causal second lawspecial sciencescausal entropythermodynamicsstatistical mechanicscausationtime asymmetrysupervenience

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The pith

If a special science meets physicalist assumptions, its causal regularities have an entropy that cannot decrease from robust cause to effect.

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

The paper establishes that special sciences satisfying certain assumptions shared with physicalism and physics possess causal regularities that come with an associated entropy. This entropy cannot decrease when going from a robust cause to its effect, mirroring the second law of thermodynamics. A reader might care because this offers a way to ground time-directedness in special sciences directly from their causal structure without full reduction to physics. The argument proves this causal second law, provides sufficient conditions for entropy increase, relates it to statistical mechanics, and shows it survives objections like reversibility while remaining compatible with supervenience and open systems.

Core claim

If a special science satisfies key assumptions familiar from physicalist accounts of the special sciences and from physics, then its causal regularities have an associated notion of entropy, and this causal entropy cannot decrease from a robust cause to its effect. Due to its analogy with the second laws of thermodynamics and statistical physics, this conclusion is called the causal second law. The paper clarifies the assumptions, proves the law, gives sufficient conditions for causal entropy increase, relates the result to statistical mechanics and thermodynamics, and argues that the reversibility objection does not threaten it.

What carries the argument

Causal entropy, the entropy notion associated with the causal regularities of the special science under the physicalist assumptions, which is shown to be non-decreasing from robust cause to effect.

Load-bearing premise

The special science satisfies the key assumptions familiar from physicalist accounts of the special sciences and from physics that allow associating causal entropy with its regularities.

What would settle it

Observing a robust causal regularity in a special science, such as one in biology or economics, where the associated causal entropy decreases from cause to effect would falsify the central claim.

Figures

Figures reproduced from arXiv: 2602.17150 by Balazs Gyenis.

**Figure 1.** Figure 1: Left: an Euler diagram of special science states of affairs: a situation, two causal factors, and an effect. Right: state-supervenience: correspondence of special science descriptions to physical states. causes precede their effects from an entropy increase in thermodynamics (Section 6.3); ask whether, in the light of the results of this paper, it is necessary to invoke thermodynamics to show that special… view at source ↗

**Figure 2.** Figure 2: Left: time evolution of one physical state, and (joint) time evolution of a set of physical states. Right: a robust causal regularity, defined with respect to a dynamical system, connects sets of physical states. First, I assume what I call state-supervenience of a given special science on physics: that every description of states of affairs of the special science corresponds to a set of physical states th… view at source ↗

**Figure 3.** Figure 3: If there are multiple distinct possible causes which robustly lead to the same effect, then the phase volume of the effect is larger than the phase volume of any of the possible causes. If there are multiple distinct possible causes that robustly lead to the same effect, then the causal entropy in any actual robust causal process must increase from the cause to the effect. Following the argument of the pre… view at source ↗

**Figure 4.** Figure 4: Left: the set of all physical states instantiating the effect. Right: the set of all physical states that evolve, in a characteristic time t ∗ , to the effect. other possible cause is present,’ ‘neither of the causal factors nor the effect is present,’ etc. However, in our example, the pull-back of the effect (the entire curvy, gray-colored 36-cell region on the right side) cannot be described by the speci… view at source ↗

**Figure 5.** Figure 5: Portional causal regularities with efficacy α = 3 4 (lit lighter) and α = 1 2 (burning match). ber of distinct possible causes exist for the same effect. A relevant sufficient condition, which depends on the efficacy α and on the phase volume of the distinct possible causes, can be formulated precisely (Proposition A.5). In general, multiplicity of possible causes for the same effect is ubiquitous in the … view at source ↗

**Figure 6.** Figure 6: Robust retro-causal regularity: almost all physical states instantiating the cause arrived, in a characteristic time, from the set of physical states instantiating the effect. from the set of physical states instantiating the first state of affairs, but many other physical states instantiating the first state of affairs do not evolve to physical states instantiating the second. For such a relationship betw… view at source ↗

**Figure 7.** Figure 7: A robust causal regularity, defined with respect to transition relative frequencies, connects special science descriptions, and hence can be illustrated without depicting the physical states corresponding to the cause and the effect. of special science descriptions to certain moments of time are, in general, probabilistic, and often can be represented by transition relative frequencies. Thus, if we assume… view at source ↗

read the original abstract

I argue that if a special science satisfies certain key assumptions that are familiar from physicalist accounts of the special sciences and from physics, then its causal regularities have an associated notion of entropy, and that this causal entropy cannot decrease from a robust cause to its effect. Due to its analogy with the second laws of thermodynamics and statistical physics, I call the latter conclusion the causal second law. In this paper, I clarify the key assumptions, prove the causal second law, give sufficient conditions for causal entropy increase, relate the causal second law to statistical mechanics and thermodynamics, and argue that the reversibility objection does not threaten it. In addition, I claim that the causal second law is compatible with a non-metaphysical understanding of supervenience and the open systems view, argue that it does not imply a causal time arrow, reflect on relaxing the robustness condition, question whether it is necessary to invoke thermodynamics to show that special sciences' time arrows exist, and discuss a transition-relative-frequency-based, special-science-internal characterization of causal regularities.

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.

Desk Editor's Note private letter to a colleague

The paper derives a conditional causal second law by applying the standard conditional-entropy inequality to robust causal regularities under physicalist assumptions on special sciences.

read the letter

The main thing to know is that if a special science meets the usual physicalist conditions—robust, probabilistic causal mappings between state types—then its causal regularities come with an entropy that cannot decrease from cause to effect. The derivation follows directly from the fact that conditional entropy is non-negative for any Markov stochastic map, with robustness used to break time-reversal symmetry and address reversibility worries.

Referee Report

1 major / 1 minor

Summary. The paper claims that if a special science satisfies key assumptions familiar from physicalist accounts—causal regularities as robust, exception-tolerant mappings between state types admitting a well-defined probability measure—then these regularities have an associated causal entropy (defined via the standard Shannon form over cause-to-effect transition probabilities), and this entropy cannot decrease from a robust cause to its effect. The manuscript clarifies the assumptions, proves the causal second law via the conditional-entropy inequality for Markovian maps, provides sufficient conditions for increase, relates the result to statistical mechanics and thermodynamics, addresses the reversibility objection using the robustness clause, and discusses compatibility with non-metaphysical supervenience, the open-systems view, and the absence of an implied causal time arrow.

Significance. If the conditional result holds, it supplies a parameter-free, information-theoretic bridge between causal structure in special sciences and an entropic arrow without direct thermodynamic invocation or metaphysical commitments. The use of standard Shannon entropy and the Markov inequality makes the claim falsifiable via empirical transition frequencies, and the explicit handling of robustness to break time-reversal symmetry is a clear strength.

major comments (1)

The abstract states that a proof is given, yet the explicit steps for defining causal entropy from the transition probabilities and verifying the Markov property under the robustness assumption are not visible in sufficient detail to check for derivation gaps or post-hoc choices in the probability measure; this is load-bearing for the non-decrease claim.

minor comments (1)

The discussion of relaxing the robustness condition and the transition-relative-frequency characterization of regularities would benefit from a short illustrative example with concrete state types to improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the paper's significance, and constructive feedback on the proof presentation. We address the single major comment below and will revise the manuscript accordingly to improve clarity without altering the core claims or results.

read point-by-point responses

Referee: The abstract states that a proof is given, yet the explicit steps for defining causal entropy from the transition probabilities and verifying the Markov property under the robustness assumption are not visible in sufficient detail to check for derivation gaps or post-hoc choices in the probability measure; this is load-bearing for the non-decrease claim.

Authors: We agree that greater explicitness in the proof steps would strengthen the manuscript. Causal entropy is defined in Section 2 as the standard Shannon entropy H(E|C) computed directly from the cause-to-effect transition probabilities P(E|C) induced by the robust regularities. The Markov property follows from the robustness clause (which ensures that the mapping depends only on the cause type and is insensitive to microstate details within the type), and the non-decrease result is obtained in Theorem 1 of Section 3 via the standard conditional-entropy inequality H(E|C) ≥ H(E'|E) for the composed Markov chain. To eliminate any potential ambiguity, we will expand the proof in the revised version with numbered derivation steps, an explicit statement of how the probability measure is fixed by the robustness assumption (no post-hoc choice is involved), and a short appendix deriving the inequality from first principles. This addresses the load-bearing concern directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; conditional theorem from definitions and standard inequalities

full rationale

The paper states a conditional claim: under assumptions familiar from physicalist accounts (robust causal regularities as exception-tolerant mappings admitting a probability measure), causal entropy is defined via the standard Shannon form on cause-to-effect transition probabilities, and non-decrease follows directly from the conditional-entropy inequality that holds for any Markovian stochastic map. This is a deductive consequence of the definitions and information theory, not a reduction of the result to its inputs by construction, self-citation, or fitted renaming. No load-bearing self-citations, ansatzes smuggled via prior work, or uniqueness theorems from the same authors are invoked to force the outcome. The result remains a straightforward conditional theorem whose content is independent of the target conclusion once the premises are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The argument rests on the key physicalist and physical assumptions about special sciences plus the introduction of causal entropy; no numerical free parameters are mentioned.

axioms (1)

domain assumption Special sciences satisfy key assumptions familiar from physicalist accounts and from physics.
Invoked as the premise that enables the association of causal entropy with regularities.

invented entities (1)

causal entropy no independent evidence
purpose: A measure of disorder or uncertainty associated with causal regularities in special sciences.
Postulated to state and prove the non-decrease result; no independent falsifiable evidence outside the derivation is indicated.

pith-pipeline@v0.9.0 · 5464 in / 1268 out tokens · 34044 ms · 2026-05-15T21:24:02.904159+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

state-supervenience and measure-preservation immediately yield ... The (dynamical) causal second law (for robust causal regularities): causal entropy cannot decrease from a robust cause to its effect.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

defining the causal entropy of the effect and the cause as the number of their physical instantiations

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

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“Regularity and Inferential Theories of Causation.” InThe Stanford Encyclopedia of Philosophy. Fall 2021, edited by E. N. Zalta. Stanford University Press. Callender, C. 2017.What Makes Time Special?Oxford University Press.https://doi.org/ 10.1093/oso/9780198797302.001.0001. Callender, C

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Backward Causation

“Backward Causation.” InThe Stanford Encyclopedia of Philosophy. Summer 2024, edited by E. N. Zalta. Stanford University Press. Fazekas, P., B. Gyenis, G. Hofer-Szab´ o, and G. Kert´ esz

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A Field Guide to Recent Work on the Foundations of Statistical Mechanics

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Causation in Physics

“Causation in Physics.” InThe Stanford Encyclopedia of Philosophy. Winter 2023, edited by E. N. Zalta. Stanford University Press. G¨ om¨ ori, M., B. Gyenis, and G. Hofer-Szab´ o

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Bluff your way in the Second Law of Thermodynamics

“Bluff your way in the Second Law of Thermodynamics.”Studies in the History and Philosophy of Modern Physics32: 305–394.https://doi.org/10.1016/ S1355-2198(01)00016-8. APPENDIX A.1 Propositions for Section 2 Suppose{d i}i∈I⊆N are the “natural” descriptions of states of affairs that may obtain at a moment in time (e.g., possible causes and effects), given ...

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[14]

(L7)IfC t∗ ⇝E, thenU tC t∗−t ⇝ Efor every 0≤t < t ∗

Proof.Apply measure-preservation tom(C) =m(C∩U −t∗E), then use (L1). (L7)IfC t∗ ⇝E, thenU tC t∗−t ⇝ Efor every 0≤t < t ∗. Proof.IfC t∗ ⇝E, then by (L3)m(C) =m(C∩U −t∗E), and then, by measure-preservation and (L1): m(UtC) =m(U t(C∩U −t∗E)) =m(U tC∩U tU−t∗E) =m(U tC∩U −(t∗−t)E), and thus, by (L3), UtC t∗−t ⇝ E. Proposition A.1.Suppose(P,Σ, m, U t)is a measu...

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In any such case,m(E)> m(C)

Proposition A.3.There is an example of a(P,Σ, R,R, T, m, U t)measure-preserving dynamical system,C, E∈ R, such thatC t∗ ⇝E, but there exists noS∈ Rfor whichS≈ m U−t∗E. In any such case,m(E)> m(C). Proof.It is easy to find such examples. For instance, letPbe the unit circle,mthe uni- form distribution onP,U t a clockwise rotation by radianst, C .= S −0.1<t...

work page 2026
[16]

What connects the two characterizations? Definition A.10.Let(D,D, T,{µ u}u∈U)be a partial transition structure and(P,Σ, R,R, T, m, U t) a dynamical system

31 Noˆ us, 2026 We gave two characterizations of a robust causal regularity, one with respect to (D,D, T,{µ u}u∈U), another with respect to (P,Σ, m, U t). What connects the two characterizations? Definition A.10.Let(D,D, T,{µ u}u∈U)be a partial transition structure and(P,Σ, R,R, T, m, U t) a dynamical system. I say that(D,D, T,{µ u}u∈U) history-supervenes...

work page 2026