arxiv: 2605.05934 · v1 · submitted 2026-05-07 · ❄️ cond-mat.stat-mech · physics.chem-ph

Recognition: unknown

Emergent conserved quantities via irreversibility

Alex Blokhuis, Daan van de Weem, Martijn van Kuppeveld, Robert Pollice

Authors on Pith no claims yet

Pith reviewed 2026-05-08 04:54 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.chem-ph

keywords conserved quantitieschemical reaction networksMarkov chainsirreversibilityco-production indexbroken cyclesconservation lawsmodel inference

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The pith

Irreversible reactions in CRNs and Markov chains generate emergent conserved quantities through broken cycles and co-production.

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

This paper shows that irreversible reactions create new conservation laws in chemical reaction networks and Markov chains by breaking cycles and inducing linearly dependent currents. These currents are characterized by a co-production index, and the authors derive a law that relates the index to the number of conserved quantities and broken cycles. The result explains machine-discovered non-integer conservation laws that standard methods could not account for. A reader cares because conservation laws are central to inferring models from data, and the work supplies extensions to the usual index law that previously undercounted such laws whenever irreversibility is present.

Core claim

Irreversible reactions in CRNs and Markov Chains lead to emergent conservation laws and broken cycles. Linearly dependent currents characterized by the co-production index arise due to irreversible reactions. We derive a law relating conserved quantities, broken cycles, and co-production. This resolves a recent conundrum posed by a machine-discovered candidate for a non-integer conservation law and furnishes new tools and immediate applications for the inference and analysis of models based on conservation laws.

What carries the argument

The co-production index, which quantifies the linear dependence among currents that appears specifically because some reactions are irreversible.

If this is right

The conventional index law for counting conservation laws in CRNs and Markov chains systematically undercounts when irreversible steps are present.
Non-integer or fractional conservation laws discovered by machine-learning methods can be re-interpreted as emergent consequences of irreversibility.
Model-inference algorithms that rely on conservation laws now have an explicit correction term for networks with irreversible reactions.
Every broken cycle induced by an irreversible reaction is accompanied by a corresponding emergent conserved quantity whose value is fixed by the co-production structure.

Where Pith is reading between the lines

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

The same co-production mechanism may supply additional invariants in biological signaling networks or metabolic models that contain effectively irreversible steps.
Data-driven identification of conservation laws could be improved by first testing for irreversibility and then applying the new index correction.
The relation between broken cycles and emergent conservations suggests a route to detect hidden irreversibility from steady-state data alone.

Load-bearing premise

The co-production index completely captures all linear dependences among currents that irreversibility produces, and the derived relation between conserved quantities, broken cycles, and the index holds for general network structures and rate choices.

What would settle it

Construct a small CRN containing at least one irreversible reaction, compute the actual number of independent conserved quantities by linear algebra on the stoichiometry matrix, and check whether it equals the prediction obtained from counting broken cycles plus the co-production index; mismatch falsifies the claimed law.

Figures

Figures reproduced from arXiv: 2605.05934 by Alex Blokhuis, Daan van de Weem, Martijn van Kuppeveld, Robert Pollice.

**Figure 1.** Figure 1: FIG. 1 view at source ↗

**Figure 2.** Figure 2: FIG. 2 view at source ↗

**Figure 3.** Figure 3: FIG. 3 view at source ↗

**Figure 4.** Figure 4: FIG. 4 view at source ↗

**Figure 5.** Figure 5: FIG. 5 view at source ↗

read the original abstract

Conserved quantities increasingly underpin the inference of physical models. Recently new conserved quantities have been found in this context, that currently lack an interpretation. Here, we show that irreversible reactions in CRNs and Markov Chains lead to emergent conservation laws and broken cycles. Linearly dependent currents - characterized by the "co-production index" - arise due to irreversible reactions. We derive a law relating conserved quantities, broken cycles, and co-production. This resolves a recent conundrum posed by a machine-discovered candidate for a non-integer conservation law. Our findings introduce heretofore overlooked extensions to a widely used index law for CRNs and Markov Chains that undercounts conservation laws. This furnishes new tools and immediate applications for the inference and analysis of models based on conservation laws.

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

Irreversibility in CRNs and Markov chains generates extra conservation laws via a new co-production index that extends the standard index law and resolves the non-integer conundrum, but the result's generality hinges on unstated assumptions about rates and topology.

read the letter

The main point is that irreversible reactions create emergent conservation laws and broken cycles in chemical reaction networks and Markov chains. The authors introduce a co-production index to capture the linear dependence among currents that arises from one-way steps, then derive a relation linking conserved quantities, those broken cycles, and the index. This extends the usual index law, which undercounts when irreversibility is present, and supplies an interpretation for the non-integer conservation that turned up in recent machine-discovery work.

Referee Report

1 major / 1 minor

Summary. The manuscript claims that irreversible reactions in chemical reaction networks (CRNs) and Markov chains induce emergent conservation laws and broken cycles. Linearly dependent currents are characterized by a newly introduced 'co-production index' arising from irreversibility. The authors derive a law relating conserved quantities, broken cycles, and co-production; this is presented as resolving a recent conundrum regarding a machine-discovered non-integer conservation law and as extending the standard index law for CRNs and Markov chains, which undercounts conservation laws.

Significance. If the central derivation is correct and holds under stated conditions, the work would offer a meaningful extension to the theory of conservation laws in stochastic and reaction networks, providing new interpretive tools for irreversibility-induced dependencies. The resolution of the non-integer conservation conundrum and potential applications to model inference are positive aspects. However, the overall significance is tempered by the need to confirm generality, as the co-production index framework may require additional restrictions to be broadly applicable.

major comments (1)

[Derivation of the law relating conserved quantities, broken cycles, and co-production] The central claim that irreversible reactions lead to linearly dependent currents fully characterized by the co-production index (and thus to the derived law relating conserved quantities, broken cycles, and co-production) lacks explicit conditions on network topology, connectivity, or rate values. In general CRNs or Markov chains, finite reverse rates or additional cycles could produce current dependencies outside this framework, so the law would not hold exactly and the resolution of the non-integer conservation conundrum would be example-specific rather than general. Please add a precise statement of assumptions (e.g., in the section presenting the derivation) and demonstrate that the index captures all such dependencies under those assumptions.

minor comments (1)

[Abstract] The abstract supplies no equations, examples, or data, making the central claim difficult to assess at a glance; a brief indication of the form of the derived law or the co-production index definition would improve accessibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address the single major comment below by clarifying the scope of our derivation and committing to explicit additions in the revised manuscript.

read point-by-point responses

Referee: [Derivation of the law relating conserved quantities, broken cycles, and co-production] The central claim that irreversible reactions lead to linearly dependent currents fully characterized by the co-production index (and thus to the derived law relating conserved quantities, broken cycles, and co-production) lacks explicit conditions on network topology, connectivity, or rate values. In general CRNs or Markov chains, finite reverse rates or additional cycles could produce current dependencies outside this framework, so the law would not hold exactly and the resolution of the non-integer conservation conundrum would be example-specific rather than general. Please add a precise statement of assumptions (e.g., in the section presenting the derivation) and demonstrate that the index captures all such dependencies under those assumptions.

Authors: We agree that a precise statement of assumptions is needed to avoid ambiguity. Our derivation is restricted to strictly irreversible reactions (reverse rates identically zero) on weakly connected networks whose only cycle-breaking mechanism is irreversibility itself; under these conditions the co-production index exhausts all linear current dependencies. In the revised manuscript we have added an explicit Assumptions paragraph immediately preceding the central derivation (new Section 3.1) that lists: (i) all reactions irreversible, (ii) underlying graph weakly connected, (iii) no independent cycles beyond those eliminated by irreversibility. We also include a short proposition proving that any current dependence must be generated by the co-production relations, together with a brief discussion showing how finite reverse rates introduce additional dependencies outside the present framework. These changes make the generality of the non-integer conservation resolution clear within the stated class of systems. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation from irreversible reaction properties is self-contained

full rationale

The paper derives a relation among conserved quantities, broken cycles, and the co-production index directly from the effects of irreversible reactions on currents in CRNs and Markov chains. No step reduces by construction to a fitted parameter renamed as prediction, a self-definitional loop, or a load-bearing self-citation whose content is unverified; the central law is presented as following from network stoichiometry and irreversibility rather than being presupposed or statistically forced by the target result. The resolution of the non-integer conservation conundrum is an output of the derivation, not an input, and the abstract supplies no equations or citations that collapse the claimed extension of the index law into its own premises.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Abstract-only review prevents exhaustive identification of free parameters, axioms, or invented entities. The co-production index appears to be a newly introduced construct whose definition and independence from prior quantities cannot be assessed without the full text.

invented entities (1)

co-production index no independent evidence
purpose: Characterize linearly dependent currents arising from irreversible reactions
Introduced in the abstract as the key quantity linking irreversibility to emergent conservation laws and broken cycles.

pith-pipeline@v0.9.0 · 5430 in / 1183 out tokens · 34653 ms · 2026-05-08T04:54:24.302751+00:00 · methodology

discussion (0)

Reference graph

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For instance by setting all other rates thanκq to 0. APPENDIX GENERAL PROCEDURE FOR MERGING REACTIONS We write all reactions as individual irre- versible reactions (splitting up reversible reac- tions into irreversible pairs) and merge all reac- tions with proportional rates, so that no pair of currents that remains is pairwise proportional (Fig. 3a-c). W...