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REVIEW 3 major objections 7 minor 1 cited by

MaCE: General Mass Conserving Dynamics for Cellular Automata

T0 review · 3 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The MaCE rule attaches exact mass conservation to existing cellular automata, turning them into reservoirs of solitons and low-level evolutionary dynamics.

desk verdict A clean mass-conserving update rule with a solid PDE limit, but the empirical 'more interesting behaviours' claim lacks a matched baseline against plain Lenia. read the letter →

arxiv 2507.12306 v1 pith:YGMECJGU submitted 2025-07-16 nlin.CG cs.NEnlin.AO

Vassilis Papadopoulos , Etienne Guichard This is my paper · ORCID
classification nlin.CGcs.NEnlin.AO MSC 37B1568Q80
keywords massconservationcellularautomataLeniaMaCEupdaterulesoftmaxredistributioncontinuouslimitsolitonsartificiallife
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

MaCE is a one-line update rule that can be bolted onto nearly any existing cellular automaton to make it conserve total mass exactly. The rule has each cell hand out its mass to its neighbours in proportions set by an exponentiated 'affinity' score, and because every cell's mass is simply redistributed, the global total cannot change. The paper shows that in the smooth continuum this rule is a combination of diffusion and advection along affinity gradients, with a parameter β controlling their relative strength. When attached to a continuous cellular automaton such as Lenia, to neural cellular automata, and to discrete totalistic cellular automata, it produces abundant moving solitons, diverse morphologies within a single parameter set, and preliminary signs of resource-driven selection. The reason to care is that it offers a general, local, parallelizable route to resource-limited artificial-life worlds.

What carries the argument

The load-bearing object is the redistribution kernel $I(x',x)=e^{\beta A(x)}1_{x'\in N_x}/Z(x')$, a normalized softmax over the reflexive $3\times3$ Moore neighbourhood. $Z(x')$ is the sum of $e^{\beta A}$ over the neighbours of the donating cell, so the rule fits the general mass-conserving form $\rho^{t+1}(x)=\sum_{x'}I(x',x)\rho^t(x')$, and the reflexivity of the neighbourhood is exactly what lets the double sum in the mass-conservation proof interchange. The continuous limit is obtained by Taylor expansion in space and time, using smoothness of $\rho$ and $A$; the resulting PDE identifies $\beta$ as an inverse-temperature-like parameter and shows that the stability threshold $3\Delta t/\Delta x^2=1$ is the largest permissible time step for the discretization.

What would settle it

Run the MaCE update on a smooth two-dimensional Gaussian mass bump with a constant affinity field; if total mass drifts by more than machine precision across many steps, the exact-conservation claim fails, and if the bump's spreading rate does not match the $\beta=0$ diffusion constant 1 after rescaling, the continuous-limit derivation is wrong.

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Extended reading notes

Core claim

The central claim is that mass conservation can be imposed on an arbitrary cellular automaton by replacing its update with the MaCE rule: each cell redistributes all of its mass among its Moore-neighbourhood cells, with the fraction sent from donor to recipient equal to $e^{\beta A}/Z$, where $A$ is any real-valued affinity computed from the current state and $Z$ normalizes the sum over the donor's neighbourhood. The paper proves that total mass is conserved for any affinity field, and derives the continuous limit $\partial_t \rho = \Delta \rho - 2\beta\nabla\cdot(\rho\nabla A)$, showing that MaCE is a parameterized mix of diffusion and conservative advection. On Lenia and two other automaton classes, the paper reports that MaCE produces a broad variety of persistent moving structures, that roughly 83% of random parameter sets yield solitons under its sampling scheme, and that one parameter set can support many morphologically distinct solitons simultaneously. It also reports hints of intrinsic selection in mass-decaying worlds with food reintroduced, and successful extrinsic evolution toward mass-maximization and movement goals.

Load-bearing premise

The derivation of the continuous limit and the numerical-stability bound assumes the mass field and the affinity field are smooth on the grid scale, and that the affinity does not depend on the mass field in a way that tightens the stability threshold below $3\Delta t/\Delta x^2=1$.

Editorial extensions

If this is right

  • Any cellular automaton that can attach a real number to each cell can be made mass-conserving by feeding that number to MaCE as the affinity; no other changes to the model are needed.
  • The continuous limit identifies the update as diffusion with unit coefficient plus conservative advection along affinity gradients, with $\beta$ governing the advection strength.
  • Mass conservation removes the need for separate death or stabilization mechanisms in neural cellular automata, since empty cells automatically receive mass from their neighbours.
  • In the $\beta\to\infty$ limit, the rule sends each cell's entire mass to its highest-affinity neighbour, giving a discrete-state version that still conserves mass and yields glider-rich totalistic dynamics.
  • In resource-limited worlds with mass decay and food reintroduction, the resulting dynamics display competition, colonisation, and reproduction-like 'mitosis' events, and the parameter landscape is easy to optimize with evolutionary strategies.

Reading between the lines

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

  • Beyond the paper: because the conservation proof uses only the reflexivity of the neighbourhood and the normalization of the redistribution weights, MaCE should also conserve mass on any symmetric graph or higher-dimensional lattice, not just the 2D Moore grid studied here.
  • Beyond the paper: a direct quantitative check of the continuous limit would be to measure how a smooth Gaussian bump spreads at $\beta=0$; the derivation predicts pure diffusion with diffusion constant 1 after the stated rescaling, and the measured rate should match that prediction.
  • Beyond the paper: the intrinsic-evolution experiment places food in a separate channel, but the affinity computation itself is fixed; allowing the affinity kernels or their parameters to be transported along with the mass would make the system's 'genome' part of the conserved dynamics, a step toward open-ended evolution that the paper leaves to future work.
  • Beyond the paper: because mass conservation forestalls both explosion and extinction, it should compress the region of parameter space that yields nontrivial dynamics, so population-based search methods are likely to find interesting regimes with much less tuning in MaCE-modified cellular automata than in their unmodified counterparts.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. The paper introduces MaCE, an update rule that can be attached to an existing cellular automaton to make its total mass exactly conserved. Each cell partitions its mass among its Moore-neighborhood neighbors with weights proportional to exp(βA), where A is an arbitrary real-valued affinity field computed from the automaton state. The paper proves the exact conservation identity (Eq. 3), derives a continuous limit ∂tρ = Δρ − 2β∇·(ρ∇A) under smoothness assumptions on ρ and A, and presents a β→∞ variant intended for discrete-state CAs. The empirical part applies MaCE to Lenia, neural CAs, and a discrete totalistic CA, reporting abundant solitons, an 83% soliton rate in random MaCELenia runs, qualitative observations of diversity and possible intrinsic evolution, and two evolutionary optimization experiments.

Significance. The exact conservation result is clean and general: the update is column-stochastic, so mass is conserved for any affinity field, including discontinuous or learned affinities. The continuous limit, where valid, gives a simple and attractive interpretation of the rule as diffusion plus advection along the affinity gradient, and the paper is honest about the smoothness assumption. The provision of code and video links is a practical strength. If the behavioral claims were backed by proper controls, MaCE would be a genuinely useful, simple plug-in for mass conservation in continuous and discrete CA models. However, the headline claim that MaCE 'tends to produce interesting behaviours more often' is not currently supported by the reported experiments, and the discrete-CA limit as written has a mass-conservation bug in the presence of ties. The underlying mechanism is likely correct, but the empirical and discrete-formulation parts need substantial revision.

major comments (3)
  1. [MaCE Discrete CA, Eq. (8)] Equation (8) does not conserve mass when the maximum affinity in a donor cell's neighborhood is attained by more than one receiver. As written, the indicator sum gives the donor's full mass to every cell that attains the maximum, so the donor's mass is duplicated. The text states that the implementation divides mass equally among tied maxima, but that tie-breaking rule is not what Eq. (8) expresses and is not the literal β→∞ limit of Eq. (2). Since ties are unavoidable in a discrete CA with a finite state set, this is a load-bearing error in the discrete formulation; the equation should be corrected (e.g., by dividing by the number of argmax cells) and the conservation proof should be extended to the corrected rule.
  2. [MaCELenia Capabilities, Results from Random Sampling] The 83% soliton-emergence rate is reported without a plain-Lenia control under the same conditions. Because the Table 1 'smart' sampling is taken from Papadopoulos et al. (2024) and was designed to place Lenia near a phase transition where solitons are already common, the 83% figure could be a property of the sampler rather than of MaCE. The abstract's claim that MaCE 'tends to produce interesting behaviours more often' is therefore unsupported by the current evidence. Please provide matched control runs of unmodified Lenia using the same kernel/growth priors, initial conditions, and number of seeds, and define the soliton criterion quantitatively. The 'Diversity in One Parameter Set' and 'Intrinsic Evolution' subsections are likewise based on visual inspection and narrative; they should either be quantified (e.g., number of distinct morphologies, survival times, biomass dynamics) or explicitly labeled as anecdotal.
  3. [Continuous Limit and Appendix] The claim that the scheme is 'numerically stable' is only justified in the linear, β=0 case. The von Neumann analysis in the appendix is restricted to β→0, and the appendix itself concedes that when A depends on ρ the condition 3Δt/Δx² ≤ 1 may not be sufficient and that values around 0.85 can be needed. The experiments use β=8 (Table 1) with ρ-dependent affinities, so the stability of the actual reported parameter regimes is not established by the derivation. In addition, the Taylor expansion leading to Eq. (7) assumes smooth A and ρ, a condition that clearly fails for the discrete CA and β→∞ applications. The abstract and main text should qualify the continuous-limit and stability statements to smooth affinities and to the linear regime, and the stability of the nonlinear regimes actually used should be checked numerically (e.g., by monitoring high-frequency growth).
minor comments (7)
  1. [Definitions] The affinity A is introduced as A(x,ρ) but then written as Aij without an explicit statement that it is evaluated from the current state ρ^t; please state this convention explicitly.
  2. [Sec. 2, paragraph after Eq. (7)] The sentence 'the discretization breaks down above this point' overstates the case: for 3Δt/Δx² > 1 the update still conserves mass exactly; it is linear stability that is lost. Please rephrase.
  3. [MaCE Discrete CA] The statement 'we are confident that a Turing machine can be built using these dynamics' is not supported by the reported experiments; either provide a construction or remove the claim.
  4. [Intrinsic Evolution] The minimum-mass threshold for food consumption is defined only in a footnote, yet it is likely to be a sensitive parameter for the observed foraging dynamics; please give its value and report a sensitivity check.
  5. [Comparison to FlowLenia] The claim that MaCELenia dynamics are 'much faster' is made while acknowledging that the effective Δt values differ between the two schemes; please make the comparison apples-to-apples or present it as a qualitative observation.
  6. [Figures and captions] Several figures (Figs. 5–8) omit the exact parameter values used for the displayed runs, such as β, the kernel generation method, the mass decay rate, and the food reintroduction rate; adding these would aid reproducibility.
  7. [References] The citation 'ereb0s labs (2022)' is a tweet with an informal URL; please provide a stable identifier or a description of the referenced content so that readers can verify the inspiration for Eqs. (1)–(2).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: mass conservation and the PDE limit are self-contained derivations from the stated update rule; the empirical sampling choice is a baseline concern, not a circular reduction.

full rationale

The paper's central mathematical claims are not circular. Equation (3) proves exact mass conservation by directly interchanging sums and using the reflexive-neighborhood identity; this is a straightforward consequence of the column-stochastic form of the update in Eq. (2), not an assumed conclusion. The continuous limit in Eq. (7) is obtained by Taylor expansion of Eq. (5) under explicitly stated smoothness assumptions on rho and A, with the parity argument for the vanishing O(Delta x^3) term; no fitted parameter is renamed as a prediction. The stability condition 3 Delta t / Delta x^2 < 1 is at least verified for beta=0 by von Neumann analysis, and the appendix candidly notes that nonlinear A(rho) may require smaller values. The only notable self-citation is the 'smart' parameter sampling taken from Papadopoulos et al. (2024), which the appendix explicitly says 'produced better results in plain Lenia.' That sampling prior may bias the reported ~83% soliton rate, and the absence of a matched plain-Lenia control weakens the comparative claim that MaCE 'tends to produce interesting behaviours more often.' But this is a missing-baseline / evidentiary issue, not an equivalence-by-construction or a fitted-input-called-prediction; the derivation itself is self-contained. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The exact conservation proof relies on symmetric reflexivity of the neighborhood; the PDE limit relies on smoothness and Taylor expansion. The free parameters β, 3Δt/Δx^2, and α are hand-set and influence the reported behaviors. No new physical entities are required by the central method.

free parameters (3)
  • β (MaCE inverse temperature) = β=8 for MaCELenia; 0<β<1 for MaCE-NCA
    Hand-chosen control of diffusion versus advection. The soliton-rich experiments use β=8, and the paper shows that increasing β changes dynamics qualitatively.
  • 3Δt/Δx^2 = 1 (MaCELenia); ~0.85 with repulsive affinity regularization
    Chosen as the largest stable value for the update. Stability is proven for the linear β=0 case, while nonlinear affinity cases require empirical downward adjustment.
  • α (cross-channel mass-transfer rate) = 0.01 to 0.1 for MaCELenia
    Introduced for cross-channel mass sharing. The paper states good values are in this range to avoid flickering or channel collapse.
assumptions (4)
  • domain assumption The neighborhood is reflexive and symmetric: (i,j)∈N(k,l) iff (k,l)∈N(i,j).
    Used in Eq. (3) to exchange summation order in the mass-conservation proof. Holds for the Moore neighborhood used in the paper.
  • domain assumption The mass field ρ and affinity field A are smooth on the grid scale.
    Required for the Taylor expansion in the continuous-limit derivation. The appendix explicitly assumes both fields are smooth; the PDE limit does not apply to discrete or β→∞ regimes.
  • standard math Taylor expansion to leading nonvanishing order in Δx captures the discrete update.
    The paper expands the 5×5 effective neighborhood and keeps O(Δx^2) terms, with parity arguments eliminating O(Δx^3).
  • ad hoc to paper For affinity depending on ρ, the linear stability threshold 3Δt/Δx^2 ≤ 1 remains approximately sufficient in practice.
    The paper states that nonlinear affinity can produce artifacts and that regularization terms require lower values, so general numerical stability is an empirical assumption rather than a proven theorem.

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Cite this review

Pith. "Pith review of MaCE: General Mass Conserving Dynamics for Cellular Automata." pith.science (2026). https://pith.science/paper/YGMECJGU

@misc{pith2026250712306,
  author       = {Pith},
  title        = {Pith review of: MaCE: General Mass Conserving Dynamics for Cellular Automata},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YGMECJGU}},
  note         = {Machine review of arXiv:2507.12306}
}
read the original abstract

We present Mass-Conserving Evolution (MaCE), a general method for implementing mass conservation in Cellular Automata (CA). MaCE is a simple evolution rule that can be easily 'attached' to existing CAs to make them mass-conserving, which tends to produce interesting behaviours more often, as patterns can no longer explode or die out. We first show that MaCE is numerically stable and admits a simple continuous limit. We then test MaCE on Lenia, and through several experiments, we demonstrate that it produces a wide variety of interesting behaviours, starting from the variety and abundance of solitons up to hints of intrinsic evolution in resource-constrained environments. Finally, we showcase the versatility of MaCE by applying it to Neural-CAs and discrete CAs, and discuss promising research directions opened up by this scheme.

Figures

Figures reproduced from arXiv: 2507.12306 by the authors.

Figure 1
Figure 1. Display of the effect of β, growing from 0 to 10 going from left to right. At low β, diffusion dominates, while increasing β generates sharper dynamics and more solitons. This limit is very useful in the case where we want to use MaCE with an automaton with a discrete set of possible values, which precludes using (2) since it necessitates a con￾tinuous division of mass. Eq.(8), however, only requires that the states… view at source ↗
Figure 2
Figure 2. Example random kernels generated with different meth￾ods (top:Gaussian, Bottom:Fourier). Fourier series kernels are more flexible, but the parameter space is harder to navigate. computing the Growth function and Kernels. The first generation method is close to the ’classic’ Lenia. Each Kernel is rotationally symmetric, and its shape is deter￾mined by three sets of three real numbers (βk, µk, σk) which define the sha… view at source ↗
Figure 3
Figure 3. displays two examples of systems with and with￾out solitons. On the left (Figure 3a), we see a connected mass with little differentiation between sections. On the right (Figure 3b), we see a plethora of solitons forming, with clear separation between them. After 100 random genera￾tions, we found that ∼ 83% of the time the system has soli￾ton emergence. Note that the sampling method we used does not sam￾ple all param… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Left: snapshot of MaceLenia world after 100 steps. Right: selection of solitons found in one parameter set [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 6
Figure 6. Figure 6: depicts an instance of a dominant soliton that undergoes ”mitosis”. Here, we performed a slightly differ￾ent experiment; mass still decays over time, but is instantly added back as food where it decayed (shown in the white 5We introduced the minimum mass threshold to c…
Figure 5
Figure 5. Figure 5: Intrinsic evolution experiment, time increases to the right [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 7
Figure 7. Figure 7: Time evolution of the final fittest individual for a system evolved to maximize mass in a harsh environment. dividual of the evolutionary process. Since mass decay is extremely harsh, to survive the system must produce fast￾moving solitons that replicate easily. At t =…
Figure 8
Figure 8. Figure 8: Time evolution of the final fittest individual for a system evolved to move to the bottom right corner [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 10
Figure 10. Figure 10: Left: display of glider diversity from random initial￾ization circle. We display red trails that help distinguish the gliders and their speeds and directions. Right: Close view of gliders. We display the last and next-to-last states in green and red, respectively. Ove…
Figure 11
Figure 11. Figure 11: Snapshots of FlowLenia vs MaCELenia after 1500 steps, Lenia parameters generated with the ’random’ sampling. [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: Snapshots of FlowLenia vs MaCELenia after 1500 steps, Lenia parameters generated with the ’smart’ sampling. In [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

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Forward citations

Cited by 1 Pith paper

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    A continuous, resource-limited Game of Life self-organizes into dividing, gliding cell-like patterns at a dilute-to-dense transition.

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