{"id":"03f6d737-e7b8-43e3-9acf-315245129e8f","arxiv_id":"2507.12306","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"MaCE makes arbitrary cellular automata exactly mass-conserving through a local softmax redistribution of mass, with continuous limit ∂tρ = Δρ - 2β∇·(ρ∇A), and yields abundant solitons and gliders in Lenia, NCA, and discrete CA experiments.","lead":"MaCE is a simple update rule that can be attached to almost any cellular automaton to make its total mass exactly constant, moving mass toward cells with high computed affinity. It has a clean continuous limit, and in tests on Lenia, neural, and discrete cellular automata it produces many drifting solitons, gliders, and resource-competition behaviors.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The ~83% soliton rate in MaCELenia is reported without a plain-Lenia control under the same random sampling; the central practical claim that MaCE makes interesting behaviours more frequent is therefore not yet established.","rationale":"The reader's weakest_assumption concerns smoothness and stability of the continuous limit. I do not think that is the most load-bearing issue: the discrete update in Eq. (2) is a nonnegative column-stochastic matrix, so exact mass conservation implies all cell values remain bounded by the conserved total mass for any A, smooth or not; the smoothness condition is explicitly stated for the PDE limit, and the paper does not claim a continuous limit for beta -> infinity or for discrete CAs. The weak point is instead the evaluation of the central practical benefit. The abstract's main justification for MaCE is that mass conservation makes interesting behaviours more frequent, and the only quantitative evidence is the 83% soliton count, which is subjective and uncontrolled. If plain Lenia under the same smart sampling yields similar counts, MaCE's headline advantage disappears even though its conservation property is real. The reader's rationale also mentions the missing baseline, so my overall CONDITIONAL verdict matches theirs; I simply locate the load-bearing concern differently from the formal weakest_assumption.","tokens_in":893,"tokens_out":1733,"duration_ms":207856,"concrete_test":"Pre-register a soliton criterion (e.g., a connected mass component above a fixed density threshold that translates by more than one kernel radius over a 100-step window) and a random seed protocol. Run the same 100 parameter draws and initial conditions from Table 1 twice: once with MaCELenia (Eq. 2, beta=8, 3*dt/dx^2=1) and once with plain Lenia's standard non-conserving update using identical kernels, growth functions, and grid size. Have the videos labeled blind by at least two raters, or by the automated detector, and compute the soliton-rate difference with a bootstrap confidence interval. If plain Lenia's rate is within the interval around MaCELenia's ~83%, the claim that MaCE makes solitons more frequent fails; if it is substantially lower, the claim survives.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Exact mass conservation (Eq. 3) and the smooth-continuum derivation are solid: the discrete update in Eq. (2) is a nonnegative column-stochastic matrix, so total mass is fixed and no unbounded growth is possible for arbitrary A. The load-bearing gap is the empirical claim in the abstract and the MaCELenia Capabilities section that MaCE 'tends to produce interesting behaviours more often'. The only quantitative support is that roughly 83% of 100 random MaCELenia runs show solitons, judged by visual inspection. There is no matched run of unmodified Lenia using the same Table 1 'smart' sampling, which was itself designed to place rules near the Lenia phase-transition where solitons are already common. Without such a baseline, the 83% number could measure the sampler rather than MaCE; the 'diversity in one parameter set' and 'hints of intrinsic evolution' sections are also qualitative and lack pre-registered metrics. Thus the practical advantage that motivates the method is unsupported, even though the conservation mechanism itself is correct.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14498,"tokens_out":9153,"duration_ms":107593,"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":[{"comment":"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.","section":"MaCE Discrete CA, Eq. (8)"},{"comment":"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.","section":"MaCELenia Capabilities, Results from Random Sampling"},{"comment":"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).","section":"Continuous Limit and Appendix"}],"minor_comments":[{"comment":"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.","section":"Definitions"},{"comment":"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.","section":"Sec. 2, paragraph after Eq. (7)"},{"comment":"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.","section":"MaCE Discrete CA"},{"comment":"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.","section":"Intrinsic Evolution"},{"comment":"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.","section":"Comparison to FlowLenia"},{"comment":"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.","section":"Figures and captions"},{"comment":"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).","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The mathematical core—exact mass conservation and the smooth-field continuous limit—appears sound and is worth publishing. The main risks are the unsupported empirical baseline for the headline 'more interesting behaviours' claim and the incorrect Eq. (8) for the discrete-CA limit. The paper also overstates numerical stability for nonlinear, finite-β regimes. These are fixable within the manuscript's scope, so I recommend major revision rather than rejection. Please ensure the authors provide plain-Lenia control statistics and correct the discrete-CA update before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe core of this paper is solid: the MaCE update is a neat, locally computable way to enforce exact mass conservation, and the proof in Eq. (3) is straightforward and correct for reflexive neighborhoods. The continuous limit ∂tρ = Δρ − 2β∇·(ρ∇A) is the real contribution—it demystifies the rule as diffusion plus advection along the affinity gradient, and the stability condition 3Δt/Δx² ≤ 1 is sensible. The cross-channel extension and the discrete β→∞ limit are also useful additions.\n\nThe soft spot is the empirical case. The abstract claims MaCE 'tends to produce interesting behaviours more often,' but the only quantitative support is the ~83% soliton rate from 100 random MaCELenia runs, judged by visual inspection. There is no matched run of unmodified Lenia under the same 'smart' sampling, which was itself designed to sit near the phase transition where solitons are common. Without that baseline, the 83% could be measuring the sampler rather than MaCE. The diversity and 'hints of intrinsic evolution' sections are qualitative—impressive to watch, but not yet evidence. The Turing-completeness remark ('we are confident that a Turing machine can be built') is an overclaim in an otherwise careful paper; there is no construction or citation to support it.\n\nTwo fairness notes: the authors are upfront about the provenance issue—the core redistribution equation is credited to an unreachable 2022 tweet—and they explicitly flag the smoothness assumptions and the caveat that 3Δt/Δx² may need to be lower for nonlinear affinities. So the limitations are not hidden.\n\nWhere does this leave us? The method is real and worth knowing. The conservation mechanism and continuous limit are formally grounded. The empirical case needs matched baselines and better soliton counting before the 'more interesting behaviours' claim is established. This is a paper for anyone working on continuous CAs, Lenia, or resource-constrained ALife. A serious referee should engage with it; the required work is revision and added experiments, not rejection.","headline":"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.","tokens_in":15036,"tokens_out":1898,"would_cite":true,"duration_ms":20436,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B15","68Q80"],"pacs":[],"model":"deepseek-v4-flash","headline":"The MaCE rule attaches exact mass conservation to existing cellular automata, turning them into reservoirs of solitons and low-level evolutionary dynamics.","keywords":["mass conservation","cellular automata","Lenia","MaCE update rule","softmax redistribution","continuous limit","solitons","artificial life"],"falsifier":"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.","tokens_in":14063,"feed_emoji":"🧬","tokens_out":9467,"duration_ms":98977,"temperature":0.7,"pith_summary":"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.","feed_headline":"MaCE update: exact mass conservation for any cellular automaton","feed_subtitle":"Attach it to Lenia or neural CAs and moving solitons emerge; the continuum limit is diffusion plus affinity advection.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines Lenia, the continuous cellular automaton that MaCELenia modifies by using its growth values as affinities.","marker":"Chan (2020a)"},{"why":"Provides the MultiChannel Lenia formulation whose growth functions and kernels MaCELenia inherits.","marker":"Chan (2020b)"},{"why":"Introduces the affinity-based mass-conservation idea and the mass-conserving Lenia baseline that MaCE is compared with and simplified from.","marker":"Plantec et al. (2023)"},{"why":"Defines the Neural Cellular Automaton that MaCE-NCA makes mass-conserving by replacing the RGB-channel updates.","marker":"Mordvintsev et al. (2020)"},{"why":"Supplies the claim that conservation of matter increases evolutionary activity, the main motivation for MaCE.","marker":"Hickinbotham and Stepney (2015)"},{"why":"Provides the 'smart' parameter-sampling method used in the soliton-survey experiments and in MaCELenia's search space.","marker":"Papadopoulos et al. (2024)"},{"why":"Supplies the evolution strategies used in the extrinsic-evolution experiments.","marker":"Beyer and Schwefel (2002)"},{"why":"Provides the complexity classification used to characterize the discrete MaCE automaton's glider-rich dynamics as class IV.","marker":"Wolfram (1984)"}],"fun_headline_variants":["MaCE rule: add mass conservation to any cellular automaton","Attach MaCE to Lenia, get solitons and diffusion-advection","Mass-conserving CA rule yields solitons and hints of evolution","MaCE: force any CA to conserve mass, then watch it evolve","Simple rule makes any cellular automaton mass-conserving"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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 $3\\Delta t/\\Delta x^2=1$.","fun_headline_variants_meta":{"raw":{"variants":["MaCE rule: add mass conservation to any cellular automaton","Attach MaCE to Lenia, get solitons and diffusion-advection","Mass-conserving CA rule yields solitons and hints of evolution","MaCE: force any CA to conserve mass, then watch it evolve","Simple rule makes any cellular automaton mass-conserving"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00016,"raw_usage":{"total_tokens":1220,"prompt_tokens":918,"completion_tokens":302,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":209}},"tokens_in":534,"tokens_out":302,"duration_ms":3783,"temperature":1.0,"reasoning_tokens":209,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:49:12.901358+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the affinity-based mass-conservation idea and the mass-conserving Lenia baseline that MaCE is compared with and simplified from."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Neural Cellular Automaton that MaCE-NCA makes mass-conserving by replacing the RGB-channel updates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the claim that conservation of matter increases evolutionary activity, the main motivation for MaCE."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the 'smart' parameter-sampling method used in the soliton-survey experiments and in MaCELenia's search space."},{"cited_title":"and Schwefel, H.-P","cited_arxiv_id":null,"evidence_quote":"Supplies the evolution strategies used in the extrinsic-evolution experiments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the complexity classification used to characterize the discrete MaCE automaton's glider-rich dynamics as class IV."}],"review_version":1}