{"id":"40b90c7f-b887-4012-a215-310320c58a72","arxiv_id":"2411.18087","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"An adaptive genetic algorithm with graph-based crossover reproduces known Lennard-Jones cluster minima for N = 2 to 20, but the paper's claimed accuracy advantage is not supported.","lead":"The paper proposes an adaptive genetic algorithm for finding the lowest-energy structures of small atomic clusters and tests it on Lennard-Jones clusters with 2 to 20 atoms. The reported energies match known values to six decimals, but the method as written contains an internal contradiction in its selection step and lacks the reproducibility details needed to support the claim of superior accuracy.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (4) assigns the highest selection probability to the worst individual and the lowest to the fittest, inverting the stated survival-of-the-fittest mechanism; the reported convergence cannot be attributed to the algorithm as written.","rationale":"The paper's central claim is that the adaptive genetic algorithm finds the lowest-energy Lennard-Jones cluster structures and that its results are more accurate than those from other methods. For this claim to hold, the selection operator must bias reproduction toward fitter, lower-energy individuals. The reader's weakest assumption identifies exactly this point, and my reading confirms it: Eq. (4) in Sec. II.C is internally inconsistent with the stated selection goal because the normalized fitness F assigns 1 to the best and 0 to the worst, while the Boltzmann factor e^{-βF} then favors F=0 over F=1. This is not a matter of disagreement with external consensus; it is an internal mathematical contradiction in a core algorithmic component. The convergence plots and Table I therefore cannot be taken as evidence for the method as described unless the implemented selection differs from Eq. (4), which would mean the paper does not describe the tested algorithm. I also note the 'more accurate than the literature' claim conflates precision with accuracy: the Wales-Doye reference values are truncated to six decimals, so additional printed digits do not establish superior accuracy, and no statistical analysis over multiple runs is provided to support the claim. These issues together undermine the central claim, so rejection is appropriate. I do not see the reported energies themselves as suspect; the known global minima for N=2 to 20 are reproduced, which is consistent with a correctly implemented optimizer. The problem is that the written method cannot be the one that produced them.","tokens_in":7205,"tokens_out":2717,"duration_ms":25889,"concrete_test":"Obtain the authors' implementation or re-implement the GA exactly as described and run it for N=15 and N=20 using three selection rules: (a) Eq. (4) as written, (b) p ∝ e^{+βF}, and (c) p ∝ e^{-βE}, with the same population size, mutation schedule, and 6000-generation limit. Record whether each version reaches -52.322627 and -77.177042. If version (a) converges as reported, then the text's fitness definition is misstated; if (a) fails while (b) or (c) succeeds, the central claim is unsupported by the written algorithm. A minimal sanity check is to compute P(worst)/P(best) for a two-individual population under Eq. (4), which is e^{β} > 1 for β>0, confirming inversion.","verdict_should_be":"REJECT","load_bearing_attack":"Section II.C, Eq. (4) defines p(x_i) = Z^{-1} e^{-βF(x_i)} with F normalized so the best individual has F=1 and the worst has F=0. For any β>0, e^{-β·1} < e^{-β·0}, so the worst individual has the largest selection probability and the fittest has the smallest. This directly contradicts the sentence immediately above Eq. (4), which states that higher fitness means a higher chance of being selected, and it also contradicts the claim in Sec. II.C that increasing β increases the likelihood of lower-energy systems being chosen. The correct Boltzmann-style form for a fitness to be maximized is p ∝ e^{+βF}, or p ∝ e^{-βE} using the raw energy. Additionally, Eq. (5) has an index error: it sums over j but uses x_i in the exponent. Because selection is the core mechanism that biases reproduction toward good solutions, an inverted selection operator means the written algorithm is not expected to converge to the reported global minima. The results in Table I can be attributed to the method only if the implemented code used a different selection rule, but no code or parameter schedule is provided. The 'machine precision' claim is also an overstatement: matching Wales-Doye to six decimals and printing more digits demonstrates precision, not accuracy, and no multiple-run statistics or uncertainties are reported. The load-bearing weakness is the Eq. (4) inconsistency.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an adaptive genetic algorithm (AGA) for locating the lowest-energy structures of Lennard-Jones clusters. The algorithm combines a Poisson-disk initialization, a normalized energy-based fitness function, Boltzmann selection, elitism, a graph-based crossover that forms child clusters via minimum-edge-sum walks on the union graph of two parents, and Gaussian mutation with possibly generation-dependent standard deviations. Results are reported for clusters with N = 2 to 20 particles and are compared with the benchmark values of Wales and Doye. The authors claim that the AGA achieves energies that are 'more accurate' than those in the literature and that the precision of the results is limited only by machine precision.","tokens_in":7518,"tokens_out":3061,"duration_ms":29728,"significance":"If the method worked as described, it would be a useful demonstration of a genetic algorithm for cluster structure optimization. However, the scope is very modest: the global minima for N ≤ 20 Lennard-Jones clusters are long-established benchmarks, and the paper provides no timing, scalability, or comparison against other modern methods. The paper's main strengths are that it reproduces the known benchmark energies to six decimal places and that the algorithm is conceptually simple. Yet the significance is undermined by a load-bearing internal inconsistency in the selection operator, a conflation of precision and accuracy in the central claim, and the absence of any specification of the supposedly adaptive parameters. As presented, the paper does not support its claims of novelty or superior accuracy.","major_comments":[{"comment":"The selection probability is defined as p(x_i) = Z^{-1} e^{-β F(x_i)}, with F normalized so that the fittest individual has F = 1 and the least fit has F = 0. For any β > 0, this assigns the smallest selection probability to the fittest individual and the largest to the least fit, directly contradicting the sentence immediately above Eq. (4) and the subsequent discussion that increasing β increases the likelihood of lower-energy systems being chosen. Selection is the core mechanism that biases reproduction toward fitter individuals, so the algorithm as written would be expected to drift away from the global minimum, not converge to it. The reported results can be attributed to this method only if the implemented code used a different selection rule, but no code or correction is provided.","section":"II.C, Eq. (4)"},{"comment":"The normalization constant Z is written as a sum over j of e^{-β F(x_i)}, using x_i inside the sum. It should sum over x_j to be consistent with the definition of Z as the denominator of p(x_i). This is an indexing error, but it is part of the same selection formula whose sign is already inverted, and both issues must be corrected before the method can be reproduced.","section":"II.C, Eq. (5)"},{"comment":"The claim that the AGA results are 'more accurate' than those of Wales and Doye and that precision is 'only governed by machine precision' is not supported. The values in Table I agree with the Wales-Doye benchmark to six decimal places and then show additional digits, but this is simply higher numerical precision in evaluating the energy of a structure, not evidence of greater accuracy. No energy lower than the literature global minimum is reported, no multiple-run statistics or uncertainties are given, and no independent verification of the printed digits beyond the benchmark comparison is presented. The conclusion overstates what the data show.","section":"III, Table I and Conclusion"},{"comment":"The paper describes the algorithm as 'adaptive' because β and σ_i can change, but it never specifies the adaptation schedule, initialization values, update rules, or even the ranges used in the reported runs. Since the adaptation mechanism is the central methodological novelty claimed in the title and abstract, omitting these details makes the results irreproducible and prevents any assessment of whether the adaptive behavior contributes to the reported convergence. The population size, maximum generation count, mutation probability, and Poisson-disk minimum distance r0 are also never stated, so the experimental setup is incomplete.","section":"II.C, II.E, and III"}],"minor_comments":[{"comment":"The index error in the normalization sum should be fixed by writing Z = sum_j e^{-β F(x_j)}; this is a typographical/indexing issue separate from the sign error in Eq. (4).","section":"II.C, Eq. (5)"},{"comment":"The phrase 'Poisson disk dampling algorithm' appears to be a typo for 'Poisson disk sampling algorithm'.","section":"II.A"},{"comment":"The crossover description says to 'Generate an N-vertex walk with a minimum edge sum' on a complete graph of 2N vertices, but no algorithm or complexity bound is given for this step, and it is unclear how the random starting vertex interacts with the minimum-edge-sum requirement. This is an underspecified operation that needs a concrete definition for reproducibility.","section":"II.D"},{"comment":"The 'Convergence Point' column is not defined; the text mentions convergence rates but does not state the tolerance or stopping criterion used to identify when a run has converged.","section":"III, Table II"},{"comment":"The convergence plots are described qualitatively, but no axis labels, population sizes, parameter values, or multiple-run ensembles are shown, so the reader cannot judge the robustness of the convergence behavior.","section":"III, Fig. 3"},{"comment":"The reference formatting is inconsistent, e.g., Ref. [16] is given as 'D. Wales and J. Doye (1997), J. Phys. Chem. A, 101, 5111 (1997)' with a doubled year; the reference list should be standardized.","section":"Introduction"}],"recommendation":"reject","confidential_remarks":"The selection operator error in Eq. (4) is genuinely load-bearing: it inverts the fitness bias, so the written algorithm cannot explain the reported convergence. Without code or a parameter schedule, the discrepancy cannot be resolved within the manuscript. The paper is also a very limited computational study (N ≤ 20, no performance comparison), and the precision-versus-accuracy claim would need substantial reframing even if the algorithm were corrected. I recommend rejection, though the authors could resubmit a corrected and more complete version after fixing the selection formula and documenting the adaptive parameters."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is a mixed bag. The genuinely new bits are the Poisson disk initialization and the graph-based crossover that builds a child as an N-vertex walk with minimum edge sum. Those are real ideas, and the authors are honest about comparing against Wales–Doye and earlier GA work. The numerical results match the known Lennard-Jones global minima for N=2..20, so the implementation almost certainly works.\n\nBut the written algorithm does not. The selection rule in Eq. (4) is p(x_i) ∝ e^{-βF(x_i)} with F normalized to 1 for the best and 0 for the worst. That assigns the highest probability to the worst individual. The text above the equation claims the opposite. This is not a minor typo; it is the core survival-of-the-fittest mechanism. Eq. (5) also has an index slip, summing over j but using x_i. If the code used e^{+βF} or e^{-βE}, then the paper misdescribes its own method, and the reported convergence cannot be attributed to the algorithm as written. No code or parameter schedules are provided, so we cannot check.\n\nThe 'machine precision' claim is also overblown. Matching Wales–Doye to six decimals and printing more digits shows precision, not accuracy. The energies are the same known minima. No multiple-run statistics or uncertainties are reported, so the convergence plots in Figure 3 are anecdotal.\n\nOn balance, the central claim is not supported. The sign error is load-bearing, and the overstatement about accuracy weakens the conclusion. But the paper is not nonsense: the initialization and crossover ideas are genuinely different from what is cited, and the error is fixable in principle. If the authors clarified the selection rule, added code or at least parameter schedules, and ran multiple trials, a revised version could be useful to the cluster-structure community.\n\nFor peer review, I would send it out rather than desk-reject, because the algorithmic combination is worth checking and the error is correctable. But I would not cite it in its current form, and I wouldn't bring it to reading group except as a cautionary example.","headline":"Eq. (4) inverts the fitness ranking, breaking the method as written; the Poisson-disk/graph-crossover ideas are still worth a referee's time.","tokens_in":8018,"tokens_out":2656,"would_cite":false,"duration_ms":23407,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims an adaptive genetic algorithm finds the lowest-energy Lennard-Jones cluster structures for N=2 to 20 and reports energies whose only limit is machine precision.","keywords":["adaptive genetic algorithm","Lennard-Jones clusters","global optimization","atomic cluster structure","Boltzmann selection","basin hopping","machine precision","genetic crossover"],"falsifier":"Instrument the selection operator in a fresh implementation exactly as Eq. (4) is written and count how often each fitness rank is selected; if the least-fit individuals are chosen most often yet the N=2 to 20 minima still converge, then the published formula cannot describe the tested method, and if the code uses the opposite sign, the machine-precision claim must be re-verified for the corrected selector.","tokens_in":6980,"feed_emoji":"⚛️","tokens_out":8142,"duration_ms":62700,"temperature":0.7,"pith_summary":"The paper proposes an adaptive genetic algorithm (AGA) for locating the global minimum energy structure of atomic clusters and demonstrates it on Lennard-Jones clusters of 2 to 20 particles. The method combines a normalized fitness function, Boltzmann-style selection, elitism, a graph-based crossover that builds child clusters from parent geometries, and Gaussian mutation whose step size adapts during the search. The authors report that the AGA recovers the known lowest-energy clusters and reproduces the Wales–Doye benchmark energies to at least six decimal places, while carrying additional digits so that precision is limited only by floating-point arithmetic. If the claim is right, the algorithm offers a generally applicable optimizer for pair-potential many-body problems that is competitive with basin hopping.","feed_headline":"Genetic algorithm finds Lennard-Jones minima to machine precision","feed_subtitle":"Adaptive GA matches Wales–Doye cluster energies for N=2–20 at full machine precision.","key_machinery":"The load-bearing components are: the normalized fitness F(x_i) = (E_max - E(x_i))/(E_max - E_min), which maps the best energy in a generation to 1 and the worst to 0; Boltzmann selection with probability p(x_i) = $Z^{{-1}}$ $e^{{-β F(x_i)}}$, whose free parameter β is used to sharpen the search; a graph-theoretic crossover in which the union of two parent clusters is represented as a complete graph whose edges carry two-particle energies, and a child is an N-vertex walk with small edge sum; and adaptive Gaussian mutation with variance that can shrink as the population matures. Elitism carries 10% of the fittest individuals unchanged. Together these operators are meant to let the population explore broadly early and fine-tune late, which is the 'adaptive' mechanism the paper credits for the reported convergence.","core_discovery":"On the paper's own terms, the central discovery is that a fully adaptive genetic algorithm—one that adjusts its mutation scale and selection pressure during the run—can locate the global minima of Lennard-Jones clusters for all sizes from N = 2 through N = 20. The energies computed by the AGA agree with the Wales–Doye reference values to the six decimal places that the reference table reports, and the authors provide additional digits, asserting that the residual error is at the level of machine precision. Because the same framework is described for any pairwise potential, the result is presented as a general optimization method for many-body configuration problems, not just a Lennard-Jones-specific solver.","pith_inferences":["A natural next test is to apply the same AGA to N=21–30 Lennard-Jones clusters, where structural transitions and deeper local-minima traps appear, to see whether the machine-precision claim survives beyond the demonstrated range.","One could define explicit schedules for β and the mutation variance in terms of population diversity, turning the qualitative 'adaptive' description into a reproducible protocol and allowing fair compute-budget comparisons with basin hopping and simulated annealing.","The graph-theoretic crossover could be generalized to non-pairwise potentials by weighting edges with incremental many-body energy contributions, although the paper does not attempt this."],"forward_implications":["If the AGA works as claimed, it provides reference-quality energies for Lennard-Jones clusters with more significant digits than the standard benchmark table, which can be used to test other optimizers.","Because the method is stated for any pairwise potential, the same code could be adapted to Morse, Sutton–Chen, or embedded-atom clusters without changing the genetic machinery.","The reported convergence data (roughly 40, 150, 5000, and 6000 generations for N=5, 10, 15, 20) give a concrete scaling baseline that future adaptive searches could improve upon.","The graph-based crossover offers a reusable way to combine two three-dimensional geometries while preserving favorable local packing from both parents."],"supporting_citations":[{"why":"Provides the basin-hopping benchmark energies for Lennard-Jones clusters with up to 110 particles, which the paper compares against for N=2–20.","marker":"[16]"},{"why":"A genetic-algorithm study whose minimum energies for selected cluster sizes match the present AGA results to the last decimal, supporting the accuracy claim.","marker":"[25]"},{"why":"Another genetic-algorithm method for Lennard-Jones optimization; the paper states selective cluster sizes match exactly with this work as well.","marker":"[26]"},{"why":"The Poisson-disk sampling algorithm adapted to initialize clusters with a minimum interparticle separation.","marker":"[27]"},{"why":"Supplies the adaptive genetic algorithm concept that motivates changing mutation and selection parameters during the search.","marker":"[32]"},{"why":"Source for elitism and the fitness-based selection framework used in the implementation.","marker":"[29]"}],"fun_headline_variants":["Adaptive GA nails Lennard-Jones minima to machine precision","Machine-precision cluster energies from adaptive GA","Adaptive GA finds all Lennard-Jones minima N=2 to 20","Self-tuning GA solves atomic cluster optimization exactly","AGA achieves machine-precision minima for Lennard-Jones clusters"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reported convergence assumes the selection step actually prefers lower-energy, higher-fitness clusters, but the written Boltzmann formula gives the worst cluster the largest selection probability, so the algorithm that was run must differ from the one described in Eq. (4).","fun_headline_variants_meta":{"raw":{"variants":["Adaptive GA nails Lennard-Jones minima to machine precision","Machine-precision cluster energies from adaptive GA","Adaptive GA finds all Lennard-Jones minima N=2 to 20","Self-tuning GA solves atomic cluster optimization exactly","AGA achieves machine-precision minima for Lennard-Jones clusters"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000948,"raw_usage":{"total_tokens":3946,"prompt_tokens":745,"completion_tokens":3201,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":361,"completion_tokens_details":{"reasoning_tokens":3122}},"tokens_in":361,"tokens_out":3201,"duration_ms":20488,"temperature":1.0,"reasoning_tokens":3122,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:31:19.109113+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Instrument the selection operator in a fresh implementation exactly as Eq. (4) is written and count how often each fitness rank is selected; if the least-fit individuals are chosen most often yet the N=2 to 20 minima still converge, then the published formula cannot describe the tested method, and if the code uses the opposite sign, the machine-precision claim must be re-verified for the corrected selector.","supporting_citations":[{"cited_title":"It should be noted that the values for selective cluster sizes match exactly, to the last decimal, with those found in Ref.[25] and [26] using a genetic algorithm approach","cited_arxiv_id":null,"evidence_quote":"Provides the basin-hopping benchmark energies for Lennard-Jones clusters with up to 110 particles, which the paper compares against for N=2–20."},{"cited_title":"Alharbi and I","cited_arxiv_id":null,"evidence_quote":"A genetic-algorithm study whose minimum energies for selected cluster sizes match the present AGA results to the last decimal, supporting the accuracy claim."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Another genetic-algorithm method for Lennard-Jones optimization; the paper states selective cluster sizes match exactly with this work as well."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The Poisson-disk sampling algorithm adapted to initialize clusters with a minimum interparticle separation."},{"cited_title":"Bridson, Fast Poisson disk sampling in arbitrary di- mensions, SIGGRAPH sketches, 10(1), 1 (2007)","cited_arxiv_id":null,"evidence_quote":"Supplies the adaptive genetic algorithm concept that motivates changing mutation and selection parameters during the search."},{"cited_title":"Barr´ on, S","cited_arxiv_id":null,"evidence_quote":"Source for elitism and the fitness-based selection framework used in the implementation."}],"review_version":1}