{"id":"7fc5b795-0bcb-4627-9516-cb4e011d1980","arxiv_id":"2501.17166","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"A proposed hyperfactorial and superfactorial formula assigns complexity percentages to swarm algorithms as a proxy for CO2 emissions, but without data or validation the percentages are not measurements.","lead":"This paper proposes a mathematical formula that multiplies complexity terms to estimate carbon emissions of swarm intelligence algorithms. The authors report complexity percentages for 30-plus algorithms, but provide no data, code, or validation, so the claimed measurements cannot be reproduced or trusted.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (1)-(4) cannot produce Table I's percentages: the hyperfactorial/superfactorial terms have no calibrated link to energy, and no parameter values or normalization are given.","rationale":"The paper's intended contribution is to measure and rank the carbon footprint of swarm algorithms, and its strongest claim is that the proposed formula quantifies CO2 emissions. The load-bearing premise is that H(Np) and sf(Ni), along with the h_k, t_l, and b_m factor products, scale with actual computational energy consumption. The reader identified this exact assumption as the weakest point. My independent read confirms it: Eq. (1)-(4) is presented with no calibration, no derivation from physical energy models, and no experimental validation. The super-exponential scaling of the hyperfactorial and superfactorial terms makes the formula's outputs astronomically sensitive to Np and Ni, which is not how real computational energy behaves. Additionally, the reported 'Comp (%)' values require a normalization from the formula's kg-CO2 output to a percentage, but no normalization procedure is given, and no parameter values for any algorithm are reported. The claim in Section III that energy consumption was 'measuring with precision' is unsupported by any data, code, or error analysis. This is not a disagreement with an external consensus; it is an internal failure of support and reproducibility. The verdict of REJECT is appropriate. No machine-checked proof, shipped code, or externally validated prediction is present to independently support the formula. The manuscript's own characterization of the approach as a prototype and its call for future refinement further indicate that the strong 'quantified' claim is premature. I therefore agree with the reader's verdict and see no reason to adjust it.","tokens_in":6587,"tokens_out":3437,"duration_ms":34621,"concrete_test":"Recompute the PSO row of Table I using Eq. (1)-(4) with the paper's experimental inputs. Since no inputs are reported, take a minimal plausible setting: Np=30, Ni=100, all h_k, t_l, b_m factors set to 1, tunit=1 h, Ph=0.1 kW, eta=1, er=0.5 kg/kWh. Compute the kg-CO2 value, then apply the paper's normalization to percentages. If the result is not 5.83% or no normalization is specified, the table is not an output of the formula. As a sensitivity probe, repeat with Np=31; the formula jumps by a factor of 31^31, which would require an implausible 46-order-of-magnitude change in measured energy for one additional particle.","verdict_should_be":"REJECT","load_bearing_attack":"The central claim is that Eq. (1)-(4) quantifies CO2 emissions of swarm algorithms, with Table I ranking algorithms by 'Comp (%)' values from 5.25% to 7.87%. For this claim to hold, three conditions must be met: (a) the formula must correspond to actual energy consumption, (b) all inputs must be specified, and (c) the conversion from the kg-CO2 output of Eq. (1)-(4) to the reported percentages must be defined. None of these conditions are met. H(Np) and sf(Ni) are introduced in Section II.B without derivation, calibration, or comparison to measured energy. They are dimensionless combinatorial products that grow super-exponentially: H(Np+1)/H(Np) = (Np+1)^(Np+1), so changing the particle count from 30 to 31 multiplies the formula by roughly 10^46. Physical energy consumption does not have this sensitivity, so the formula cannot be a faithful proxy for computational work. The Discussion states that experiments were performed and energy consumption was 'measuring ... with precision', but no parameter values, no normalization rule, and no measurement data are reported. The Table I percentages are therefore not reproducible from the paper, and unless the omitted inputs are supplied, the ranking is an arbitrary index rather than a measurement. The manuscript itself calls the approach 'a prototype' and says future work should refine the formula, which further undercuts the claim that CO2 emissions were quantified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a deterministic formula, Eq. (1)-(4) in Section II.B, for quantifying the CO2 emissions of swarm-intelligence algorithms. The formula multiplies a hyperfactorial of the number of particles, a superfactorial of the number of iterations, products of hyperparameter, topology, and boundary-handling factors, and a hardware/energy term (tunit, Ph, η, er). From this, Table I reports \"Comp (%)\" values between 5.25% and 7.87% for 35 algorithms and claims that simpler stochastic algorithms have lower emissions while hybrid algorithms have higher emissions. The stated contribution is an unprecedented framework for measuring the environmental impact of swarm algorithms in ICT. The paper gives no measured emissions, no parameter values, no normalization rule, no benchmark comparison, and no error analysis.","tokens_in":6946,"tokens_out":2980,"duration_ms":30508,"significance":"If the proposed formula were validated against measured energy consumption, a complexity-based proxy for comparing the carbon footprints of swarm algorithms would be a useful practical tool for green AI. The paper also usefully draws attention to the multiplicity of algorithmic factors, hyperparameters, topologies, and boundary-handling choices that can affect computational cost. However, the central claim is unsupported: Eq. (1)-(4) is introduced without derivation or calibration, the inputs to the formula are never specified, and Table I is presented without any reproducible protocol. As it stands, the manuscript provides an illustrative index defined by an arbitrary formula, not a measurement of CO2 emissions. The lack of empirical grounding, reproducibility, and internal consistency between the \"prototype\" language and the strong claims in the abstract prevents the result from being assessed at the level required for publication.","major_comments":[{"comment":"The core claim that Eq. (1)-(4) quantifies CO2 emissions is not supported because the formula is ad hoc and uncalibrated. H(Np) and sf(Ni) are dimensionless combinatorial functions with super-exponential growth; for instance, increasing the particle count from Np to Np+1 multiplies H(Np) by (Np+1)^(Np+1), which for Np=30 is an astronomically large factor. No argument or empirical comparison establishes that energy consumption follows such growth. The factors hk, tl, and bm are also left completely unspecified in both value and units, yet they multiply a quantity asserted to be measured in kg CO2. Without a derivation, a calibration step, or a comparison to measured energy, the formula is an arbitrary index rather than a model of emissions.","section":"II.B, Eq. (1)-(4)"},{"comment":"The \"Comp (%)\" values in Table I are not reproducible from the paper. No values are given for Np, Ni, hk, tl, bm, tunit, Ph, η, or er for any of the 35 algorithms, and no normalization rule is specified that converts the kg-CO2 output of Eq. (1)-(4) into percentages. The reader therefore cannot reconstruct any entry of the table, test the ranking, or verify that the differences (e.g., 5.25% for Bat Algorithm vs. 7.87% for Fast Bacterial Swarming) reflect the formula rather than unspecified choices. This is a load-bearing gap because the entire empirical conclusion of the paper rests on these percentages.","section":"Table I"},{"comment":"Section III states that \"computational experiments were conducted under controlled conditions, measuring energy consumption with precision,\" but the manuscript reports no measurement protocol, no hardware configuration, no raw energy measurements, no comparison between formula outputs and measured values, and no error bars. Since the central claim is that the authors \"quantified the environmental impact of various swarm algorithms,\" the absence of any experimental data makes that claim untestable. Either the experiments and data must be reported in full, or the claims must be reduced to a theoretical proposal.","section":"III"},{"comment":"The paper itself describes the formula as \"a prototype\" (Section II.B) and concludes that \"future work should focus on refining the formula for real-world applications\" (Section IV). This language is incompatible with the abstract's assertion that complexity percentages \"ranged from 5.25% to 7.87%\" and that the authors \"were able to quantify the environmental impact of various swarm algorithms.\" The strong claims need to be either supported by validation or explicitly downgraded to an illustrative complexity index with no claim of quantitative accuracy.","section":"II.B and IV"}],"minor_comments":[{"comment":"The text contains numerous unusual and nonstandard phrasings such as \"apodictic indication,\" \"asyndetic integration,\" \"veracious influence on carbonic effluvia,\" and \"consanguineous relationship.\" These should be replaced with plain scientific language for clarity.","section":"Abstract and Introduction"},{"comment":"The equation is split across four numbered lines, making it appear as four separate statements. It should be presented as a single equation with the definition of all symbols immediately following.","section":"Eq. (1)-(4)"},{"comment":"The table caption and the table body are not fully self-contained: some algorithm names such as \"Bees Algorithms\" and \"Wolf Search\" are ambiguous, the abbreviations are only partially expanded, and the exact meaning of \"Comp (%)\" is not defined in the caption or the text.","section":"Table I"},{"comment":"The abstract states that the work constructs \"a convex optimization problem,\" but no convex optimization problem is formulated or solved anywhere in the paper. Either provide the formulation or remove this claim.","section":"Abstract and Methodology"},{"comment":"Reference [2] is cited as \"Itu-T\" and would be better cited as the ITU-T L.1470 recommendation with the standard institutional authorship. Several other references are cited vaguely in the introduction without clear point-by-point support; consider tightening the related-work discussion.","section":"References"}],"recommendation":"reject","confidential_remarks":"The manuscript does not meet the evidentiary standard for a journal publication: the central formula is uncalibrated, the reported percentages are unreproducible, and the claimed experiments are never described. The problems are not merely local; they concern the paper's main contribution. A rejection with an invitation to resubmit after substantial additional work (real measurements, parameter disclosure, validation against energy data, and a clearly defined normalization) seems appropriate. I also note that several strong claims in the abstract and discussion are not supported by the body of the paper, which would need to be corrected in any future version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: this is a desk reject, but not because the topic is bad. The paper introduces a hyperfactorial/superfactorial formula for CO2 emissions and applies it to 30 swarm algorithms, producing a neat table. The table itself is easy to read and the categorization (stochastic, multi-agent, hybrid, nature-inspired) is sensible. I don't think the formula or the table exist elsewhere, so as a new definition it's a new artifact. The literature review touches the right ICT-carbon themes. That's the extent of the credit.\n\nThe problem is that the central empirical claim is unsupported and, as written, tautological. Eq. (1)-(4) multiplies H(Np), sf(Ni), hyperparameter/topology/boundary factors, tunit, Ph, eta, er. No values are given for any of these. No normalization formula is given. Table I's 'Comp (%)' values cannot be reconstructed. The paper says experiments measured energy 'with precision,' but no measurements, code, or parameter settings are in the manuscript. So the table isn't a measurement; it's a normalized output of the same formula. The conclusion that hybrid algorithms emit more follows from the structure of Eq. (1)-(4), not from independent observation. That's fitting presented as prediction.\n\nThe stress-test note has the right arithmetic: H(Np+1)/H(Np) = (Np+1)^(Np+1). Moving from 30 to 31 particles multiplies the formula by about 10^46. Physical energy doesn't behave that way. So the hyperfactorial cannot be a faithful proxy for computational work unless calibrated, and no calibration is offered. The manuscript itself calls the approach 'a prototype' and says future work should refine the formula. That's honest but it undercuts the abstract's claim that emissions were quantified.\n\nThere are smaller issues: the prose is overwrought ('apodictic,' 'asyndetic,' 'carbonic effluvia'), and some citations are only loosely connected. None of that matters compared to the missing data.\n\nWho is this for? Maybe someone collecting exotic carbon-accounting formulas, or a referee checking whether the field has any validation standards. The topic is legitimate, but this paper doesn't clear the bar for empirical work. I would not cite it and I would not send it to peer review; it needs either real energy measurements with the formula fitted/validated, or a clear reframing as a proposed index with no empirical claims. My recommendation: desk reject, with an invitation to resubmit if they supply parameters, normalization, code, and at least one comparison against measured energy.","headline":"A novel-looking complexity formula for swarm-algorithm CO2 that is actually an uncalibrated index; Table I is a normalized output of the formula, not a measurement.","tokens_in":7430,"tokens_out":3078,"would_cite":false,"duration_ms":26967,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A formula deriving swarm algorithms' CO2 emissions from their computational complexity, ranking them from 5.25% to 7.87%.","keywords":["swarm intelligence","carbon footprint","ICT energy consumption","algorithmic complexity","hyperfactorial function","superfactorial function","green AI","computational complexity"],"falsifier":"Run PSO, the Firefly Algorithm, and Fast Bacterial Swarming on identical hardware for identical optimization tasks, meter the energy in kWh, convert it to kg CO2 using regional grid intensity, and compare against the paper's predictions; a mismatch in ranking or ratio would show the formula is not measuring emissions.","tokens_in":6435,"feed_emoji":"🌍","tokens_out":6908,"duration_ms":63062,"temperature":0.7,"pith_summary":"The paper aims to establish that the carbon dioxide emissions of swarm-intelligence computations can be computed deterministically from algorithmic complexity, without direct energy metering. This matters because ICT accounts for roughly 6 percent of global emissions, and algorithm choice is a practical lever for reducing that share. The proposed formula multiplies hyperfactorial and superfactorial functions of particle and iteration counts by hyperparameter, topology, boundary-handling, hardware, and regional factors. Applied to more than 30 swarm algorithms, it yields complexity percentages from 5.25% for the Bat Algorithm to 7.87% for Fast Bacterial Swarming, which the authors read as evidence that simpler stochastic methods have lower carbon impact than hybrids. If the formula is right, practitioners could rank algorithms by expected emissions before running them.","feed_headline":"New formula ranks swarm algorithms by CO2 impact","feed_subtitle":"From Bat Algorithm at 5.25% to Fast Bacterial Swarming at 7.87%, simpler search means lower emissions.","key_machinery":"The load-bearing object is the product $H(N_p) \\cdot sf(N_i)$, with $H(N_p)=\\prod_{i=1}^{N_p} i^i$ and $sf(N_i)=\\prod_{j=1}^{N_i} j!$; these hyperfactorial and superfactorial functions convert particle and iteration counts into a sharply growing measure of computational work. The formula multiplies this measure by factor groups for hyperparameters, swarm topologies, and boundary-handling methods, and then by $t_{\\mathrm{unit}} P_h \\eta e_r$, so that algorithmic parameters alone yield a carbon figure. The mechanism's role is to replace empirical energy measurement with a parameter-derived complexity score that can be compared across algorithms on a percentage scale.","core_discovery":"The paper's central claim is that the CO2 emissions of a swarm algorithm equal the product of a hyperfactorial of particle count, a superfactorial of iteration count, factor groups for hyperparameters, topologies, and boundary handling, unit computation time, hardware power, utilization, and regional emission factor, as stated in Eq. (1)–(4). Normalizing these results as percentages produces a unified scale on which the Bat Algorithm scores 5.25% and Fast Bacterial Swarming scores 7.87%. The authors report that this ranking shows algorithmic simplicity correlates with lower environmental impact: hybrid algorithms carry higher computational overhead, while stochastic and random-search methods are less resource-intensive. They further claim that this integration of swarm characteristics and CO2 emissions surpasses the threshold of existing models.","pith_inferences":["A natural extension not pursued in the paper is to calibrate the hyperfactorial and superfactorial exponents against metered energy data; until then, the percentages should be read as complexity indices, not validated emissions constants.","The same parameter-to-emissions structure could be adapted to non-swarm machine-learning models by replacing the hyperfactorial-superfactorial product with an operation-count or FLOP-based complexity measure.","Because the paper configures all algorithms with standard parameters, a sensitivity analysis varying hyperparameters would reveal whether the ranking is stable or an artifact of the chosen settings.","If calibrated, the formula could feed cloud scheduling policies that route jobs to lower-footprint algorithms or greener regions, a practical consequence the paper leaves implicit."],"forward_implications":["Algorithmic simplicity becomes a usable proxy for carbon impact: the paper's ranking places stochastic and random-search methods below hybrid ones.","ICT teams could compare swarm algorithms on a single complexity-percentage axis without first running them on metered hardware.","Hybrid algorithms, despite their performance advantages, carry an environmental overhead that should be weighed when selecting an optimizer.","The formula gives a first deterministic route from hyperparameter settings to an emissions estimate, a step toward carbon-aware algorithm selection.","Refinement of the formula for real-world applications is the paper's stated next step, implying the current percentages are a starting scale rather than settled constants."],"supporting_citations":[{"why":"Supplies the 2022 global CO2 baseline of 36 gigatonnes that motivates the paper's environmental framing.","marker":"[1]"},{"why":"Establishes ICT's share of global emissions at roughly 6 percent, the problem the paper targets.","marker":"[2]"},{"why":"Provides the energy-consumption factors (cloud duration, hardware, region) that the formula's hardware and regional terms draw on.","marker":"[6]"},{"why":"Defines the swarm hyperparameters (acceleration coefficients, inertia weight, stopping criteria) that enter as the hk factors.","marker":"[14]"},{"why":"Gives the median 72-hour training and tuning time used as the computation-duration context.","marker":"[15]"},{"why":"Supplies the catalogue of nature-inspired and hybrid algorithms evaluated and reported in Table I.","marker":"[16]–[18]"}],"fun_headline_variants":["Simplicity cuts CO2: swarm algorithm complexity studied","Swarm algorithm CO2 emissions ranked: Bat vs Bacterial","CO2 cost: Bat 5.25%, Bacterial 7.87% in swarm study","Swarm intelligence ranks CO2 footprint of algorithms","Simple swarm algorithms carry smaller carbon footprint"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that Eq. (1)–(4) is proportional to actual computational energy consumption, so if that proportionality fails, the percentages in Table I are an arbitrary index rather than carbon emissions.","fun_headline_variants_meta":{"raw":{"variants":["Simplicity cuts CO2: swarm algorithm complexity studied","Swarm algorithm CO2 emissions ranked: Bat vs Bacterial","CO2 cost: Bat 5.25%, Bacterial 7.87% in swarm study","Swarm intelligence ranks CO2 footprint of algorithms","Simple swarm algorithms carry smaller carbon footprint"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001013,"raw_usage":{"total_tokens":4246,"prompt_tokens":883,"completion_tokens":3363,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":499,"completion_tokens_details":{"reasoning_tokens":3281}},"tokens_in":499,"tokens_out":3363,"duration_ms":23993,"temperature":1.0,"reasoning_tokens":3281,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T18:30:12.354022+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run PSO, the Firefly Algorithm, and Fast Bacterial Swarming on identical hardware for identical optimization tasks, meter the energy in kWh, convert it to kg CO2 using regional grid intensity, and compare against the paper's predictions; a mismatch in ranking or ratio would show the formula is not measuring emissions.","supporting_citations":[{"cited_title":"Monitoring globa l carbon emissions in 2022,","cited_arxiv_id":null,"evidence_quote":"Supplies the 2022 global CO2 baseline of 36 gigatonnes that motivates the paper's environmental framing."},{"cited_title":"Greenhouse gas emissions trajectories for th e information and communication technology sector compatible with the un fccc paris agreement,","cited_arxiv_id":null,"evidence_quote":"Establishes ICT's share of global emissions at roughly 6 percent, the problem the paper targets."},{"cited_title":"Sim- plified swarm optimization for hyperparameters of convolut ional neural networks,","cited_arxiv_id":null,"evidence_quote":"Defines the swarm hyperparameters (acceleration coefficients, inertia weight, stopping criteria) that enter as the hk factors."}],"review_version":1}