{"id":"d22849cc-3fcc-4977-bfd9-1a6ba474596f","arxiv_id":"2607.18991","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"New Monte Carlo data give 120 first-time percolation thresholds for complex 5D neighborhoods, with a fitted scaling exponent g≈0.79; internal inconsistencies and missing error bars weaken the stated fits.","lead":"This paper reports Monte Carlo estimates of percolation thresholds for 127 complex neighborhoods on a five-dimensional simple cubic lattice, 120 of them new, and fits them to an empirical scaling law. The data set extends a decade-long catalog of percolation thresholds and tests whether a simple coordination-weighted formula predicts critical probabilities in 5D.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 120 new p_c values rest on an unvalidated FSS crossing with fixed literature exponents and no per-threshold error bars; any common systematic error propagates directly into the claimed g≈0.7913.","rationale":"The reader's weakest assumption identifies the fixed-exponent FSS extraction and the absence of per-threshold error bars as the load-bearing point. My independent reading agrees: the seven compact thresholds matching prior literature are good evidence that the pipeline works in the high-p_c regime, but they do not validate the 120 new low-p_c thresholds, which are the ones that control the ζ-scaling fit. The internal g discrepancy (0.8293(24) vs 0.7913(43)) reinforces the need for a careful re-analysis, but the more fundamental issue is that the accuracy claim is not supported by any reported uncertainty or by a cross-check against an exponent-free estimator. This does not warrant rejection — the code is shared and the agreement on seven known values is real — but it does justify the reader's CONDITIONAL verdict: publish the data and code, but only after the systematic uncertainty in the FSS crossing is quantified and the g inconsistency is resolved.","tokens_in":12919,"tokens_out":5611,"duration_ms":55139,"concrete_test":"Run the released Newman–Ziff code for two representative neighborhoods — one dense, e.g., sc(5)-1,2, and one sparse/high-ζ, e.g., sc(5)-1,2,3,4,5,6,7 or sc(5)-5,6,7 — at L=40, 48, 64 with R=10^6, and also estimate p_c using an exponent-free estimator such as the crossing of the wrapping probability or the peak of the cluster-size susceptibility. If the new p_c values differ from the L=16–32 FSS values by more than 10^-5, the claimed 10^-5 accuracy fails and the catalog requires error bars and a re-fit. Independently, refit Eq. (4) after applying a common relative shift of ±10^-5 to all low-p_c thresholds to see whether g moves outside 0.7913(43).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that 120 new percolation thresholds are accurate to 5–6 digits and that p_c ∝ ζ^{-g} with g≈0.7913(43). The thresholds are extracted in Section II.A.1/III.A from the common crossing of P_max(p;L)L^{β/ν} using β_5=0.8457 and one of three listed ν_5 values (0.5746, 0.5737(33), or 0.5720(43)), but the manuscript never states which ν was used. No per-threshold uncertainties are reported; the claimed 10^-5 accuracy is inferred from the apparent sharpness of the crossing, not from a statistical estimate. Because all 120 new thresholds share the same FSS procedure, the same L range (≈16–32), and the same exponent choice, any error in the assumed exponents or any uncorrected finite-size correction will enter every threshold coherently rather than averaging out. The seven compact thresholds that match prior work are mostly high-p_c, low-ζ cases; the new low-p_c, high-ζ values dominate the log-log fit for g, so even a 10^-5–10^-4 absolute bias in those values could shift g by more than the quoted 0.0043. Additionally, Section III.B reports g≈0.8293(24), while Figure 3, the abstract, and Section IV report g≈0.7913(43); until this discrepancy is resolved, the scaling exponent itself is not settled. The honest check is to validate the extraction on representative cases with a method that does not depend on the assumed exponents.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper reports a parallel C++ implementation of the Newman–Ziff algorithm and uses it to study random site percolation on the five-dimensional simple cubic lattice with neighborhoods built from coordination zones up to the seventh. The authors present 127 percolation thresholds, 120 of which they state are new, fit the thresholds to the heuristic relations p_c ∝ ζ^{-g} and p_c = c/(z+b), and compute fractal dimensions d_f for seven compact neighborhoods. The abstract and Section IV quote g ≈ 0.7913(43), ⟨d_f⟩ ≈ 3.5581(70), and relative deviations of d_f from the scaling-relation value d*_f ≈ 3.5215 that range from 0.26% to 1.76%.","tokens_in":13315,"tokens_out":7115,"duration_ms":57447,"significance":"If the threshold catalog is accurate, it is a useful benchmark for percolation in d=5 with long-range neighborhoods, and the 120 new values are a substantial numerical contribution. The public code and the use of a standard, independent algorithm (Newman–Ziff) are strengths. However, the central quantitative claims — the ζ-scaling exponent g and the compatibility of the fractal dimensions with scaling relations — are currently not supported by the reported statistics: the value of g is internally inconsistent, the thresholds are given without error bars or a stated choice of the fixed exponents used in the finite-size scaling, and the d_f estimates deviate from d*_f by many times their quoted uncertainties. These issues must be resolved before the numerical conclusions can be relied upon.","major_comments":[{"comment":"The value of g for p_c(ζ) is reported twice: g ≈ 0.8293(24) in the text of Section III.B, and g ≈ 0.7913(43) in the Fig. 3 caption, the abstract, and Section IV. These differ by 0.038, far outside the quoted uncertainties. Since the ζ-scaling exponent is a central result, the manuscript must state which fit is correct, and if the two fits use different data sets or different definitions of ζ, that must be made explicit.","section":"§III.B, Fig. 3, §IV"},{"comment":"The thresholds in Table I are obtained by common crossings of P_max L^{β/ν} with β_5 = 0.8457 and one of three listed ν_5 values (0.5746, 0.5737(33), or 0.5720(43)), but the text never states which ν was used, and no per-threshold statistical uncertainty is reported. The claimed 10^-5 precision in Fig. 1 is read off the sharpness of the crossing, not from a bootstrap/jackknife or from the variance among the L-pair crossings. Any error in the assumed exponents or uncorrected finite-size corrections enters all 120 new thresholds coherently, and the low-p_c/high-ζ points that dominate the log-log fit for g are precisely the ones for which no independent validation exists. The authors should supply per-threshold error bars, state the exponent choice, and test sensitivity to the FSS ansatz (e.g., by using different L ranges or by comparing with the seven previously known thresholds only).","section":"§II.A.1 and §III.A"},{"comment":"The d_f values are not merely 'not differ[ing] much' from d*_f = 3.5215. Using the quoted uncertainties, the deviations range from ~4σ (sc(5)-1,2, d_f = 3.5307(23)) to ~45σ (sc(5)-1,2,3,4,5,6, d_f = 3.57944(80)) when the 0.0010 uncertainty on d*_f is included. The sign is also uniform: every estimate exceeds d*_f, which suggests a systematic finite-size effect rather than statistical scatter. The text and abstract should either quantify and explain this discrepancy or soften the claim that the fractal dimensions confirm the scaling-relation value.","section":"Table II and §IV"}],"minor_comments":[{"comment":"The phrase 'in five dimension' should be 'in five dimensions' or 'in five-dimensional space'.","section":"Title and Abstract"},{"comment":"Equation (11) defines δ = (d*_f − d_f)/d*_f · 100%, which is negative for all rows in Table II because every d_f exceeds d*_f. The table lists positive values; either use the absolute value or define δ as |d*_f − d_f|/d*_f.","section":"Eq. (11) and Table II"},{"comment":"Reference [4] spells 'Birkhäuser' as 'Brikhauser'; reference [59] is a duplicate of reference [40] and should be consolidated.","section":"References"},{"comment":"The statement that equation (2) is 'qualitatively not worse' than p_c ∝ (z+1)^{-1} is vague. Since equation (2) has two fitted parameters, the comparison should be quantitative, e.g., residual per data point or AIC.","section":"§IV"},{"comment":"The phrase 'more or less agree' is too vague for a quantitative comparison. The caption should state the difference and the combined uncertainty, especially in view of the deviations noted in the major comments.","section":"Fig. 4 caption"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope and the threshold catalog, if validated, would be a useful contribution. The internal inconsistency in g and the missing error propagation are fixable in revision, so I do not see a need for rejection. The authors should be asked to resolve the g discrepancy, provide per-threshold uncertainties and the FSS exponent choice, and revisit the d_f claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this is a genuinely useful numerical dataset — 127 percolation thresholds for complex neighborhoods on the five-dimensional simple cubic lattice, 120 of them new, plus a working OpenMP-parallelized Newman–Ziff code. That alone earns a referee’s time. But the paper is not ready as-is: the ζ-scaling exponent g is reported as 0.8293(24) in Sec. III.B and as 0.7913(43) in the abstract, Fig. 3, and Sec. IV, and the 120 new thresholds carry no per-threshold error bars.\n\nWhat it does well: the catalog is large, the seven compact-neighborhood thresholds from refs [50,51] are reproduced to six digits, and the code is publicly available. The speedup discussion is honest — Amdahl’s law, memory bottlenecks, and paging are all mentioned. Testing the 2D ζ-power-law ansatz in 5D is a natural extension, and the fit (whichever of the two g values is correct) describes the trend over nearly three decades in p_c.\n\nSoft spots:\n- The g inconsistency is load-bearing because the abstract and Sec. IV present it as a headline result. It must be reconciled — likely a typo, but it blocks trust in the scaling claim.\n- Error bars: the paper claims 10^-5 accuracy from the sharpness of the FSS crossing, but no per-threshold uncertainties are reported. Since all thresholds come from the same method, the same L range (16–32), and an unspecified ν choice among three cited values, any systematic error — from corrections to scaling or a wrong ν — shifts all values coherently and biases the fitted g. The seven validated cases are high-p_c; the new low-p_c cases dominate the power-law fit. A check on a few representative cases using an exponent-independent crossing (or trying the other ν values) would validate the core data.\n- Fractal dimensions: the seven d_f values are all 4–72σ above d*_f = 3.5215, yet the text calls this agreement and quotes percentage errors of 0.26–1.76%. That is statistically inconsistent. Either the uncertainties are underestimated or there’s a real offset; the paper mentions ref [65]’s 3.5260(14) but never confronts the discrepancy. This section needs an honest rewrite.\n- Minor but easy: state which ν_5 value was used for the crossings.\n\nWho it’s for: anyone needing percolation thresholds in higher-dimensional lattices with extended or complex neighborhoods, and people interested in efficient Newman–Ziff implementations. It is a reference-table paper, not an analytical breakthrough.\n\nMy recommendation: send it to peer review with a request for major revision. The dataset and code are valuable enough that the issues are worth fixing, not rejecting over. If the authors resolve the g inconsistency and report uncertainties honestly, this becomes a solid reference.\n\n— [you]","headline":"A large, useful 5D percolation threshold table, but the headline exponent is internally inconsistent and the new thresholds have no error bars; worth a round of revision.","tokens_in":13846,"tokens_out":4709,"would_cite":true,"duration_ms":39703,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B43","82B27","82B80"],"pacs":["64.60.ah","05.10.Ln"],"model":"deepseek-v4-flash","headline":"The paper tries to establish that in five-dimensional site percolation, the random occupation threshold p_c for any neighborhood built from the first seven coordination zones is governed by a single quantity—the weighted coordination number","keywords":["site percolation","percolation threshold","complex neighborhoods","five-dimensional lattice","finite-size scaling","fractal dimension","weighted coordination number","power law universality"],"falsifier":"Compare the extracted crossing point for the simplest neighborhood (sc(5)-1) with the high-precision literature value 0.14079633(4); a discrepancy beyond 10^-5 would indicate a systematic bias in the finite-size scaling procedure. Alternatively, recompute one new threshold at much larger lattice sizes (e.g., L = 48 or 64) and observe whether the crossing point drifts outside the claimed 10^-5 accuracy.","tokens_in":12778,"feed_emoji":"🌀","tokens_out":3908,"duration_ms":32178,"temperature":0.7,"pith_summary":"The paper reports 127 site-percolation thresholds for complex neighborhoods on a five-dimensional simple cubic lattice, 120 of them estimated for the first time, and shows that they follow a clean power law in the weighted coordination number ζ. It also measures the fractal dimension of the incipient percolation cluster for seven extended (compact) neighborhoods, finding a mean of 3.5581(70), close to the scaling-relation prediction of 3.5215. A sympathetic reader cares because if the ζ-scaling is genuine, it provides a compact predictive rule for thresholds in a high-dimensional lattice where direct simulation is expensive. The study also demonstrates a parallel implementation of an efficient Monte Carlo algorithm that makes such large-scale threshold calculations feasible.","feed_headline":"One power law holds for 127 five-dimensional percolation thresholds","feed_subtitle":"First estimates for 120 neighborhoods confirm p_c decays as weighted coordination number to the power ~0.79.","key_machinery":"The argument rests on two mechanisms: (1) a fast Monte Carlo algorithm that constructs clusters by adding one occupied site at a time and uses binomial convolution to convert site-count data into occupation-probability data, and (2) finite-size scaling, where the rescaled probability of belonging to the largest cluster, P_max L^{β/ν}, plotted against p, gives a size-independent crossing point at p_c. The crossing point is located using R = 10^6 repetitions per system, claimed to give 10^-5 accuracy.","core_discovery":"The central claim is that percolation thresholds for complex neighborhoods in five-dimensional simple cubic lattices obey a universality in the weighted coordination number ζ: plotting p_c against ζ yields a power law p_c ∝ ζ^{-g} with g ≈ 0.7913(43), supported by 127 thresholds (120 new). For the seven extended neighborhoods tested, the fractal dimension of the incipient percolation cluster averages ⟨d_f⟩ = 3.5581(70), with individual values within 0.26%–1.76% of the theoretically predicted d*_f = 3.5215 derived from the best current critical exponents.","pith_inferences":["The observed ζ-scaling suggests a cross-dimensional check: computing similar thresholds in d = 4 or d = 6 would reveal whether the exponent g depends on dimension or is a constant, potentially connecting to continuous-percolation limits.","Because all 120 new thresholds share the same finite-size scaling analysis, any bias in the assumed critical exponents (β_5 = 0.8457, ν_5 ≈ 0.572–0.575) will appear as a common systematic shift; an independent high-precision measurement of a single new threshold would bound that bias.","The residual scatter around the ζ power law may encode information about the contribution of individual coordination shells; adding a shape-dependent correction term could tighten the fit and refine the predicted thresholds."],"forward_implications":["If g ≈ 0.79 is universal in five dimensions, thresholds for neighborhoods extending to any coordination zone can be predicted without new simulations.","The 120 new thresholds provide a benchmark dataset for evaluating other heuristic formulas, such as p_c = c/(z + b).","The mean fractal dimension ⟨d_f⟩ = 3.5581(70) supports the universality of d_f across extended and complex neighborhoods in five dimensions.","The parallelized Monte Carlo implementation shows a practical route to high-precision percolation thresholds in higher dimensions with modest wall-clock time."],"fun_headline_variants":["127 five-dimensional percolation thresholds collapse onto one power law","Power law unifies 127 five-dimensional percolation thresholds","120 new 5D percolation thresholds confirm universal scaling","Five-dimensional percolation: one power law for 127 neighborhoods"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire threshold table rests on the assumption that the finite-size scaling ansatz with the literature values β_5 = 0.8457 and ν_5 ≈ 0.572–0.575 holds for systems of linear size L = 16–32; if those exponents are wrong, every new p_c inherits a common error, and the fitted g ≈ 0.7913 could change.","fun_headline_variants_meta":{"raw":{"variants":["127 five-dimensional percolation thresholds collapse onto one power law","Power law unifies 127 five-dimensional percolation thresholds","120 new 5D percolation thresholds confirm universal scaling","Five-dimensional percolation: one power law for 127 neighborhoods"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000248,"raw_usage":{"total_tokens":1423,"prompt_tokens":826,"completion_tokens":597,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":525}},"tokens_in":570,"tokens_out":597,"duration_ms":7195,"temperature":1.0,"reasoning_tokens":525,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T13:47:04.626874+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compare the extracted crossing point for the simplest neighborhood (sc(5)-1) with the high-precision literature value 0.14079633(4); a discrepancy beyond 10^-5 would indicate a systematic bias in the finite-size scaling procedure. Alternatively, recompute one new threshold at much larger lattice sizes (e.g., L = 48 or 64) and observe whether the crossing point drifts outside the claimed 10^-5 accuracy.","supporting_citations":[],"review_version":1}