{"id":"6c4a992f-b684-483c-b9cc-6e17e33d9eb1","arxiv_id":"2607.24975","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"Bond percolation of strongly-connected clusters on directed square lattices is in one new 2D universality class, distinct from ordinary percolation, across Manhattan, L, ice, and random-diode orientations.","lead":"Strongly-connected clusters on directed 2D lattices form a new percolation universality class, distinct from ordinary undirected percolation, shared by several bond-direction patterns. High-precision thresholds, exponents, and wrapping probabilities are reported with public code.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"No fatal flaw: hyperscaling/FSS assumption is the weakest link as the reader flagged, but pc is independently anchored; the real residual risk is unquantified corrections to scaling in the β, γ fits (L=64–1024, no correction term).","rationale":"The reader identified the correct weakest assumption (standard FSS/hyperscaling plus the leading-correction ansatz), and my independent read lands in the same place, with one refinement: the potential circularity in ν is weaker than it first appears because the out-component method pins pc using only exactly known ordinary-percolation exponents, giving a ν-independent anchor that agrees with the wrapping-based estimates. The residual honest concern is that the exponent fits (Sec. IV.C, Appendix B) use no correction-to-scaling term over a single decade of system sizes (L=64–1024), in a paper that elsewhere demonstrates corrections are non-negligible (the 2σ pav vs pc-c discrepancy). This caps the trust one should place in the fourth decimal of β and γ, but it does not threaten the headline claim: the new universality class is established by universal wrapping amplitudes that differ from ordinary percolation by enormous statistical margins, cross-arrangement consistency on four lattices, the exact pc=1 anchor on the random diode model, and concordance with two independent prior groups. Algorithms are standard (Tarjan/Kosaraju), code is public, and internal consistency checks (hyperscaling giving 1.3330(3), scaling collapses in Figs. 9 and 12) all behave. ACCEPT is the right verdict; the proposed direct-ν measurement is a cheap, decisive check using data the authors' pipeline already generates.","tokens_in":21820,"tokens_out":2569,"duration_ms":88417,"concrete_test":"Measure ν directly, bypassing hyperscaling, from the wrapping-probability transition width: with the existing Eq. (5) machinery, compute dR_L/dp at pc on the Manhattan and L-lattices for L=32…1024 and fit max dR_L/dp ∝ L^{1/ν}. Then re-fit β and γ including a correction term with ω scanned over 0.5–2 (or fixed ω=1) on L=64–2048. If the direct ν agrees with 1.333 within ~0.005 and the corrected β, γ stay within the quoted error bars, the hyperscaling concern is settled; if direct ν deviates or the corrected exponents drift by >2σ, the claimed exponent set and df=1.8035(1) need revision (though the distinct-class conclusion survives).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (a single new universality class with β=0.2618(3), γ=2.1423(4), ν≃4/3) rests on two pillars: (i) exponent fits at pc from log-log slopes of largest-cluster size S(L) and average cluster size q(L), and (ii) the hyperscaling relation dν=2β+γ used to infer ν=1.3330(3) and df=1.8035. Three points matter for load-bearing assessment. First, the pc values at which the exponent fits are performed are not hostage to the ν=4/3 assumption: the out-component method (Sec. IV.A) uses only the exactly known ordinary-percolation exponents τ=187/91, σ=36/91 and gives pc=0.6971571(5), agreeing with the wrapping-based cell-to-cell value 0.697160(2). So the mild circularity—assume ν=4/3 to get pc, then \"derive\" ν=1.333 from β,γ measured at that pc—is substantially defused by this independent anchor, and pc enters the S∼L^{−β/ν} fits only as the location of the critical point, where the exponent fit is first-order insensitive to pc error of order 1e−5. Second, the genuinely soft spot is the exponent fitting procedure itself: Appendix B states the exponents come from straight-line fits to the five points L=64…1024 with no correction-to-scaling term (S = a L^{−β/ν}(1 + b L^{−ω})), even though the paper itself invokes corrections to explain the 2σ pc discrepancy between pav and pc-c estimators. If a correction amplitude of similar relative size acts on S(L) or q(L), fitted slopes over one decade of L can shift at the level of the claimed precision (±0.0003 in β), and the quoted error bars reflect only the spread across lattices plus statistics, not systematics. Third, the distinctness from ordinary percolation—the core novelty—does not depend on any of this: the wrapping probabilities (0.750 vs 0.521) are universal amplitude-level quantities differing by >100σ, and agreement with independent groups (de Noronha et al., Wang & Li) corroborates the class. So the concern is precision of the exponent set, not existence of the class.","agreement_with_reader":"agree"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The authors study bond percolation of strongly connected components on four globally isotropic directed square lattices: Manhattan, L-lattice, random diode, and square ice. Using breadth-first search, Tarjan's algorithm, an incremental SCC algorithm, wrapping-based binary searches, and a cluster algorithm for ice configurations, they estimate thresholds, crossing probabilities, critical exponents, cluster-size scaling, and hull dimensions. The principal conclusion is that all four arrangements share a strongly connected percolation universality class distinct from ordinary undirected percolation, with β=0.2618(3), γ=2.1423(4), ν≃4/3, d_f=1.8035(1), and universal wrapping probabilities near 0.861, 0.750, and 0.639. Independent threshold estimates and scaling collapses provide substantial support for the central qualitative claim.","tokens_in":22264,"tokens_out":1959,"duration_ms":67866,"significance":"If the numerical conclusions hold, the paper establishes a useful benchmark universality class for strongly-connected percolation: a common exponent set across four directed square-lattice arrangements, clearly distinct from ordinary percolation, together with threshold constants, wrapping amplitudes, cluster-size scaling, and hull scaling. Particularly strong features are the independent out-component determination of p_c using exact ordinary-percolation exponents, the exact random-diode threshold, the agreement of several algorithmic estimators, the explicit scaling collapses, and the release of simulation code. The result is also falsifiable through the reported high-precision exponents and wrapping probabilities. The main qualification is that the smallest quoted errors currently appear to exclude potentially relevant finite-size systematics.","major_comments":[{"comment":"§IV.C and Appendix B, Fig. 10/Table III: the headline uncertainties β=0.2618(3) and γ=2.1423(4) appear to be statistical errors from pure power-law fits over L=64–1024. Appendix B explicitly says that the threshold extrapolations do not allow for corrections to scaling, and no correction term is reported for the exponent fits either. Yet §IV.A invokes corrections to scaling to explain the roughly 2σ discrepancy between p_av and p_c-c. If comparable corrections affect S(L) or q(L), they can shift one-decade log-log slopes at or above the quoted precision. The qualitative distinction from ordinary percolation is unambiguous, but the stated precision requires a systematic-error analysis: vary the fit window, include a correction-to-scaling term, report local-slope estimates, and propagate threshold uncertainty. The final error bars should include both statistical and systematic components.","section":"§IV.C, Fig. 10, Table III, Appendix B"},{"comment":"The central universality-class inference uses ordinary 2D hyperscaling, dν=2β+γ, to infer ν in Eq. (9), while §IV.A already assumes ν=4/3 in the p_av and p_c-c extrapolations. The circularity is substantially reduced by the independent out-component determination p_c=0.6971571(5), which agrees with p_c-c=0.697160(2) and uses only exact ordinary-percolation τ and σ. Nevertheless, the manuscript should make the logical ordering explicit and provide a robustness check for the exponent and wrapping data as p_c is varied over the independently supported interval. This would separate the measured hyperscaling consistency from an assumed value of ν.","section":"§IV.A, §IV.C, Eqs. (7)–(10)"},{"comment":"Table II and the final paragraph of §IV.B report inconsistent combined crossing probabilities. The table's average row gives 0.86118(06), 0.75001(07), and 0.63889(10), whereas the text gives 0.86117(6), 0.75001(7), and 0.63867(11). The difference in the both-direction value is about two quoted errors and is larger than a harmless rounding discrepancy. Since these universal amplitudes are among the paper's principal predictions and are explicitly contrasted with Cardy–Pinson values, the correct numbers and their combination procedure must be stated consistently.","section":"§IV.B, Table II"}],"minor_comments":[{"comment":"§IV.B: the one-direction crossing estimate is described as \"suspiciously close to 0.75.\" If retained, this observation should be presented as exploratory; the present evidence does not establish an exact value, and the phrase may invite overinterpretation.","section":"§IV.B"},{"comment":"Table I: for consistency with the other preferred estimates and with the out-component result p_c=0.6971571(5), please clarify exactly how the final \"Average\" cell-to-cell entry combines the three wrapping definitions and whether correlations between horizontal/vertical measurements are included.","section":"§IV.A, Table I"},{"comment":"Fig. 8 uses an L^{-1} extrapolation for R_L(p_c), while threshold fits elsewhere use L^{-1/ν} or L^{-1-1/ν}. Please state the rationale and whether changing the assumed correction exponent materially changes Table II.","section":"§IV.B, Fig. 8"},{"comment":"§III.E: the Potts-cluster update is said to require a constant number of sweeps and four sweeps are used. A short diagnostic (for example an autocorrelation estimate or independence check) would strengthen confidence that residual ice-configuration correlations are negligible at the quoted precision.","section":"§III.E"},{"comment":"§IV.D: the hull fit yielding d_h=1.3331(2) should specify the fitted L range and give the same correction-to-scaling or fit-window sensitivity requested for β and γ. The coincidence with 4/3 is currently suggestive rather than established.","section":"§IV.D, Fig. 14"},{"comment":"The reproducible code link and appendix descriptions are valuable. For long-term reproducibility, please also state the random-number generator, seeding policy, code version or archive DOI, and total run statistics for the principal tables.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The work appears timely and within the journal's scope. Given the unusually small exponent uncertainties, I recommend that the editor request a clear accounting of systematic fit uncertainty, in addition to purely statistical errors."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is simple: strongly-connected bond percolation on isotropic directed square lattices is in one universality class, clearly distinct from ordinary percolation, and the four arrangements they study (Manhattan, L, ice, random diode) share it. That is the real addition beyond the earlier resistor-diode papers.\n\nWhat they do well is the multi-method campaign. Out-component scaling with the exact ordinary τ and σ pins pc independently of any new exponents; wrapping pav and cell-to-cell, Tarjan at pc, and the incremental full-p curves all line up. Wrapping probabilities at threshold (roughly 0.861 / 0.750 / 0.639) sit more than 100σ away from the Cardy–Pinson numbers, so the class distinction does not rest on delicate slope fits. Hull dimension consistent with 4/3 is a nice extra. Public C code is a genuine plus.\n\nSoft spots are real but limited. The β and γ values come from bare log-log slopes on L = 64–1024 with no correction-to-scaling term, even though the authors themselves invoke corrections to explain the 2σ pav vs cell-to-cell pc split. Quoted errors (±0.0003 on β) therefore look a touch tight; the central values could shift at that level. Hyperscaling is assumed rather than tested, but pc itself is not hostage to it, and the existence of the class does not depend on the last digits of ν or df. Prior literature already pointed at a distinct class; novelty is the pure-directed scope, arrangement universality, wrapping amplitudes, and algorithms.\n\nThis is for people who need benchmark numbers or who work on directed networks and percolation. It deserves a serious referee. I would engage with it and expect it to be cited as the clean 2D reference for the pure-directed case.","headline":"Clean high-precision numerics establishing one new 2D universality class for strongly-connected clusters on pure directed lattices; the class is real, the quoted exponent digits are a bit optimistic.","tokens_in":23177,"tokens_out":487,"would_cite":true,"duration_ms":13129,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Strongly-connected percolation on directed lattices forms one universality class distinct from ordinary undirected percolation.","keywords":["strongly-connected percolation","directed lattices","universality class","critical exponents","wrapping probabilities","Manhattan lattice","bond percolation","fractal dimension"],"falsifier":"On large Manhattan or L-lattices, measure the one-direction wrapping probability at the estimated threshold and test whether it converges to ~0.75, or recompute β from largest-cluster scaling beyond L = 1024 and check whether it stays near 0.262 rather than ordinary percolation’s 5/36.","tokens_in":22823,"feed_emoji":"🔀","tokens_out":873,"duration_ms":27663,"temperature":0.7,"pith_summary":"This paper shows that when bonds on a two-dimensional square lattice point in directions, the clusters in which every site can reach every other along directed paths—strongly-connected clusters—percolate with critical exponents and wrapping probabilities that differ sharply from ordinary undirected percolation. High-precision simulations on four globally isotropic arrangements (Manhattan, L-lattice, random diode, and square ice) produce a shared set of exponents, fractal dimensions, crossing probabilities, and thresholds. A reader who cares about networks, traffic, the web, or metabolic graphs cares because those systems are built from directed reachability, yet their large-scale critical geometry was only partly mapped. The work also supplies practical algorithms for locating thresholds and maintaining clusters as bonds are added. The shared numbers across arrangements imply that local direction patterns wash out at large scales, leaving a single new class.","feed_headline":"Directed lattices share a new percolation class","feed_subtitle":"Four isotropic bond setups yield the same exponents, distinct from ordinary undirected percolation.","key_machinery":"Strongly-connected clusters (sites mutually reachable by directed paths), located by Tarjan’s algorithm, binary-search threshold finding, and finite-size wrapping detection on periodic lattices. These tools extract universal exponents and crossing numbers that do not depend on the local direction pattern.","core_discovery":"Bond percolation of strongly-connected clusters on the two-dimensional square lattice, for several globally isotropic bond-direction arrangements, belongs to one universality class distinct from ordinary undirected percolation. The measured values are β = 0.2618(3), γ = 2.1423(4), ν ≃ 4/3, fractal dimension df = 1.8035(1), and wrapping probabilities at threshold of about 0.861 (any direction), 0.750 (one direction), and 0.639 (both)—all incompatible with ordinary percolation.","pith_inferences":["The same class likely covers other isotropic directed models the paper only conjectures about, such as two-neighbor or randomly-oriented Manhattan lattices.","The numerics leave open an exact value of 3/4 for one-direction wrapping probability as an analytic target.","Higher-dimensional and non-square lattices are the natural next test of whether the class survives or splits."],"forward_implications":["Critical exponents and wrapping numbers for strongly-connected percolation can be treated as universal across isotropic directed square lattices.","Site percolation on the Manhattan lattice shares the same threshold and exponents as bond percolation on the L-lattice.","Hulls of wrapping strongly-connected clusters are consistent with fractal dimension exactly 4/3.","High-precision thresholds are established: pc ≈ 0.697160 (Manhattan), 0.740193 (L-lattice), 0.708838 (ice), and exactly 1 for random diodes."],"fun_headline_variants":["Directed lattices share one new strongly-connected percolation class","Isotropic bond setups yield shared exponents unlike ordinary percolation","Strongly-connected clusters on directed grids form distinct universality class","Four directed bond arrangements show same percolation exponents","2D directed lattices: strongly-connected percolation leaves ordinary class"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"Ordinary two-dimensional finite-size scaling and hyperscaling still hold for these directed clusters, so the correlation-length exponent and fractal dimension can be read directly from the measured β and γ.","fun_headline_variants_meta":{"raw":{"variants":["Directed lattices share one new strongly-connected percolation class","Isotropic bond setups yield shared exponents unlike ordinary percolation","Strongly-connected clusters on directed grids form distinct universality class","Four directed bond arrangements show same percolation exponents","2D directed lattices: strongly-connected percolation leaves ordinary class"]},"model":"grok-4.5","effort":"low","cost_usd":0.00382,"raw_usage":{"total_tokens":1131,"prompt_tokens":687,"num_sources_used":0,"completion_tokens":84,"cost_in_usd_ticks":38204000,"prompt_tokens_details":{"text_tokens":687,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":360,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":687,"tokens_out":84,"duration_ms":6287,"temperature":1.0,"reasoning_tokens":360,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T04:29:26.470179+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"On large Manhattan or L-lattices, measure the one-direction wrapping probability at the estimated threshold and test whether it converges to ~0.75, or recompute β from largest-cluster scaling beyond L = 1024 and check whether it stays near 0.262 rather than ordinary percolation’s 5/36.","supporting_citations":[],"review_version":1}