{"id":"e3a627f3-0883-4929-8760-0af4f05dc69b","arxiv_id":"2504.16510","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Active monopolar forces on critical percolation networks reproduce the thermal subdiffusion exponent 1-d_s/2, whereas dipolar forces cause slow collapse, saturation, and, in rigid triangular lattices, a sharp collapse transition.","lead":"This paper simulates active spring networks built from disordered fractal clusters (critical percolation) under random active forces. It finds that monopolar forces produce subdiffusion with an exponent set by the network's spectral dimension, while dipolar forces shrink the network and can drive collapse above a modified rigidity threshold.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reported active monopole MSD exponent 0.316±0.003 in Fig. 4(a) conflicts with the central prediction ν=1−d_s/2=0.335; the headline claim needs this discrepancy resolved.","rationale":"I focused on the strongest claim as summarized by the reader: the active monopole MSD exponent is solely controlled by d_s. The paper's own numbers in Fig. 4(a) are the primary evidence for this, and they disagree with the predicted value by roughly twenty times the reported statistical error. The thermal case matches, so the issue is not the measurement of d_s or the general theoretical framework; it is the active case specifically. This is a concrete, addressable problem rather than a matter of taste. I do not claim the paper is wrong; a late-time asymptotic fit may well recover 0.335. But the paper currently does not show that, and the abstract asserts the stronger conclusion. The reader's conditional verdict is appropriate; the Maxwell-criterion concern raised by the reader is real but secondary, because it affects the dipolar collapse/rigidity extension rather than the headline fractal subdiffusion result. Hence I keep the verdict unchanged while disagreeing with the reader's choice of weakest assumption.","tokens_in":15678,"tokens_out":12407,"duration_ms":130968,"concrete_test":"Rerun the athermal monopole simulation on a larger critical percolation cluster (e.g., L=200) with τ=100, discard an initial transient of at least τ_N/10, and fit the MSD over sliding windows such as [10^3,10^4], [10^4,10^5], and [10^5,10^6]. Verify that the exponent is stable and equals 1−d_s/2=0.335 within statistical error, and repeat for f=0.5 and f=2 and for at least two independent percolation realizations. If the fitted exponent remains near 0.316, the central claim is contradicted; if it drifts toward 0.335 as the window moves right, the reported fit was a crossover artifact.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central result is that for monopolar active noise the intermediate-time MSD exponent is ν=1−d_s/2, 'solely controlled by the spectral dimension' (Abstract; Eq. 14). With d_s=1.33 (Fig. 2), this predicts ν=0.335. The thermal fit (Fig. 3) gives 0.336±0.004, consistent. However, the athermal monopole fit (Fig. 4(a)) is reported as 0.31633±0.002541, a 7.5σ departure from 0.335. Because the claim is specifically that the exponent depends only on d_s, a 6% systematic offset in the data that supports that claim is load-bearing. The paper neither acknowledges this offset nor explains it; possible sources (fit window including the t≈τ crossover or the long-time ballistic/diffusive regimes, finite lattice size, or artifacts of subtracting the empirical mean displacement in Eq. 6) are all testable. Until one of these is demonstrated, the headline scaling is not established by the presented numerics.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reports Langevin dynamics simulations of bead-spring networks built from critical bond-percolation clusters on square and triangular lattices, driven by thermal noise and by random-telegraph active forces. It claims that for force monopoles the intermediate-time MSD is subdiffusive with exponent ν = 1 − d_s/2, controlled solely by the spectral dimension d_s, and that the network reaches a dynamically swelled steady state. For force dipoles it reports a quasi-steady-state saturation regime followed by a second ballistic rise and then diffusion at low dipole fraction, persistent or fluctuating rotations, and a collapse transition in triangular lattices above a dilution-dependent threshold. The paper further proposes a modified Maxwell criterion, Eqs. (B7)–(B9), in which ON-state active dipoles contribute additional constraints and shift the rigidity percolation point to p'_rigid = p_rigid/(1 + P ϕ). The authors are candid about limitations: the dipole network does not reach a true steady state, excluded volume and hydrodynamic interactions are absent, and the monopole CM drift is only approximately removed by Eq. (6).","tokens_in":15930,"tokens_out":3954,"duration_ms":42182,"significance":"If the central claim holds, the paper gives a clean, falsifiable prediction: for active monopolar noise on disordered fractals, the anomalous MSD exponent depends only on d_s, not on force amplitude, correlation time, or other fractal dimensions. The thermal control case matches the theory well (ν = 0.336 ± 0.004 against 1 − d_s/2 = 0.335), and d_s = 1.33 is measured independently from the connectivity matrix. The dipole phenomenology, including the shifted rigidity boundary and rotational motion, is a plausible extension of earlier work on the Sierpinski gasket to disordered percolation clusters. The paper does not provide machine-checked proofs or a fully parameter-free derivation, since the analytic MSD formulas (Eqs. 10–11) are taken from the authors' prior paper; nonetheless, the numerical confrontation with those formulas is a useful contribution. The main value would be in extending active-network results from deterministic to disordered fractals and in motivating experiments on chromatin and actomyosin networks, provided the quantitative discrepancies noted below are resolved.","major_comments":[{"comment":"The headline prediction ν = 1 − d_s/2 = 0.335 (with d_s = 1.33 from Fig. 2) is not reproduced by the athermal monopole fit, which is reported as ν = 0.31633 ± 0.002541. This is a statistical offset of about 7.5σ from the predicted value, the paper does not acknowledge or explain it, and the claim that the exponent is 'solely controlled' by d_s is therefore not established by the presented numerics. Please report the exact fit window, the sensitivity of ν to the fit boundaries (especially the t ≈ τ crossover and the late-time ballistic/diffusive regimes), and the effect of the mean-displacement subtraction in Eq. (6). Without this analysis, the central scaling result rests on a fit that is, on its face, inconsistent with the theory at the level of the fitted uncertainty.","section":"§VI B, Fig. 4(a), Eq. (14)"},{"comment":"The modified Maxwell criterion treats each ON-state active dipole as an additional mechanical constraint, adding P ϕM to the number of constraints. A prescribed motor force is not a fixed-length geometric constraint; whether active stresses stabilize floppy modes must follow from the force balance and dynamics rather than from constraint counting. As written, Eq. (15) is an assumption, and consequently the predicted dilution boundary f_c = 0 at p'_rigid and the conclusion that active disordered solids must be poised above a shifted isostatic point are unsupported. Please provide a derivation of the stability condition, or a direct numerical test that is independent of f and P (e.g., measuring the rigidity onset for different force amplitudes and ON probabilities, or comparing the actual constraint count with the stability threshold) to distinguish constraint counting from stress-induced stabilization.","section":"Appendix B, Eqs. (B7)–(B9)"},{"comment":"The dipole MSD is analyzed in a pseudo-steady state while R_g is continuously decreasing by 2–5% over the simulation. The control in Fig. 5(b) shows δR_g < sqrt(MSD) only for lag times shorter than about 10^6, but the reported saturation, second ballistic rise, and diffusive regime extend to longer times, and the rotation analysis in Fig. 8 covers even longer intervals. The paper should demonstrate that the time-averaged MSD is insensitive to the ongoing shrinkage over the full lag-time window, or restrict the fitting to lag times where the pseudo-steady-state condition is quantitatively verified. As it stands, the interpretation of the long-time dipole regimes as intrinsic network dynamics rather than as artifacts of a slowly collapsing reference configuration remains open.","section":"§VI C, Figs. 5–6"}],"minor_comments":[{"comment":"Eq. (14) states the regime as 't << τ', but the surrounding text and Eq. (11) describe the regime τ << t << τ_N; the inequality in Eq. (14) is a typo and should be corrected.","section":"§V, Eq. (14)"},{"comment":"The expression 't_tot − t' lacks parentheses in both the prefactor and the summation upper limit, making the intended denominator (t_tot − t) unclear.","section":"§IV C, Eqs. (5)–(7)"},{"comment":"The text writes 'ν = 0.33(5)' for the predicted value 1 − d_s/2 = 0.335; the notation is ambiguous and should be written as 0.335 or 0.33(5) with an explicit convention.","section":"§VI A"},{"comment":"The text says a 'second ballistic regime (not shown)' due to CM drift, then says this regime is replaced by linear scaling after applying Eq. (6). Please clarify which data points are excluded from the fit and how the CM subtraction affects the fitted window.","section":"§VI B"},{"comment":"The power-law fit is labeled 'g(x) ∼ x^{−0.54}' while the horizontal axis of the inset appears to be the dipole fraction ϕ; please define x explicitly (presumably x = ϕ).","section":"Fig. 7(a) inset"},{"comment":"The long-time ballistic-like rise for ϕ = 0.2 is attributed in §VI C to 'effectively free motion of dangling ends', while the abstract and conclusion attribute the two second-rise behaviors to persistent versus fluctuating local rotations; these explanations should be reconciled or distinguished in the text.","section":"§VI C and §VII"}],"recommendation":"major_revision","confidential_remarks":"The decisive issue for the referee process is the monopole exponent mismatch: the paper's central quantitative claim is contradicted by its own fit at high formal significance, and the authors do not discuss the offset. This should be addressed with a concrete fit-window and finite-size analysis before publication. The Appendix B Maxwell-counting extension is likely to be contested by readers; if it remains a stated assumption rather than a derived result, the associated collapse-boundary predictions should be presented as conjectures. The dipole pseudo-steady-state control is also worth strengthening, as the rotation analysis in Fig. 8 extends well beyond the lag times where the δR_g control is demonstrated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a worthwhile computational study with a load-bearing numerical discrepancy at its center. The paper extends the spectral-dimension scaling ν=1−d_s/2 from the Sierpinski gasket to disordered critical percolation clusters for active monopoles, and it adds new phenomenology for active dipoles: slow collapse below the rigidity threshold, a sharp collapse transition in triangular lattices, and spontaneous rotations. Those are real, new observations, and the comparison to their own analytic theory (Ref. [13]) is a good idea. The thermal case is clean: d_s=1.33 gives predicted 0.335, and they fit 0.336±0.004.\n\nBut the athermal monopole fit in Fig. 4(a) gives 0.3163±0.0025, about 6% below the prediction and off by ~7σ. The abstract says the exponent is \"solely controlled\" by d_s, so this offset sits on the central claim. The paper never mentions the difference. It could easily be a fit-window artifact—the crossover at t=τ and the late-time ballistic CM regime bracket the anomalous window—but until they show that, the headline scaling is not actually demonstrated by their data. This is the first thing a referee should ask for.\n\nSecond, the modified Maxwell criterion in Appendix B counts each ON dipole as an extra mechanical constraint. That is an assumption, not a derivation: a prescribed active force does not fix a bond length, and the proposed p'_rigid has no dependence on force amplitude, even though the collapse force f_c clearly does. The authors use Eq. (15) to place the f_c=0 points at p=0.6 and 0.55, so the predicted rigidity boundary rests on that counting. The collapse data themselves are interesting and should stand on their own; the interpretation needs more support, or at least an acknowledgment that it is an effective rule.\n\nSmaller issues: the dipole MSD is measured in a pseudo-steady state while R_g drifts; the authors check this, and it is probably fine, but the second ballistic rise and rotation interpretation are more speculative. No system sizes, number of realizations, or code/data are given, which makes the numbers hard to check.\n\nWho it is for: soft-matter and active-matter folks who care about percolating networks, chromatin subdiffusion, or motor-driven gel collapse. It deserves a serious referee, but the referee should insist on a resolved exponent fit and a frank discussion of the constraint-counting assumption before the ambitious claims are accepted.","headline":"The central monopole-exponent claim is undercut by the paper's own fitted exponent (0.316 vs. 0.335), but the disordered-percolation phenomenology is genuinely new and worth sending to review.","tokens_in":16440,"tokens_out":5809,"would_cite":false,"duration_ms":61413,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"On critical percolation clusters used as disordered fractals, active force monopoles produce subdiffusive motion whose exponent is fixed by the network's spectral dimension, while active force dipoles drive slow collapse, sharp collapse…","keywords":["active matter","fractal networks","spectral dimension","anomalous diffusion","mean square displacement","percolation","force dipoles","rigidity percolation"],"falsifier":"Simulate a bond-percolation cluster with force monopoles at fixed $d_s$ while varying $f$ and $\\tau$ (for example $f=0.5$ versus $2$, $\\tau=1$ versus $100$): if the fitted intermediate exponent moves away from $1-d_s/2$, or the prefactor does not scale as $f^2\\tau$, the spectral-dimension-only claim fails. Independently, measure the collapse threshold of diluted triangular lattices with active dipoles at several $\\phi$: if $f_c$ does not extrapolate to zero at $p'_{\\mathrm{rigid}}=p_{\\mathrm{rigid}}/(1+P\\phi)$, the modified Maxwell criterion fails.","tokens_in":15448,"feed_emoji":"🕸️","tokens_out":9556,"duration_ms":84117,"temperature":0.7,"pith_summary":"This paper asks how disordered fractal spring networks—realized as the infinite cluster at bond percolation on square and triangular lattices—respond to stochastic active forces. It tries to establish that under force monopoles the intermediate-time mean-square displacement is governed entirely by the spectral dimension $d_s$ of the fractal, with $\\mathrm{MSD}\\sim t^{1-d_s/2}$, the same exponent that thermal noise produces. For force dipoles, it argues that the same fractal clusters never settle: they shrink continuously and the MSD shows a saturation plateau followed by either persistent or fluctuating local rotations. The work matters because such networks are minimal models for actomyosin gels, chromatin, and other active biopolymer systems, where molecular motors deliver localized forces rather than uniform thermal kicks.","feed_headline":"Active monopoles on fractal networks share one spectral exponent","feed_subtitle":"The subdiffusion exponent depends only on the network's spectral dimension; dipole forces instead collapse and rotate the network.","key_machinery":"The engine of the argument is the spectral dimension $d_s$, the exponent in the vibrational density of states $g(\\omega)\\sim\\omega^{d_s-1}$ of the fractal; for the critical percolation cluster used here, $d_s=1.33$. The analytic MSD is a normal-mode sum over these vibrational states, and in the intermediate window $\\tau\\ll t\\ll\\tau_N$ the sum reduces to $\\mathrm{MSD}\\sim t^{1-d_s/2}$. For dipoles, the load-bearing counting device is Maxwell's isostatic criterion, modified so that a dipole in its ON state adds one mechanical constraint per active bond, shifting the rigidity threshold to $p'_{\\mathrm{rigid}}=p_{\\mathrm{rigid}}/(1+P\\phi)$.","core_discovery":"The central claim is that on a disordered fractal at the percolation threshold, the anomalous exponent of the time-averaged mean-square displacement under active monopolar noise is $\\nu=1-d_s/2$, identical to the thermal exponent and independent of force amplitude and correlation time. Dipolar active forces behave differently: on critical percolation clusters they produce no true steady state, the network collapses slowly, and after a saturation regime the MSD rises ballistically then diffusively for a low fraction of dipoles, or diffusively for full dipole occupancy, depending on whether local node rotations are persistent or fluctuating. On mechanically stable triangular lattices above the isostatic point, contractile dipoles produce a sharp collapse transition above a critical force $f_c\\sim \\phi^{-0.54}$, and the paper derives a modified Maxwell criterion $p'_{\\mathrm{rigid}}=p_{\\mathrm{rigid}}/(1+P\\phi)$ in which active dipole links act as extra constraints that shift rigidity percolation to lower bond dilution. A direct corollary is that active disordered solids should be poised above the isostatic point to remain stable against even weak, rare dipolar forces.","pith_inferences":["Inference beyond the paper: the exponent identity $\\nu=1-d_s/2$ for monopoles offers a null test for chromatin data—if the subdiffusion exponent is unchanged after ATP depletion, the active noise may be effectively monopolar, whereas dipolar motor action should show saturation or collapse signatures instead of a shifted exponent.","Inference beyond the paper: the modified Maxwell criterion makes a quantitative prediction that motor density $P\\phi$ lowers the rigidity threshold; this could be tested by measuring the elastic plateau or collapse onset of a reconstituted actin-myosin network as motor concentration is varied.","Inference beyond the paper: the slow collapse of sub-isostatic clusters under arbitrarily weak dipoles implies that biological networks maintained by weak motors must be above isostaticity or stabilized by bending rigidity, both of which are absent from the bead-spring model.","Inference beyond the paper: because the paper attributes rotation to residual anisotropy of a finite dipole realization, a testable extension is that the mean angular velocity decays with increasing system size while angular-velocity fluctuations persist, which could be checked in larger simulations or confined active gels."],"forward_implications":["For monopolar active forces, the subdiffusion exponent in the intermediate regime depends only on $d_s$ and not on the force amplitude, correlation time, or dipole fraction, so the same exponent should appear across very different activity levels.","Thermal and active monopole noise give the same anomalous exponent, so distinguishing active from passive fluctuations in gels or chromosomes requires amplitude or correlation-time measurements, not the exponent itself.","Contractile dipoles on sub-isostatic fractal clusters cause progressive collapse; the 'steady state' is only a pseudo-steady state on intermediate times, and the long-time state is collapsed.","In rigid triangular networks there is a threshold force for collapse that grows roughly as $\\phi^{-0.54}$; below threshold the network size is stable, and active dipoles shift the rigidity point to $p'_{\\mathrm{rigid}}=p_{\\mathrm{rigid}}/(1+P\\phi)$.","Persistent unidirectional rotation appears in fully connected networks with a low dipole fraction, while near the rigidity threshold rotation becomes direction-reversing and fluctuation-dominated."],"supporting_citations":[{"why":"Supplies the analytical normal-mode MSD expressions for thermal and active forces and the deterministic-fractal precedent that this paper extends to disorder.","marker":"[13]"},{"why":"Provides the percolation-theory definitions and the fractal dimensions $d_f$ and $d_s$ used to characterize the critical cluster.","marker":"[17]"},{"why":"Establishes the fracton density-of-states scaling $g(\\omega)\\sim \\omega^{d_s-1}$ that the spectral-dimension argument rests on.","marker":"[18]"},{"why":"Supplies the random-telegraph process used to model stochastic active forces.","marker":"[20]"},{"why":"Provides the cluster-labeling algorithm used to isolate the infinite percolation cluster.","marker":"[21]"},{"why":"Gives the thermal MSD result on fractals, $\\mathrm{MSD}\\sim t^{1-d_s/2}$, which the active monopole result is matched against.","marker":"[22-25]"},{"why":"Formulates the Maxwell isostatic criterion that underlies the modified rigidity-threshold formula for active dipoles.","marker":"[26]"}],"fun_headline_variants":["Monopolar noise sets universal subdiffusion exponent on fractals","Active monopoles on percolation clusters: one exponent rules all","Dipoles collapse fractal networks; rigidity shifts with activity","Fractal active matter: monopoles diffuse, dipoles collapse","Universal exponent for active monopoles on disordered fractals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that an active dipole in its ON state can be counted as an extra mechanical constraint, like a strut, in Maxwell rigidity counting; a prescribed motor force is not literally a geometric constraint, so the shifted threshold $p'_{\\mathrm{rigid}}=p_{\\mathrm{rigid}}/(1+P\\phi)$ rests on that counting.","fun_headline_variants_meta":{"raw":{"variants":["Monopolar noise sets universal subdiffusion exponent on fractals","Active monopoles on percolation clusters: one exponent rules all","Dipoles collapse fractal networks; rigidity shifts with activity","Fractal active matter: monopoles diffuse, dipoles collapse","Universal exponent for active monopoles on disordered fractals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0002,"raw_usage":{"total_tokens":1463,"prompt_tokens":1123,"completion_tokens":340,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":739,"completion_tokens_details":{"reasoning_tokens":255}},"tokens_in":739,"tokens_out":340,"duration_ms":3584,"temperature":1.0,"reasoning_tokens":255,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:02:01.717976+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a bond-percolation cluster with force monopoles at fixed $d_s$ while varying $f$ and $\\tau$ (for example $f=0.5$ versus $2$, $\\tau=1$ versus $100$): if the fitted intermediate exponent moves away from $1-d_s/2$, or the prefactor does not scale as $f^2\\tau$, the spectral-dimension-only claim fails. Independently, measure the collapse threshold of diluted triangular lattices with active dipoles at several $\\phi$: if $f_c$ does not extrapolate to zero at $p'_{\\mathrm{rigid}}=p_{\\mathrm{rigid}}/(1+P\\phi)$, the modified Maxwell criterion fails.","supporting_citations":[{"cited_title":"isostatic point","cited_arxiv_id":null,"evidence_quote":"Supplies the analytical normal-mode MSD expressions for thermal and active forces and the deterministic-fractal precedent that this paper extends to disorder."},{"cited_title":"Anomalous diffusion in fractal globules","cited_arxiv_id":null,"evidence_quote":"Provides the percolation-theory definitions and the fractal dimensions $d_f$ and $d_s$ used to characterize the critical cluster."},{"cited_title":"Active fractal net- works with stochastic force monopoles and force dipoles unravel subdiffusion of chromosomal loci","cited_arxiv_id":null,"evidence_quote":"Establishes the fracton density-of-states scaling $g(\\omega)\\sim \\omega^{d_s-1}$ that the spectral-dimension argument rests on."},{"cited_title":"Spatial confinement of active microtubule networks induces large-scale ro- tational cytoplasmic flow","cited_arxiv_id":null,"evidence_quote":"Supplies the random-telegraph process used to model stochastic active forces."},{"cited_title":"Actomyosin contractility rotates the cell nucleus","cited_arxiv_id":null,"evidence_quote":"Provides the cluster-labeling algorithm used to isolate the infinite percolation cluster."},{"cited_title":"Percolation and cluster dis- tribution","cited_arxiv_id":null,"evidence_quote":"Formulates the Maxwell isostatic criterion that underlies the modified rigidity-threshold formula for active dipoles."}],"review_version":1}