{"id":"28afdc26-345c-46bf-84e0-e99ad3f98a35","arxiv_id":"2506.23687","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Factorizable q-state active Potts models (q=4,6,8) produce new factorized-symmetry dynamic modes, including q=6 spiral waves of three states with scaling exponents modified from equilibrium.","lead":"Computer simulations show that active Potts models with factorizable state numbers (4, 6, or 8) develop new dynamic patterns, including spiral waves built from only three of six states. The patterns and the changed phase-transition exponents are new nonequilibrium phenomena in a minimal statistical-physics model.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Pot6a exponent shift may be an FSS artifact: fits include the non-collapsing x ≥ 10 wing.","rationale":"The reader's weakest_assumption identifies exactly this FSS asymptotic-regime concern, and I agree it is the most load-bearing issue. The qualitative factorized modes (W3, M3, M2, W4) are supported by direct snapshots, time series, and multiple system sizes, so they are likely robust. The modified-exponent claim, however, is a precise quantitative statement highlighted in the abstract and summary, and it depends on fits whose collapse the paper explicitly acknowledges to fail for large scaling arguments. Since the reported difference is small relative to the fit sensitivity, a concrete refit restricted to the collapsing region, or with corrections to scaling, would settle whether the claim survives. The long-term stability of factorized modes is secondary; the W3/M6 transition is analyzed in steady state, and the M3 no-switching observation up to 10^9 steps is consistent with the paper's qualitative claims. Therefore the reader's CONDITIONAL verdict is appropriate and unchanged.","tokens_in":18424,"tokens_out":5975,"duration_ms":65027,"concrete_test":"Re-extract the Pot6a h = 1 scaling exponents using only data in |(J_k,[k+2]−J_c)L^{1/ν}| < 10 (or < 5), and also fit with a correction-to-scaling term, e.g., R_3 L^{β/ν} = f(x)(1 + a L^{-ω} g(x)). If 1/ν shifts by more than ~0.05 toward the h = 0 value, the claimed exponent modification is a finite-size artifact. Independently, compute 1/ν via the Binder-cumulant quotient method for L = 46, 64, 90; a decreasing trend toward 0.95 would corroborate that concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The quantitative central claim—that the W3–M6 transition in Pot6a has exponents modified from equilibrium (1/ν = 1.12 ± 0.02 vs 0.95 ± 0.02 for h = 1 vs h = 0)—rests entirely on the finite-size scaling analysis of Sec. III A 1. The paper itself notes (Fig. 6 and text) that for (J_k,[k+2]−J_c)L^{1/ν} ≳ 10 the Binder cumulant, susceptibility, and order-parameter curves do not collapse for either h = 0 or h = 1, attributing this to crossover. The reported exponents come from quartic fits to these curves for L ≤ 90; if the fitting window includes the non-collapsing wing, the extracted exponents are effective, L-dependent values rather than true asymptotic critical exponents. The claimed difference, 0.17, is only about 8σ of the quoted statistical errors, so a modest contamination of the fits by the non-collapsing region could erase it. The h = 0 baseline (1/ν = 0.95) is itself not a standard Potts fixed point (contrast Pot6b, which gives 1/ν ≈ 1.24 close to the 3-state Potts value), so the comparison relies wholly on the quality of the collapse. Absent released code or data, this is the most load-bearing weakness in the paper's central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reports Monte Carlo simulations of q-state active Potts models (q=4, 5, 6, 8) on a two-dimensional square lattice, with cyclic flipping energies h and state-dependent nearest-neighbor contact energies J_{s,s'}. It maps dynamic modes over several parameter slices: homogeneous cycling of q states (HCq), wave modes of q states (Wq), intermediate wave modes (WI), and mixed phases (Mq). For q=6, with factorizable 6 = 2 × 3, the paper identifies additional factorized-symmetry modes: waves of three states (W3), homogeneous cycling of three states (HC3), mixed phases of three states (M3), and mixed two-state variants (M2W3, M2HC3). Using finite-size scaling of the order parameter, susceptibility, and Binder cumulant for the three-fold symmetry, it reports critical exponents for the W3–M6 transition at h=0 and h=1 and for the HC3–M3 transition under a different parameter slice. The principal claims are that factorized-symmetry spatiotemporal modes emerge under cyclic driving and that the W3–M6 transition exponents are modified from equilibrium (1/ν = 1.12 ± 0.02 versus 0.95 ± 0.02 at h=1 versus h=0). The paper also maps four- and eight-state models and discusses connections to the Ashkin–Teller model and lattice Lotka–Volterra models.","tokens_in":18762,"tokens_out":8803,"duration_ms":94551,"significance":"The qualitative phenomenology is valuable: the observation that factorizable q enables modes with reduced state symmetry is a natural but nontrivial extension of the author's earlier work on three- and four-state active Potts models, and the systematic phase diagrams over multiple parameter slices give a useful map. Strengths include extensive Monte Carlo sampling with system-size checks (L=22–256, multiple independent runs), explicit mode-classification metrics, and careful comparison with known equilibrium and nonequilibrium results, including the mapping to the Avni clock model. If the exponent shift is confirmed, it would be an interesting example of nonequilibrium criticality in a lattice model with stable spatiotemporal patterns, distinct from absorbing-state Lotka–Volterra models. The main caveat is that the quantitative exponent claim rests on finite-size scaling fits whose quality is not fully documented, and no code or raw data are provided for independent verification.","major_comments":[{"comment":"The central quantitative claim is the difference in 1/ν between h=1 (1.12 ± 0.02) and h=0 (0.95 ± 0.02) for the W3–M6 transition in Pot6a. The paper explicitly notes that for (J_{k,[k+2]} − J_c)L^{1/ν} ≳ 10 the scaling curves do not collapse for either h=0 or h=1, and the fits use L ≤ 90. If the quartic fits include this non-collapsing wing, the quoted exponents are effective, L-dependent values rather than asymptotic critical exponents; the difference of about 0.17 is only roughly 8σ of the quoted statistical errors, so a modest contamination could erase it. The h=0 baseline is also not equal to the exact three-state Potts value 1/ν = 6/5, so the comparison cannot be anchored to a known fixed point. I ask the authors to specify the exact fitting window in the scaling variable, to show the stability of Table I when the x ≳ 10 region is excluded and when L=128 is included, and to provide a quantitative collapse criterion or goodness-of-fit measure. Without this, the modified-exponent claim is not established.","section":"Sec. III A 1, Fig. 6, Table I"},{"comment":"The stability of the factorized mixed phase M3 is asserted from the absence of odd–even switching in simulation periods up to about 10^9 MC steps. This is a finite-time bound, whereas the Introduction claims dynamics in the long-term limit (t → ∞) that are independent of initial states. Please provide a quantitative lower bound on the switching time, report whether any odd–even switching event is observed in the longest runs, and discuss the system-size dependence of the switching time. Otherwise the M3 mode could be a long-lived metastable state rather than a stable phase of the infinite system.","section":"Sec. III A 2, Fig. 8"}],"minor_comments":[{"comment":"The section heading 'F our-State Potts Model' contains a typo; it should read 'Four-State Potts Model'.","section":"Section III C heading"},{"comment":"Reference 60, 'J. Zeininger and et al.,' should be 'J. Zeininger et al.'","section":"References"},{"comment":"The symbol s_n is used both for the local state index s and for the order-parameter magnitude in Eq. (3); using a different notation, such as ρ_n, would avoid confusion.","section":"Eqs. (2)–(4)"},{"comment":"The mode definitions rely on operational thresholds (N_s/N > 0.95, N_s/N > 0.05, and τ_HC3/τ_1 > 10). It would be helpful to state whether the reported phase boundaries are stable under moderate changes of these thresholds.","section":"Sec. III A 1 and Sec. III A 2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of the journal. The main risk is the finite-size scaling analysis behind the modified-exponent claim; the authors should be required to report the fitting window and provide a robustness analysis of Table I. Because the paper relies heavily on Monte Carlo data, depositing the code and raw scaling data would materially help referees and readers. If the exponent shift survives the requested checks, the result is a solid contribution; if not, the qualitative mode phenomenology can still support a paper, but the abstract and conclusions would need to be refocused."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid simulation paper that usefully extends the active Potts program to q=4,5,6,8 and finds genuinely new spatiotemporal modes when q factors. The mode classification is careful and the phase diagrams are believable. The one claim I'd push back on is the modified scaling exponent for the W3-M6 transition in Pot6a; the FSS analysis is too fragile to support that at face value.\n\nWhat's new: W3, M2W3, M2HC3 for q=6 and W4 for q=8 are not in the earlier literature. The link to equilibrium factorized phases (Ashkin-Teller) is nicely drawn. The paper is also honest about limitations: it notes the poor collapse for large scaling variable and the finite simulation time for M3 stability. That's good practice.\n\nWhere it's soft: The headline quantitative result—1/nu going from 0.95 (h=0) to 1.12 (h=1) at the W3-M6 transition—rests on quartic fits for L<=90, and the paper itself says the curves don't collapse for (J-Jc)L^(1/nu) >= 10. If the fits include that region, the exponents are effective, not universal. The h=0 baseline is also not a standard Potts value; it's a factorized transition, so calling it 'equilibrium' doesn't make it a clean reference. Without released code or raw data, I can't tell how sensitive those exponents are to the fit window. That's the load-bearing weakness.\n\nThe rest of the paper holds up. The mode identities are based on multiple diagnostics (order parameters, contact probabilities, lifetimes), and the system-size checks at L=128 and 256 give reasonable confidence. The q=5 case provides a nice negative control: no factorized modes, as expected.\n\nVerdict: worth refereeing. The phenomenology is a real contribution, and the exponent question is interesting even if the current evidence is provisional. A good referee should ask for a more careful FSS treatment—e.g., fit only the collapsing region, show the exponents as a function of L_min, and perhaps release the raw data. If the exponent shift survives, that's a solid addition; if it doesn't, the mode classification still stands on its own.","headline":"New factorized-symmetry modes are the real story; the claimed exponent shift needs stronger FSS evidence.","tokens_in":19274,"tokens_out":4458,"would_cite":true,"duration_ms":43934,"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":"In a six-state active Potts model, dynamics can reorganize into three-state spiral waves, and this factorized-symmetry mode has transition exponents different from equilibrium.","keywords":["active Potts model","factorized symmetry","dynamic phase transition","spiral waves","cyclic flipping energy","Monte Carlo simulation","finite-size scaling","nonequilibrium dynamics"],"falsifier":"Repeat the Pot6a W3–M6 scaling analysis at $L=128$ and $L=256$ with longer sampling: if the fitted $1/\\nu$ drifts back toward the equilibrium value $0.95$ as $L$ grows, or if the fourth-order cumulant curves fail to collapse once the crossover to W6 is included, the reported exponent modification is a finite-size artifact. Similarly, extending the Pot6b HC3/M3 runs beyond $10^9$ Monte Carlo steps and finding odd-even switching would show the factorized modes are transient rather than stable.","tokens_in":18209,"feed_emoji":"🌀","tokens_out":14320,"duration_ms":137393,"temperature":0.7,"pith_summary":"This paper studies the long-time nonequilibrium dynamics of $q$-state Potts models ($q=4,5,6,8$) on a two-dimensional square lattice driven by a cyclic flipping energy $h$. With standard contact energies, every $q$ shows the same two families: cyclic changes among $q$ homogeneous phases at low $h$ and coexisting $q$-state waves at high $h$. The paper's central claim is that when $q$ is factorizable, extra dynamic modes appear in which a subset of states organizes coherently: for $q=6$ in particular, a spiral wave of three states and cyclic changes among three homogeneous phases. It further claims that the transition from the six-state mixed phase to the three-state wave mode has scaling exponents modified from equilibrium values ($1/\\nu = 1.12 \\pm 0.02$ vs $0.95 \\pm 0.02$), while the transition between two static mixed phases does not. This matters because it shows far-from-equilibrium fluctuations can turn equilibrium factorized phases into stable dynamic patterns, indicating a distinct nonequilibrium universality class.","feed_headline":"Six-state Potts model runs spiral waves that skip half its states","feed_subtitle":"Cyclic flipping energies let three-state spiral waves emerge from a six-state lattice, with shifted transition exponents.","key_machinery":"The machinery is the active Potts model: a $q$-state Potts model on a square lattice in which local single-site flips are biased by a cyclic flipping energy $h$, so detailed balance holds pairwise but not around the full state cycle. Mode selection is governed by the contact energies between non-nearest states: increasing $J_{k,[k+2]}$ (or $J_{k,[k+3]}$) stabilizes contacts between states of the same factor (odd/even, diagonal pairs, or residue classes) and destabilizes full symmetric cycling. The quantitative workhorse is finite-size scaling of the $n$-fold order parameter $R_n$ (which measures $n$-fold rotational symmetry in state space), susceptibility $\\chi_n$, and a fourth-order cumulant $U_n$, whose collapse yields the exponents reported for the W3–M6 and HC3–M3 transitions.","core_discovery":"For $q=6$, with $J_{k,[k+3]}=J_{k,[k+2]}$ (Pot6a), the paper finds five modes: homogeneous cycling HC6, intermediate waves WI, six-state waves W6, three-state spiral waves W3, and a six-state mixed phase M6. In W3, only three odd- or even-numbered states form large domains, with the other three states confined to boundary clusters; the waves switch stochastically between odd and even triplets. The W3–M6 transition is continuous, and finite-size scaling gives $J_c = 1.214 \\pm 0.001$ and exponents $1/\\nu = 1.12 \\pm 0.02$, $\\gamma/\\nu = 1.732 \\pm 0.007$, $\\beta/\\nu = 0.132 \\pm 0.002$ at $h=1$, compared with $0.95 \\pm 0.02$, $1.742 \\pm 0.002$, and $0.1233 \\pm 0.0004$ at equilibrium. For $J_{k,[k+3]}=0$ (Pot6b), the factorized modes are homogeneous cycling of three states (HC3) and mixing of three states (M3); their transition exponents are close to the three-state Potts values within error. The same factorization picture appears for $q=4$ (HC2/M2) and $q=8$ (W4), and the paper frames all of these as dynamic symmetry factorization of $q=2\\times3$, $2\\times2$, and $2\\times4$.","pith_inferences":["The paper does not simulate composite $q$ beyond 8; a natural extension is to test $q=9$, $10$, or $12$, where tuning the appropriate second-neighbor contact energy should produce residue-class waves analogous to W3 and M2W3.","The observed state-skipping suggests a nucleation-rate competition mechanism; measuring nucleation and growth rates of states $k+1$ and $k+2$ directly could turn this qualitative picture into a quantitative prediction.","The four-state phase diagram at $J_{k,[k+2]}=-2$ maps onto a nonreciprocal two-spin clock model; probing intermediate $J_{k,[k+2]}$ could reveal whether the HC4/W4 coexistence seen in the symmetric model is generic or special to that line."],"forward_implications":["Factorizable $q$ Potts models become minimal lattice settings where equilibrium factorized phases turn into dynamic waves and cycling modes under a cyclic nonreciprocal drive.","The W3–M6 exponent shift implies the active drive can change the universality class of a continuous transition to a propagating wave mode, not just move its location.","The absence of a detectable exponent shift for HC3–M3 suggests only genuinely propagating wave modes, not static mixed phases, feel the nonequilibrium drive.","The long-lived factorized modes distinguish these models from cyclic-competition lattice models, which relax to a single absorbing state in the long-time limit."],"supporting_citations":[{"why":"Supplies the equilibrium Potts transition point and the exponent values the nonequilibrium exponents are compared against.","marker":"[16]"},{"why":"Supplies the equilibrium factorized phases of the four-state model and the crossover explanation for imperfect data collapse.","marker":"[21]"},{"why":"Supplies the extended multicolor model factorized phases used to frame the $q=2^n$ and $q=3n$ cases.","marker":"[22]"},{"why":"Introduced the cyclic-flipping-energy Potts dynamics whose homogeneous-cycling and wave modes this paper extends.","marker":"[34]"},{"why":"Gives the four-state active Potts results and the phase-fraction diagnostic used to classify modes.","marker":"[36]"},{"why":"Gives the nonreciprocal four-state model with modified scaling exponents that this paper compares and contrasts.","marker":"[38, 39]"},{"why":"Explains symmetry factorization in the six-state model, used to interpret the W3 spiral wave.","marker":"[55]"}],"fun_headline_variants":["Six-state Potts model reveals spiral waves that skip states","Skip-state dynamics emerge in factorizable Potts models","Three-state spiral waves emerge from six-state Potts lattice","Odd-even state skipping rewrites Potts wave dynamics","Factorizable Potts models show cyclic and skipping wave modes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the finite-size scaling fits on lattices up to $L=90$ are already in the asymptotic regime, despite the imperfect collapse for $(J_{k,[k+2]}-J_c)L^{1/\\nu} \\gtrsim 10$, and that the factorized modes are true long-time attractors rather than transients persisting only up to about $10^9$ Monte Carlo steps.","fun_headline_variants_meta":{"raw":{"variants":["Six-state Potts model reveals spiral waves that skip states","Skip-state dynamics emerge in factorizable Potts models","Three-state spiral waves emerge from six-state Potts lattice","Odd-even state skipping rewrites Potts wave dynamics","Factorizable Potts models show cyclic and skipping wave modes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000532,"raw_usage":{"total_tokens":2609,"prompt_tokens":1039,"completion_tokens":1570,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":655,"completion_tokens_details":{"reasoning_tokens":1505}},"tokens_in":655,"tokens_out":1570,"duration_ms":11920,"temperature":1.0,"reasoning_tokens":1505,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:34:31.985719+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the Pot6a W3–M6 scaling analysis at $L=128$ and $L=256$ with longer sampling: if the fitted $1/\\nu$ drifts back toward the equilibrium value $0.95$ as $L$ grows, or if the fourth-order cumulant curves fail to collapse once the crossover to W6 is included, the reported exponent modification is a finite-size artifact. Similarly, extending the Pot6b HC3/M3 runs beyond $10^9$ Monte Carlo steps and finding odd-even switching would show the factorized modes are transient rather than stable.","supporting_citations":[{"cited_title":"Guislain and E","cited_arxiv_id":null,"evidence_quote":"Supplies the equilibrium Potts transition point and the exponent values the nonequilibrium exponents are compared against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced the cyclic-flipping-energy Potts dynamics whose homogeneous-cycling and wave modes this paper extends."},{"cited_title":"Akinci, Phys","cited_arxiv_id":null,"evidence_quote":"Gives the four-state active Potts results and the phase-fraction diagnostic used to classify modes."},{"cited_title":"Rulquin and J","cited_arxiv_id":null,"evidence_quote":"Explains symmetry factorization in the six-state model, used to interpret the W3 spiral wave."}],"review_version":1}