{"id":"74174080-78ce-4f4a-b843-f9127591bdcd","arxiv_id":"2608.05496","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Irreversible one-flip threshold dynamics on a lattice freezes balanced random configurations into bicontinuous labyrinth absorbing states, with a frozen-pattern window that narrows as 1/σ under a standardized-bias control.","lead":"By letting each spin in a Gaussian-kernel threshold dynamics flip at most once, this paper shows that balanced random mixtures freeze into stable two-phase labyrinth patterns instead of coarsening to consensus. The result gives a mechanistic explanation for why irreversible commitment processes can get stuck in mixed states, with a measured signature that reversible dynamics cannot produce.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unproved σ-independence of the cascade onset z*0 is the load-bearing assumption: Eq. (17) and the scale-free control claim stand or fall with it.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing point: the σ-independence of z*0 is numerically supported but not proved, and the central scaling law rests on it. I agree with that assessment. The qualitative finding—that irreversible one-flip dynamics freezes balanced bicontinuous labyrinths from random initial conditions—is well supported by direct simulation, by the comparison with reversible relaxation, and by the intervention experiments (order shuffling, frustration ablation, single-pass comparison). The exact t=0 standardization (Eq. 16), the closed-form pinning constant, and the curvature-drive calibration are solid. The remaining vulnerability is the dynamical onset assumption: it is not a static geometric fact, since Sec. V D shows no static excursion set of the initial field predicts survival, so there is no known argument from the initial-condition statistics alone that guarantees z*0 is exactly σ-independent. The fourfold range in σ is real evidence but does not rule out a slow drift that would break Eq. (17) at larger scales. This does not warrant rejection, because all accessible tests currently support the assumption; it also does not warrant acceptance as a proven law. The verdict should remain CONDITIONAL, pending either a proof of fixed z*0 or a substantial extension of the numerical range with a finite-size check. The absence of released code and data is a secondary reproducibility concern, not the central logical weakness.","tokens_in":17507,"tokens_out":3978,"duration_ms":41401,"concrete_test":"Recompute the onset protocol at σ=32 and σ=64 with at least 100 realizations per point, keeping L=40σ, and separately at σ=16 with L=52σ to test finite-size convergence. Regress z*0 on σ including the new points under both operational definitions; if the slope exceeds about 0.005 per unit σ, or if the crossing shifts by more than 0.1 between L=26σ and L=52σ at fixed σ, then the fixed-z*0 assumption and Eq. (17) are not supported at larger scales.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative law, Δp0 = z*0/(4√π σ), depends on the assumption that cascade onset occurs at a fixed standardized bias z*0 independent of σ. The paper states this explicitly in Sec. V B and supports it numerically over σ∈[4,16] with L=26σ and 15 realizations per point, giving slopes −0.004±0.003 per unit σ for both operational definitions. This is genuine evidence but not proof, and the accessible range may not constrain the σ→∞ limit: because z0 controls only the t=0 field statistics, not the nonlinear one-flip dynamics, a slow drift in z*0 could arise from lattice-scale effects, from the finite ratio L/σ used for the collapse, or from the changing role of the O(1/σ^2) corrections in Eqs. (14)–(15). The paper's own Sec. V D closes the natural static-percolation route to proving fixed z*0, so the boundary is a purely dynamical numerical observation. If z*0 drifts at σ=32, 64, or at larger L/σ, then the abstract's claim of scale-independent onset and the 1/σ window law fail qualitatively, even if a direct fit over σ∈[3,8] still appears consistent with α≈1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies an asynchronous majority-threshold dynamics on Z^2 with a Gaussian kernel and a one-flip (\"irreversible\") constraint, and contrasts it with the reversible MBO-type relaxation of the same rule. The main results are: (i) exact lattice pinning constants for flat interfaces and a closed-form critical stripe width; (ii) a curvature drive for droplets with coefficient sqrt(2π)σ^3 and a universal mobility function g(z) describing the eligible boundary fraction; (iii) the identification of a composition window around p0=1/2 in which random initial conditions freeze into balanced bicontinuous labyrinths, with a window half-width Δp0 ∝ 1/σ controlled by the standardized initial bias z0 ≈ 2√π(2p0−1)σ; (iv) a characterization of the frozen morphology as arrested coarsening with sublinear feature width; and (v) an intervention-based decomposition showing that the coarse domains and wall positions are set deterministically, with a secondary first-passage race and frustrated interfacial sites locking, but not placing, the walls.","tokens_in":17829,"tokens_out":13287,"duration_ms":122847,"significance":"The paper's strengths are its exact single-interface calculations, the clean t=0 standardization of the initial field, the careful mobility collapse, and the unusually explicit segregation of proved statements from numerically supported assumptions. The irreversibility mechanism and the frustration signature are conceptually clear and transferable. If the 1/σ window law holds beyond the tested range, the paper provides a generic route to absorbing-state labyrinth patterns in two-sided committed dynamics. The main limitation is the unproved σ-independence of the cascade-onset threshold z*0; the paper acknowledges this, but because Eq. (17) and the abstract's window-law statement depend on it, the quantitative central claim is conditional on numerical extrapolation.","major_comments":[{"comment":"The central law Δp0 = z*0/(4√π σ) rests on the assumption, stated in Sec. V B, that cascade onset occurs at a fixed value z*0 independent of σ. The evidence in Sec. V C (slopes −0.004±0.003 over σ∈[4,16], 15 realizations per point) is genuine, and the matched-z0 collapse is visually good, but it does not bound a slow drift at larger σ or larger L/σ. The independent direct fit α=1.05±0.09 over σ∈[3,8] is consistent with 1/σ, but an effective exponent near 1 would also result from a slowly varying z*0(σ), for example z*0 ∝ σ^ε with small ε. Because the abstract and Sec. VIII present the 1/σ law as a quantitative result, I ask that the manuscript either extend the z*0 measurements to σ=32 and 64 with a few L/σ values, or explicitly rephrase the law as an empirical statement valid over the tested fourfold range, with the asymptotic form left as a stated conjecture.","section":"Sec. V B/C, Eq. (17)"},{"comment":"The σ-independence regression is performed at a single box ratio L=26σ and with 15 realizations per point. Since the onset is a smooth crossover, the measured z*0 could in principle depend on L/σ; the paper does not report a finite-size check for z*0, although the morphology section does such a check for ℓ_w. A brief statement of the dependence of z*0 on L/σ at a representative σ would close an otherwise open route to explaining the apparent σ-independence by a coincidental finite-size effect.","section":"Sec. V C"}],"minor_comments":[{"comment":"There is a typo in \"irrerversible model\" in the paragraph introducing the reversible variant.","section":"Sec. II"},{"comment":"The word \"wihout\" should read \"without\" in \"leaves the boundary slightly non-circular wihout introducing fine-scale structure.\"","section":"Sec. IV D"},{"comment":"The phrase \"far from what the the initial field alone implies\" contains a duplicated \"the.\"","section":"Sec. VII C"},{"comment":"The title \"Sharp metastability threhold\" contains a misspelling; it should be \"threshold.\"","section":"Reference [12]"},{"comment":"The phrase \"described by σ ≈0.3\" is an incomplete shorthand for the feature-width law; it should read \"described by an exponent ℓ_w ∝ σ^0.30\" or similar, both in the text and in the Fig. 4 caption.","section":"Sec. VI"},{"comment":"The heading \"An exact control variable\" is potentially misleading because Eq. (16) is the large-σ, near-balance limit of the exact ratio E[m]/√Var[m]; consider renaming it \"A control variable\" or \"An asymptotically exact control variable.\"","section":"Sec. V A"},{"comment":"The paper gives detailed numerical protocols but no data- or code-availability statement; a sentence on availability of simulation code would aid reproducibility.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The recommendation is driven by the gap between the strength of the claim in the abstract and Sec. VIII and the numerical basis of the z*0-independence assumption. The paper is transparent about this gap, and the additional simulations I request are feasible; if the editor prefers to accept conditional numerical scaling laws without asymptotic extrapolation, a minor revision with a rephrased abstract could suffice. The manuscript is within the journal's scope and I see no citation or attribution problems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Levy has a genuinely new result here. Making majority threshold dynamics irreversible—each spin flips at most once—changes the absorbing state from consensus or striped pinned states to balanced, bicontinuous frozen labyrinths. That qualitative finding is well supported: the reversible and irreversible dynamics differ only by the one-flip constraint, and the difference in outcome is stark. The exact standardization of the initial bias, z0 ≈ 2√π(2p0−1)σ, is clean, and the collapse of the magnetization curves against z0 across σ∈[4,16] is convincing. The droplet mobility collapse onto a single g(z) is also nice, and the intervention experiments—fixed initial condition with varying update order, ablation of frustrated sites—give a credible mechanistic account: coarse domains are deterministic, update order only fine-tunes wall positions, frustration is a passive interfacial signature. Credit is due for treating the flat-interface pinning and curvature drive as calibration rather than new physics, and for stating assumptions plainly.\n\nThe main quantitative claim, Δp0 ∝ 1/σ, rests on the assumption that cascade onset happens at a fixed z0* independent of σ. The paper says this explicitly in Sec. V B, and the numerical support over σ∈[4,16] is real: weighted regressions give slopes −0.004±0.003 per unit σ for both operational definitions. That is genuine evidence, but it is not a proof, and the paper's Sec. V D closes the natural percolation route to a proof. If z0* drifts at larger σ or larger L/σ, the 1/σ law and the scale-independent onset claim fail qualitatively. This is a real soft spot, but it is not hidden and not manufactured. The paper also does not ship code or data; the Gillespie protocols are detailed enough to reproduce, but code would remove doubt. Minor: ℓ_w ≈ 10.6 σ^0.30 is explicitly presented as a finite-range description, not an asymptotic exponent, and the robust observation is that ℓ_w/σ falls below unity near σ≈28. The ring offset C(σ) and registration spread δ are numerical characterizations rather than derivations; acceptable in a first paper.\n\nWho is this for? Statistical mechanicians interested in irreversible dynamics, arrested coarsening, and pattern formation; also people studying competing contagions or commitment dynamics, where the frustration signature is a transferable observable. It deserves a serious referee. I would send it to a good condensed-matter/stat-mech journal, not desk-reject. I would cite it as a minimal mechanism for persistent mixed states under irreversible local alignment, while noting the fixed-z* assumption if I rely on the exact 1/σ form.","headline":"A clean, honest minimal model showing irreversible one-flip threshold dynamics freezes balanced labyrinths; the 1/σ window law is conditional on an unproved but clearly flagged σ-independence of cascade onset.","tokens_in":18305,"tokens_out":2972,"would_cite":true,"duration_ms":27411,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Adding a one-flip cap to majority threshold dynamics freezes balanced, non-consensus labyrinth patterns where the reversible rule would coarsen to consensus or stripes.","keywords":["irreversible threshold dynamics","one-flip dynamics","labyrinth patterns","arrested coarsening","frustrated spins","Gaussian kernel lattice model","standardized initial bias","composition window"],"falsifier":"Run the one-flip dynamics at $\\sigma=32$ and $64$, tuning $p_0$ to keep $z_0$ matched, and measure the cascade onset $z_0^*$ from the same magnetization or giant-cluster crossings used here; if the crossing moves by more than the bootstrap scatter observed at $\\sigma\\le16$, the claimed $\\sigma$-independence and the derived $1/\\sigma$ law are wrong.","tokens_in":17264,"feed_emoji":"🌀","tokens_out":7031,"duration_ms":65563,"temperature":0.7,"pith_summary":"The paper studies what happens when the usual threshold dynamics of Merriman–Bence–Osher type is made irreversible: on $\\mathbb{Z}^2$, each spin may flip out of its local Gaussian-weighted minority at most once. From Bernoulli initial conditions near equal balance, this one-flip rule freezes balanced, non-consensus labyrinth patterns, whereas the same rule without the one-flip cap coarsens to consensus or pinned stripes. The pattern survives only inside a composition window around $p_0=1/2$ whose half-width scales as $1/\\sigma$, because the initial bias enters dynamics only through the standardized quantity $z_0\\simeq 2\\sqrt{\\pi}(2p_0-1)\\sigma$, with onset at a $\\sigma$-independent value $z_0^*\\approx0.31$\\textendash$0.36$. If correct, the paper establishes a generic mechanism by which permanent, two-sided commitment arrests coarsening in local alignment systems, leaving a measurable frustrated-spin signature that reversible relaxation cannot produce.","feed_headline":"One allowed flip per spin turns consensus into labyrinths","feed_subtitle":"In a majority-threshold lattice model, capping each spin at one flip arrests coarsening into balanced bicontinuous patterns.","key_machinery":"The load-bearing objects are the standardized initial bias and the frustrated spin. The bias, $z_0=\\mathbb{E}[m]/\\sqrt{\\operatorname{Var}[m]}\\simeq 2\\sqrt{\\pi}(2p_0-1)\\sigma$, collapses the entire initial excursion-set geometry of the random field onto one scale-free variable, and the paper's window law follows from assuming cascade onset occurs at a fixed $z_0^*$. The frustrated spin, a site that has used its single flip and sits with $s_i m_i<0$, is the signature of irreversibility: no reversible relaxation can stop with a spin strictly against its local field, and in the frozen state these sites decorate the domain walls and lock them in place.","core_discovery":"Adding a one-flip cap to asynchronous Gaussian threshold dynamics changes the absorbing-state selection. From random initial conditions close to balance, the system freezes into smooth bicontinuous labyrinths with low residual magnetization, instead of the near-consensus or flat pinned states reached by reversible descent. The control variable is the standardized initial bias $z_0 = \\mathbb{E}[m]/\\sqrt{\\operatorname{Var}[m]} \\simeq 2\\sqrt{\\pi}(2p_0-1)\\sigma$; the cascade boundary sits at a fixed $z_0^*$ independent of $\\sigma$ over a fourfold range, so the patterned window narrows as $1/\\sigma$. The frozen morphology is arrested coarsening: no wavelength is selected, both phases span at balance, and the feature width grows sublinearly with $\\sigma$, falling below the interaction range at large scales. The paper traces the mechanism by intervention: the coarse domain layout is deterministic, walls are placed by deterministic front advance with only a secondary first-passage race, and frustrated spins\\textemdash spent sites whose spin opposes their local field\\textemdash lock the walls without placing them.","pith_inferences":["An extension the paper leaves implicit: if $z_0^*$ is indeed $\\sigma$-independent for the Gaussian kernel, the same $1/\\sigma$ window should occur for any positive, integrable kernel; only the numerical prefactor in $z_0$ would change.","The measured $\\ell_w\\propto\\sigma^{0.3}$ may be a crossover rather than an asymptotic exponent; testing at $\\sigma\\ge32$ would show whether $\\ell_w/\\sigma$ keeps declining or saturates, which distinguishes true arrested coarsening from a slowly growing selected scale.","The deterministic-layout result suggests pattern reproducibility in committed-agent models is governed by initial conditions alone; a testable prediction is that duplicates of the same initial condition under independent clocks agree site-wise almost everywhere except near walls.","The wall-placement race invites a quantitative comparison with first-passage percolation: at fixed large $\\sigma$, the scatter of individual-order walls about the consensus wall should follow the same scaling as competing-growth interfaces."],"forward_implications":["The half-width of the patterned composition window obeys $\\Delta p_0 = z_0^*/(4\\sqrt{\\pi}\\sigma)\\propto 1/\\sigma$, with $z_0^*\\approx0.31$\\textendash$0.36$ depending on the crossing convention.","At balance the absorbing labyrinth is bicontinuous with both phases spanning in the great majority of large boxes, and its structure factor has no finite-wavenumber peak, ruling out wavelength selection.","The frozen feature width $\\ell_w$ grows sublinearly, $\\ell_w\\approx10.6\\,\\sigma^{0.30}$, and $\\ell_w/\\sigma$ passes below unity near $\\sigma=28$, so the frozen length is not set by the interaction range.","Holding the initial condition fixed and varying update order shows the coarse domains are deterministic; the update-order race fine-tunes only the last fraction of $\\sigma$ of the wall positions, and frustration ablation leaves the coarse layout intact.","Frustrated-site density distinguishes irreversible from reversible absorption: reversible dynamics must stop with every spin aligned with its local field, while the one-flip absorbing state carries a finite density of frustrated spins, 97\\textendash98% of them on the walls."],"supporting_citations":[{"why":"Defines the reversible threshold-dynamics (MBO) scheme whose mean-curvature flow and lattice pinning serve as the model's calibration.","marker":"[1]"},{"why":"Supplies the propagation-failure analogue that the exact flat-interface pinning constant generalizes.","marker":"[2]"},{"why":"Provides the zero-temperature Ising freezing comparison: reversible freezing yields only flat stripes and never frustrated spins.","marker":"[3]"},{"why":"Gives the competing-grown-front and first-passage-percolation counterpart for the secondary wall-placement race.","marker":"[13]"},{"why":"Establishes the game-theoretic context of coexistent equilibria that the balanced frozen states go beyond.","marker":"[17]"}],"fun_headline_variants":["One flip per spin freezes balanced spins into mazes","Irreversible rule traps balanced spins in freeform mazes","Frozen labyrinths from a one-flip cap on threshold dynamics","Arrested coarsening: one-time flips create bicontinuous mazes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the premise, supported only numerically for interaction ranges $\\sigma=4$ through $16$, that cascade onset occurs at a fixed value $z_0^*$ of the standardized initial bias independent of $\\sigma$; if $z_0^*$ drifts at larger $\\sigma$ or system size, the $1/\\sigma$ window law and the scale-free control claim would fail.","fun_headline_variants_meta":{"raw":{"variants":["One flip per spin freezes balanced spins into mazes","Irreversible rule traps balanced spins in freeform mazes","Frozen labyrinths from a one-flip cap on threshold dynamics","Arrested coarsening: one-time flips create bicontinuous mazes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000305,"raw_usage":{"total_tokens":1802,"prompt_tokens":1051,"completion_tokens":751,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":667,"completion_tokens_details":{"reasoning_tokens":674}},"tokens_in":667,"tokens_out":751,"duration_ms":6484,"temperature":1.0,"reasoning_tokens":674,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T12:01:46.256402+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the one-flip dynamics at $\\sigma=32$ and $64$, tuning $p_0$ to keep $z_0$ matched, and measure the cascade onset $z_0^*$ from the same magnetization or giant-cluster crossings used here; if the crossing moves by more than the bootstrap scatter observed at $\\sigma\\le16$, the claimed $\\sigma$-independence and the derived $1/\\sigma$ law are wrong.","supporting_citations":[{"cited_title":"Throughout, the front remains close to circular and does not develop concave pockets that could arrest it; small boundary fluctuations stay bounded rather than amplifying (App","cited_arxiv_id":null,"evidence_quote":"Defines the reversible threshold-dynamics (MBO) scheme whose mean-curvature flow and lattice pinning serve as the model's calibration."},{"cited_title":"The standardized initial bias is therefore z0 = E[m]p Var[m] − − − − − − − − → σ≫1, p0→ 1 2 2√π(2p 0 −1)σ,(16) withO(1/σ 2) corrections from the exact forms (14)–(15)","cited_arxiv_id":null,"evidence_quote":"Supplies the propagation-failure analogue that the exact flat-interface pinning constant generalizes."},{"cited_title":"Since the frozen state is already smooth at that scale (Sec","cited_arxiv_id":null,"evidence_quote":"Provides the zero-temperature Ising freezing comparison: reversible freezing yields only flat stripes and never frustrated spins."},{"cited_title":"Rational group decision making: A random field Ising model at T = 0.Physica A: Statistical Mechan- ics and its Applications, 238(1-4):66–80, April 1997","cited_arxiv_id":null,"evidence_quote":"Gives the competing-grown-front and first-passage-percolation counterpart for the secondary wall-placement race."},{"cited_title":"Sharp metastability threhold for two-dimensional bootstrap percolation.Probability The- ory and Related Fields, 125(2):195–224, 2003","cited_arxiv_id":null,"evidence_quote":"Establishes the game-theoretic context of coexistent equilibria that the balanced frozen states go beyond."}],"review_version":1}