{"id":"6f845631-aa19-48c8-b800-0447b35f4026","arxiv_id":"2608.03604","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Asymmetric division plus positive feedback can produce asymptotically stable 1:2 and 2:3 temporal clustering in yeast cell-cycle population models.","lead":"Budding yeast cells that divide unevenly can lock into stable repeating birthday patterns, with a few groups of mothers and a few groups of daughters. A new proof shows these 1:2 and 2:3 patterns are mathematically stable when fast-growing cells speed each other up.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"2:3 stability theorem rests on deferred derivations; eq. (10) is not verifiable from the main text.","rationale":"The reader's stated weakest assumption (Section 3.2, d1>g) is real but not load-bearing. For d1>g, let delta=d1-g>0. D1 hits r2 at t_a=(r2-s1+d0-d1)/alpha+s1-d0, M hits r2 at t_a+delta/alpha, and D1 hits 1 at t*=1-r2+(r2-s1+d0-d1)/alpha+s1-d0. Then D0(t*)=1-r2+s1+(r2-s1+d0-d1)/alpha and M(t*)=1-delta/alpha; in the circle chart near 0 this is -delta/alpha, giving exactly the Jacobian in eq. (7). So the omitted case can be filled and does not threaten Theorem 3.4. However, the 2:3 result has a genuine support gap: the main text defers the key derivations to supplementary material and states the Jacobian matrices without derivation. Since the central claim includes the 2:3 PCS, Theorem 4.4 should be regarded as conditional on those calculations. This matches the reader's CONDITIONAL verdict; I would keep it unchanged rather than escalate, because the omitted 1:2 case checks out and the 2:3 derivations are plausibly routine though unverified. The appropriate test is an independent re-derivation of eq. (10).","tokens_in":881,"tokens_out":798,"duration_ms":246255,"concrete_test":"Independently re-derive the 2:3 maps from the event order r1s1r2s21 using the Section 3.2 method: write the piecewise-linear trajectories of D0, D1, D2, M1, M2 between event times, solve for the return time and relabeled coordinates, and compute the Jacobian at (d0,d1,d2,m2)=(0,g,w,g,w). Compare each entry with DF_f and DF_s in eq. (10) and recompute the characteristic polynomials and eigenvalue moduli for alpha>1. Separately verify Lemma 4.1's formulas g=(alpha+(1-alpha)(r2-s1))/(1+2alpha) and w=1-g satisfy the strict event-order inequalities of Lemma 4.2. If any entry or eigenvalue bound differs, Theorem 4.4 is not supported as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing weakness is not the omitted d1>g branch (which direct calculation shows yields the same 1:2 switching Jacobian modulo the 0~1 identification), but the fact that the 2:3 half of the central claim is not proven in the manuscript as written. Lemma 4.1's proof, Lemma 4.2's proof, and the full derivation of F_f and F_s for the 2:3 mode are all deferred to Supplementary Materials; Section 4.2 says 'We will forgo the explicit derivation' and immediately presents the Jacobian matrices (10). Lemma 4.3 and Theorem 4.4 depend entirely on these unshown formulas. An algebraic slip in event timing or in the mod-1 treatment of a cluster crossing 1 would change DF_f/DF_s and could place eigenvalues on or outside the unit circle. The reader cannot verify eq. (10) from the information in the main text, so the asymptotic stability of the 2:3 PCS is conditional on supplementary calculations that are not stated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a population model of cell cycle dynamics with asymmetric division and positive feedback. It introduces fixed (C_f) and switching (C_s) cluster models and defines 1:2 and 2:3 perfectly clustered solutions (PCSs). For each mode it claims existence and local asymptotic stability under positive feedback. The 1:2 case is analyzed in the main text; the 2:3 case relies on deferred proofs, and the Jacobian matrices in Eq. (10) are asserted without derivation. The paper also reports numerical simulations of spontaneous clustering.","tokens_in":16788,"tokens_out":12946,"duration_ms":120856,"significance":"If the results hold, they would provide the first proof that positive feedback can stabilize temporal clustering when division is asymmetric, complementing earlier negative-feedback results and the known fact that symmetric division with positive feedback only stabilizes synchrony. The 1:2 linear-stability argument is transparent, and the application of Anderson's theorem to the switching-model characteristic polynomial is elegant. The numerical simulations are useful illustrations. However, the contribution is conditional because the 2:3 half of the central theorem is not verifiable from the submitted text, and the 1:2 switching-model stability proof omits one branch of initial conditions.","major_comments":[{"comment":"The 2:3 existence and stability proof is incomplete as submitted. Lemma 4.1's proof is deferred to Supplementary Materials; Lemma 4.2's proof is likewise deferred; and Section 4.2 states 'We will forgo the explicit derivation' before presenting the Jacobians (10). Lemma 4.3 and Theorem 4.4 rest entirely on these unshown formulas. The supplementary materials are not included with the manuscript, so the reader cannot verify the event timings, the mod-1 treatment, or the resulting matrices. An algebraic slip in the event order could move eigenvalues outside the unit circle. The authors must include the full derivations in the main text or an accessible appendix/supplement before the claim can be accepted.","section":"Section 4 (Lemmas 4.1, 4.2, 4.3; Theorem 4.4; Eq. (10))"},{"comment":"The switching-model map F_s is derived only under the assumption d1 < g. The text says 'both cases give the same result' but does not provide the d1 > g calculation. In the d1 > g case, D1 leaves R before M, so the ordering of the intervals used in the derivation is reversed. Although a direct calculation does yield the same Jacobian up to the mod-1 identification, the manuscript should present this case explicitly or provide a symmetry argument; as written, the proof of Theorem 3.4 for C_s is incomplete.","section":"Section 3.2 (Fs derivation, Eq. (7))"}],"minor_comments":[{"comment":"Typos: 'essense' (Sec. 2.1) should be 'essence'; 'assymetric' (Sec. 1) should be 'asymmetric'. Also 'theorem 3.1' near the end of Sec. 3.1 should be 'Lemma 3.1', and 'theorem 4.1' in Sec. 4.1 should be 'Lemma 4.1'.","section":"Throughout"},{"comment":"The connection between P and F is only described verbally. The text should state explicitly that P is a power of F (e.g., P = F^{p+q} for C_s and P = F^{lcm(p,q)} for C_f), so that the eigenvalues of DP are powers of the eigenvalues of DF. This would make the stability argument in Lemma 3.3 fully transparent.","section":"Section 2.3"},{"comment":"The statement 'under positive feedback with α^{1/5} > 1' is unclear; presumably it means α = 1 + ρ(1/5) > 1. Please rephrase.","section":"Lemma 4.3"},{"comment":"The notation '1∼0' in the figures is not defined in the main text. A brief explanation that the circle is identified with [0,1) with 1 identified with 0 would help.","section":"Figures 2 and 4"},{"comment":"The statement says the MATLAB script is in the Supplementary Materials, but no supplementary file is included in the submitted manuscript. The code and any supplementary proofs should be uploaded for review.","section":"Data & Code Availability Statement"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends heavily on prior work by overlapping authors ([2], [18], [25]) and explicitly defers key derivations to 'Supplementary Materials' that are not provided. The 2:3 theorem is currently a claim that 'similar calculations work,' which is not acceptable for a rigorous journal. The abstract and introduction overstate what is actually proven in the main text. The editor should require the supplementary materials and a full derivation of Eq. (10) before further consideration."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere's my take on arXiv:2608.03604. The core result is real and new: positive feedback plus asymmetric division can make 1:2 and 2:3 perfectly clustered solutions (PCS) asymptotically stable in the fixed and switching cluster models. That directly contrasts with Young et al. [25], where positive feedback under symmetric division only stabilizes synchrony. This matters for anyone working on cell cycle clustering or population oscillators.\n\nThe paper does several things well. The framework is clean, the explicit formula for g in the 1:2 case is derived carefully from first principles, and the eigenvalue analysis for the 1:2 Jacobians is transparent. For the 2:3 case, using Eneström-Kakeya bounds to place all eigenvalues strictly inside the unit disk is neat, provided the Jacobian matrices are correct. These are not curve-fits or simulation back-engineering; the parameters and maps come directly from the model, so the circularity burden is low.\n\nThe soft spots are real, though not fatal. Section 3.2 only derives the switching-model map under the assumption d1 < g, then asserts the d1 > g case gives the same result without showing it. A direct calculation likely confirms it, but the written proof is incomplete as is. More concerning, the 2:3 half of the central result is not self-contained: Lemma 4.1, Lemma 4.2, and the entire derivation of the Jacobian matrices in eq. (10) are deferred to supplementary materials. Section 4.2 explicitly says 'We will forgo the explicit derivation.' That means Theorems 4.3 and 4.4 rest on formulas the reader cannot verify from the main text. This is a load-bearing gap even though the formulas look plausible.\n\nI also note the paper only proves stability in the smooth fixed/switching models, not in the biologically more relevant alternating model c_a. The authors acknowledge this openly, and that limitation is acceptable if clearly stated.\n\nOverall: the paper is a genuine step forward and the main idea holds up, but the 2:3 stability proof needs to be actually included before the theorem claims are fully supported. I would send it to peer review, with the request that the supplementary derivations be folded into the main text or at least supplied in full. If the 2:3 Jacobians check out, this is a solid contribution.\n\nRecommendation: send to a serious referee; this is not desk-reject material.","headline":"Positive feedback plus asymmetric division can indeed stabilize clustered solutions, but the 2:3 stability proof is not self-contained in the main text.","tokens_in":17249,"tokens_out":2571,"would_cite":true,"duration_ms":24560,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37N25","92B25","34C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Asymmetric division plus positive feedback makes 1:2 and 2:3 mother–daughter temporal clusters asymptotically stable in a budding-yeast cell-cycle model.","keywords":["temporal clustering","asymmetric division","positive feedback","cell cycle model","perfectly clustered solution","yeast autonomous oscillations","phase synchrony","stability"],"falsifier":"Compute the exact event-time return map for the switching model C_s with initial perturbations on both sides of the daughter-ahead/daughter-behind boundary (d1 = g); if any parameter set that satisfies the lemmas gives a Jacobian eigenvalue outside the unit disk on the daughter-ahead side, the asymptotic stability claim for C_s fails.","tokens_in":16477,"feed_emoji":"🧬","tokens_out":9639,"duration_ms":104670,"temperature":0.7,"pith_summary":"The paper sets out to show that asymmetric division plus positive feedback in a budding-yeast cell-cycle model can make temporal clustering stable, not just full synchrony. It proves that 1:2 and 2:3 perfectly clustered solutions—populations split into p mother clusters and q daughter clusters with p ≤ q—exist and are asymptotically stable in both the fixed and switching cluster models. This matters because earlier results said positive feedback under symmetric division can only stabilize the synchronized state; the mother–daughter asymmetry gives the population a second phase offset that lets several cohorts coexist at once. If the proof is right, unequal division is enough to explain the stable multi-clustered oscillations seen in bioreactor yeast cultures.","feed_headline":"Asymmetric division plus positive feedback stabilizes yeast clusters","feed_subtitle":"Proof shows 1:2 and 2:3 mother–daughter temporal clusters form from random cell populations and persist.","key_machinery":"The load-bearing object is the Perfectly Clustered Solution (PCS): a p:q periodic state with p synchronized mother–daughter pairs plus q−p lone daughter clusters, all clusters equal in size. The argument runs through a specific event order r1 s1 r2 s21, in which a synchronized mother–daughter pair crosses the response region R while the lone daughter cluster enters the signaling region S, so positive feedback compresses the pair. The technical machinery is the pair of Poincaré return maps F_f (fixed model) and F_s (switching model) built from exact solutions of the piecewise-constant-speed ODE; the contraction factor 1/α, where α=1+ρ(1/3) or 1+ρ(1/5) is the boosted speed, appears directly in","core_discovery":"The central claim is Theorem 3.4 and Theorem 4.4: with positive feedback, the cluster models C_f and C_s admit a 1:2 PCS and a 2:3 PCS with the event order r1 s1 r2 s2 1, and each is asymptotically stable. A PCS is an exact periodic solution in which every mother cluster is synchronized with a daughter cluster, all clusters have equal cell numbers, and the mother cell jumps from 1 to g>0 while the daughter restarts at 0. The proof derives exact parameter formulas for g (and for the second mother position w in 2:3), constructs the Poincaré return maps F_f and F_s on the section {M=g}, linearizes them, and shows that every eigenvalue lies inside the unit circle, using root-location bounds for","pith_inferences":["The same compression mechanism—a lone daughter cluster in S speeding up a mother–daughter pair in R—should generalize to other p:q modes and other event orders, so the 1:2 and 2:3 cases are plausibly the first members of a family.","An observable consequence for yeast bioreactors: cell-cycle-related oscillations should contain unequal, phase-locked mother and daughter subpopulations, with mother cohorts leading, whenever positive feedback is the organizing force.","Because the proof stops at the smoothed cluster models, a numerical study of the discontinuous alternating model c_a could map where its basins agree with c_f/c_s and reveal whether any extra structures appear.","Because the pattern depends on exact timing of S and R crossings, shifting the signaling or responsive region widths should turn clustering on and off; this gives an experimental handle on the mechanism."],"forward_implications":["Full synchrony is not the only stable outcome of positive feedback: once division is asymmetric, multiclustered mother–daughter states can attract nearby trajectories.","Small perturbations of a 1:2 or 2:3 PCS decay, so in the presence of weak noise the population should return to the same clustered pattern rather than dispersing.","Because the PCS is asymptotically stable, it is structurally stable: an open set of parameter values near the exact ones in Lemmas 3.1 and 4.1 supports qualitatively similar stable clustered solutions.","Stability in the ideal cluster models carries over to the corresponding full cell models, linking the idealized proof to populations of individual cells.","Random initial conditions in simulations self-organize spontaneously into the same p:q patterns, matching the prediction that these states are attractors."],"supporting_citations":[{"why":"Introduces the cluster-model framework and proves negative-feedback asymmetric-division stability that this paper extends to positive feedback.","marker":"[2]"},{"why":"Defines the fixed and switching cell-cycle models that smooth over the mother jump; the cluster models C_f and C_s are derived from them.","marker":"[1]"},{"why":"Supplies the base cell-cycle feedback model and the previous result that positive feedback under symmetric division stabilizes only synchrony.","marker":"[25]"},{"why":"Root-location bounds used to place eigenvalues of the 2:3 switching-model Jacobian inside the unit circle.","marker":"[3]"},{"why":"Proposal that asymmetric division yields p repeating mother patterns and q daughter patterns, p ≤ q, motivating the PCS definition.","marker":"[4]"},{"why":"Transfer result used to conclude stability in cluster models implies stability in the corresponding cell models.","marker":"[18]"}],"fun_headline_variants":["Yeast clusters self-organize via asymmetric division and feedback","Asymmetric division and feedback lock yeast into stable clusters","Stable temporal clusters emerge from asymmetric division and feedback","Proof: asymmetric division and feedback stabilize 1:2, 2:3 yeast clusters","Mother-daughter clusters: asymmetric division and feedback make them stable"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"For the switching model, the stability calculation assumes the perturbed daughter cluster begins slightly behind its paired mother cluster; the opposite ordering is asserted to give the same result without being derived, and the Jacobian used may not be valid on the other side of that ordering switch.","fun_headline_variants_meta":{"raw":{"variants":["Yeast clusters self-organize via asymmetric division and feedback","Asymmetric division and feedback lock yeast into stable clusters","Stable temporal clusters emerge from asymmetric division and feedback","Proof: asymmetric division and feedback stabilize 1:2, 2:3 yeast clusters","Mother-daughter clusters: asymmetric division and feedback make them stable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001945,"raw_usage":{"total_tokens":7409,"prompt_tokens":674,"completion_tokens":6735,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":418,"completion_tokens_details":{"reasoning_tokens":6658}},"tokens_in":418,"tokens_out":6735,"duration_ms":43362,"temperature":1.0,"reasoning_tokens":6658,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T16:09:41.846757+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the exact event-time return map for the switching model C_s with initial perturbations on both sides of the daughter-ahead/daughter-behind boundary (d1 = g); if any parameter set that satisfies the lemmas gives a Jacobian eigenvalue outside the unit disk on the daughter-ahead side, the asymptotic stability claim for C_s fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the cluster-model framework and proves negative-feedback asymmetric-division stability that this paper extends to positive feedback."},{"cited_title":"Algoud,Asymmetric Division of Cell Population Structure Models With Negative Feedback, Ph.D","cited_arxiv_id":null,"evidence_quote":"Defines the fixed and switching cell-cycle models that smooth over the mother jump; the cluster models C_f and C_s are derived from them."},{"cited_title":"Young, B","cited_arxiv_id":null,"evidence_quote":"Supplies the base cell-cycle feedback model and the previous result that positive feedback under symmetric division stabilizes only synchrony."},{"cited_title":"Anderson, E.B","cited_arxiv_id":null,"evidence_quote":"Root-location bounds used to place eigenvalues of the 2:3 switching-model Jacobian inside the unit circle."},{"cited_title":"Bellgardt,Analysis of synchronous growth of baker’s yeast","cited_arxiv_id":null,"evidence_quote":"Proposal that asymmetric division yields p repeating mother patterns and q daughter patterns, p ≤ q, motivating the PCS definition."},{"cited_title":"Moses,Dynamical systems in biological modeling: clustering in the cell division cycle of yeast, Ph.D","cited_arxiv_id":null,"evidence_quote":"Transfer result used to conclude stability in cluster models implies stability in the corresponding cell models."}],"review_version":1}