{"id":"e1ca677e-71fc-4ccd-8433-6690e77e1db3","arxiv_id":"1908.09392","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Branching cell complexes can undergo more than two percolation transitions, and their upper percolation transition spans several universality classes, linked by an exact mapping to interdependent percolation in multiplex networks.","lead":"This paper studies what happens when links are randomly removed from branched networks built by gluing polygons together, a model for curved or non-flat network geometry. It finds extra, intermediate phase transitions in these networks and shows the main transition can be discontinuous, BKT-like, or a newly identified anomalous continuous type.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The higher-order critical-point conditions are internally inconsistent: Eq. (25) gives r1=5r2 while m=3 requires r1=4r2, and Eq. (33) cannot produce the s=4 ratios used in Figs. 7-10; the anomalous-scaling claim needs these fixed and Fig. 10 rechecked.","rationale":"The reader's conditional verdict is the right calibration. The intermediate transitions in Sec. VI follow from direct root analysis of the exact fixed-point equation and survive any correction of the higher-order formulas. The central unresolved issue is the quantitative support for the headline anomalous scaling: the printed Eqs. (25) and (33) contradict the m=3 examples that are supposed to realize the critical points of order s. Because the paper's own Fig. 10 shows s=5,6 deviations, the missing piece is a corrected derivation of the critical manifold and a rerun of the scaling test. I do not see a basis for rejection: the s=4 example appears to lie on the correct manifold, and the RG mechanism is coherent. I also checked the reader's flagged source-term assumption; while unproved, a bounded ∂F_n/∂x cannot change the leading exponential rate in Eq. (46), so it is less load-bearing than the formula mismatch. Thus no change to the conditional verdict.","tokens_in":20744,"tokens_out":26624,"duration_ms":270792,"concrete_test":"Set m=3 and (r1,r2,r3)=(8/11,2/11,1/11). Compute f(T)=T−1+(1−p_c)R(1−T^2) and its derivatives at p_c=1−1/(2r1)=5/16; verify that f'(1)=f''(1)=f'''(1)=0 and f''''(1)>0, i.e., a true order-4 point. Then plug the same ratios into the printed Eq. (33); if the identity fails, replace Eq. (33) with the composition condition R(1−(1−x)^{m−1})=r1(m−1)x+O(x^s), which is what setting f^{(j)}(1)=0 actually requires. Recompute the Fig. 10 curves for s=4,5,6 by iterating Eq. (42) to n=2·10^4 and plot −ln P∞ against Δp with the predicted σ=(s−3)/(s−1); if the s=5,6 residuals persist after using the corrected manifold, they are not due to the formula mismatch.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing weak point is the definition of the higher-order critical points on which the new universality class P∞∼exp[−A/(Δp)^σ], σ=(s−3)/(s−1), is built. These definitions are not internally consistent. Differentiating f(T)=T−1+(1−p)R(1−T^{m−1}) at p_c=1−1/[(m−1)r1] gives, for m=3, f''(1)∝4r2−r1 and, once f''=0, f'''(1)∝r2−2r3. The order-4 example (r1,r2,r3)=(8/11,2/11,1/11) used in Figs. 7 and 10 is exactly this manifold (r1=4r2, r2=2r3). But Eq. (25) prints r1=(2m−1)/(m−2)r2=5r2, and Eq. (33), under the natural readings of the fraction bars, gives for m=3, k=2 a ratio r2/r1 of π/2 or 1/2, not 1/4. Thus the manuscript's general critical-point formulas do not describe the m=3 examples that generate its headline anomalous exponent. Fig. 10 already reports that the s=5 and s=6 numerical exponents deviate from the theory; without corrected conditions one cannot distinguish a finite-size or continuum artefact from off-manifold parameters. The Sec. VIII subleading-source assumption is also asserted, but it is less decisive: a bounded inhomogeneous term cannot change the leading exponential growth rate used for ψ; the mismatch in the printed critical conditions is the more direct obstruction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies link percolation on random branching simplicial and cell complexes, where each link is incident to k m-polygons drawn from a distribution r_k. The authors derive a one-dimensional recursion for the percolation probability T, show that its fixed-point equation is formally analogous to that of interdependent percolation in a correlated multiplex network, and use this mapping plus a renormalization-group treatment to characterize the possible phase transitions. The main claims are: (i) such complexes can display one or more intermediate hybrid transitions in addition to the lower and upper percolation thresholds, at which T and the fractal exponent ψ jump while remaining below one; and (ii) the upper transition can be discontinuous, BKT-like, or continuous with an anomalous singular dependence P∞ ∼ exp[−A/(Δp)^σ] with σ = (s−3)/(s−1) at a critical point of order s. The central recursive equations and the RG framework are clearly presented, and the numerical phase diagrams are self-consistent. However, the paper contains inconsistencies in the stated conditions for the higher-order critical points on which claim (ii) is built.","tokens_in":21064,"tokens_out":11208,"duration_ms":103986,"significance":"If the claims are correct, the paper identifies new universality classes in a solvable hierarchical network model and establishes a conceptually interesting bridge between percolation on branching cell complexes and interdependent percolation in multiplex networks. The derivation is self-contained: Eq. (10) follows from the model, no parameters are fitted to data, and the RG scaling calculation is explicit. The prediction of anomalous continuous transitions with exponent σ = (s−3)/(s−1) is a falsifiable statement and is a genuinely new result for this class of models, if it survives the corrections to the critical-point conditions. The numerical confirmation for s = 4 in Fig. 10 is encouraging, but the deviations reported for s = 5, 6 cannot be interpreted until the critical-manifold conditions are fixed and the numerical parameters are verified to lie on the true manifold.","major_comments":[{"comment":"The condition for a second-order (transcritical) critical point at T_c = 1 is misstated. Differentiating f(T) = T − 1 + (1−p)R(1−T^{m−1}) at p_c = 1 − 1/[(m−1)r1] gives f''(1) = [2(m−1)r2 − (m−2)r1]/r1, so the condition f''(1) < 0 is r1 > 2(m−1)/(m−2) r2, not r1 > (2m−1)/(m−2) r2 as printed. For m = 3 the printed inequality requires r1 > 5r2, which would exclude the s = 4 example (r1,r2,r3) = (8/11,2/11,1/11) used in Figs. 7 and 10, since that example satisfies r1 = 4r2.","section":"Sec. IV, Eq. (19)"},{"comment":"The tricritical condition is printed as r1 = (2m−1)/(m−2) r2, but the correct condition f''(1) = 0 gives r1 = 2(m−1)/(m−2) r2, i.e., r1 = 4r2 for m = 3. The paper itself later uses the correct form in Sec. IX D (the line 'when r1 = 2r2(m−1)/(m−2), pc = ...'), so the two parts of the manuscript contradict each other. This inconsistency must be resolved because the tricritical manifold determines where the universality class of the upper transition changes.","section":"Sec. IV, Eq. (25)"},{"comment":"The stated formula for r_k at a critical point of order s is not compatible with the example used in the paper. For m = 3 and k = 2, Eq. (33) gives r2 = Γ(1/2)Γ(1/2)/Γ(3) r1 = (π/2)r1, whereas vanishing of f''(1) requires r2 = r1/4. The coefficients (0.727273, 0.181818, 0.0909091) shown for the s = 4 case in Figs. 7 and 10 satisfy the algebraic relations r1 = 4r2 and r2 = 2r3, not Eq. (33). Consequently, the s = 5 and s = 6 parameter sets used in Fig. 10 may not lie on the true critical manifold, and the reported deviations from the theoretical σ could be an artifact of off-manifold parameters. The authors should either correct Eq. (33) to the actual solution of Eqs. (31)–(32) or provide the explicit parameter values used and verify them against the corrected conditions.","section":"Sec. IV, Eq. (33)"},{"comment":"The numerical evidence for the anomalous exponent σ = (s−3)/(s−1) is incomplete. The s = 4 data agree with σ = 1/3, but the s = 5 and s = 6 exponents are reported to deviate from the prediction, and the paper attributes this to finite sizes or the continuous approximation. Given that the critical-point conditions in Sec. IV are misstated, the s = 5 and s = 6 runs may have been performed at parameters that are not critical points of order s. The authors should recompute these cases using the corrected conditions and then reassess whether the deviations persist; without this, the claim that the anomalous scaling generalizes beyond s = 4 is not supported.","section":"Sec. IX D and Fig. 10"},{"comment":"The derivation of the fractal exponent assumes that the non-homogeneous term ∂F_n/∂x in Eq. (42) is subleading compared with the homogeneous term, so that M_n is proportional to ∏ λ_n for large n. This is stated without proof. The conclusion is plausible—a bounded forcing term cannot change the exponential growth rate—but the manuscript should justify the claim, for example by bounding ∂F_n/∂x uniformly in n and noting that the contribution from the first source term already grows at the rate ∏ λ_n. Since Eq. (47) and the subsequent RG expression for P∞ rest on this step, a short argument or a reference to a derivation is needed.","section":"Sec. VIII A, Eq. (42)"}],"minor_comments":[{"comment":"The phase enumeration is numbered (1), (3), (3); the second item should read (2).","section":"Sec. III"},{"comment":"The formula M_n = (\\bar N_n)^{\\psi_n} is typeset in a way that could be misread as (\\bar N_n)\\psi_n; please clarify with an explicit exponent.","section":"Eq. (3)"},{"comment":"The figure shows the theoretical prediction as a thin dotted line, but it is unclear whether this line corresponds to s = 4 only or is meant for all s. For a useful comparison, the predicted slopes for s = 5 and s = 6 should be included or clearly described in the caption.","section":"Fig. 10"}],"recommendation":"major_revision","confidential_remarks":"The central difficulty is concentrated in Sec. IV: Eqs. (19), (25), and (33) are inconsistent with the manuscript's own examples and with the correct derivative conditions, while Sec. IX D uses the correct condition. This suggests a systemic derivation error rather than a single typo. Because the anomalous-scaling claim and Fig. 10 depend on these conditions, the authors need to correct them and re-examine the numerical evidence for s = 5, 6. If the corrected conditions remove the agreement for the reported parameter sets, the manuscript's central claim about the new universality class would be substantially weakened; if the agreement is restored after recomputation, the paper would be a solid contribution. The paper fits the journal's scope well."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nHere is my honest read of 1908.09392. Two things to know. First, the paper has real content: an exact equation-level mapping between link percolation on branching cell complexes and interdependent percolation in maximally correlated multiplex networks, plus a clear demonstration that multimodal branching distributions r_k produce intermediate hybrid transitions where both the percolation probability and the fractal exponent jump while remaining below one. The recursive equations are derived cleanly, no parameters are fitted, and the phase diagrams are self-consistent. Second, the headline new universality class, P_inf ~ exp[-A/(Delta p)^sigma] at higher-order critical points, rests on critical-point conditions that are internally inconsistent as printed.\n\nThe problem is concrete. For m=3, the tricritical condition obtained by setting f''(1)=0 is r1=4r2, but Eq. (25) prints r1=5r2. And Eq. (33), under any natural reading of the fraction bars, does not reproduce the ratios (8/11, 2/11, 1/11) that the paper itself uses for the order-4 critical point in Figs. 7 and 10. Since those examples are the evidence for the anomalous scaling, the paper's central claim is not yet backed by its own formulas. This looks like a fixable error in the closed-form parametrization rather than a flaw in the underlying RG expansion, which is set up correctly in terms of derivatives of f. But it has to be fixed, and the s=5,6 numerics rechecked, before the claim stands.\n\nTwo smaller concerns. The numerical verification is deterministic iteration of the recursion, not independent stochastic simulation; that is fine as a consistency check but weaker than a direct Monte Carlo test. And the assertion in Sec. VIII that the inhomogeneous term in Eq. (42) is subleading is stated without proof. I think that one is likely harmless for the leading exponential growth rate, but a sentence of justification would close the gap. The mapping to multiplex percolation uses Ref. [46] from the same group, but the relation is derived here rather than assumed, so the self-citation is not a problem.\n\nWho benefits: anyone working on percolation in hierarchical or non-amenable networks, and the multiplex-percolation community will care about the formal bridge. The intermediate transitions and the mapping are solid contributions worth publishing on their own. I would send this to referees, with a clear request that the authors correct Eqs. (25) and (33) and re-examine Fig. 10 after doing so.\n\nBest.","headline":"Solid exact mapping and intermediate-transition results, but the anomalous scaling claim rests on critical-point formulas that are internally inconsistent as printed.","tokens_in":21645,"tokens_out":4758,"would_cite":true,"duration_ms":46865,"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":"Branching cell complexes can have more than two percolation transitions.","keywords":["percolation","branching simplicial complexes","cell complexes","non-amenable graphs","phase transitions","multiplex networks","renormalization group","fractal exponent"],"falsifier":"Iterate the exact generating-function recursions (Eq. (38)) for a branching simplicial complex with $r_k=0.62\\delta_{k,1}+0.07\\delta_{k,2}+0.31\\delta_{k,20}$ to very large $n$ (e.g., $n=10^4$) and compute $T$ and $\\psi_n$ from Eq. (47). If no discontinuity appears near $p\\approx 0.0395$ with $T<1$ and $\\psi<1$ on both sides, the intermediate-transition claim fails. Separately, at a critical point of order $s=5$, plot $-\\ln P_\\infty$ versus $\\ln(p-p_c)$ over several decades; the predicted slope is $\\sigma=1/2$, so a cleanly different slope would falsify the anomalous scaling.","tokens_in":20488,"feed_emoji":"🕸️","tokens_out":11871,"duration_ms":108532,"temperature":0.7,"pith_summary":"Link percolation on two-dimensional branching simplicial and cell complexes—networks built by repeatedly gluing triangles or polygons onto links—is governed by a single recursive equation whose fixed points can bifurcate more than once. The paper establishes that, in addition to the usual lower threshold at $p^*=0$ and upper threshold at $p_c$, a multimodal distribution of the number of polygons glued per link produces one or more intermediate phase transitions. At each intermediate transition both the percolation probability and the fractal exponent jump discontinuously, yet the largest connected component remains smaller than the full network volume on both sides. The paper also proves an exact mapping between this percolation problem and the mutually connected giant component in correlated multiplex networks, and a renormalization-group analysis shows that the upper transition can be discontinuous, of Berezinskii-Kosterlitz-Thouless type, or continuous with anomalous exponential scaling. A sympathetic reader would care because this unifies two previously separate sources of explosive percolation and predicts a much richer phase structure for network geometry.","feed_headline":"Extra percolation jumps appear in branching cell complexes","feed_subtitle":"A mapping to multiplex networks predicts intermediate phase transitions and new universality classes.","key_machinery":"The central object is the recursive percolation map $T_{n+1}=1-(1-p)\\sum_{k\\ge 1}r_k(1-T_n^{m-1})^k$, encoding the probability that the two initial nodes stay connected after $n$ rounds of gluing and link removal. Its fixed-point equation $f(T)=0$ is studied with bifurcation theory; the type of bifurcation (transcritical, saddle-node, or pitchfork of order $s$) sets the upper-transition universality class, while multi-modal $r_k$ creates additional saddle-node bifurcations that give intermediate transitions. The fractal exponent is obtained from the largest eigenvalue $\\lambda_n$ of the Jacobian of the generating-function recursion via $\\psi_n=\\ln\\lambda_n/\\ln[\\langle k\\rangle(m-1)]$, and the RG flow of the map near $(p_c,T_c)$ yields the order-parameter scaling. The exact mapping identifies $T$ with $1-S'$ (one minus the MCGC reach probability), $p$ with $1-\\tilde p$, polygon side count $m$ with $\\kappa+1$, and $r_k$ with the multiplex activity distribution $P(B)$, transferring known results from correlated multiplex networks.","core_discovery":"The central claim is that branching cell complexes are not limited to the two percolation transitions previously known for non-amenable graphs. When the branching distribution $r_k$ has more than one mode, the fixed-point equation for the percolation probability develops additional hybrid (saddle-node) bifurcations below the upper threshold; at these, $T$ and the fractal exponent $\\psi$ both jump while remaining below one, so the giant component stays sub-extensive. At the upper threshold the transition can take several universality classes depending on the bifurcation: transcritical bifurcations give a discontinuous transition, saddle-node bifurcations give a BKT singularity $P_\\infty\\sim\\exp[-\\alpha/(\\Delta p)^{1/2}]$, and pitchfork critical points of order $s$ give continuous transitions with $P_\\infty\\sim\\exp[-A/(\\Delta p)^\\sigma]$, $\\sigma=(s-3)/(s-1)$, generalizing BKT to arbitrary $\\sigma$. These results follow from analyzing the recursive map $T_{n+1}=1-(1-p)\\sum_k r_k(1-T_n^{m-1})^k$ and from an exact mapping, for a modified construction, to the mutually connected giant component equation of multiplex networks with maximally correlated degrees.","pith_inferences":["The mapping suggests that the intermediate transitions in branching complexes are the geometric analogue of cascading-failure events in interdependent networks; if so, intuitions about fragility from multiplex percolation could transfer to designing or avoiding multi-stage failure in hierarchical materials.","The anomalous scaling family $P_\\infty\\sim\\exp[-A/(\\Delta p)^\\sigma]$ with $\\sigma=(s-3)/(s-1)$ interpolates between BKT and exponential singularities; this family may appear in other hierarchical or non-amenable geometries where an order-$s$ pitchfork bifurcation controls percolation.","These complexes are two-dimensional; the same recursive-equation approach might extend to higher-dimensional branching simplicial complexes, where new intermediate transitions or modified anomalous scaling could appear.","The paper's phase diagrams come from fixed-point analysis; a finite-size scaling theory for the intermediate jumps (how $M_n$ grows with $n$ exactly at $p_c^*$) is not developed and would be a natural next test of whether the discontinuities survive finite-size corrections."],"forward_implications":["A multimodal branching distribution $r_k$ produces one or more intermediate discontinuous transitions at which $T$ and $\\psi$ jump but remain below 1; the paper gives explicit phase diagrams for trimodal distributions on triangles.","The universality class of the upper percolation transition is controlled by the topology of the branching complex: discontinuous, BKT, and anomalous continuous transitions can all occur as $r_k$ varies.","At a critical point of order $s$, the percolation probability approaches its critical value as $|\\Delta T|\\sim|\\Delta p|^{1/(s-1)}$, so the effective critical exponent can be tuned by the branching distribution.","Because of the exact mapping, any transition found for these branching cell complexes has a counterpart in correlated multiplex networks with perfect inter-layer degree correlation, and vice versa.","The lower percolation threshold is always $p^*=0$ in these complexes, so every intermediate transition occurs while the largest component is still sub-extensive."],"supporting_citations":[{"why":"Supplies the correlated multiplex network whose MCGC equation is mapped exactly to the branching-cell-complex equation, transferring the multiple-transition phenomenology.","marker":"[46]"},{"why":"Develops the generating-function and renormalization-group formalism for percolation on 2D cell complexes that this paper adapts to branching complexes.","marker":"[13]"},{"why":"Establishes discontinuous percolation on Farey graphs and hyperbolic structures, the reference upper-transition class this paper generalizes.","marker":"[32]"},{"why":"Provides the flower-network RG result for continuous percolation at tricritical points, which the paper extends to higher-order critical points.","marker":"[38]"},{"why":"Supplies the RG treatment of BKT and discontinuous transitions used in the order-parameter analysis.","marker":"[40]"},{"why":"Proves that non-amenable graphs have lower and upper percolation thresholds, setting the two-transition baseline for these complexes.","marker":"[47]"},{"why":"Introduces the mutually connected giant component in interdependent networks, the phenomenon that the mapping connects to percolation on branching complexes.","marker":"[27]"}],"fun_headline_variants":["Branching cell complexes host extra percolation transitions","More than two percolation jumps in branching complexes","BKT and discontinuous percolation in branching complexes","Intermediate percolation transitions found in branching cell complexes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that, in the recursion for the component size $M_n$, the term that does not depend on earlier component sizes is exponentially smaller than the term that does; if both grew at the same exponential rate, the formula for the fractal exponent and the predicted universality classes would need revision.","fun_headline_variants_meta":{"raw":{"variants":["Branching cell complexes host extra percolation transitions","More than two percolation jumps in branching complexes","BKT and discontinuous percolation in branching complexes","Intermediate percolation transitions found in branching cell complexes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000294,"raw_usage":{"total_tokens":1746,"prompt_tokens":1018,"completion_tokens":728,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":634,"completion_tokens_details":{"reasoning_tokens":665}},"tokens_in":634,"tokens_out":728,"duration_ms":6567,"temperature":1.0,"reasoning_tokens":665,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:15:24.804200+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Iterate the exact generating-function recursions (Eq. (38)) for a branching simplicial complex with $r_k=0.62\\delta_{k,1}+0.07\\delta_{k,2}+0.31\\delta_{k,20}$ to very large $n$ (e.g., $n=10^4$) and compute $T$ and $\\psi_n$ from Eq. (47). If no discontinuity appears near $p\\approx 0.0395$ with $T<1$ and $\\psi<1$ on both sides, the intermediate-transition claim fails. Separately, at a critical point of order $s=5$, plot $-\\ln P_\\infty$ versus $\\ln(p-p_c)$ over several decades; the predicted slope is $\\sigma=1/2$, so a cleanly different slope would falsify the anomalous scaling.","supporting_citations":[{"cited_title":"Kryven and G","cited_arxiv_id":null,"evidence_quote":"Supplies the correlated multiplex network whose MCGC equation is mapped exactly to the branching-cell-complex equation, transferring the multiple-transition phenomenology."},{"cited_title":"Kryven, R","cited_arxiv_id":null,"evidence_quote":"Develops the generating-function and renormalization-group formalism for percolation on 2D cell complexes that this paper adapts to branching complexes."},{"cited_title":"Boettcher, V","cited_arxiv_id":null,"evidence_quote":"Establishes discontinuous percolation on Farey graphs and hyperbolic structures, the reference upper-transition class this paper generalizes."},{"cited_title":"Nogawa and T","cited_arxiv_id":null,"evidence_quote":"Provides the flower-network RG result for continuous percolation at tricritical points, which the paper extends to higher-order critical points."},{"cited_title":"Nogawa, J","cited_arxiv_id":null,"evidence_quote":"Supplies the RG treatment of BKT and discontinuous transitions used in the order-parameter analysis."},{"cited_title":"Lyons, Jour","cited_arxiv_id":null,"evidence_quote":"Proves that non-amenable graphs have lower and upper percolation thresholds, setting the two-transition baseline for these complexes."}],"review_version":1}