{"id":"7da0a10c-6dd5-41dc-86a0-b110ed9027c1","arxiv_id":"2607.12561","paper_version":2,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every non-isotrivial one-parameter family of rational maps on the Riemann sphere, the gonality of the dynatomic curves tends to infinity, resolving the Gonality Conjecture.","lead":"This paper proves a long-standing conjecture: for any genuinely varying one-parameter family of holomorphic maps of the Riemann sphere, the \"dynatomic curves\" formed by points of period n become more and more complicated as n grows. The proof ties together arithmetic height theory and holomorphic bifurcation theory, and yields new uniform bounds on preperiodic points over number and function fields.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The smallness-to-vanishing bridge depends on an unproved generalization: Proposition 6.1 is quoted from an abelian-scheme proof, and Proposition 7.4 needs it for arbitrary families.","rationale":"The reader's weakest_assumption identifies exactly the step I find least secure. I reviewed the route from Theorem 1.2 to the contradiction: (i) smallness is expressed in Moriwaki heights; (ii) Proposition 7.4 converts this into vanishing of S∧T_f∧ω_1^{m-1}; (iii) Proposition 7.5 slices this to get the conditions in Theorem 4.9; (iv) Theorem 4.9 uses the one-dimensional bifurcation theory to contradict the choice of a bifurcation parameter. Step (ii) rests solely on Proposition 6.1. The paper explicitly acknowledges that the inequality is imported (Section 6.1) and that no proof is included. This is not a question of novelty or of disagreement with established results; it is a verifiable gap in the written argument. Other deferred points (Remark 2.21, the sketched Section 9 genericity) are less central for the N=1 theorem: the lifted current's representation-independence is explicitly not used, and Section 9's periodic genericity only affects N≥3. Thus the single load-bearing concern is Proposition 6.1. I do not think this forces rejection: the result is plausible and the surrounding machinery is coherent, but acceptance should remain conditional until the comparison is either supplied or independently certified.","tokens_in":50339,"tokens_out":19409,"duration_ms":203995,"concrete_test":"Check [JSX26, Prop. 2.7] line by line: identify every place where the abelian scheme structure (Mordell-Weil, isogenies, Neron models, or multiplication maps) is used. Then re-derive Prop. 6.1 for the exact setup of Section 7 (U=X×P^N with f, L=L_{2,f}, H a big/nef class). If the proof uses any abelian-specific input that cannot be replaced by the existence of the canonical height and the Northcott-type finiteness, the 'proof works in general' assertion is unsupported. As a minimal analytic sanity check, compute h_geom and h_Moriwaki for horizontal curves in the model case X=P^1, U=P^1×P^1, L=O(1,1), H=O(1) and verify the inequality with a constant independent of the curve.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 6.1 (Section 6.1) asserts h_geom(Z) ≤ C h_Moriwaki(Z) for all K-subvarieties when H is big and nef, citing [JSX26, Prop. 2.7] where π:U→X is an abelian scheme, with the remark 'its proof works in general.' This inequality is load-bearing: in Proposition 7.4 (Section 7.2), smallness of (Γ_n) gives h_Moriwaki→0; the proof needs h_geom→0 to conclude S_n∧T_f∧ω_1^{m-1}→0 and hence S∧T_f∧ω_1^{m-1}=0. That vanishing is then used in Proposition 7.5 to get S_θ∧T_f=0 and S_θ∧R_s=μ_s, which are the inputs to Theorem 4.9 and the contradiction at a bifurcation point. No derivation is supplied in the paper; the cited proposition is in a different setting. If the comparison fails, or holds only up to a positive additive constant, a small sequence need not have vanishing geometric height, and the bridge from arithmetic smallness to the dynamical contradiction breaks. This is an internal gap, not a disagreement with consensus.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper aims to prove the Gonality Conjecture in arithmetic dynamics: for any non-isotrivial one-parameter algebraic family of endomorphisms of P^1 over C which is not the flexible Lattès family, every small sequence of distinct horizontal curves has gonality tending to infinity and genus growing superlinearly in the degree. In particular, distinct dynatomic curves have gonality tending to infinity, resolving Conjecture 1.1 when combined with previous work of Nguyen–Saito in the flexible Lattès case. The authors also prove higher-dimensional analogues under Assumption A, and derive uniform boundedness results for iterated preimages over number fields and geometric uniform boundedness for preperiodic points over function fields. The proof combines Moriwaki heights and arithmetic equidistribution, a new woven-current approximation theorem for curves of bounded gonality, and bifurcation theory.","tokens_in":50640,"tokens_out":14447,"duration_ms":145495,"significance":"If the main results are correct, this is a major advance: it resolves a conjecture open for general one-parameter families, previously known only for the unicritical and flexible Lattès families, and it gives a new bifurcation-theoretic mechanism for gonality and genus growth. The paper also contributes an independent analytic result, Theorem 3.9, extending woven-current approximation from bounded genus to bounded gonality, and it connects arithmetic equidistribution with bifurcation currents in a novel way. The applications to uniform boundedness of preimages and preperiodic points are substantial. However, the central proof currently relies on a load-bearing height comparison that is only cited from a different setting, and on a transition to the woven approximation that is not fully justified. The significance is therefore high but conditional on filling these gaps.","major_comments":[{"comment":"Proposition 6.1 asserts h_geom(Z) ≤ C h_Moriwaki(Z) for every K-subvariety Z under H big and nef, in the full generality of a flat projective morphism π:U→X. The only reference is [JSX26, Prop. 2.7], whose stated setting is an abelian scheme; the text says 'its proof works in general' without supplying the proof. This inequality is load-bearing: Proposition 7.4 uses it to infer from hhat_f(V_n)→0 that hhat_geom(O(V_n))→0, and hence S∧T_f∧ω_1^{m-1}=0. That vanishing is the essential input for Proposition 7.5 and the contradiction via Theorem 4.9. If the comparison fails, or holds only up to an additive constant, the bridge from arithmetic smallness to the dynamical contradiction breaks. The authors should either give a complete proof in the required generality or cite a statement proved in that generality.","section":"§6.1, Proposition 6.1"},{"comment":"After (7.6), the text states 'By Proposition 7.5, we have ... S_θ∧R_s=μ_s for μ_bif,θ-a.e. s∈π^{-1}(θ)'. But Proposition 7.5(2) is conditional: it only gives this equality when S_θ∧R_s is admissible, and the proposition does not prove admissibility for μ_bif,θ-almost every s. Later the proof replaces S_θ by a uniformly woven approximation S_r from Theorem 3.6 and applies Lemma 7.7 to obtain s with S_r∧R_s admissible and positive. To apply Theorem 4.9, condition (2) requires S_r∧R_t ≤ μ_t for every t for which S_r∧R_t is admissible. The inequality S_r ≤ S_θ does not imply admissibility of S_θ∧R_t from admissibility of S_r∧R_t, so the desired bound does not follow from Proposition 7.5 as written. This is a load-bearing step in the contradiction; it needs a direct argument proving either the a.e. equality including admissibility, or the bound for S_r.","section":"§7.3, proof of Theorem 1.4"}],"minor_comments":[{"comment":"Typos: 'whcih' and 'subdivion' occur in the paragraph defining the subdivision Q; 'f∪' appears in the proof of Proposition 2.10. These are harmless but should be corrected.","section":"§2.1"},{"comment":"The adelic bifurcation line bundle L_bif and its geometric part eL_bif are used for Proposition 6.10. The descent from End_{d,N} to M_{d,N} and the Deligne pairing construction would benefit from a precise reference in the quasi-projective setting; the current text only sketches the construction.","section":"§6.4"},{"comment":"In the proof of Proposition 7.1, the phrase 'Pick a smooth projective compactification Y ... such that id and f extend to morphisms p_1,p_2' should clarify that resolution/blow-ups are needed; as written the extension of a rational dynamical system to a projective model is not automatic.","section":"§7.1"},{"comment":"The references [Mor00a] and [Mor00b] appear to refer to the same article; they should be consolidated.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is ambitious and, if repaired, would be a strong contribution. The main concern is that Proposition 6.1 is attributed to [JSX26], a self-citation to a preprint whose proof is set in the abelian-scheme case; the assertion that the proof extends is not demonstrated. Given that this comparison is the bridge from arithmetic smallness to the dynamical contradiction, I would not recommend acceptance until a full proof in the needed generality is supplied. The a.e. admissibility issue in Proposition 7.5/Theorem 1.4 also needs a concrete argument. The rest of the architecture is coherent and the analytic results, especially Theorem 3.9, are valuable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Big result, real gap. Ji and Xie prove the Gonality Conjecture for dynatomic curves in every non-isotrivial one-parameter P^1 family outside the flexible Lattès family, and more: any small sequence of horizontal curves has gonality tending to infinity and genus/degree tending to infinity. That is a genuine breakthrough. Previous methods only handled the unicritical and Lattès families. The higher-dimensional version under Assumption A is a bonus. The architecture — smallness → equidistribution → woven-current approximation → bifurcation — is coherent, and the analytic machinery is substantial. Theorem 3.9, replacing bounded genus by bounded gonality in the woven-current approximation, is a useful contribution in its own right.\n\nThe soft spot the stress-test flags is real. Proposition 6.1 compares geometric height to Moriwaki height, h_geom ≤ C h_Moriwaki. It is quoted from [JSX26], where the setting is an abelian scheme, with only the comment that the proof works in general. Here it is load-bearing: Proposition 7.4 needs it to convert arithmetic smallness into vanishing of S∧T_f∧ω^{m-1}_1, which drives the contradiction. If the comparison fails, or holds only up to an additive constant, the bridge from arithmetic smallness to the dynamical contradiction breaks. The paper does not supply the proof. That is an internal gap, not a disagreement with consensus.\n\nSmaller issues: Remark 2.21 defers representation-independence of the lifted woven current; they say they do not use it, which is fine, but the corner is untidy. The proofs of Assumption A in Section 9 — split Lattès and very general moduli curve — are sketched rather than fully argued; plausible, but expansion needed.\n\nOverall, this is a serious preprint from people who know the area. The main theorem is not a repackaging; it is a new mechanism. The citation pattern is honest — self-citations are to prior rigidity and height results, not to the target. The circularity burden is low. But Proposition 6.1 needs independent verification before I would certify correctness.\n\nWho gets value: anyone in arithmetic dynamics, complex dynamics, or height theory. It deserves a serious referee — a good editor should send it out, with a request to verify Proposition 6.1 in the needed generality and to expand Section 9.","headline":"The paper is a genuine breakthrough — Gonality Conjecture for non-isotrivial P^1 families — but its load-bearing comparison between geometric and Moriwaki height is quoted rather than proved.","tokens_in":51095,"tokens_out":1785,"would_cite":true,"duration_ms":19110,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37F10","37P30","14H51","11G50"],"pacs":[],"model":"deepseek-v4-flash","headline":"In any genuinely varying family of rational maps on the projective line, the curves parameterizing preperiodic points must acquire large gonality and superlinear genus growth; this proves the Gonality Conjecture and yields uniform boundedne","keywords":["gonality","dynatomic curves","preperiodic points","arithmetic dynamics","uniform boundedness","woven currents","bifurcation currents","canonical height"],"falsifier":"A direct falsifier would be a non-isotrivial one-parameter family of rational maps, not of the flexible elliptic-curve type, together with a sequence of distinct dynatomic curves of bounded gonality; the theorem predicts none exists. A more local falsifier: find any non-abelian family where the geometric-height ≤ constant·canonical-height inequality fails, which would sever the proof at its first arithmetic step even if the analytic mechanism stands.","tokens_in":50202,"feed_emoji":"🌀","tokens_out":6100,"duration_ms":62442,"temperature":0.7,"pith_summary":"This paper proves that in any non-isotrivial one-parameter family of rational maps on the projective line, the curves that track preperiodic points—the dynatomic curves—must become increasingly complicated: their gonality tends to infinity, and their genus grows faster than linearly in their degree over the parameter curve. The only exception is the flexible family built from elliptic-curve endomorphisms, where the growth was already established by older methods. This resolves the Gonality Conjecture in arithmetic dynamics. The mechanism is new: when such a family is not stably trivial, its bifurcation locus is non-empty, and the paper shows that arithmetic smallness of a sequence of curves cannot coexist with bounded geometric complexity in the presence of bifurcation. As applications, the paper proves uniform bounds for iterated preimages over number fields and for preperiodic points over complex function fields, for every one-parameter family on the line.","feed_headline":"Bifurcation forces preperiodic curves to grow complex","feed_subtitle":"In any genuinely varying family of maps on the projective line, gonality of dynatomic curves tends to infinity—settling a conjecture.","key_machinery":"The central object is the uniformly woven current: a positive closed current expressed as an integral of currents of integration over analytic curves with uniformly bounded local volumes. The paper develops structure theorems showing that limits of algebraic curves with bounded normalized genus, or with bounded gonality after passing to symmetric products, are uniformly woven. The load-bearing analytic result is Theorem 4.9: a uniformly woven current satisfying S ∧ T_f = 0 and whose admissible slices are bounded by the maximal entropy measure can only be supported over stable parameters. Since every non-isotrivial non-flexible one-dimensional family has a non-empty bifurcation locus, the exi","core_discovery":"Theorem 1.2 states that if f is a non-isotrivial one-parameter algebraic family of endomorphisms of P^1 of degree d at least 2 over C, not among the flexible elliptic-curve families, and K is any field finitely generated over Q over which f is defined, then every sequence of distinct horizontal curves over K that is small for the canonical dynamical height satisfies gonality tending to infinity and genus/degree tending to infinity. Dynatomic curves and preimage curves are automatically small, so the Gonality Conjecture follows. The higher-dimensional generalization, Theorem 1.4, gives the same conclusion for generic small horizontal curves in any one-parameter family of endomorphisms of P^N","pith_inferences":["The proof's decisive geometric step—bounded gonality implies a uniformly woven limit after passing to symmetric products—does not use dynamics and could serve as a general criterion in other contexts where a rigidity theorem forbids woven currents.","The reliance on a height-comparison inequality imported from an abelian-scheme setting is the point most worth checking; if that comparison fails in general, the arithmetic input would need replacement, while the analytic bifurcation mechanism would remain intact.","A natural testbed for further progress is a one-parameter family in dimensions two and higher where no classification of stable families exists; any gonality-growth theorem there would need a substitute for the rigidity that is available in dimension one."],"forward_implications":["The Gonality Conjecture for dynatomic curves in dimension one is now a theorem: in any non-isotrivial family of rational maps other than the flexible elliptic-curve families, distinct dynatomic curves have gonality tending to infinity.","For any non-isotrivial one-parameter family of maps on the projective line over a number field, the number of rational iterated preimages of a marked point of bounded degree is uniformly bounded independently of the parameter.","The Geometric Uniform Boundedness Conjecture holds along every algebraic curve in the moduli space of degree-d maps of the line: for each bounded gonality of the field, the number of preperiodic points is uniformly bounded, with no exceptional subvariety in dimension one.","In higher dimensions, the same conclusions hold for any one-parameter family satisfying Assumption A, namely a non-empty bifurcation locus and, in dimension at least three, periodic genericity.","The normalized genus, genus/degree, also diverges for every small horizontal sequence, a stronger statement reflecting a genuinely dynamical source of ramification rather than the linear growth typical of abelian-scheme towers."],"fun_headline_variants":["Gonality Conjecture proved for arithmetic dynamics","Bifurcation forces gonality and genus to blow up","Small dynatomic curves have gonality tending to infinity","Genus and gonality grow under bifurcation","Uniform boundedness from bifurcation and currents"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is a comparison inequality bounding the geometric height by a constant multiple of the canonical height; the paper cites this inequality from a setting where it was proved for abelian schemes and asserts without proof that the argument works in full generality, so if it fails outside that setting the bridge from arithmetic smallness to the dynamical contradiction collapses.","fun_headline_variants_meta":{"raw":{"variants":["Gonality Conjecture proved for arithmetic dynamics","Bifurcation forces gonality and genus to blow up","Small dynatomic curves have gonality tending to infinity","Genus and gonality grow under bifurcation","Uniform boundedness from bifurcation and currents"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000155,"raw_usage":{"total_tokens":1023,"prompt_tokens":689,"completion_tokens":334,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":433,"completion_tokens_details":{"reasoning_tokens":274}},"tokens_in":433,"tokens_out":334,"duration_ms":4022,"temperature":1.0,"reasoning_tokens":274,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T06:28:17.919806+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct falsifier would be a non-isotrivial one-parameter family of rational maps, not of the flexible elliptic-curve type, together with a sequence of distinct dynatomic curves of bounded gonality; the theorem predicts none exists. A more local falsifier: find any non-abelian family where the geometric-height ≤ constant·canonical-height inequality fails, which would sever the proof at its first arithmetic step even if the analytic mechanism stands.","supporting_citations":[],"review_version":2}