{"id":"3e4f4c7d-8ef1-420f-b2fb-59d3984d9802","arxiv_id":"2507.17643","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For surjective endomorphisms of normal projective varieties over characteristic zero fields, the arithmetic degree of any Zariski dense orbit must be a cohomological Lyapunov multiplier; if the first dynamical degree exceeds the second, the Kawaguchi-Silverman conjecture follows.","lead":"The paper proves that if a point moves under a surjective self-map of a projective variety and its orbit fills the whole variety, then the exponential growth rate of the point's height must be one of the map's cohomological Lyapunov multipliers. This restricts the possible arithmetic degrees and yields the Kawaguchi-Silverman conjecture in the case where the first dynamical degree is strictly larger than the second.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof applies [KS16a, Theorem 5] to a Jordan block Λ that can contain eigenvalues μ_i(f) < 1, where the normalized height sequence need not converge, so the canonical height vector used in the contradiction is not justified.","rationale":"The paper's central claim is that a Zariski-dense orbit has arithmetic degree equal to one of the cohomological Lyapunov multipliers. The overall strategy is plausible: decompose a big class into a part coming from multipliers larger than α and a part from smaller multipliers, then compare height growth. The reader identified the dependence on the unpublished theorems [Xieb] used as Theorem 2.2 and Theorem 2.3 as the weakest assumption. That is a legitimate external-dependency concern. However, a more internal load-bearing issue appears in the application of [KS16a, Theorem 5] to produce canonical height vectors for the lifting L. The matrix Λ can have diagonal entries μ_i(f) < 1 when the Albanese variety is trivial and some cohomological Lyapunov multiplier is below 1, e.g. a rational surface automorphism with λ1 > 1. For such an eigenvalue, the normalized height sequence h_D(f^n) μ^{-n} generically diverges because h_D(f^n) is bounded but not tending to zero fast enough. The paper does not address this, and the subsequent formula for ĥ_{tilde M1}(f^n) is essential for the contradiction. This is a gap in the proof as written, although it seems repairable by restricting E to multipliers exceeding α_f(x), since the branch under consideration has α_f(x) > √λ1(g). The concrete test would settle whether [KS16a, Theorem 5] in fact applies as stated; if the theorem already handles all positive eigenvalues, the concern evaporates, and the principal risk remains the external [Xieb] theorems. Therefore I recommend keeping the conditional verdict, with the condition expanded to include a justification or amendment of the canonical-height argument for the μ < 1 case.","tokens_in":10213,"tokens_out":24675,"duration_ms":281153,"concrete_test":"Check the precise hypotheses of [KS16a, Theorem 5]: if it assumes all eigenvalues of Λ are > 1, then test the excluded case on a rational surface automorphism with λ1(f) > 1 and trivial Albanese. Take a divisor class D with f^*D = μ D, μ = 1/λ1(f) < 1, and a Zariski-dense point x; compute or bound h_D(f^n x) μ^n for a Weil/Moriwaki height h_D. If h_D(f^n x) tends to a nonzero constant, then h_D(f^n x) μ^{-n} diverges, confirming that the cited convergence theorem cannot justify ĥ_L for this coordinate.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Theorem 1.2, E is defined as ⊕_{μ_i(f) > √λ1(g)} E_{μ_i}. When the Albanese variety is trivial, λ1(g) is set to 0, so E includes all positive cohomological Lyapunov multipliers, including values μ_i(f) < 1. Such values occur, for example, for a rational surface automorphism with λ1(f) > 1: then λ2(f) = 1, so μ2(f) = 1/λ1(f) < 1. The proof then applies [KS16a, Theorem 5] to the full matrix Λ of the Jordan form of f^* on E and asserts that h_L ∘ f^n Λ^{-n} converges pointwise to a canonical height vector ĥ_L satisfying ĥ_L ∘ f = ĥ_L Λ and ĥ_L = h_L + O(1). For a coordinate with eigenvalue 0 < μ < 1, the height sequence a_n := h_D(f^n x) obeys a_{n+1} = μ a_n + O(1); generically it converges to a nonzero bounded limit rather than decaying. Hence a_n Λ^{-n} = a_n μ^{-n} diverges. The standard canonical-height construction for Jordan blocks requires the relevant eigenvalues to exceed 1; the paper gives no argument covering μ < 1. Since ĥ_{tilde M1}(f^n) = Σ c_{i,k} n^k μ_i^n is then used for the lower bound on h_{eB}(f^{n_j}), the contradiction in the main theorem is unsupported in the trivial-Albanese, μ_i < 1 case. The gap appears repairable by restricting E to ⊕_{μ_i(f) > α_f(x)} E_{μ_i}, because α_f(x) > √λ1(g) in the nontrivial branch, but the proof as written does not do this.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies surjective endomorphisms f:X→X of normal projective varieties over finitely generated fields of characteristic zero. It proves (Theorem 1.2) that for any point x whose f-orbit is Zariski dense, the arithmetic degree α_f(x) is one of the cohomological Lyapunov multipliers μ_i(f)=λ_i(f)/λ_{i-1}(f), and in particular α_f(x)≥1. Corollary 1.3 deduces the Kawaguchi–Silverman conjecture when λ_1(f)>λ_2(f). Corollary 1.5 applies Theorem 1.2 to a dynamical Mordell–Lang type statement for product endomorphisms. The proof uses Moriwaki heights, a lift of the generalized eigenspaces of f^* on N^1(X)_R to Pic(X)_R, and the big-cone characterization of Lyapunov multipliers imported from [Xieb], then runs a height-growth comparison on a big divisor along a subsequence.","tokens_in":10593,"tokens_out":19052,"duration_ms":207572,"significance":"If the proof is repaired, this is a significant strengthening of the known result of Kawaguchi–Silverman that α_f(x) is the modulus of an eigenvalue of f^* on N^1(X)_R; it pins the arithmetic degree to the specific cohomological Lyapunov multipliers. The corollaries are concrete and testable, and the DML application is a nice demonstration of the height method. The paper is generally well structured and the central arithmetic-degree conclusion is not assumed; it is derived from external results, the most important being the big-cone characterization in [Xieb]. The main reservation is that the proof as written has a local but load-bearing gap in the canonical-height step when the Albanese variety is trivial; this is repairable by restricting the relevant eigenspace sum to multipliers exceeding α_f(x).","major_comments":[{"comment":"The definition E:=⊕_{μ_i(f)>√λ1(g)}E_{μ_i(f)} includes every positive μ_i(f) when Alb(X) is trivial, because then λ1(g)=0. This can include multipliers <1; e.g. a surface automorphism with λ1(f)>1 has μ2(f)=λ2(f)/λ1(f)=1/λ1(f)<1. The proof then applies [KS16a, Theorem 5] to the full Jordan matrix Λ and states that h_L∘f^n Λ^{-n} converges pointwise to a canonical height satisfying ĥ_L∘f=ĥ_LΛ and ĥ_L=h_L+O(1). For a coordinate with eigenvalue 0<μ<1, the recurrence a_{n+1}=μa_n+b_n with b_n bounded forces a_n=h_D(f^n x) to be bounded along every orbit, so a_n μ^{-n} does not converge and typically diverges. Hence the canonical-height construction, and the formula ĥ_{M1}(f^n(x))=Σ c_{i,k}n^kμ_i^n on which the lower bound h_{eB}(f^{n_j})≥C μ_{i0}^{n_j} depends, is not justified in the trivial-Albanese case. The gap is local: because the contradiction branch has α_f(x)>λ1(g), one can replace E by ⊕_{μ_i(f)>α_f(x)}E_{μ_i(f)}; Lemma 3.1 still applies, and all remaining eigenvalues in Λ are >1, where the canonical-height step is standard. As written, however, the main theorem is not proved.","section":"Section 3, proof of Theorem 1.2, definition of E and application of [KS16a, Theorem 5]"},{"comment":"The two key structural facts about cohomological Lyapunov multipliers—the big-cone characterization and the existence of a big class in the sum of the corresponding generalized eigenspaces—are quoted from the unpublished preprint [Xieb] (Theorems 1.3 and 1.4). Every step of the proof of Theorem 1.2 and Corollary 1.5 depends on these facts, so the main result is conditional on a source that is not yet publicly refereed. The authors should state this dependence explicitly and, if possible, include or append proofs, or at least confirm that [Xieb] is available in a citable final form. This does not by itself invalidate the argument, but it is a load-bearing incompleteness for a journal submission.","section":"Section 2, Theorems 2.2 and 2.3"}],"minor_comments":[{"comment":"There is a typo in the sentence \"under the weaker assumption that the the orbit is Zariski dense\": \"the\" is repeated.","section":"Section 1"},{"comment":"The convergence estimate \"the norm of (f^*)^{n-1}M Λ^{-n} is O(n^{r+2g}(ρ0/μ)^n)\" is asserted from the linear-recurrence property without further detail; since this estimate is a key input to the lifting result, a short justification would improve readability.","section":"Section 3, Lemma 3.1"},{"comment":"The sentence \"Since p1^*L1 is always big, we may choose L1 appropriately such that p1^*L1=A+E\" is slightly confusing: one chooses L1 ample, then p1^*L1 is big and the decomposition into an ample part A and effective part E exists. Rephrasing would avoid the impression that L1 is being chosen after the decomposition.","section":"Section 4, proof of Corollary 1.5"},{"comment":"In the sentence about [XY25], \"the second-named and third-named authors\" should be \"the second- and third-named authors\".","section":"Acknowledgements and references"}],"recommendation":"major_revision","confidential_remarks":"The canonical-height gap in the proof of Theorem 1.2 is the main obstacle; it appears genuinely local and repairable, so I would not recommend rejection. The heavy reliance on [Xieb], an unpublished preprint from the same group, is worth monitoring: if [Xieb] is not yet publicly vetted, the journal may want assurance that Theorems 2.2 and 2.3 have been independently checked."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is a genuine step forward: for a surjective endomorphism of a normal projective variety over a finitely generated characteristic-zero field, the arithmetic degree of a Zariski dense orbit is a cohomological Lyapunov multiplier, and when lambda_1 > lambda_2 the Kawaguchi–Silverman conjecture follows. That goes beyond KS16a (eigenvalue only) and Mat25 (generic orbits). The proof idea is attractive: lift the relevant part of N^1 into Pic using a spectral radius bound on Pic^0, then use the big-cone intersection of eigenspaces to build a big divisor whose height growth pins down the arithmetic degree. Lemma 3.1 is a nice construction, and the DML corollary is a useful extra.\n\nThe weak spot that matters sits in the contradiction argument. E is defined as the sum of generalized eigenspaces for mu_i(f) > sqrt(lambda_1(g)). When the Albanese is trivial, sqrt(lambda_1(g)) = 0, so E includes every positive multiplier, including mu_i(f) < 1. For a surface automorphism with lambda_1 > 1, the eigenvalue 1/lambda_1 < 1 is there. The proof then applies [KS16a, Theorem 5] to the full Jordan matrix Lambda and concludes that h_L ∘ f^n Lambda^{-n} converges to a canonical height vector. For a coordinate with 0 < mu < 1, the height along the orbit is bounded, so dividing by mu^n blows up; that theorem is only valid for eigenvalues above 1. So the canonical height vector used later is not justified in those cases. The repair is straightforward: take E = direct sum over mu_i(f) > alpha_f(x) instead. In the nontrivial branch alpha_f(x) > sqrt(lambda_1(g)), so the lifting and growth estimates still work, and all eigenvalues in E are >1. But the proof as written does not do this.\n\nThere is also a dependency on [Xieb], an unpublished preprint of the same group, for Theorems 2.2 and 2.3. Those are load-bearing and should be checked carefully. This is a real citation-pattern soft spot, not a fatal one.\n\nOverall, the strategy is plausible, the main theorem is significant, and the flaws I see are repairable rather than fatal. I would send this to a serious referee, with specific instructions to verify the canonical-height step and the hypotheses on [Xieb].","headline":"Real strengthening of Kawaguchi–Silverman with a repairable gap in the canonical-height step; worth refereeing seriously.","tokens_in":11150,"tokens_out":9698,"would_cite":false,"duration_ms":101242,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37P30","37P55","14G40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a surjective endomorphism of a normal projective variety over a finitely generated field of characteristic zero, if the orbit of a point is Zariski dense, then the arithmetic degree of the point must be one of the cohomological…","keywords":["arithmetic degree","cohomological Lyapunov multiplier","dynamical degree","Kawaguchi–Silverman conjecture","Zariski dense orbit","dynamical Mordell–Lang conjecture","Moriwaki height","endomorphism of projective variety"],"falsifier":"Compute, for a concrete surjective endomorphism of a normal projective variety over a number field, both the arithmetic degree of a Zariski dense orbit, for instance via heights of iterates, and the list $\\{\\mu_1,\\dots,\\mu_d\\}$; find an orbit whose $\\alpha$ is a real eigenvalue of $f^*$ on $N^1(X)_{\\mathbb{R}}$ but is not among the $\\mu_i$. Alternatively, exhibit an endomorphism for which the claimed identity $\\{\\mu_i\\}=\\{\\alpha:\\mathrm{Im}(f^*-\\alpha)\\cap\\mathrm{Big}(X)=\\emptyset\\}$ fails, since the proof imports that identity.","tokens_in":10002,"feed_emoji":"📈","tokens_out":6418,"duration_ms":59379,"temperature":0.7,"pith_summary":"This paper proves a precise restriction on arithmetic degrees in algebraic dynamics. For a surjective endomorphism $f$ of a normal projective variety over a finitely generated field of characteristic zero, if the orbit of a point $x$ is Zariski dense, then its arithmetic degree $\\alpha_f(x)$ must be one of the cohomological Lyapunov multipliers $\\mu_i(f)=\\lambda_i(f)/\\lambda_{i-1}(f)$, the ratios of consecutive dynamical degrees. Since the first multiplier $\\mu_1(f)=\\lambda_1(f)$ is the dynamical degree, the theorem says that dense orbits grow at one of a short list of intrinsic rates rather than at an arbitrary eigenvalue of the numerical pull-back. A corollary settles the Kawaguchi–Silverman conjecture whenever the two largest dynamical degrees differ, and another corollary gives a dynamical Mordell–Lang statement by comparing growth rates of two dense orbits.","feed_headline":"Dense orbits grow at Lyapunov-multiplier rates","feed_subtitle":"Zariski dense orbits must grow at one of finitely many dynamical ratios; a Kawaguchi–Silverman corollary follows.","key_machinery":"The carrying object is the cohomological Lyapunov multiplier $\\mu_i(f)=\\lambda_i(f)/\\lambda_{i-1}(f)$, the ratio of consecutive dynamical degrees of $f$, which are real eigenvalues of the numerical pull-back $f^*:N^1(X)_{\\mathbb{R}}\\to N^1(X)_{\\mathbb{R}}$. The key steps are a lifting lemma that lifts the generalized eigenspaces for multipliers above the spectral radius of $f^*$ on $\\mathrm{Pic}^0(X)$ to $\\mathrm{Pic}(X)_{\\mathbb{R}}$ in an $f^*$-equivariant way, and a big-cone property from a companion paper ensuring that a big divisor class lies in the span of the eigenspaces of the multipliers. Heights along the orbit are then controlled by canonical heights built from Jordan blocks, while the part belonging to smaller multipliers grows too slowly to match $\\alpha$.","core_discovery":"The central claim is Theorem 1.2: under the Zariski-density assumption, $\\alpha_f(x)\\in\\{\\mu_1(f),\\dots,\\mu_d(f)\\}\\cap\\mathbb{R}_{\\geq 1}$. The proof assumes for contradiction that $\\alpha$ lies strictly between $\\mu_\\ell$ and $\\mu_{\\ell+1}$, splits a big divisor class coming from the intersection of the big cone with the sum of generalized eigenspaces of $f^*$ corresponding to multipliers above $\\alpha$, and shows that one part of the height grows at rate $\\alpha$ while the other part grows too fast or too slow, forcing a contradiction. The corollary is that if $\\lambda_1(f)>\\lambda_2(f)$, then $\\alpha_f(x)=\\lambda_1(f)$ for every Zariski dense orbit, exactly as the Kawaguchi–Silverman conjecture predicts.","pith_inferences":["The same height-splitting strategy might be adapted to dominant rational maps, where Zariski dense orbits are harder to control but the multiplier list is still defined.","A testable computational extension is to compute the multiplier list and the arithmetic degree for concrete low-dimensional examples, such as endomorphisms of abelian surfaces or cyclic covers, and check the equality $\\alpha=\\mu_i$.","For the dynamical Mordell–Lang application, the disjointness hypothesis is stronger than needed: since the proof only uses $\\alpha_f(x)\\neq\\alpha_g(y)$, any condition implying different arithmetic degrees would give the same conclusion, and Theorem 1.2 suggests checking the multiplier lists is the natural route."],"forward_implications":["The arithmetic degree of every Zariski dense orbit is forced into a finite list of dynamical invariants, so existence of the limit is paired with a strong membership statement.","When $\\lambda_1(f)>\\lambda_2(f)$, the Kawaguchi–Silverman conjecture holds: every Zariski dense orbit has arithmetic degree $\\lambda_1(f)$.","If the multiplier lists of two endomorphisms are disjoint, no positive-dimensional subvariety of the product can contain a dense intersection of the product orbit, because the two coordinates grow at incompatible rates (Corollary 1.5).","The result strengthens the earlier description of $\\alpha_f(x)$ as an eigenvalue of $f^*$ on $N^1(X)_{\\mathbb{R}}$ by selecting the specific eigenvalues that are Lyapunov multipliers.","Since the arguments run over finitely generated fields via Moriwaki heights, the theorem applies over arbitrary characteristic-zero fields, including $\\mathbb{C}$."],"supporting_citations":[{"why":"Supplies the two load-bearing structural theorems: the cohomological Lyapunov multipliers are exactly the real eigenvalues $\\alpha$ whose image under $f^*-\\alpha$ misses the big cone, and the sum of the corresponding generalized eigenspaces meets the big cone.","marker":"[Xieb]"},{"why":"Supplies existence of the arithmetic degree, canonical heights for Jordan blocks, and the description of $\\alpha_f(x)$ as an eigenvalue of the numerical pull-back, all of which the proof builds on.","marker":"[KS16a]"},{"why":"States the Kawaguchi–Silverman conjecture that the paper's Theorem 1.2 and Corollary 1.3 address.","marker":"[KS16b]"},{"why":"Provides Moriwaki heights over finitely generated fields, the height machinery used to define arithmetic degrees and control height growth.","marker":"[Mor00]"},{"why":"Provides the proposition that the Albanese morphism is surjective when the orbit is Zariski dense, used at the start of the proof of Theorem 1.2.","marker":"[LM21]"},{"why":"Proves the analogous membership statement for generic orbits, the result that this paper extends from generic orbits to Zariski dense orbits.","marker":"[Mat25]"},{"why":"Supplies the height-argument framework toward dynamical Mordell–Lang and the elementary lemma on linear recurrence sequences used in the height-growth estimates.","marker":"[XY25]"}],"fun_headline_variants":["Arithmetic degrees match Lyapunov multipliers","Dense orbit growth pinned to Lyapunov multipliers","Zariski dense orbits take Lyapunov multiplier rates","Arithmetic degree forced to cohomological Lyapunov multiplier","Lyapunov multipliers govern dense orbit arithmetic degree"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on two cited structural theorems about cohomological Lyapunov multipliers: that they are exactly the real eigenvalues $\\alpha$ of the numerical pull-back whose image misses the big cone, and that the sum of the corresponding generalized eigenspaces meets the big cone; if either statement fails under the stated hypotheses, the conclusion no longer follows.","fun_headline_variants_meta":{"raw":{"variants":["Arithmetic degrees match Lyapunov multipliers","Dense orbit growth pinned to Lyapunov multipliers","Zariski dense orbits take Lyapunov multiplier rates","Arithmetic degree forced to cohomological Lyapunov multiplier","Lyapunov multipliers govern dense orbit arithmetic degree"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00018,"raw_usage":{"total_tokens":1207,"prompt_tokens":753,"completion_tokens":454,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":369,"completion_tokens_details":{"reasoning_tokens":378}},"tokens_in":369,"tokens_out":454,"duration_ms":4313,"temperature":1.0,"reasoning_tokens":378,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:44:27.077953+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a concrete surjective endomorphism of a normal projective variety over a number field, both the arithmetic degree of a Zariski dense orbit, for instance via heights of iterates, and the list $\\{\\mu_1,\\dots,\\mu_d\\}$; find an orbit whose $\\alpha$ is a real eigenvalue of $f^*$ on $N^1(X)_{\\mathbb{R}}$ but is not among the $\\mu_i$. Alternatively, exhibit an endomorphism for which the claimed identity $\\{\\mu_i\\}=\\{\\alpha:\\mathrm{Im}(f^*-\\alpha)\\cap\\mathrm{Big}(X)=\\emptyset\\}$ fails, since the proof imports that identity.","supporting_citations":[],"review_version":1}