{"id":"ffb5bc64-a256-434e-9dcd-e8c38bcb04c0","arxiv_id":"2504.01563","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Height arguments establish finiteness and structural results for return sets in the dynamical Mordell-Lang problem for endomorphisms of projective varieties and products, plus examples of complex behavior on zero-entropy tori.","lead":"This paper uses height arguments to prove that return sets of dense orbits into curves are finite under non-integer cohomological Lyapunov multiplier conditions, and that product systems inherit return set structure from the base when one factor's degree exceeds the other's dynamical degree. A smart generalist might read it to see how arithmetic complexity measures resolve orbit intersection questions in algebraic dynamics.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Height functoriality and Northcott properties in arbitrary characteristic are the least secure step linking non-integer multipliers to finiteness of return sets","rationale":"The reader's identification of the height properties as the weakest assumption matches the load-bearing step in the argument; the abstract-only assessment already flags the exact point where the derivation is least secure, and no stronger internal inconsistency is visible from the given material.","tokens_in":1741,"tokens_out":395,"duration_ms":78256,"concrete_test":"Extract the precise height inequality used in the proof of (i) (the step immediately after invoking the non-integer multiplier) and test it on the explicit example of f: ℙ¹ → ℙ¹, f(z) = z² over an algebraically closed field of characteristic 2; compute the canonical height of a dense orbit point and check whether the inequality still produces a contradiction when the return set is infinite.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Result (i) derives finiteness of {n | f^n(x) ∈ C} from the assumption that some cohomological Lyapunov multiplier λ of f^k is not an integer, via height inequalities of the form h(f^{kn}(x)) ≥ c·|λ|^n h(x) + O(1) that force unbounded height on the return set (contradicting Northcott on C). This step invokes standard functoriality of Weil heights under pullback by endomorphisms and Northcott-type finiteness, stated to hold over any field. In positive characteristic these properties are not automatic: if f is inseparable, or if heights are defined via intersection theory on a model over a function field, the inequality can fail to be strict or the finiteness on C can fail even when λ ∉ ℤ. The paper applies the same height machinery to both (i) and (ii), so the gap is load-bearing for the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper uses height arguments to establish two results on the dynamical Mordell-Lang problem over fields of arbitrary characteristic. Result (i) asserts that if f is an endomorphism of a projective variety and some cohomological Lyapunov multiplier of an iterate f^k is non-integral, then the return set {n | f^n(x) in C} is finite for a dense orbit and a curve C. Result (ii) states that for a product endomorphism f × g on X × C with deg(g) exceeding the first dynamical degree of f, the return sets of the product system take the same form as those of (X, f); this is applied to split self-maps of products of curves with pairwise distinct degrees. The paper also constructs examples of complicated (non-finite, non-periodic) return sets for certain zero-entropy endomorphisms of tori and split maps where the height method does not apply.","tokens_in":1976,"tokens_out":601,"duration_ms":49192,"significance":"If the central claims hold, the results supply new sufficient conditions for finiteness of return sets that are valid in arbitrary characteristic and clarify the structure of return sets under degree comparisons. The explicit examples of intricate return sets even for zero-entropy maps on tori provide concrete illustrations that go beyond prior conjectures in [CGSZ21] and [XY25], helping delineate the boundaries of the dynamical Mordell-Lang problem.","major_comments":[{"comment":"The derivation of the key height inequality h(f^{kn}(x)) ≥ c · |λ|^n h(x) + O(1) (used to obtain unbounded heights on the return set, contradicting Northcott finiteness on C) in the proof of result (i) invokes functoriality of Weil heights under pullback by endomorphisms and Northcott-type properties over arbitrary fields. These properties are not automatic in positive characteristic when f is inseparable or when heights arise from intersection theory on models over function fields; the inequality may fail to be strict, rendering the finiteness conclusion load-bearing on an unverified step.","section":"height arguments for result (i)"},{"comment":"Result (ii) and its application to split self-maps of products of curves likewise rely on the same height machinery to compare dynamical degrees and control return sets; any gap in the height inequalities therefore propagates to the claim that the return sets 'have the same form' as those of (X, f).","section":"proof of result (ii) and application to split maps"}],"minor_comments":[{"comment":"The abstract and introduction should explicitly state the precise definition of 'cohomological Lyapunov multiplier' and the normalization of dynamical degrees used throughout.","section":"abstract and §1"},{"comment":"Clarify whether the examples in the final section are constructed over finite fields or function fields, as this affects the interpretation of the height failures.","section":"examples section"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and constructive comments. Below we respond point by point to the major comments.","responses":[{"response":"The functoriality of Weil heights under pullback holds for any endomorphism of a projective variety over an arbitrary field, as it follows directly from the definition of the height associated to an ample line bundle (the pullback multiplies the height by the degree on the Picard group, with the error term controlled by the geometry of the variety). This is standard in arithmetic geometry and does not require separability; the cohomological Lyapunov multiplier is precisely the spectral radius governing this growth, independent of characteristic. Northcott finiteness likewise applies to points of bounded height over finitely generated fields (including function fields), via the usual reduction to number fields or the absolute height. The non-integral multiplier ensures the lower bound produces unbounded heights on the return set. We will add a clarifying paragraph citing these standard facts to make the argument fully explicit in positive characteristic.","revision_made":"partial","referee_comment":"[height arguments for result (i)] The derivation of the key height inequality h(f^{kn}(x)) ≥ c · |λ|^n h(x) + O(1) (used to obtain unbounded heights on the return set, contradicting Northcott finiteness on C) in the proof of result (i) invokes functoriality of Weil heights under pullback by endomorphisms and Northcott-type properties over arbitrary fields. These properties are not automatic in positive characteristic when f is inseparable or when heights arise from intersection theory on models over function fields; the inequality may fail to be strict, rendering the finiteness conclusion load-bearing on an unverified step."},{"response":"Because the height inequalities are valid in arbitrary characteristic as explained above, the comparison of dynamical degrees in result (ii) proceeds without issue: the degree condition on g ensures that the height growth on the product is dominated in a way that reduces the return-set structure to that of (X, f). The application to split self-maps of products of curves with pairwise distinct degrees satisfies the hypothesis directly, so the conclusion remains unchanged. No further revision is required for this section.","revision_made":"no","referee_comment":"[proof of result (ii) and application to split maps] Result (ii) and its application to split self-maps of products of curves likewise rely on the same height machinery to compare dynamical degrees and control return sets; any gap in the height inequalities therefore propagates to the claim that the return sets 'have the same form' as those of (X, f)."}],"tokens_in":1550,"tokens_out":557,"duration_ms":116385,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper states two results on the dynamical Mordell-Lang problem. Result (i) claims that for an endomorphism of a projective variety, if some cohomological Lyapunov multiplier of an iterate is not an integer, then the return set of a dense orbit into a curve is finite. Result (ii) says that for a product map f × g on X × C, if deg(g) exceeds the first dynamical degree of f, the return sets of the product system match those of (X, f). They apply this to split self-maps on products of curves with distinct degrees and give examples on tori with zero entropy where return sets are more intricate than some earlier expectations. These statements are not direct reductions of the cited prior work, so the specific claims are new. The height arguments are the main tool, and the examples are concrete enough to be useful for seeing where the conditions matter. The weakest link is the invocation of standard height functoriality and Northcott finiteness over arbitrary fields, including positive characteristic. When the map is inseparable or the height is defined via intersection theory on a model over a function field, the inequality h(f^{kn}(x)) ≥ c |λ|^n h(x) + O(1) can lose strictness, and finiteness on the curve may not follow. The abstract treats these properties as holding without extra verification, so the proofs will need to show explicitly that the non-integer multiplier still produces the required growth. The examples section is fine as far as it goes and does not rely on the same step. This is work for people already inside arithmetic dynamics who follow the Mordell-Lang variants. It is narrow but the statements are precise enough that a referee can check the height estimates and the examples directly. I would send it to peer review rather than desk reject.","headline":"Two conditional height-based theorems on return-set finiteness and product reductions, plus torus examples, but the arbitrary-characteristic height inequalities are the part that needs checking.","tokens_in":2453,"tokens_out":441,"would_cite":false,"duration_ms":26927,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Standard Weil/Moriwaki height growth and Northcott finiteness on projective varieties; no J-cost, φ-ladder or recognition-cost structure","alignment":"orthogonal","rationale":"The paper's core machinery (dynamical degrees λ_i(f), cohomological Lyapunov multipliers μ_i(f)=λ_i/λ_{i-1}, height inequalities h(f^{kn}(x)) ≥ c|μ|^n h(x)+O(1), Northcott on curves) is classical arithmetic geometry (Weil heights over function fields, Moriwaki heights, Albanese reduction). It never invokes the RS recognition cost J(x)=½(x+x^{-1})-1, φ-ladders, 8-tick periodicity, or any theorem from the supplied Lean corpus (reality_from_one_distinction, AbsoluteFloorClosure, Cost.FunctionalEquation, AlexanderDuality, etc.). The results are therefore in a domain RS has no opinion on.","tokens_in":65741,"confidence":"high","tokens_out":210,"duration_ms":9051,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"If no cohomological Lyapunov multiplier of an endomorphism iteration is an integer, then the return set of a dense orbit into a curve is finite.","keywords":["dynamical Mordell-Lang","height functions","endomorphisms","projective varieties","Lyapunov multipliers","return sets","arbitrary characteristic"],"falsifier":"An explicit endomorphism of a projective variety in positive characteristic where all Lyapunov multipliers are non-integers yet some dense orbit returns infinitely often to a curve.","tokens_in":2647,"feed_emoji":"","tokens_out":669,"duration_ms":38006,"temperature":0.7,"pith_summary":"The paper establishes two results on the dynamical Mordell-Lang problem using height arguments that work in arbitrary characteristic. First, for an endomorphism of a projective variety, the set of times when a dense orbit returns to a curve is finite whenever no cohomological Lyapunov multiplier is an integer. Second, in a product system where the curve factor has degree exceeding the dynamical degree of the variety factor, the return sets match the form of those in the variety system alone. These allow treatment of split maps on products of curves with distinct factor degrees, while counterexamples in other cases show return sets can be more complex than previously thought even for zero-entropy maps on tori.","feed_headline":"Non-integer multipliers force finite returns to curves","feed_subtitle":"Height arguments establish this for endomorphisms in any characteristic and reduce product cases to the base system when degrees differ.","key_machinery":"Height functions on projective varieties and the inequalities they satisfy when Lyapunov multipliers are non-integral, which bound the number of returns to subvarieties.","core_discovery":"For an endomorphism f of a projective variety X over a field of arbitrary characteristic, if any cohomological Lyapunov multiplier of any iterate is not an integer, then for a dense orbit under f and a curve C in X, the return set {n | f^n(x) in C} is finite. Additionally, for a product endomorphism f × g on X × C with deg(g) > δ1(f), the return sets of the product system have the same form as those of (X, f).","pith_inferences":["The results suggest that height methods can resolve more cases of dynamical Mordell-Lang in positive characteristic.","The examples of complicated return sets on tori indicate that zero entropy alone does not imply simple dynamics for returns.","One might test the results by computing multipliers and returns explicitly on low-dimensional varieties like elliptic curves or abelian varieties."],"forward_implications":["The dynamical Mordell-Lang conjecture holds in the case where multipliers avoid integers.","Return sets for product systems reduce to the base case under the degree condition.","Split self-maps on products of curves with pairwise distinct degrees have controlled return sets.","In cases without the multiplier condition, return sets can be infinite and complicated even with zero entropy."],"fun_headline_variants":["Non-integer Lyapunov multipliers force finite curve returns","Heights establish finite returns for non-integer multipliers","Degree gap in products preserves return structure of base","Dynamical Mordell-Lang advanced via heights in all characteristics"],"cache_read_input_tokens":64,"weakest_assumption_plain":"Height functions on projective varieties satisfy the usual functoriality and Northcott properties even when the base field has arbitrary characteristic.","fun_headline_variants_meta":{"raw":{"variants":["Non-integer Lyapunov multipliers force finite curve returns","Heights establish finite returns for non-integer multipliers","Degree gap in products preserves return structure of base","Dynamical Mordell-Lang advanced via heights in all characteristics"]},"model":"grok-4.3","cost_usd":0.006119,"raw_usage":{"total_tokens":2825,"prompt_tokens":701,"num_sources_used":0,"completion_tokens":59,"cost_in_usd_ticks":61190500,"prompt_tokens_details":{"text_tokens":701,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2065,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":701,"tokens_out":59,"duration_ms":22145,"temperature":1.0,"reasoning_tokens":2065,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-22T22:15:22.455631+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit endomorphism of a projective variety in positive characteristic where all Lyapunov multipliers are non-integers yet some dense orbit returns infinitely often to a curve.","supporting_citations":[],"review_version":1}