{"id":"e7d04c6c-f99e-486a-ab7f-98cff83868e1","arxiv_id":"1908.06231","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For an endomorphism of a p-adic variety that extends to an integral model, every integral periodic point has primitive period at most the special fiber size times a p-power factor times (|k|^{d'} - 1), with d' the maximum cotangent dimension in the special fiber.","lead":"This paper proves an explicit upper bound on how long a periodic point can take to repeat for endomorphisms of varieties over p-adic fields, using only data of the special fiber. The bound is uniform and works even when the variety or its reduction is singular, which earlier bounds did not cover uniformly.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2 is false as stated in dimension zero: d'=0 makes the right-hand side of (1.1) zero for any zero-dimensional variety.","rationale":"The Reader's conditional verdict is reasonable, and their flagged flatness issue is a genuine gap in the written proof. My stress-test adds a sharper, statement-level objection: the theorem as printed has a degenerate but legitimate counterexample in dimension zero. Because this is easily repaired by assuming dim X > 0 or by inserting max(1, |k|^{d'}-1), and because the intended positive-dimensional cases appear to be handled by a standard Fakhruddin-style argument once the flatness of A and the replacement of sigma by sigma^r are made explicit, I do not recommend rejecting the paper. I would keep the verdict conditional: the theorem statement should be amended, the construction of A should be specified, and the transition to the r-th power before applying h(m) subset m^2 should be stated. The reliance on Fakhruddin's Proposition 3 and Darafsheh's Corollary 2 is standard and not independently verified here, but it is not the main obstacle.","tokens_in":5893,"tokens_out":41068,"duration_ms":450285,"concrete_test":"Evaluate (1.1) for X = Spec K, X = Spec R, f = id, and P the unique R-point. Here d'=0, |\\bar X(k)|=1, and |k|^{d'}-1=0, so the claimed upper bound is 0 while the primitive period is 1. This single calculation settles that the theorem statement as written is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The stated theorem claims a bound for all projective varieties over K, but the factor |k|^{d'}-1 is zero when d'=0. Let X = Spec K, let X = Spec R be the obvious model, and let f be the identity. Then the unique R-point P has primitive period n=1, the special fiber has one k-point with cotangent dimension 0, so |\\bar X(k)|=1, |k|^{d'}=1, p^{v(p)-1} or 2^{v(2)} is finite, and the claimed right-hand side of (1.1) is 0. The asserted inequality 1 <= 0 fails. The same issue occurs for any nontrivial permutation of a finite zero-dimensional model. This is not a proof gap: the statement needs either an explicit positive-dimension hypothesis or a replacement of |k|^{d'}-1 by max(1, |k|^{d'}-1). Separately, the proof's soft spot identified by the Reader is real: the flatness and non-zero-divisor property of the orbit algebra A are asserted rather than proved. This is repairable by defining A as the scheme-theoretic image of the finite disjoint union of the t orbit sections; evaluation then embeds A into R^t, and since R is a DVR, A is torsion-free and finite flat. But the paper does not supply this argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an explicit uniform bound for the primitive period of an R-point that is periodic under an endomorphism f of a projective K-variety X, assuming f extends to a model X over the ring of integers R of a p-adic field K. The bound is expressed in terms of |\\bar X(k)|, the maximum cotangent-space dimension d' on the special fiber, and p-adic data. The proof follows Fakhruddin's three-step strategy: bound the period of the reduction, bound the order of the induced action on the cotangent space, and show the remaining factor is a p-power with controlled valuation. The paper also gives a weak Neron model corollary and a cubic polynomial example.","tokens_in":6158,"tokens_out":10220,"duration_ms":94951,"significance":"If the technical gaps are repaired, the result is a useful contribution: it gives a fully explicit, reduction-only bound for primitive periods of integral periodic points, covering singular models and cases of bad reduction that earlier good-reduction results exclude. The strategy is transparent and the reliance on external results (Fakhruddin, Darafsheh, Poonen) is clearly stated; there is no circularity. The examples illustrate the scope of the method. However, the current manuscript contains several load-bearing gaps and one false statement in the main theorem, so the result is not yet established as written.","major_comments":[{"comment":"The theorem is false as stated when d'=0. Take X=Spec K, X=Spec R, f=id. Then P has primitive period n=1, |\\bar X(k)|=1, d'=0, and the right-hand side of (1.1) is zero, so the asserted inequality 1≤0 fails. The same issue affects Theorem 1.3. The statement needs either an explicit hypothesis d'≥1 or the replacement of |k|^{d'}-1 by max(1, |k|^{d'}-1).","section":"§1, Theorem 1.2, display (1.1)"},{"comment":"The proof writes σ=id+h and asserts h(m)⊂m^2 for the original σ. This is false; in Example 3.1, with A=Z_3[x]/(x(x-3)), m=(x,3), and σ(x)=3-x, one obtains h(x)=3-2x, which is not in m^2=(3x,9). The argument only works after replacing σ by σ^r, where r is the order of the induced map on m/m^2, so that the linear part is trivial. This replacement is missing, and the subsequent claim that the order s of σ is a p-power is false for the original σ (in Example 3.1 the order is 2 when p=3).","section":"§3, Step 3"},{"comment":"The assertions that A is finite flat over R and that π is not a zero divisor in A are used in Step 2 (for example, to identify dim_k(π)/(mπ,π^2) with dim_k A/(m,π)=1) but are not proved. The orbit subscheme should be defined as the scheme-theoretic image of the finite union of the orbit sections; since R is a DVR, the resulting quotient A embeds into R^t, is torsion-free, and hence is finite flat over R. The paper should supply this argument, as the current text merely states the needed properties.","section":"§3, paragraph 'Recall that A is local of finite rank over R...'"},{"comment":"The contradiction argument that s is a p-power omits the choice of a nonzero h(a). One must choose a∈m with h(a)≠0 (which exists when s≠1) before applying the binomial expansion; then ν(s h(a))=ν(h(a)) while all remaining terms have strictly larger valuation, giving the contradiction. As written, if h(a)=0 the equation gives no contradiction, so the conclusion '0 ∉ m^{ν(h(a))+1}' is not justified.","section":"§3, Step 3, p-power order proof"}],"minor_comments":[{"comment":"The sentence 'σ is the identity precisely when \\barσ is' is not true in general; an R-algebra automorphism can be nontrivial while acting trivially on m/m^2, a p-order effect that Step 3 is meant to control. Please rephrase to describe the exact sequence whose kernel is a p-group.","section":"§3, Step 2"},{"comment":"Please spell out how Corollary 2 of [Dar05] yields r ≤ |k|^{d'}-1. Since the induced linear map fixes the line spanned by π, the relevant group is an affine group of dimension d', not simply GL_{d'}(k), so the bound is not immediate from the usual general-linear-group order bound.","section":"§3, Step 2"},{"comment":"There is a typographical spacing error in 'ove r' in the abstract.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The paper's strategy is sound and the flaws appear repairable, but the false d'=0 case and the missing replacement of σ by σ^r are substantial. The authors should also add the flatness argument for A and tighten the p-power proof before the manuscript is reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on Huang's paper. The main theorem is false as stated, but the fix is small and the method is worth engaging.\n\nWhat's new: the paper extends the Fakhruddin-Hutz-Bell-Ghioca-Tucker line to endomorphisms that merely extend to a model, allowing singular special fibers and weak Neron models. That's a genuine widening, and Example 2.3 with the cubic polynomial is a nice illustration.\n\nThe problem: Theorem 1.2 has a zero-dimensional counterexample. Take X = Spec K, X = Spec R, f = id. Then d' = 0, |Xbar(k)| = 1, and the claimed RHS is 0, but the primitive period is 1. Same issue for any permutation of a finite zero-dimensional model. The statement needs either a positive-dimension assumption or max(1, |k|^{d'} - 1) in the formula.\n\nThe proof has two soft spots the reader flagged, and they are real. Step 3 asserts h(m) subset m^2 for the original sigma; in Example 3.1, h(x) = -2x + 3 is not in m^2. You need to pass to sigma^r first, where r is the order on the cotangent space, then the small-displacement condition holds. The paper skips that. Also, the orbit algebra A is asserted to be finite flat local with pi not a zero divisor; that's not proved. The scheme-theoretic image of the orbit sections embeds A into R^t, making A torsion-free and finite flat, so the argument can be repaired, but it needs writing out.\n\nAre these fatal? The d' = 0 bug is a statement-level error, and the proof gaps are fillable. The overall strategy is standard and the theorem, with the right hypotheses, is plausible. The paper's citations look appropriate and no circularity. I didn't verify Fakhruddin's or Darafsheh's propositions independently, but they are external and the reliance is clearly stated.\n\nWho is this for? People working on Morton-Silverman and p-adic dynamics. It's an incremental step, not a breakthrough. If the author fixes the statement and tightens the proof, it's a solid small result.\n\nFor peer review: yes, send it out. A good referee can help the author repair the statement and the proof. Desk rejection would be wrong because the core idea is sound and the result, once fixed, is useful.","headline":"The main theorem is false as stated at d'=0, but the proof method is sound and the statement is repairable with a small hypothesis fix.","tokens_in":6695,"tokens_out":3905,"would_cite":false,"duration_ms":36761,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37P20","14G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Primitive periods of integral periodic points under endomorphisms of varieties are bounded explicitly by data of the special fiber","keywords":["arithmetic dynamics","primitive period","p-adic periodic points","uniform boundedness","weak Neron model","special fiber","cotangent space","endomorphisms of varieties"],"falsifier":"Compute the reduced orbit ring $A$ for an integral periodic point of a p-adic endomorphism on a singular model with bad reduction and check whether the uniformizer $\\pi$ is a zero divisor; a single example with $\\dim_k((\\pi)/(m\\pi,\\pi^2)) \\ne 1$ would invalidate the Step 2 bound on the cotangent order, and any example with primitive period larger than the stated bound would refute the theorem directly.","tokens_in":5687,"feed_emoji":"🔁","tokens_out":12402,"duration_ms":108636,"temperature":0.7,"pith_summary":"The paper proves an explicit uniform upper bound for the primitive period of a periodic point that is defined over the ring of integers of a p-adic field and lies on a variety with an endomorphism extending to an integral model. The bound depends only on reduction data: the number of points in the special fiber, the residue-field size, and the maximum cotangent-space dimension at special-fiber points. This matters because a uniform bound on rational preperiodic points is a known open problem in arithmetic dynamics, and bounding periods is a direct route toward it. The result works for morphisms without good reduction and for singular models, so it applies more broadly than earlier period bounds.","feed_headline":"Special fiber fixes p-adic period bound","feed_subtitle":"For endomorphisms of varieties, primitive periods are read off from residue field and cotangent data of the fiber.","key_machinery":"The argument is carried by the reduced orbit subscheme $\\operatorname{Spec} A$ attached to the orbit of $P$ under a suitable iterate $g$. The ring $A$ is local of finite rank over $R$, with maximal ideal $m$, and $g$ induces an $R$-endomorphism $\\sigma$ of $A$. The action on the cotangent space $m/m^2$ (the maximal ideal modulo its square) controls the non-$p$ part of the period, while the decomposition $\\sigma = \\mathrm{id} + h$ with $h(m)\\subseteq m^2$ controls the $p$-power part through a binomial expansion. A filtration of $m$ gives $\\dim_k(m/m^2)\\le d'+1$, with the span of $\\pi$ invariant under $\\sigma$, so the order of the linear action is at most $|k|^{d'}-1$; the binomial argument shows the remaining order is a $p$-power bounded by the valuation of $p$.","core_discovery":"The central claim is Theorem 1.2. Let $K$ be a finite extension of $\\mathbb{Q}_p$, $R$ its ring of integers, $k$ its residue field, and $\\pi$ a uniformizer. If $P\\in \\mathcal{X}(R)$ is periodic under an endomorphism $f$ that extends to an integral model $\\mathcal{X}$, and $d'$ is the maximum dimension of the cotangent spaces at points of the special fiber $\\bar{\\mathcal{X}}(k)$, then the primitive period $n$ of $P$ satisfies $n \\le |\\bar{\\mathcal{X}}(k)|\\, p^{v(p)-1}(|k|^{d'}-1)$ for $p>2$ and $n \\le |\\bar{\\mathcal{X}}(k)|\\, 2^{v(2)}(|k|^{d'}-1)$ for $p=2$. The proof splits the period as $n = n_0 r t$: $n_0$ is the period of the reduction of $P$ in the special fiber, $r$ is the order of the induced map on the cotangent space, and $t$ is a $p$-power. It bounds $n_0$ by $|\\bar{\\mathcal{X}}(k)|$, $r$ by $|k|^{d'}-1$, and $t$ by $v(p)-1$ (or $v(2)$ when $p=2$). This extends a known method to singular models and maps without good reduction.","pith_inferences":["A direct computational test of the proof's core assumption is to form the reduced orbit ring $A$ for a periodic point on a singular model with bad reduction and check whether $\\pi$ is a zero divisor; if any example gives $\\dim_k((\\pi)/(m\\pi,\\pi^2))>1$, the Step 2 bound on $r$ would be unsupported.","The same three-factor decomposition might extend from periodic points to periodic subvarieties, replacing the cotangent space of a point by infinitesimal data along the orbit; the paper does not pursue this.","The bound addresses periods, not the full uniform-boundedness conjecture: bounding the primitive period of a single point does not by itself bound the number of preperiodic points, since the special fiber may support many distinct orbits. Combining this period control with a count of periodic points of bounded period on the reduction would be the natural next step."],"forward_implications":["For $K=\\mathbb{Q}_p$ the bound becomes $n \\le |\\bar{\\mathcal{X}}(\\mathbb{F}_p)|(p^{d'}-1)$ for odd $p$ and $n \\le |\\bar{\\mathcal{X}}(\\mathbb{F}_p)|\\,2(2^{d'}-1)$ for $p=2$.","Any $K$-rational periodic point on a variety admitting a weak Neron model satisfies the same bound, because such a model is bijective on $K$-points.","For cubic polynomial maps with no $K$-rational repelling fixed point, the worked example yields $n \\le (|k|+1)p^{v(p)-1}(|k|-1)$.","Because the model is not required to be nonsingular and the morphism only needs to extend to the model, the bound covers singular and non-projective examples beyond earlier constructions."],"supporting_citations":[{"why":"Supplies the decomposition of a period into reduction order, cotangent order, and p-power that the proof adapts.","marker":"[Fak01]"},{"why":"Provides the binomial-expansion argument used to prove the remaining factor is a p-power.","marker":"[Poo14]"},{"why":"Gives the bound on the order of the induced linear map on the cotangent space, $r \\le |k|^{d'}-1$.","marker":"[Dar05]"},{"why":"Underlies the example of cubic polynomial maps admitting weak Neron models with controlled special fiber.","marker":"[BH12]"}],"fun_headline_variants":["Special fiber sets explicit bound on primitive periods","Cotangent data plus special fiber limit periods","p-adic period bound from residue field reduction","Explicit period bounds for variety endomorphisms","Uniform period bound: reduction and cotangent spaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof relies on the orbit ring $A$ being free over $R$, so that the uniformizer $\\pi$ is not a zero divisor; the paper states this property but does not prove it.","fun_headline_variants_meta":{"raw":{"variants":["Special fiber sets explicit bound on primitive periods","Cotangent data plus special fiber limit periods","p-adic period bound from residue field reduction","Explicit period bounds for variety endomorphisms","Uniform period bound: reduction and cotangent spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000432,"raw_usage":{"total_tokens":2206,"prompt_tokens":947,"completion_tokens":1259,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":563,"completion_tokens_details":{"reasoning_tokens":1188}},"tokens_in":563,"tokens_out":1259,"duration_ms":12872,"temperature":1.0,"reasoning_tokens":1188,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:55:34.284156+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the reduced orbit ring $A$ for an integral periodic point of a p-adic endomorphism on a singular model with bad reduction and check whether the uniformizer $\\pi$ is a zero divisor; a single example with $\\dim_k((\\pi)/(m\\pi,\\pi^2)) \\ne 1$ would invalidate the Step 2 bound on the cotangent order, and any example with primitive period larger than the stated bound would refute the theorem directly.","supporting_citations":[],"review_version":1}