{"id":"78f03aec-9bea-4906-86f0-663cd0a896b9","arxiv_id":"2608.08262","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"For monocritical arithmetically T-effective toric metrics on semiabelian compactifications, generic small sequences equidistribute and minimal-height subvarieties are special.","lead":"New theorems in arithmetic geometry show that points of minimal height on semiabelian varieties spread out according to explicit measures for a class of non-canonical toric metrics. The same results give a Bogomolov rigidity statement: any subvariety carrying a Zariski-dense set of minimal-height points must be a translate of an algebraic subgroup.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 35's dimension-count scaling O(n) is asserted without derivation; a model computation on (P^1)^2 suggests the quadratic core grows like n^2, which would invalidate Lemmas 36, 57-59 and the §6.4 parameter choice.","rationale":"The reader's weakest_assumption identifies exactly the same estimate as the load-bearing point: Lemma 35's normalized scaling of the quadratic error core. My independent reading agrees that every compressed estimate (Lemmas 36, 57, 58, 59) and the final parameter choice in §6.4 depend on this scaling, and that the lemma is argued by a dimension count rather than a written derivation. The concern is not a disagreement with the quasi-canonical or canonical theorems themselves, which are plausible and well-sourced; it is a precise gap in the technical core of the proof. The proposed model computation is deliberately chosen to be the minimal nontrivial toric case where the dimension count can be tested: t=2, g=0, X=(P^1)^2, with a toric test perturbation supported on two coordinate walls. In that case the pulled-back wall weights scale linearly in n, so the mixed product of two walls scales like n^2 unless some cancellation from the reference bundle occurs. If the computation confirms n^2 growth, Lemma 35 is false and the error budget in Theorem 1.3 collapses; if it confirms O(n), the paper must supply the missing cancellation argument. Since the reader already returned a CONDITIONAL verdict, this stress-test does not move the verdict; it sharpens the condition that must be met before the proof can be accepted.","tokens_in":61509,"tokens_out":20631,"duration_ms":213225,"concrete_test":"Implement Lemma 35 in the explicit model: K=Q, G=G_m^2, X_Σ=(P^1)^2 with fan four rays, D=O(1,1) with canonical toric metric, N trivial (g=0, d=2). Let f be the toric test metric on the trivial bundle given, on the standard affine charts, by local potentials max{0,u_1} and max{0,u_2} regularized smoothly with compact support in the torus; put F_n=φ_n^*O(f) with φ_n(z_1,z_2)=(z_1^n,z_2^n). Compute the normalized quadratic core Q_n = (F_n^2·L_n^{d-1})/L_n^d using the toric Monge-Ampère/arithmetic intersection formalism (or a direct model computation at one non-archimedean place with a vertical divisor whose tropical support is the two walls). If |Q_n| ~ n^2 for large n, Lemma 35 is false. If |Q_n| ≤ C n, identify the cancellation that the dimension count must be hiding and write it out; the same check with f replaced by f∘[m] tests the O(nm) claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main theorem rests on the error estimates of §5.3-§6.3. The single most load-bearing step is Lemma 35, which claims that after normalization by L_n^d, the quadratic intersection of two algebraically trivial perturbations F_n = φ_n^*O(f) retains at most one factor n (and at most one further factor m after compression). The proof is not a derivation but a dimension count: it says two n-slope factors cannot survive because more than t toric factors vanish. This does not follow. The Chow-ring vanishing bounds the number of algebraic divisor classes, not the metric coefficients of F_n; the n-dependence of F_n enters through the slopes of the pulled-back local potentials, which are not counted by toric dimension. In the model case G=G_m^2, X=(P^1)^2, D=O(1,1), L_n^d=2, let f be a toric test metric whose local potentials have walls along the two coordinate directions. Then f_n(u,v)=f(nu,nv) has wall weights multiplied by n in each direction, so the r=2 term F_n^2·L_n^{d-1}/L_n^d is of order n^2 times the mixed volume of the dual cells, not O(n). Lemma 36 then carries the wrong α_{*,n}; Lemma 57's O(nm) becomes O((nm)^2); Lemma 59 and the parameter choice in §6.4 (m=n^{1/2}, λ=n^{-7/4}) no longer make the error vanish. Since these lemmas are the only control on the quadratic error in the Minkowski-variation argument, Theorem 1.3 is not established unless Lemma 35 is proved or replaced.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves generic equidistribution for adelic line bundles of the form L = G(D) ⊗ π*N on toric compactifications of semiabelian varieties over number fields, where D is a big T-effective toric divisor equipped with a semipositive, monocritical, arithmetically T-effective toric metric. It further derives a Bogomolov rigidity theorem for the same metric class. The proof splits into canonical, quasi-canonical (via explicit compression paths), and monocritical stages, then transports Kühne's local-trivialization and difference-morphism argument to general toric compactifications for the Bogomolov application.","tokens_in":61900,"tokens_out":7227,"duration_ms":65314,"significance":"If correct, this extends Kühne's canonical semiabelian equidistribution theorem to a class of non-canonical toric metrics, and gives the corresponding Bogomolov rigidity. The manuscript is unusually careful with source attribution, explicitly flags the generic/strict distinction, and provides a detailed local-trivialization construction at all places. The main theorems are crisp and falsifiable. The paper also ships, at the level of a written proof, a systematic source ledger for the external inputs it uses.","major_comments":[{"comment":"The proof of Lemma 35 asserts that after normalization by L_n^d the quadratic core of the error terms retains at most one explicit factor n (and at most one further factor m after compression), but this is justified by a dimension count rather than a written derivation. The dimension count bounds the number of algebraic divisor classes in a top intersection; it does not control the size of the metric slopes of the pulled-back test metric F_n = φ_n^*O(f), which enter through the local potentials of f. In the model case G = G_m^2, X = (P^1)^2, D = O(1,1), a toric test metric f with walls along the two coordinate directions gives f_n(u,v) = f(nu,nv), so the r = 2 term F_n^2·L_n^{d-1}/L_n^d is of order n^2 times the mixed volume of the dual cells, not O(n). If this model computation is representative, then Lemmas 36, 57, 58, and 59 inherit the wrong scaling, and the parameter choice m(n) = n^{1/2}, λ = n^{-7/4} in §6.4 no longer makes the Minkowski error vanish. Since these estimates are the only control on the quadratic error in the variation argument, Theorems 1.3 and 1.4 are not established unless Lemma 35 is proved or replaced by a correct scaling estimate.","section":"§5.3, Lemma 35 (used in Lemmas 36, 57–59, §6.4)"}],"minor_comments":[{"comment":"There are several spacing typos in the title and headers, e.g. 'V arieties' and occasional 'Kuhne' instead of 'Kühne'.","section":"Title page"},{"comment":"Auxiliary statements are numbered inconsistently: some are 'Proposition 12', 'Lemma 13', while others are 'Theorem 4.1', 'Proposition 4.17'. This makes cross-referencing needlessly difficult.","section":"Throughout"},{"comment":"The term 'quadratic core' is never defined precisely. Since the lemma is load-bearing, the statement should formalize the exact intersection products and normalizations involved.","section":"§5.3, Lemma 35"},{"comment":"The phrase 'extends verbatim' is too strong given that the proof supplies nontrivial replacements for Kühne's Lemmas 18–20; suggesting a black-box transfer obscures the actual verification performed.","section":"§9.5, Proposition 96"}],"recommendation":"major_revision","confidential_remarks":"This is a serious paper with a clear extension programme. However, the central error-scaling estimate (Lemma 35) is not proved, and the dimension-count argument given does not address the metric-slope dependence. The model computation in the stress test, which I checked on (P^1)^2, indicates that the claimed O(n) scaling may be false as stated. I recommend a major revision in which the author either proves a correct O(n) scaling under the monocritical / arithmetically T-effective hypotheses, or adjusts the parameter choices and subsequent estimates to accommodate a larger error. The overall architecture is coherent conditional on this estimate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this paper is not padding: the compression-path mechanism and the monocritical/arithmetically T-effective metric class are genuinely new relative to Kühne, BGPS, and Yuan–Zhang, and the paper is unusually careful about what it does and does not prove. Second, the main theorem rests on Lemma 35, and the proof of that lemma is not really a proof. It is a dimension count that bounds the number of toric divisor classes, not the metric slopes that carry the n-dependence. The stress-test model on (P^1)^2 is concrete and worrying: f_n(u,v)=f(nu,nv) multiplies wall weights by n in each coordinate, so the quadratic core F_n^2 · L^{d-1}/L^d should grow like n^2, not n. If that computation is right, Lemmas 36, 57–59 and the §6.4 parameter choice collapse, and Theorem 1.3 is not established.\n\nWhat the paper does well deserves credit. The source ledgers are honest and specific, and Remark 61 explicitly flags the generic/strict distinction and refuses to import strict equidistribution without circularity. The paper also states its scope clearly: it is not a new proof of the canonical theorem and not a theorem for all semipositive metrics. Proposition 4.27's valuation/height package and the arithmetic T-effectivity definition look coherent, and the reduction from monocritical to quasi-canonical in Section 8 is a sensible use of the BGPS KR machinery.\n\nThe other soft spot is Proposition 96, which asserts that Kühne's local-trivialization argument extends verbatim to arbitrary toric compactifications. Kühne worked with (P^1)^t; the extension to a general fan requires the corner-current package of §9.4–9.5. There are several \"this is exactly Kühne's computation\" transitions without a written check, so even granting Lemma 35, I would want the verbatim claim verified line by line. The final restriction package is truncated in the input, so I could not assess the last step of the Bogomolov argument.\n\nWho is this for? Arithmetic geometers working on equidistribution, heights, and Bogomolov. A serious referee should definitely see it, because the overall architecture is important and the gap is a specific, checkable estimate. As it stands, I would not cite Theorem 1.3 in my own work, and the paper needs heavy revision before I would trust it. But this is a conditional reject, not a dismiss: the framework is sound enough that fixing or replacing Lemma 35 could make the main theorems true.\n\nRecommendation: send to peer review, with a referee who is explicitly asked to verify Lemma 35 and Proposition 96.","headline":"A careful, honest extension of Kühne's semiabelian equidistribution to a controlled non-canonical toric metric class, but the load-bearing error-scaling Lemma 35 is asserted rather than proved, and the stress-test computation suggests it may be wrong.","tokens_in":62415,"tokens_out":1773,"would_cite":false,"duration_ms":20015,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G50","14G40","14M25","14K15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that generic small sequences on a compactified semiabelian variety equidistribute at every place when the toric metric is big, T-effective, monocritical, and arithmetically T-effective, and that the…","keywords":["equidistribution","semiabelian varieties","toric metrized divisors","monocritical metrics","arithmetic T-effectivity","Bogomolov theorem","adelic line bundles","heights"],"falsifier":"Write out the first-variation error expansion explicitly for the split example $G = \\mathbb{G}_m^2 \\times E$ with $\\mathbb{P}^1 \\times \\mathbb{P}^1$ compactification and a non-canonical monocritical metric, and check whether the normalized quadratic error term is bounded by $O(nm)$ as Lemma 57 claims; an explicit computation showing $O(n^2m)$ or $O(nm^2)$ would break the equidistribution proof.","tokens_in":61295,"feed_emoji":"📐","tokens_out":10354,"duration_ms":91388,"temperature":0.7,"pith_summary":"The paper aims to prove that small algebraic points on a semiabelian variety with a toric compactification spread out uniformly at every place, even when the metric on the toric part is not the canonical one. The main theorem says this holds whenever the toric metrized divisor is big, T-effective, monocritical, and arithmetically T-effective; in the quasi-canonical case the limit measure is the explicitly normalized mixed curvature measure of the toric bundle and the pulled-back abelian line bundle. The paper also proves the corresponding Bogomolov statement: a subvariety whose essential minimum equals that of the whole variety must be special, meaning a translate of a connected algebraic subgroup by a torsion point, with the converse holding under a special-point hypothesis. The payoff is that earlier canonical semiabelian equidistribution is extended to a controlled class of non-canonical toric metrics, and the proof works by compressing the metric toward a quasi-canonical model along explicit paths.","feed_headline":"Generic small points equidistribute for non-canonical toric metrics","feed_subtitle":"Proof via explicit compression paths plus a quasi-canonical replacement; Bogomolov follows for this metric class.","key_machinery":"The load-bearing mechanism is the explicit compression path $\\overline{D}_m = m^{-1}[m]^*\\overline{D}$ for a quasi-canonical metric, whose local support functions take the normal form $\\Psi_{\\overline{D}_m,v}(u) = \\Psi_{\\overline{D}}(u - u_v/m) - \\gamma_v/m$, together with the auxiliary $n$-division tower $G_n$ on which height, intersection, and measure estimates are made. The crucial scaling lemma (Lemma 35) asserts that after normalization by $L_n^d$, the quadratic core of the error terms retains at most one explicit factor $n$, and after horizontal compression by $[m]$ at most one additional factor $m$, so the compressed quadratic errors are $O(n)$ and $O(nm)$. This dimension-count bookkeeping feeds every later compressed estimate. The monocritical case is then reduced to the quasi-canonical case by replacing the metric with $\\Psi_{\\overline{D}',v}(z) = \\Psi_{\\overline{D}}(z - u_v)$, where $(u_v)_v$ is the critical tropical vector; this replacement preserves smallness of sequences and forces the limiting valuation measure to be the Dirac mass at the critical vector.","core_discovery":"The central claim is Theorem 1.3: for a semiabelian variety $G$ over a number field $K$ with split toric part of dimension $t$ and abelian quotient $A$ of dimension $g$, let $\\mathbb{G}$ be the compactification attached to a proper toric variety $X_\\Sigma$, and let $\\overline{D}$ be a toric metrized $\\mathbb{R}$-divisor on $X_\\Sigma$. If $\\overline{D}$ is big, $T$-effective, monocritical, and arithmetically $T$-effective, then the adelic line bundle $\\overline{L} = \\mathcal{G}(\\overline{D}) \\otimes \\pi^*\\overline{N}$ has the $v$-adic equidistribution property at every place $v$: every generic $\\overline{L}$-small sequence of $K$-points has $v$-adic Galois orbit measures converging weakly, with limit equal, in the quasi-canonical case, to $\\hat{c}_1(\\mathcal{G}(\\overline{D})_v)^t \\wedge \\hat{c}_1(\\pi^*\\overline{N}_v)^g$ divided by $\\mathcal{G}(\\overline{D})^t(\\pi^*N)^g$. Theorem 1.4 draws the Bogomolov consequence: under the same hypotheses, an irreducible subvariety $X$ with $\\mu^{\\mathrm{ess}}_{\\overline{L}}(X) = \\mu^{\\mathrm{ess}}_{\\overline{L}}(\\mathbb{G})$ is special, and the converse holds when $\\mathbb{G}(K)$ has $\\overline{L}$-special points. The proof isolates the asymptotic estimates used in the canonical semiabelian case and reinterprets them as estimates along explicit compression paths, giving a comparison mechanism between canonical, quasi-canonical, and more general toric metrics.","pith_inferences":["The compression method suggests a general recipe for other arithmetic equidistribution problems: if a reference metric is approached by a one-parameter family of metrics with controlled error growth after normalization, equidistribution transfers; testing this on semipositive toric metrics that are not monocritical would show whether the unique-critical-vector condition can be relaxed to a finite ","The explicit parameter choice $m(n) = \\lfloor n^a\\rfloor$, $\\lambda = n^{-(3+a)/2}$, $0 < a < 1$, predicts convergence rates $O(n^{-(1-a)/2})$; a numerical check on a split example such as $\\mathbb{G}_m^2 \\times E$ could test whether this rate is sharp or merely an artifact of the proof.","The paper deliberately leaves arbitrary semipositive toric metrics untouched; if the monocritical condition fails, the limiting measure might be a mixture of Dirac masses at several critical vectors, and the Bogomolov implication would need a separate argument."],"forward_implications":["Every generic $\\overline{L}$-small sequence in $\\mathbb{G}(\\overline{K})$ has $v$-adic Galois orbit measures converging weakly at every place $v$ under the stated conditions on $\\overline{D}$.","When the toric metric is additionally quasi-canonical, the limiting measure is the normalized mixed Chern measure $\\hat{c}_1(\\mathcal{G}(\\overline{D})_v)^t \\wedge \\hat{c}_1(\\pi^*\\overline{N}_v)^g / (\\mathcal{G}(\\overline{D})^t(\\pi^*N)^g)$.","Equality of essential minima, $\\mu^{\\mathrm{ess}}_{\\overline{L}}(X) = \\mu^{\\mathrm{ess}}_{\\overline{L}}(\\mathbb{G})$, forces $X$ to be special, and the converse holds when $\\mathbb{G}(K)$ has $\\overline{L}$-special points.","Strict quasi-canonical equidistribution holds on the minimal translates that arise in the Bogomolov restriction argument, supplying the non-circular upgrade used in the proof."],"supporting_citations":[{"why":"Supplies the canonical semiabelian equidistribution theorem and the local-trivialization estimates this paper extends.","marker":"[1]"},{"why":"Provides the adelic line bundle framework and the generic equidistribution theorem whose quasi-canonical condition is relaxed here.","marker":"[18]"},{"why":"Isolates monocritical toric metrics and their critical tropical data, used in the reduction to the quasi-canonical model.","marker":"[15]"},{"why":"Gives the toric successive-minima formula $\\mu^{\\mathrm{ess}}_{\\overline{D}}(X_\\Sigma) = \\mu^{\\mathrm{abs}}_{\\overline{D}}(T)$ used throughout the height comparisons.","marker":"[17]"},{"why":"Classifies toric metrized divisors by roof functions and effectivity, supplying the definitions of $T$-effectivity and semipositivity.","marker":"[16]"},{"why":"Yuan's arithmetic bigness theorem is the ultimate source behind the quadratic arithmetic volume comparison in the Kühne–Yuan–Ikoma package.","marker":"[31]"},{"why":"Provides the abelian Bogomolov theorem used in the base reduction for the Bogomolov application.","marker":"[29]"},{"why":"Supplies tropical compactifications and the corner-current/Monge–Ampère description used for arbitrary fan charts in the Haar-slice formula.","marker":"[12]"},{"why":"Supplies Raynaud uniformization and theta functions for the non-archimedean local trivialization and the theta–Néron comparison.","marker":"[14]"}],"fun_headline_variants":["Compression paths prove equidistribution for non-canonical toric metrics","Beyond quasi-canonical: equidistribution via compression paths","Monocritical T-effective toric metrics yield generic equidistribution","Semiabelian equidistribution extends via explicit compression paths","Bogomolov via quasi-canonical replacement for toric metric class"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on a dimension-count estimate saying that the quadratic error terms, after rescaling by the top self-intersection, grow at most linearly in the division parameter $n$ and at most linearly in the compression parameter $m$; if the true growth were faster, the errors would not vanish.","fun_headline_variants_meta":{"raw":{"variants":["Compression paths prove equidistribution for non-canonical toric metrics","Beyond quasi-canonical: equidistribution via compression paths","Monocritical T-effective toric metrics yield generic equidistribution","Semiabelian equidistribution extends via explicit compression paths","Bogomolov via quasi-canonical replacement for toric metric class"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000214,"raw_usage":{"total_tokens":1516,"prompt_tokens":1127,"completion_tokens":389,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":743,"completion_tokens_details":{"reasoning_tokens":313}},"tokens_in":743,"tokens_out":389,"duration_ms":7121,"temperature":1.0,"reasoning_tokens":313,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:13:18.367979+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Write out the first-variation error expansion explicitly for the split example $G = \\mathbb{G}_m^2 \\times E$ with $\\mathbb{P}^1 \\times \\mathbb{P}^1$ compactification and a non-canonical monocritical metric, and check whether the normalized quadratic error term is bounded by $O(nm)$ as Lemma 57 claims; an explicit computation showing $O(n^2m)$ or $O(nm^2)$ would break the equidistribution proof.","supporting_citations":[{"cited_title":"Points of small height on semiabelian varieties.J","cited_arxiv_id":null,"evidence_quote":"Supplies the canonical semiabelian equidistribution theorem and the local-trivialization estimates this paper extends."},{"cited_title":"The distribution of Galois orbits of points of small height in toric varieties.Amer","cited_arxiv_id":null,"evidence_quote":"Isolates monocritical toric metrics and their critical tropical data, used in the reduction to the quasi-canonical model."},{"cited_title":"Successive minima of toric height func- tions.Ann","cited_arxiv_id":null,"evidence_quote":"Gives the toric successive-minima formula $\\mu^{\\mathrm{ess}}_{\\overline{D}}(X_\\Sigma) = \\mu^{\\mathrm{abs}}_{\\overline{D}}(T)$ used throughout the height comparisons."},{"cited_title":"Arithmetic positivity on toric varieties.J","cited_arxiv_id":null,"evidence_quote":"Classifies toric metrized divisors by roof functions and effectivity, supplying the definitions of $T$-effectivity and semipositivity."},{"cited_title":"Big line bundles over arithmetic varieties.Invent","cited_arxiv_id":null,"evidence_quote":"Yuan's arithmetic bigness theorem is the ultimate source behind the quadratic arithmetic volume comparison in the Kühne–Yuan–Ikoma package."},{"cited_title":"Equidistribution of small points on abelian varieties.Ann","cited_arxiv_id":null,"evidence_quote":"Provides the abelian Bogomolov theorem used in the base reduction for the Bogomolov application."},{"cited_title":"Metrics, measures and heights","cited_arxiv_id":null,"evidence_quote":"Supplies tropical compactifications and the corner-current/Monge–Ampère description used for arbitrary fan charts in the Haar-slice formula."},{"cited_title":"Non-Archimedean and tropical theta functions.Math","cited_arxiv_id":null,"evidence_quote":"Supplies Raynaud uniformization and theta functions for the non-archimedean local trivialization and the theta–Néron comparison."}],"review_version":1}