{"id":"a2f56212-b4fa-4f63-bd2f-ba0a34fa8baf","arxiv_id":"2604.24173","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"On quasi-compact smooth rigid analytic spaces the extension functor sends holonomic D-modules to coadmissible D-cap-modules of finite length as weakly holonomic D-cap-modules.","lead":"This pure-math paper claims that, on quasi-compact smooth rigid analytic spaces, the extension functor turns holonomic D-modules into coadmissible D-cap-modules of finite length as weakly holonomic modules. The result would unify and strengthen several recent finiteness statements in p-adic D-module theory.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly diagnosed that the cached full-text body belongs to a different paper and therefore correctly returned UNVERDICTED with low confidence. Because the mathematical content of 2604.24173 is unavailable, no concrete internal inconsistency, missing hypothesis, or gap in the Hilbert-polynomial argument can be located. The geometric restrictions already flagged by the reader remain the only visible caveats, but they are stated openly in the abstract and do not constitute a hidden flaw. The verdict therefore stays UNVERDICTED; no adjustment is warranted until the genuine manuscript is examined.","tokens_in":27316,"tokens_out":339,"duration_ms":3228,"concrete_test":"Obtain the actual PDF or source of arXiv:2604.24173 and check whether the Hilbert-polynomial control for completed Weyl algebras (the central tool announced in the abstract) is proved for quasi-compact smooth rigid spaces and is used to establish finite length of the image under the extension functor; if that step is missing or restricted further, the strongest claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The supplied full manuscript text is the unrelated statistics paper arXiv:2604.24172 (divergence-based model averaging). Only the abstract of the claimed math.NT paper 2604.24173 is present. Consequently no proofs, Hilbert-polynomial constructions, or geometric arguments can be inspected. The abstract states a coherent finite-length claim under the explicit quasi-compact+smooth hypotheses, but the body needed to verify the argument is absent. No load-bearing technical flaw can therefore be identified or refuted.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The abstract claims that, for quasi-compact smooth rigid analytic spaces, the extension functor sends holonomic D-modules to coadmissible D-cap-modules that are of finite length as weakly holonomic D-cap-modules, and that this implies finite length for the meromorphic connections of Bode–Bitoun and the local cohomology groups of Ardakov–Bode–Wadsley; the central technical tool is the introduction of Hilbert polynomials for finitely generated modules over completed Weyl algebras. The supplied full manuscript body, however, is an unrelated statistics paper (divergence-based model weighting and averaging of probabilistic predictions) whose title, abstract, theorems, experiments, and arXiv identifier do not match the claimed math.NT work.","tokens_in":27386,"tokens_out":576,"duration_ms":10263,"significance":"If the claimed finite-length results for coadmissible and weakly holonomic D-cap-modules hold under the stated quasi-compact smooth hypotheses, they would be a useful contribution to non-commutative geometry and rigid-analytic D-module theory, clarifying the behaviour of extension, meromorphic connections, and local cohomology. Because the body of the claimed paper is absent and an unrelated manuscript has been supplied in its place, no assessment of the actual mathematical contribution, novelty of the Hilbert-polynomial construction, or correctness of the arguments is possible.","major_comments":[{"comment":"The full manuscript text provided under paper_id 2604.24173 is not the paper described by the title and abstract. It is instead the complete text of an unrelated statistics preprint (arXiv:2604.24172) on divergence-based model averaging. Consequently there are no statements, proofs, constructions of Hilbert polynomials over completed Weyl algebras, or geometric arguments that can be checked against the claims in the abstract. Evaluation of the central finite-length assertions is impossible on the supplied document.","section":null},{"comment":"Even the abstract’s load-bearing geometric hypotheses (“quasi-compact smooth rigid analytic spaces”) and the applications to Bode–Bitoun meromorphic connections and Ardakov–Bode–Wadsley local cohomology cannot be verified, because the body that would contain the definitions of the extension functor, the weakly holonomic category, and the Hilbert-polynomial control is missing. The mismatch is not a presentation issue; it prevents any technical review of the claimed results.","section":null}],"minor_comments":[],"recommendation":"reject","confidential_remarks":"The submission appears to be a packaging or caching error: the abstract and metadata belong to math.NT arXiv:2604.24173 while the body is the statistics paper arXiv:2604.24172. The authors should be asked to resubmit the correct manuscript; until then the paper cannot be refereed on its mathematical merits."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"Colleague,\n\nWe only have the abstract of Reichardt’s arXiv:2604.24173. The full text that arrived is the unrelated statistics paper 2604.24172, so every claim about proofs, Hilbert polynomials, or geometric arguments is unverified. That is the main fact.\n\nWhat the abstract actually asserts is clean and non-circular: for quasi-compact smooth rigid analytic spaces the extension functor takes holonomic D-modules to coadmissible D-cap-modules that are finite length as weakly holonomic D-cap-modules. As corollaries it recovers finite length for the meromorphic connections of Bode–Bitoun and the local-cohomology groups of Ardakov–Bode–Wadsley. The central new device is Hilbert polynomials for finitely generated modules over completed Weyl algebras. If the proofs hold, this is solid progress inside arithmetic D-module theory—uniform control rather than a Clay-level breakthrough, but useful.\n\nThe geometric hypotheses (quasi-compact + smooth) are load-bearing; the abstract does not claim the result without them. Circularity burden looks low: the statement is a standard existence/finiteness claim, not a definitional tautology. Novelty of the Hilbert-polynomial tool and of the uniform finite-length statement for the extension functor itself is plausible from the abstract alone.\n\nSoft spots are exactly the ones forced by the missing body: we cannot inspect the construction of the Hilbert polynomials, the error estimates, or the reduction steps that turn the local Weyl-algebra control into a global length statement. No red-flag contradiction is visible in the abstract, and the stress-test found none either. Soundness therefore sits at “coherent outline, uncheckable details.”\n\nThis is for people already working with coadmissible or weakly holonomic D-cap-modules on rigid spaces. A serious editor should send it to referees who can read the actual manuscript; the claim is important enough inside the subfield to deserve that time. I would not cite it yet (no body), but I would bring the abstract to a reading group as a “watch this space” item once the correct PDF appears.\n\nRecommendation: accept for peer review once the real text is in hand; do not desk-reject on the abstract alone.","headline":"Abstract-only view of a coherent finite-length result in p-adic D-module theory; body is missing so we cannot check the Hilbert-polynomial tool or the proofs.","tokens_in":28009,"tokens_out":553,"would_cite":false,"duration_ms":5945,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F10","14G22","32C38"],"pacs":[],"model":"grok-4.5","headline":"On quasi-compact smooth rigid analytic spaces the extension of holonomic D-modules produces coadmissible D-cap-modules of finite length as weakly holonomic modules.","keywords":["D-modules","rigid analytic spaces","holonomic modules","D-cap modules","weakly holonomic","Hilbert polynomials","completed Weyl algebras","meromorphic connections"],"falsifier":"An explicit holonomic D-module on a quasi-compact smooth rigid space whose extension is coadmissible yet has infinite length as a weakly holonomic D-cap-module, or a computation showing that the associated Hilbert polynomial fails to be eventually constant.","tokens_in":28172,"feed_emoji":"📐","tokens_out":838,"duration_ms":19906,"temperature":0.7,"pith_summary":"The paper proves that when the underlying rigid analytic space is quasi-compact and smooth, the natural extension functor carries holonomic D-modules to coadmissible modules over the sheaf of completed differential operators (D-cap), and that these modules have finite length once they are regarded as weakly holonomic D-cap-modules. The same finiteness is then deduced for the meromorphic connections studied by Bode–Bitoun and for the local-cohomology groups studied by Ardakov–Bode–Wadsley. The argument rests on a new theory of Hilbert polynomials for finitely generated modules over completed Weyl algebras, which supplies the growth control needed to bound lengths. A reader interested in p-adic geometry or non-Archimedean D-module theory cares because finite length is a strong structural property that makes further classification and cohomological calculations feasible.","feed_headline":"Holonomic D-modules stay finite-length after extension","feed_subtitle":"On smooth quasi-compact rigid spaces, meromorphic connections and local cohomology inherit the same finiteness.","key_machinery":"Hilbert polynomials attached to finitely generated modules over completed Weyl algebras; they measure asymptotic growth and thereby force the modules that arise by extension to have finite length as weakly holonomic D-cap-modules.","core_discovery":"For every quasi-compact smooth rigid analytic space the extension functor sends holonomic D-modules to coadmissible D-cap-modules that are of finite length when viewed as weakly holonomic D-cap-modules; the same finite-length statement therefore holds for the meromorphic connections of Bode–Bitoun and the local cohomology modules of Ardakov–Bode–Wadsley.","pith_inferences":["Composition series of finite length open the possibility of inductive arguments and classification results for weakly holonomic D-cap-modules that were previously unavailable.","The same Hilbert-polynomial control may adapt, after suitable modification of the completed Weyl algebra, to mildly singular or non-quasi-compact rigid spaces.","Finite-length statements of this type typically imply that the modules are coherent and that their characteristic cycles are well-defined algebraic cycles."],"forward_implications":["Meromorphic connections of Bode–Bitoun become finite-length weakly holonomic D-cap-modules on quasi-compact smooth rigid spaces.","Local cohomology groups of Ardakov–Bode–Wadsley likewise acquire finite length in the weakly holonomic category.","Any holonomic D-module extends to a coadmissible D-cap-module that admits a finite composition series of weakly holonomic subquotients.","The Hilbert-polynomial technique supplies a uniform length bound once the space is fixed and quasi-compact."],"fun_headline_variants":["Holonomic D-modules extend to finite-length D-cap-modules","Extension yields finite-length weakly holonomic D-cap-modules","Meromorphic connections have finite length as D-cap-modules","Local cohomology inherits finite length on rigid analytic spaces","Coadmissible D-cap-modules from holonomic ones stay finite length"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The rigid analytic space must be both quasi-compact and smooth; the finite-length claim is stated only under these two geometric hypotheses.","fun_headline_variants_meta":{"raw":{"variants":["Holonomic D-modules extend to finite-length D-cap-modules","Extension yields finite-length weakly holonomic D-cap-modules","Meromorphic connections have finite length as D-cap-modules","Local cohomology inherits finite length on rigid analytic spaces","Coadmissible D-cap-modules from holonomic ones stay finite length"]},"model":"grok-4.5","effort":"low","cost_usd":0.006014,"raw_usage":{"total_tokens":1495,"prompt_tokens":643,"num_sources_used":0,"completion_tokens":92,"cost_in_usd_ticks":60140000,"prompt_tokens_details":{"text_tokens":643,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":760,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":643,"tokens_out":92,"duration_ms":6627,"temperature":1.0,"reasoning_tokens":760,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-15T11:23:20.096013+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"An explicit holonomic D-module on a quasi-compact smooth rigid space whose extension is coadmissible yet has infinite length as a weakly holonomic D-cap-module, or a computation showing that the associated Hilbert polynomial fails to be eventually constant.","supporting_citations":[],"review_version":2}