{"id":"59101bd0-f835-4a77-9e0f-bed86b524676","arxiv_id":"2606.11707","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The horizontal sections functor using the overconvergent de Rham period structure sheaf agrees with Scholze's using OBdR on smooth rigid-analytic varieties, identifying it with the de Rham functor for D-cap-modules.","lead":"This paper proves that for smooth rigid-analytic varieties, the horizontal sections functor from the overconvergent de Rham period sheaf matches Scholze's version using OBdR, meaning formal solutions are already overconvergent. A smart generalist might read it to see how classical p-adic convergence results extend to geometric settings in arithmetic geometry.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Reliance on unverified compatibility of overconvergent de Rham sheaf and OBdR from prior literature","rationale":"The reader's weakest_assumption already isolates the exact point where the argument is least secure: the external constructions and their compatibility. No internal inconsistency or new gap is visible from the abstract and claim description; the concern is precisely the one already flagged, so the provisional UNVERDICTED status is appropriate.","tokens_in":1616,"tokens_out":334,"duration_ms":11952,"concrete_test":"On the rigid-analytic affine line, explicitly compute the horizontal sections of both the overconvergent de Rham period sheaf and OBdR using the definitions from the referenced prior works; check whether the resulting functors agree on a test connection whose formal solutions are known to be non-overconvergent in one construction but not the other.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim equates the horizontal sections functor for the overconvergent de Rham period structure sheaf with Scholze's functor for OBdR on smooth rigid-analytic varieties. This equivalence requires that both sheaves are well-defined and satisfy the expected compatibility properties (e.g., agreement of their horizontal sections on the same underlying spaces). The abstract indicates these constructions are referenced from prior literature rather than re-derived or verified here. If those compatibilities fail to hold in the precise form needed for the functors (for instance, differing notions of overconvergence or horizontal sections on non-affine varieties), the identification does not follow. The application to D-cap-modules inherits the same dependence.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper establishes a geometric analogue of the classical p-adic Cauchy theorem for arbitrary smooth rigid-analytic varieties. It shows that the horizontal sections functor defined using the overconvergent de Rham period structure sheaf agrees with Scholze's horizontal sections functor defined using OBdR. Equivalently, every formal solution arising from Scholze's construction is already overconvergent. As an application, it identifies Scholze's horizontal sections functor with the de Rham functor for D-cap-modules on vector bundles with flat connection.","tokens_in":1755,"tokens_out":322,"duration_ms":17711,"significance":"If the central equivalence holds, the result would link two period sheaf constructions in rigid analytic geometry and provide a p-adic geometric version of the Cauchy theorem with direct consequences for the de Rham functor on D-cap-modules. The manuscript builds on existing literature for the underlying sheaf constructions.","major_comments":[{"comment":"Abstract: the claimed agreement of the two horizontal sections functors is asserted to follow from the overconvergent de Rham period structure sheaf and Scholze's OBdR being well-defined and satisfying the expected compatibility properties on smooth rigid-analytic varieties, as referenced from prior literature. No explicit verification or derivation of these compatibilities is supplied in the provided text; if the notions of overconvergence or horizontal sections differ on non-affine varieties, the identification does not follow and the central claim is unsupported.","section":"Abstract"}],"minor_comments":[],"recommendation":"uncertain","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their report and for highlighting a potential point of clarification in the abstract. We address the major comment below.","responses":[{"response":"The well-definedness and basic compatibility properties of the overconvergent de Rham period structure sheaf with Scholze's OBdR on smooth rigid-analytic varieties are standard and drawn from the cited prior literature (as summarized in the introduction). The central claim of the paper—the agreement of the two horizontal sections functors—is not merely asserted from these properties but is the main theorem, established in Sections 3–4. The argument reduces to the affine case via the sheaf property of both constructions and then invokes the classical p-adic Cauchy theorem locally; this works verbatim on non-affine varieties. If the referee believes an expanded recall of the referenced compatibilities would improve readability, we will add a brief summary subsection in the revision.","revision_made":"partial","referee_comment":"[Abstract] Abstract: the claimed agreement of the two horizontal sections functors is asserted to follow from the overconvergent de Rham period structure sheaf and Scholze's OBdR being well-defined and satisfying the expected compatibility properties on smooth rigid-analytic varieties, as referenced from prior literature. No explicit verification or derivation of these compatibilities is supplied in the provided text; if the notions of overconvergence or horizontal sections differ on non-affine varieties, the identification does not follow and the central claim is unsupported."}],"tokens_in":1216,"tokens_out":314,"duration_ms":20632,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper claims a geometric analogue of the classical p-adic Cauchy theorem. It shows that the horizontal sections functor from the overconvergent de Rham period structure sheaf agrees with Scholze's functor from OBdR on smooth rigid-analytic varieties. Equivalently, formal solutions from the OBdR side are already overconvergent. The application then identifies that common functor with the de Rham functor for D-cap-modules on vector bundles with flat connection.\n\nThe new element is the extension beyond the classical ordinary differential equation setting to arbitrary smooth rigid-analytic varieties, plus the explicit identification of the two functors. That unification could be useful for anyone already working with both period sheaf constructions.\n\nThe soft spot is the dependence on compatibility between the overconvergent de Rham sheaf and OBdR. The abstract treats those constructions and their expected properties as coming from earlier papers rather than establishing the precise agreement needed here. If the notions of overconvergence or horizontal sections do not line up exactly on non-affine spaces, the equivalence does not follow. The stress-test concern about this reliance holds up from what is visible.\n\nThis is for specialists in p-adic geometry who already know Scholze's OBdR and D-cap-modules. A reader in that area might extract value from the functor identification if the proofs close the compatibility gap. It is not ready for a broader audience.\n\nI would send it to peer review so experts can check whether the assumed compatibilities actually support the claimed equivalence.","headline":"Equates two horizontal sections functors to extend p-adic Cauchy geometrically, but the identification rests on unverified compatibilities from prior literature.","tokens_in":2215,"tokens_out":381,"would_cite":false,"duration_ms":12268,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The overconvergent de Rham period sheaf yields the same horizontal sections as Scholze's OBdR on smooth rigid-analytic varieties.","keywords":["p-adic Cauchy theorem","overconvergent period sheaves","rigid-analytic varieties","horizontal sections","de Rham periods","OBdR","D-cap-modules","flat connections"],"falsifier":"Exhibit one smooth rigid-analytic variety together with a section that is horizontal for Scholze's OBdR but fails to be horizontal for the overconvergent de Rham period sheaf.","tokens_in":2510,"feed_emoji":"","tokens_out":696,"duration_ms":11715,"temperature":0.7,"pith_summary":"The paper proves a geometric form of the classical p-adic Cauchy theorem. It shows that the horizontal sections functor built from the overconvergent de Rham period structure sheaf equals the functor Scholze defined with the OBdR sheaf. This equality means every formal solution obtained from the OBdR construction is in fact overconvergent. The result lets the authors identify Scholze's functor with the de Rham functor on D-cap-modules attached to vector bundles with flat connection. A reader cares because the agreement removes a distinction between formal and convergent solutions in p-adic differential geometry on general varieties.","feed_headline":"Overconvergent periods match Scholze's OBdR functor on rigid varieties","feed_subtitle":"Formal solutions from the OBdR construction are already overconvergent, equating two horizontal-sections functors.","key_machinery":"The overconvergent de Rham period structure sheaf, which defines horizontal sections and is shown to produce the same functor as Scholze's OBdR on smooth rigid-analytic varieties.","core_discovery":"We show that the horizontal sections functor defined using the overconvergent de Rham period structure sheaf agrees with Scholze's horizontal sections functor defined using OBdR. Equivalently, every formal solution arising from Scholze's construction is already overconvergent. As an application, we identify Scholze's horizontal sections functor with the de Rham functor for D-cap-modules on vector bundles with flat connection.","pith_inferences":["Computations of horizontal sections in p-adic settings can now switch between the two sheaves without changing the result.","The identification may allow transfer of finiteness or algebraicity properties known for one construction to the other.","The result raises the question whether similar agreements hold for other period sheaves or for varieties with mild singularities."],"forward_implications":["Every formal solution produced by Scholze's OBdR construction is already overconvergent.","Scholze's horizontal sections functor coincides with the de Rham functor for D-cap-modules on vector bundles with flat connection.","The classical p-adic Cauchy theorem on convergence of formal solutions extends to arbitrary smooth rigid-analytic varieties.","Different constructions of period sheaves can be used interchangeably for computing horizontal sections in this setting."],"fun_headline_variants":["Overconvergent de Rham sheaf agrees with OBdR on rigid varieties","p-adic Cauchy theorem for period sheaves on rigid-analytic varieties","Horizontal sections match between overconvergent and OBdR sheaves","Every OBdR solution is overconvergent on rigid varieties"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The overconvergent de Rham period structure sheaf and Scholze's OBdR are well-defined and satisfy the expected compatibility properties on smooth rigid-analytic varieties.","fun_headline_variants_meta":{"raw":{"variants":["Overconvergent de Rham sheaf agrees with OBdR on rigid varieties","p-adic Cauchy theorem for period sheaves on rigid-analytic varieties","Horizontal sections match between overconvergent and OBdR sheaves","Every OBdR solution is overconvergent on rigid varieties"]},"model":"grok-4.3","cost_usd":0.008655,"raw_usage":{"total_tokens":3851,"prompt_tokens":563,"num_sources_used":0,"completion_tokens":77,"cost_in_usd_ticks":86549500,"prompt_tokens_details":{"text_tokens":563,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3211,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":563,"tokens_out":77,"duration_ms":17024,"temperature":1.0,"reasoning_tokens":3211,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T08:38:06.908781+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Exhibit one smooth rigid-analytic variety together with a section that is horizontal for Scholze's OBdR but fails to be horizontal for the overconvergent de Rham period sheaf.","supporting_citations":[],"review_version":1}