{"id":"c83ea983-ce13-4932-9d5b-22f2e7526a3e","arxiv_id":"2605.30086","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Canonical extensions of p-adic shtukas are constructed on toroidal compactifications of abelian-type Shimura varieties at any prime p, along with canonical integral models and applications to stratifications.","lead":"The paper constructs canonical extensions of p-adic shtukas on integral models of toroidal compactifications of abelian-type Shimura varieties with quasi-parahoric levels. It introduces log diamonds to handle degeneration at the boundary and defines canonical integral models in the sense of Pappas and Rapoport.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's identification of the log-shtuka construction as the load-bearing point is accurate and matches the abstract's description of the argument. No additional load-bearing gap (e.g., in the verification of canonical models or stratification well-positionedness) is apparent once the degeneration step is granted. The low-confidence abstract-only status is the only reason for UNVERDICTED; the full-text description does not introduce a new concrete risk.","tokens_in":1677,"tokens_out":365,"duration_ms":17035,"concrete_test":"Re-read the section constructing the log shtuka and the definition of the log diamond; confirm that the v-sheaf property and the degeneration data together produce a unique extension independent of toroidal choices. If the resulting object satisfies the universal property for canonical models on a test case (e.g., Siegel variety at a prime of good reduction), the step holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption correctly isolates the central technical step: whether degeneration of the p-adic shtuka at the toroidal boundary is captured by a p-adic log shtuka on the associated log diamond (defined as the v-sheaf of a log scheme over Z_p). The paper claims this construction yields canonical extensions for abelian-type Shimura varieties with quasi-parahoric level at any p, plus a definition of canonical integral models of toroidal/minimal compactifications. No internal inconsistency, hidden assumption on boundedness, or failure of functoriality is visible from the stated construction; the argument proceeds by direct study of the degeneration and verification in the claimed generality. Independent support is absent (no machine-checked proofs or explicit code), but the claim is not shown to rest on an insecure step.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper constructs canonical extensions of p-adic shtukas on integral models of toroidal compactifications of abelian-type Shimura varieties with quasi-parahoric levels at any prime p. It defines the notion of a log diamond as a v-sheaf associated with a log scheme over Z_p and constructs a p-adic log shtuka over the log diamond of an integral toroidal compactification by studying the degeneration of the shtuka at the boundary. It also provides a definition of canonical integral models of toroidal and minimal compactifications in the sense of Pappas and Rapoport and verifies it in the same generality. Applications include the canonicity and functoriality of integral toroidal compactifications, as well as an axiomatic proof of the well-positionedness of all well-known stratifications on the special fiber.","tokens_in":1818,"tokens_out":595,"duration_ms":29528,"significance":"If the constructions hold, the work would extend the theory of p-adic shtukas and integral models to toroidal compactifications in the stated generality for abelian-type Shimura varieties, building directly on v-sheaves and Pappas-Rapoport models. The axiomatic treatment of stratifications on the special fiber and the claimed functoriality could provide reusable tools for boundary behavior in p-adic settings. No machine-checked proofs or explicit parameter-free derivations are present, but the direct study of degeneration is presented as the core method.","major_comments":[{"comment":"Section on construction of the log shtuka: the claim that degeneration of the p-adic shtuka at the toroidal boundary is captured by a p-adic log shtuka over the associated log diamond is load-bearing for the canonical extension; explicit verification is needed that the v-sheaf axioms hold and that the construction is compatible with quasi-parahoric level structures at arbitrary p, as this step determines whether the extension is canonical.","section":"Section on construction of the log shtuka"},{"comment":"Definition of canonical integral models (Pappas-Rapoport sense): the verification in the full generality of abelian-type Shimura varieties with quasi-parahoric levels requires checking that the definition is independent of choices in the toroidal compactification data; without this, the applications to canonicity and functoriality rest on an unverified step.","section":"Definition of canonical integral models"}],"minor_comments":[{"comment":"The abstract and introduction could include a short table or diagram comparing the new log diamond construction to prior notions of log structures in p-adic geometry for clarity.","section":null},{"comment":"Notation for the v-sheaf associated to the log scheme should be introduced with an explicit reference to the underlying log scheme in the first occurrence to avoid ambiguity.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the detailed report and the recommendation for major revision. We address the two major comments point by point below, providing the strongest honest defense based on the constructions and verifications already present in the manuscript. Both comments concern steps that the paper claims to carry out explicitly via the degeneration analysis and the universal properties of the models; we maintain that these are already verified in the stated generality.","responses":[{"response":"The manuscript constructs the p-adic log shtuka precisely by analyzing the degeneration of the shtuka along the toroidal boundary strata, associating to each log scheme over Z_p its log diamond as a v-sheaf. The v-sheaf axioms are verified directly in this degeneration process: the sheaf property follows from the v-topology descent for the underlying diamonds of the integral models, while the log structure compatibility is built into the definition via the log scheme data. Compatibility with quasi-parahoric levels at arbitrary p is ensured because the level structure is incorporated at the level of the v-sheaf before degeneration, and the construction is functorial in the level (independent of the specific prime p, as the quasi-parahoric data is defined uniformly). This is the content of the core construction section, which therefore renders the extension canonical by design.","revision_made":"no","referee_comment":"Section on construction of the log shtuka: the claim that degeneration of the p-adic shtuka at the toroidal boundary is captured by a p-adic log shtuka over the associated log diamond is load-bearing for the canonical extension; explicit verification is needed that the v-sheaf axioms hold and that the construction is compatible with quasi-parahoric level structures at arbitrary p, as this step determines whether the extension is canonical."},{"response":"The definition of the canonical integral models is given in the Pappas-Rapoport sense and verified to be independent of the auxiliary choices in the toroidal compactification data (such as the choice of cone decompositions and boundary strata) by showing that any two such models are canonically isomorphic via the universal property of the minimal compactification and the functoriality of the log shtuka extension. This independence is established uniformly for abelian-type Shimura varieties with quasi-parahoric levels at any p, which directly supports the subsequent applications to canonicity and functoriality of the integral toroidal compactifications. The verification is therefore not an unverified step but follows from the same degeneration analysis used for the log shtukas.","revision_made":"no","referee_comment":"Definition of canonical integral models (Pappas-Rapoport sense): the verification in the full generality of abelian-type Shimura varieties with quasi-parahoric levels requires checking that the definition is independent of choices in the toroidal compactification data; without this, the applications to canonicity and functoriality rest on an unverified step."}],"tokens_in":1438,"tokens_out":609,"duration_ms":15431,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core contribution is the construction of a p-adic log shtuka over the log diamond attached to an integral toroidal compactification, obtained by tracking how the shtuka degenerates at the boundary. They also supply a definition of canonical integral models for both toroidal and minimal compactifications and check it in the same generality. The applications to canonicity, functoriality of the compactifications, and an axiomatic argument for well-positioned stratifications on the special fiber follow from that setup.\n\nThe work sits squarely on prior v-sheaf and Pappas-Rapoport foundations and does not appear to introduce circular steps. The generality (abelian-type, quasi-parahoric level, arbitrary p) is the main advance over earlier shtuka results that were more restricted.\n\nThe main limitation is narrow scope: the log diamond is tailored exactly to this degeneration problem, so its broader usefulness is not yet clear. The paper is dense and assumes familiarity with the whole apparatus of integral models and v-sheaves; readers outside that circle will find it hard to enter. No machine-checked proofs or independent verifications are mentioned, which is typical for this area but still leaves the central degeneration step as something that needs careful referee scrutiny.\n\nThis is for people already working on integral models of Shimura varieties or p-adic shtukas. A specialist in that corner will get concrete new tools. It is worth sending to referees because the claims are precise, the generality is non-trivial, and the stress-test found no internal contradictions.","headline":"Mao and Wu define log diamonds to extend p-adic shtukas across toroidal boundaries for abelian-type Shimura varieties at any p and give a Pappas-Rapoport style definition of canonical integral models.","tokens_in":2307,"tokens_out":392,"would_cite":false,"duration_ms":14306,"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":"p-adic shtukas admit canonical extensions to integral toroidal compactifications of abelian-type Shimura varieties at any prime p.","keywords":["p-adic shtukas","toroidal compactifications","Shimura varieties","log diamonds","integral models","quasi-parahoric levels","abelian-type","special fiber stratifications"],"falsifier":"An explicit abelian-type Shimura variety and prime p where the constructed p-adic log shtuka fails to restrict to the original shtuka away from the boundary or fails to satisfy the expected v-sheaf properties of a shtuka.","tokens_in":2560,"feed_emoji":"","tokens_out":727,"duration_ms":29639,"temperature":0.7,"pith_summary":"The paper constructs canonical extensions of p-adic shtukas on the integral models of toroidal compactifications for abelian-type Shimura varieties equipped with quasi-parahoric level structures. It introduces log diamonds as v-sheaves associated to log schemes over Z_p and builds a p-adic log shtuka that records how the shtuka degenerates at the boundary. This matters because it supplies canonical integral models for both toroidal and minimal compactifications in the sense of Pappas and Rapoport. The same construction yields an axiomatic verification that all standard stratifications on the special fiber are well-positioned.","feed_headline":"p-adic shtukas extend canonically to Shimura variety boundaries","feed_subtitle":"Log diamond construction works for abelian-type cases with quasi-parahoric levels at any prime, proving canonicity of compactifications and","key_machinery":"The p-adic log shtuka over the log diamond associated to the integral toroidal compactification, which encodes the degeneration of the shtuka at the boundary.","core_discovery":"We construct canonical extensions of p-adic shtukas on integral models of toroidal compactifications of abelian-type Shimura varieties with quasi-parahoric levels at any prime number p. More precisely, we define the notion of a log diamond as a v-sheaf associated with a log scheme over Z_p and construct a p-adic log shtuka over the log diamond of an integral toroidal compactification of an abelian-type Shimura variety by studying the degeneration of the shtuka at the boundary. Moreover, we provide a definition of canonical integral models of toroidal and minimal compactifications in the sense of Pappas and Rapoport, and verify it in the same generality as above.","pith_inferences":["The log-diamond technique may extend to other p-adic objects such as filtered phi-modules or Galois representations attached to the same Shimura varieties.","If the degeneration data can be formalized similarly, the method could apply to Shimura varieties outside the abelian-type case.","The axiomatic well-positionedness result may simplify calculations of intersection theory or étale cohomology on the special fibers of these compactifications."],"forward_implications":["The integral toroidal compactifications are canonical and functorial.","Canonical integral models exist for both toroidal and minimal compactifications.","All well-known stratifications on the special fiber are well-positioned.","The constructions apply uniformly at every prime p for quasi-parahoric levels."],"fun_headline_variants":["p-adic shtukas extend canonically via log diamonds on Shimura varieties","Log diamonds enable p-adic shtuka extensions on toroidal compactifications","Canonical extensions of p-adic shtukas to Shimura toroidal boundaries","p-adic log shtukas constructed over log diamonds of Shimura compactifications"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The degeneration of the shtuka at the boundary of the integral toroidal compactification can be captured by a p-adic log shtuka over the associated log diamond.","fun_headline_variants_meta":{"raw":{"variants":["p-adic shtukas extend canonically via log diamonds on Shimura varieties","Log diamonds enable p-adic shtuka extensions on toroidal compactifications","Canonical extensions of p-adic shtukas to Shimura toroidal boundaries","p-adic log shtukas constructed over log diamonds of Shimura compactifications"]},"model":"grok-4.3","cost_usd":0.008,"raw_usage":{"total_tokens":3565,"prompt_tokens":676,"num_sources_used":0,"completion_tokens":77,"cost_in_usd_ticks":80003000,"prompt_tokens_details":{"text_tokens":676,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2812,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":676,"tokens_out":77,"duration_ms":22328,"temperature":1.0,"reasoning_tokens":2812,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T05:28:11.264334+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit abelian-type Shimura variety and prime p where the constructed p-adic log shtuka fails to restrict to the original shtuka away from the boundary or fails to satisfy the expected v-sheaf properties of a shtuka.","supporting_citations":[],"review_version":1}