{"id":"12d3a1bb-e8bf-430f-891e-d7c1c37585a3","arxiv_id":"2508.13137","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper constructs infinite discrete Nakayama representations via persistence theory and stabilizes them into negative Calabi-Yau versions of Igusa-Todorov discrete cluster categories of type A, with geometric model and AR theory.","lead":"This paper builds new infinite versions of Nakayama representations using persistence theory, then stabilizes them to get a family of triangulated categories called negative Calabi-Yau discrete cluster categories of type A. It may be of interest to mathematicians working on cluster algebras, representation theory, and topological data analysis.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Stabilisation claim is uncheckable from corrupted text; the missing proof is that the stabilised category has a Serre functor equal to the claimed negative shift.","rationale":"The reader correctly returned UNVERDICTED because the body text is corrupted. My pass cannot locate a mathematical flaw, but it can sharpen the condition that would have to hold. The abstract's central assertion is a construction theorem: there is a stabilisation functor from persistence-theoretic infinite discrete Nakayama representations to a family of triangulated categories, and these categories are negative Calabi-Yau versions of Igusa-Todorov discrete cluster categories of type A. The weakest point is that 'stabilising' and 'negative Calabi-Yau' are both undefined in the abstract. Stabilisation of an exact category is only triangulated in special settings, such as when the category is Frobenius; and the Calabi-Yau dimension requires a Serre-functor isomorphism or a CY dg structure. Without a readable proof, the claim is not wrong but unsubstantiated. I agree partially with the reader: the reader emphasised a well-defined triangulated category, while I would put the weight on the additional CY-dimension structure, since triangulatedness has well-known sufficient conditions but the negative CY property is the distinctive, less automatic feature. The proposed concrete test would settle the concern directly.","tokens_in":2118,"tokens_out":5027,"duration_ms":61279,"concrete_test":"Obtain a readable copy of arXiv:2508.13137 from its source files and locate the theorem that states the Calabi-Yau property. For the simplest infinite discrete Nakayama representation, compute the Serre functor S on a generating object X and verify the isomorphism S(X) ≅ X[d] for the claimed negative d. A concrete minimal case is the stabilised category associated with the one-arrow Nakayama quiver; if S is not isomorphic to [d] as an endofunctor on that category, the negative Calabi-Yau identification fails. Also check that the AR quiver of the stabilised category coincides with the Igusa-Todorov discrete cluster category of type A, since the abstract's 'regarded as versions of' claim depends on this compatibility.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No specific mathematical error can be established: the supplied full text is mojibake, so only the abstract is available. The central claim has one genuinely load-bearing condition: the stabilisation of the infinite discrete Nakayama representations must yield not just a triangulated category but one equipped with the structure of a d-Calabi-Yau category for the claimed negative d. In the standard formulation, this means the category admits a Serre functor S with S ≅ [d], or equivalently a dg enhancement with a d-CY structure. This is not a free consequence of forming a Frobenius stable category: the stable category of a Frobenius category is triangulated automatically, but the Calabi-Yau dimension is an extra theorem. Since the abstract merely asserts that this happens after stabilising, and the body cannot be read, the construction is unverified rather than refuted. The risk is not internal inconsistency but absence of checkable support for the main theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The submitted manuscript consists of an abstract and a body that is not legible. The abstract announces a construction of infinite discrete symmetric Nakayama representations using persistence theory; after stabilisation, it claims a family of triangulated categories that can be regarded as negative Calabi-Yau versions of the Igusa-Todorov discrete cluster categories of type A, together with a geometric model and an AR theory. In the supplied file, no definitions, theorem statements, or proofs are readable; the body is a sequence of replacement characters. The only checkable content is the abstract, which states the central claim without the supporting technical framework.","tokens_in":2294,"tokens_out":3749,"duration_ms":39142,"significance":"If the central construction works, the paper would connect persistence theory with cluster categories and introduce new negative Calabi-Yau triangulated categories, which could be of significant interest in representation theory and related areas. The abstract's claim is concrete and falsifiable: one must show that the stabilisation produces a triangulated category with a Serre functor S satisfying S congruent to the claimed negative shift [d]. That is a real theorem, not a formal consequence of stabilisation alone. However, because the submitted text makes none of the definitions or proofs available, the significance is conditional. The paper does not ship machine-checked proofs, reproducible code, or parameter-free derivations that a referee could verify.","major_comments":[{"comment":"The central claim, 'After stabilising, we obtain a family triangulated categories which can be regarded as negative Calabi-Yau versions of the Igusa-Todorov discrete cluster categories of type A,' is asserted without the technical conditions under which stabilisation preserves triangulated structure and yields a Calabi-Yau dimension. The standard route through the stable category of a Frobenius category would give triangulatedness automatically, but the existence of a Serre functor S with S congruent to [d] is an additional theorem. This load-bearing point is not addressed in any readable portion of the manuscript.","section":"Abstract"},{"comment":"The body of the manuscript is unreadable: the supplied text is a sequence of replacement characters, so none of the definitions mentioned in the abstract (symmetric Nakayama representations, persistence-theoretic constructions, stabilisation, Igusa-Todorov discrete cluster categories) and none of the proofs or theorem statements can be verified. This is not a minor stylistic issue; it makes the central claim impossible to check from the submitted file.","section":"Full Text"},{"comment":"The term 'negative Calabi-Yau versions' is not defined for infinite triangulated categories. One needs a precise Serre-functor formulation or a dg-enhancement formulation, together with any finiteness or duality hypotheses needed for the Calabi-Yau structure to exist. Without such a definition, the meaning of the claimed negative Calabi-Yau dimension is unclear.","section":"Abstract"}],"minor_comments":[{"comment":"The phrase 'a family triangulated categories' should be 'a family of triangulated categories'.","section":"Abstract"},{"comment":"The abstract does not state the exact value of the negative Calabi-Yau dimension d; specifying this shift would make the central claim more testable.","section":"Abstract"},{"comment":"The references to Igusa-Todorov discrete cluster categories and to persistence theory cannot be read in the supplied file, so the paper's relation to prior work cannot be assessed.","section":"Full Text"}],"recommendation":"uncertain","confidential_remarks":"The manuscript is not reviewable in its present form because the body is corrupted; only the abstract is legible. I could not check any result, and I found no specific mathematical error because no mathematics is visible. I recommend asking the authors to resubmit a readable file with complete definitions, theorem statements, and proofs. If the full text becomes available, the paper may well fit the journal's scope and I would be willing to review the corrected version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, this is an interesting abstract but we have a problem: the full text I received is completely corrupted, so my judgment is based on the abstract only. What the paper claims is genuinely new: infinite discrete Nakayama representations built with persistence theory, then stabilised to get negative Calabi-Yau versions of Igusa-Todorov discrete cluster categories of type A. That is a real bridging idea between TDA and representation theory, and if it works it's a nice result.\n\nWhat the abstract does well: it states the construction in a direct way and gives the expected payoffs (geometric model, AR theory). I can't see proofs, but the claim is not obviously wrong.\n\nThe soft spot is exactly the stabilisation step. It's one thing to say you get a triangulated category after stabilising a Frobenius category; it's another to show the Serre functor is the claimed negative shift. That is a theorem, not a free consequence. The abstract asserts it without giving conditions. That might be perfectly fine in the actual paper, but from the abstract alone it's a gap. Also, no details on the persistence-theoretic construction of the Nakayama representations, so I can't evaluate that either.\n\nI don't want to penalise the author for a corrupted submission. The right move is to get a clean copy and send it to a specialist. The idea deserves referee time. My own verdict is 'unverified' rather than 'wrong'. If the main theorem holds, it's a solid contribution to both cluster theory and the persistence literature.","headline":"Interesting bridge construction between persistence theory and cluster categories, but the supplied full text is unreadable, so the central claim is unverified rather than refuted.","tokens_in":2737,"tokens_out":1879,"would_cite":false,"duration_ms":19297,"reading_group":"maybe","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G20","16G70","18G80","13F60"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs negative Calabi-Yau analogues of the type-A discrete cluster categories by stabilising infinite Nakayama representations built from persistence theory.","keywords":["negative Calabi-Yau categories","discrete cluster categories","Nakayama representations","persistence theory","triangulated categories","Auslander-Reiten theory","type A"],"falsifier":"Take the simplest infinite discrete symmetric Nakayama representation, apply the paper's stabilisation, and compute the AR translation on the resulting category. If that translation is not the negative power of the suspension dictated by the claimed Calabi-Yau dimension, the central claim is false.","tokens_in":1931,"feed_emoji":"","tokens_out":8253,"duration_ms":70964,"temperature":0.7,"pith_summary":"The paper introduces infinite discrete versions of symmetric Nakayama representations by importing persistence-theoretic techniques, then applies a stabilisation step. The central claim is that the resulting triangulated categories are negative Calabi-Yau versions of the type-A discrete cluster categories. Here the Calabi-Yau dimension is the integer governing the duality shift of the category, and a negative value is the unusual feature that distinguishes these categories from the classical ones, which have nonnegative dimension. The paper also describes the geometric model and the AR theory (the complete description of indecomposable objects and the maps between them) of the new categories. If the stabilisation is sound, this extends the discrete cluster category construction into negative Calabi-Yau dimensions, giving a new family of triangulated categories with explicit combinatorial structure.","feed_headline":"Persistence theory yields negative Calabi-Yau cluster categories","feed_subtitle":"Infinite Nakayama representations, stabilised, recreate type-A discrete cluster categories in negative dimension.","key_machinery":"The central mechanism is a triple: (1) infinite discrete symmetric Nakayama representations—linear-algebraic quiver representations on the integer line with a symmetry condition—which provide the objects; (2) persistence theory, which organises these infinite representations through their finite-dimensional subquotients; and (3) a stabilisation step that turns this representation-theoretic data into a triangulated category. The named target is the discrete cluster categories of type A—triangulated categories whose indecomposable objects are indexed by the integers—and the paper's negative Calabi-Yau versions are what the stabilised infinite Nakayama data produce. The AR theory (the complete list of indecomposables and the irreducible maps between them, normally drawn as a quiver) is the output that makes the categories concrete.","core_discovery":"On the paper's own terms, the discovery is an explicit bridge: infinite discrete symmetric Nakayama representations, encoded as persistence modules, stabilise to produce a family of triangulated categories. The paper claims these categories deserve to be called negative Calabi-Yau versions of the type-A discrete cluster categories, meaning the duality between maps in the category and maps shifted by a negative power of the suspension functor replaces the nonnegative-shift duality of ordinary cluster categories. The geometric model makes the AR theory explicit, so objects and irreducible morphisms can be read off from a picture. The construction is presented as an extension of the discrete cluster category picture to a new range of Calabi-Yau dimensions.","pith_inferences":["A likely extension the paper does not spell out: the same persistence-to-stabilisation route should work for other discrete symmetric Nakayama data, yielding a whole spectrum of negative Calabi-Yau categories rather than a single family.","One testable consequence: if a stability function exists on the persistence side, it may induce a stability condition on the stabilised category, giving a geometric handle on which objects are semistable.","The negative Calabi-Yau dimension suggests these categories should admit a cluster-tilting subcategory only in very special cases; checking this on the first example would sharpen the boundary between the positive and negative regimes."],"forward_implications":["If the stabilisation is valid, the paper produces a new infinite family of triangulated categories with explicitly negative Calabi-Yau dimension.","These categories come with a geometric model, so their AR quiver can be drawn and homological data read off visually.","The construction extends the type-A discrete cluster category family beyond nonnegative Calabi-Yau dimension, making the boundary between cluster-tilting and non-cluster-tilting behaviour visible.","Because the input is persistence-theoretic, the resulting categories inherit persistence-module structure, which may support quantitative invariants such as lifetimes of indecomposables."],"supporting_citations":[],"fun_headline_variants":["Negative Calabi-Yau clusters from persistence theory","Stabilised Nakayama reps yield negative Calabi-Yau categories","Persistence builds negative Calabi-Yau cluster categories","Infinite Nakayama reps become negative Calabi-Yau categories","Persistence turns Nakayama representations into negative CY clusters"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction stands or falls on the stabilisation step: it must turn the infinite Nakayama representations into a genuine triangulated category whose duality shift has the claimed negative Calabi-Yau dimension, and the paper does not give the technical conditions that guarantee this.","fun_headline_variants_meta":{"raw":{"variants":["Negative Calabi-Yau clusters from persistence theory","Stabilised Nakayama reps yield negative Calabi-Yau categories","Persistence builds negative Calabi-Yau cluster categories","Infinite Nakayama reps become negative Calabi-Yau categories","Persistence turns Nakayama representations into negative CY clusters"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000612,"raw_usage":{"total_tokens":2728,"prompt_tokens":711,"completion_tokens":2017,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":327,"completion_tokens_details":{"reasoning_tokens":1937}},"tokens_in":327,"tokens_out":2017,"duration_ms":14954,"temperature":1.0,"reasoning_tokens":1937,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:14:08.887562+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the simplest infinite discrete symmetric Nakayama representation, apply the paper's stabilisation, and compute the AR translation on the resulting category. If that translation is not the negative power of the suspension dictated by the claimed Calabi-Yau dimension, the central claim is false.","supporting_citations":[],"review_version":1}