{"id":"6087d151-851d-46d9-883f-b1bd29c1363e","arxiv_id":"2506.02999","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For persistence modules of circle-valued functions, the interleaving distance equals the bottleneck distance between arc barcodes on a geometric model.","lead":"This mathematics paper proves that two different ways of measuring similarity between circle-valued persistence modules always agree: the interleaving distance and the bottleneck distance. The result gives a computable summary, a barcode of arcs and loops on an annulus, that captures exactly the same information as the more abstract distance.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 3.9 is not well-posed because the Φ_M maps are defined only up to scalar, so the central isometry theorem is ambiguous as stated.","rationale":"The paper's main theorem, Theorem 7.6, asserts an equality of two distances. For that assertion to have content, both distances must be well-defined invariants of the input modules. The bottleneck distance dB is defined through explicit combinatorial operations and is unambiguous. The interleaving distance dI, however, rests on the morphisms Φ_M of Definition 2.1, which are introduced only up to an arbitrary nonzero scalar. Definition 3.7 composes these without fixing a scale, so each Φ^{2δ}_M carries an unspecified scalar. Definition 3.9 then uses these in equalities with τ^{-δ}g∘f and τ^{-δ}f∘g. Because the two composition products share the same scale factor under rescaling of f and g, independent rescalings of the Φ's cannot be simultaneously absorbed unless the relevant Hom spaces have extra degrees of freedom. In the generic one-dimensional case, the existence of an interleaving becomes dependent on the chosen scalars. Thus d_I is not a well-defined function of the isomorphism classes of M and N as stated, and Theorem 7.6 is not well-posed. The reader's verdict correctly identified this issue and also noted a missing case in the proof of Lemma 5.13 (the metric property of dB); I view the latter as secondary because the statement of dB is clear and the gap appears repairable by treating infinite-length arcs separately. The scaling ambiguity, by contrast, affects the definition of the central object itself. For this reason I agree with the reader's conditional verdict and would not change it: the result is plausible and likely repairable, but as written the main theorem lacks a precise left-hand side.","tokens_in":31088,"tokens_out":10392,"duration_ms":109828,"concrete_test":"Check whether Definition 3.9 is invariant under independent rescalings of the Φ maps. Concretely, take a pair of indecomposable modules M,N in a preprojective component, with δ=1 and Hom(M,τ^{-1}N) and Hom(N,τ^{-1}M) both one-dimensional, so f and g are unique up to scalars. Write the two interleaving equations with Φ^2_M multiplied by c_M and Φ^2_N by c_N. If the system has a solution for (c_M,c_N)=(1,1) but none for c_M=1,c_N=2 (or vice versa), then d_I depends on normalization, confirming the gap. This can be computed by hand from the AR-quiver mesh relations for a small quiver such as the equioriented A4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 2.1 states that Φ_M : M → τ^{-1}M is well defined only up to scaling by an element of k*. Definition 3.7 then defines Φ^δ_N as a composition of such morphisms, inheriting an arbitrary nonzero scalar factor. Definition 3.9 declares M,N δ-interleaved if there exist f,g with τ^{-δ}g∘f = Φ^{2δ}_M and τ^{-δ}f∘g = Φ^{2δ}_N. These equalities are not well-defined predicates because the right-hand sides are only specified up to independent nonzero scalars. If the chosen scalars for Φ^{2δ}_M and Φ^{2δ}_N are a and b, then replacing f,g by λf, μg scales both compositions by λμ; one can compensate a common rescaling of both Φ's but not independent rescalings when the relevant Hom spaces are one-dimensional. Hence d_I(M,N) as defined can depend on arbitrary choices made in Definition 2.1 (or Definition 3.7), and the isometry theorem does not have a well-defined left-hand side. Lemma 3.15, used for the triangle inequality, likewise holds only for compatible normalizations. This is a definitional gap in the paper's central object, not an internal inconsistency; it is likely repairable by fixing a normalization or by formulating the interleaving condition projectively, but as written the theorem is not a precise mathematical statement.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a generalization of the interleaving and bottleneck distances to persistence modules associated with circle-valued functions, modeled as representations of non-cyclically oriented quivers of type \\tilde A. The interleaving distance is defined using the Auslander-Reiten translate on the derived category, and the barcode is defined as a multiset of graded arcs and closed curves on the geometric model of the derived category. The centerpiece is an isometry theorem (Theorem 7.6) asserting that the interleaving distance between two \\tilde A persistence modules equals the bottleneck distance between their geometric barcodes. The paper also contains a dictionary relating its barcodes to the barcodes and Jordan blocks of Burghelea-Dey and Burghelea-Haller, and several worked examples.","tokens_in":31359,"tokens_out":8346,"duration_ms":84974,"significance":"If the main theorem is correct, the paper would provide a genuinely intrinsic isometry theorem for circle-valued persistence, avoiding the usual detour through multidimensional persistence or derived equivalences with equioriented quivers. The use of the geometric model to encode barcodes as arcs and closed curves is conceptually appealing and could be a useful bridge between representation theory and topological data analysis. The paper is ambitious and includes many useful structural observations, but the central object of the paper, the interleaving distance of Definition 3.9, is not rigorously well-posed as written, and several load-bearing proof steps appear incomplete. The result is therefore not yet established to the standard required for publication.","major_comments":[{"comment":"The predicate \"M and N are δ-interleaved\" is not well-defined. Definition 2.1 states that Φ_M : M → τ^{-1}M is well defined only up to scaling by an element of k^*. Definition 3.7 then constructs Φ^δ_N by composing such morphisms, so each Φ^{2δ}_M and Φ^{2δ}_N is determined only up to an arbitrary nonzero scalar. The equalities τ^{-δ}g∘f = Φ^{2δ}_M and τ^{-δ}f∘g = Φ^{2δ}_N in Definition 3.9 therefore are not precise mathematical statements unless a normalization is fixed. Since d_I in Definition 3.12 is defined as the infimum over such ill-defined predicates, the left-hand side of Theorem 7.6 is ambiguous. The problem propagates: Lemma 3.15, Proposition 3.16, and the stability proofs in Sections 6 and 7 all rely on equalities involving these unnormalized Φ maps. The paper needs to either choose a canonical normalization for each Φ_M (for example by fixing specific mesh morphisms) or reformulate the interleaving condition projectively, and then prove that the resulting distance is independent of any remaining choices.","section":"Definition 3.9, with Definition 2.1 and Definition 3.7"},{"comment":"The triangle inequality proof for the bottleneck distance applies Lemma 5.7 to arcs that may have infinite length. Lemma 5.7 is stated only for graded arcs or closed curves of finite length, but the barcode B can contain arcs with endpoints on different boundary components, which have infinite length. In the proof, when γ ∈ B_{2(d1+d2)} is such an infinite-length arc, the displayed equalities involving ℓ(γ) are not covered by Lemma 5.7. This is not merely a cosmetic gap: the contradiction argument used to prove B_{2(d1+d2)} ⊆ Coim(η2∘η1) depends on the length formula for the very element γ. The proof should either extend Lemma 5.7 to infinite lengths with a separate argument, or handle infinite-length arcs directly (for instance, by observing that finite endpoint-sliding operations preserve infinite length and that an infinite-length arc cannot be δ-equivalent to a finite-length arc).","section":"Lemma 5.13"},{"comment":"The functor F from the full subcategory S to finite-dimensional vector spaces is not well-defined as stated. The definition F(f) = F(Σ α_m f_m) = Σ α_m depends on the chosen basis {f_m} of each Hom-space, and no argument is given that different bases yield the same scalar or that composition of morphisms is preserved. The claim that the basis morphisms are \"compatible with composition\" is asserted but not proved, and in general a linear combination of morphisms can be represented in multiple ways. This matters because the Hall's theorem argument uses the fact that F(Φ^{2δ}_{M_I}) is a rank |I| diagonal matrix; if F is not a genuine linear functor, the rank conclusion has no basis. The proof should either construct F explicitly from a compatible family of bases, or replace this step by a direct rank argument using actual matrices with respect to fixed bases of the relevant Hom-spaces.","section":"Lemma 6.2"},{"comment":"The composition θδ ∘ η_f is not well-defined as stated. In the theorem statement, η_f is described as a matching B(M) → B(Im f), while θδ is defined immediately before as a perfect matching B(N) → B(τ^δ N). These do not compose. The proof later refers to B(τ^{-δ}N) and to inclusions of Im f into τ^{-δ}N, suggesting that the intended construction involves an additional matching induced by the inclusion Im f ↪ τ^{-δ}N and a bijection B(N) → B(τ^{-δ}N) induced by τ^{-δ}. The statement and proof need to be corrected so that the domain and codomain of the composed matching are explicit and consistent.","section":"Theorem 7.4"}],"minor_comments":[{"comment":"There is a typo: \"resrticted\" should be \"restricted\".","section":"Corollary 3.8"},{"comment":"The word \"trivalised\" should be \"trivialized\".","section":"Section 9.3"},{"comment":"The phrase \"Ausander-Reiten\" should be \"Auslander-Reiten\".","section":"Section 1.2.1"},{"comment":"Definition 5.6 refers to a \"graded arc or closed curve\" but the notion of length for a closed curve is only implicit through Definition 4.6; it would help to state explicitly that the length of (γ, λ, l) is l, including the trivial case l = 0.","section":"Definition 5.6"},{"comment":"The dictionary in Section 8 is useful, but the table entry for closed and open intervals both mapping to arcs between different boundary components could be confusing; a sentence clarifying that the grading distinguishes preprojective from preinjective objects would improve readability.","section":"Section 8"}],"recommendation":"major_revision","confidential_remarks":"The paper is not ready for acceptance in its current form because the main definition is ambiguous and several key proofs have gaps. However, the overall strategy is plausible and the gaps appear repairable within the scope of the manuscript, so I recommend major revision rather than rejection. The authors should also carefully re-check the composition in Theorem 7.4 and the functor F in Lemma 6.2, as these are not merely typographical issues."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nHere is the short version: this is a serious and likely correct isometry theorem for circle-valued persistence, but the central interleaving distance is not well-defined as written. The fix is small, the theorem is probably right, and the paper deserves another pass before you trust the details.\n\nWhat is genuinely new: it extends the isometry theorem to finite non-cyclic tilde A quivers, i.e. circle-valued persistence, complementing the Burghelea–Dey/Haller invariants and the continuous cyclic result of Gao–Zhao. The geometric model barcode—arcs and closed curves on an annulus—is a nice unifying picture, and the dictionary with Jordan blocks in Section 8 is useful. The strategy of splitting the proof into preprojective, preinjective, and regular components is sound, and the paper is honest about the literature. Credit where due: no parameter fitting, no self-citation, and the examples are informative.\n\nThe load-bearing problem is Definition 3.9. In Definition 2.1, Φ_M is defined only up to scaling by an element of k*. No normalization is fixed, and Definition 3.7 inherits that ambiguity. Since rescaling f and g scales both compositions τ^{-δ}g∘f and τ^{-δ}f∘g by the same factor, you can absorb a common rescaling of Φ^{2δ}_M and Φ^{2δ}_N, but not independent ones. The interleaving predicate, hence d_I, depends on choices made in Definition 2.1. This is repairable—fix representatives, or formulate the condition with an explicit normalization—but as written Theorem 7.6 has an ambiguous left-hand side.\n\nA second, minor soft spot: Lemma 5.13's triangle inequality applies the length formula from Lemma 5.7, which is stated for finite-length arcs, to possibly infinite-length arcs. The missing case is harmless because an arc connecting two boundary components cannot be δ-equivalent to an arc on one boundary, but the proof as printed skips that case distinction.\n\nThe regular-component proof (Theorem 7.4) is dense; I did not verify every line. The induced-matching strategy is standard and the structure is plausible.\n\nWho is this for? TDA researchers working on circle-valued persistence and representation theorists interested in gentle algebras. It deserves a serious referee, not a desk reject. I would engage with it, but I would ask the authors to fix the normalization issue before publication.\n\nMy recommendation: send it to peer review, with a request that the referee check Definition 3.9 carefully. I would not accept it as is, but the core result is worth the referee time.","headline":"A serious isometry theorem for circle-valued persistence, but the interleaving distance is not well-defined as written because the Φ maps are only defined up to scalar.","tokens_in":31878,"tokens_out":5888,"would_cite":true,"duration_ms":58551,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N31","16G20","16G70"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the interleaving distance equals the bottleneck distance for persistence modules of circle-valued functions, with barcodes drawn as graded arcs and closed curves on an annulus.","keywords":["persistent homology","circle-valued functions","interleaving distance","bottleneck distance","isometry theorem","Auslander-Reiten translate","geometric model","gentle algebras"],"falsifier":"Fix any normalization of $\\Phi_M$ in a homogeneous tube, for example set $\\Phi_M = c\\,\\varphi$ with $c=1$ and $c=2$ on the same quasi-simple $M$, and compute the minimal $\\delta$ for which $M$ is $\\delta$-interleaved with itself under Definition 3.9; the metric axioms require this distance to be zero independent of the scaling, and if the two scalings yield different interleaving predicates or different distances to a fixed module $N$, the isometry theorem fails as stated, with Example 9.1 (the winding-number comparison) as a concrete case to recompute under each scaling.","tokens_in":30899,"feed_emoji":"⭕","tokens_out":11362,"duration_ms":99725,"temperature":0.7,"pith_summary":"This paper establishes an isometry theorem for persistence modules attached to circle-valued functions: the interleaving distance between two such modules equals the bottleneck distance between their barcodes. The barcodes are not classical interval barcodes but multisets of graded arcs and closed curves on an annulus, the geometric model for the derived category of the relevant quiver. The proof runs through the Auslander-Reiten translate, showing that the interleaving relation between morphisms is mirrored exactly by endpoint-sliding operations on these arcs and curves. If correct, the result gives a computable combinatorial distance that captures the full algebraic interleaving information in the circle-valued setting, extending the classical isometry theorem for real-valued persistence and its zigzag analogues.","feed_headline":"Interleaving equals bottleneck for circle-valued persistence","feed_subtitle":"A new isometry theorem makes the algebraic distance computable from arc-and-curve barcodes on an annulus.","key_machinery":"The load-bearing object is the Auslander-Reiten translate $\\tau$ on the bounded derived category of the gentle algebra $kQ$, together with the maps $\\Phi_M: M \\to \\tau^{-1}M$ formed by composing the two arrows of an Auslander-Reiten mesh. These maps replace the classical shift and transition morphisms: Definition 3.9 declares $M,N$ $\\delta$-interleaved when morphisms $f: M \\to \\tau^{-\\delta}N$ and $g: N \\to \\tau^{-\\delta}M$ satisfy $\\tau^{-\\delta}g \\circ f = \\Phi_M^{2\\delta}$ and $\\tau^{-\\delta}f \\circ g = \\Phi_N^{2\\delta}$. The matching geometric machinery is the annulus model of $\\tilde A$: indecomposable objects correspond to graded arcs (between marked boundary points, or between the two boundary components) and to graded primitive closed curves, and the operations $s$ and $t$ move an arc's start or end to the next marked point, exactly tracking the meshes of the Auslander-Reiten quiver. The bottleneck distance is then defined by $\\delta$-shortness and $\\delta$-equivalence under these endpoint movements, and the proof shows the two distances agree component-by-component.","core_discovery":"The central discovery is Theorem 7.6: for persistence modules $M,N$ of type $\\tilde A$ (representations of a non-cyclically oriented cycle quiver, the algebraic form of circle-valued persistence), the interleaving distance defined through Auslander-Reiten translates satisfies $d_I(M,N) = d_B(B(M),B(N))$, where $B(-)$ sends a module to the multiset of graded arcs and closed curves representing its indecomposable summands in the geometric model. The paper proves the equality in two parts: for preprojective and preinjective summands, interleavings give matchings in a bipartite graph whose edges are defined by the partial order $M \\leq \\tau^{-\\delta}N \\leq \\tau^{-2\\delta}M$, and conversely such matchings give interleavings; for regular summands inside tubes, the argument uses canonical injections and induced matchings, comparing lengths in uniserial tubes with the $\\delta$-short and $\\delta$-equivalent conditions on arcs. The barcode formalism thereby absorbs the Jordan-block data of circle-valued persistence into closed curves with an eigenvalue parameter, and the bottleneck distance compares these objects by moving their endpoints along the boundary components.","pith_inferences":["The proof's reliance on the geometric model of gentle algebras suggests the same two-distance formalism could be pushed to other gentle algebras whose geometric models admit endpoint operations with the same mesh-tracking property; the annulus and disc are the cases where those operations are already explicit.","The scaling ambiguity in Definition 2.1 leaves open the possibility that different normalizations of $\\Phi_M$ produce different interleaving distances, so a natural test is to fix a normalization and check whether the isometry statement survives; if it does not, the theorem needs a canonical choice before it is fully well-posed.","A consequence the authors leave implicit is that any two circle-valued maps whose level-set modules are $\\delta$-interleaved have $\\delta$-matched annulus barcodes, giving an algebraic stability statement in the spirit of the geometric stability results discussed in Section 8."],"forward_implications":["For any two circle-valued persistence modules of type $\\tilde A$, the interleaving distance can be computed as a bottleneck distance on finite multisets of arcs and closed curves, making the algebraic distance accessible to the same matching algorithms used for classical barcodes.","The classical isometry theorem for equioriented type A quivers is a special case: on the disc model, arcs between marked points are exactly the intervals $[a,b)$, and the endpoint operations reproduce the usual $\\delta$-matching condition.","The barcode of a circle-valued module encodes not only interval-like summands but also band objects $(\\gamma,\\lambda,l)$, so the Jordan-block invariants of level-set persistence are naturally part of the same distance framework.","The equality $d_I = d_B$ gives algebraic stability in the circle-valued setting: if two modules are $\\delta$-interleaved, their barcodes are $\\delta$-matched, and vice versa, so persistent features cannot move more than $\\delta$ under an interleaving of size $\\delta$.","Because both distances are metrics on their respective spaces, the isometry identifies the two metric spaces, so either notion can be used for comparison, with the bottleneck side supplying the finite combinatorial computation."],"supporting_citations":[{"why":"Supplies the geometric model of the derived category of a gentle algebra, identifying indecomposable objects with graded arcs and closed curves and defining the endpoint operations $s$ and $t$.","marker":"[33]"},{"why":"Provides the classification of indecomposable representations of type A quivers as interval representations, the base case the barcode generalisation extends.","marker":"[21]"},{"why":"Supplies the canonical-injection and induced-matching construction used in Section 7 to turn regular-module interleavings into bottleneck matchings.","marker":"[5]"},{"why":"Introduces barcodes and Jordan blocks for circle-valued maps, the invariants whose comparison with the annulus model is worked out in Section 8.","marker":"[12]"},{"why":"Defines the classical interleaving distance that Definition 3.9 generalises to the derived-category setting via the Auslander-Reiten translate.","marker":"[16]"},{"why":"Defines the bottleneck distance whose generalisation to arcs and closed curves is Definition 5.12.","marker":"[17]"},{"why":"Shows how a derived-category equivalence yields an isometry theorem for zigzag persistence, the template this paper adapts to type $\\tilde A$.","marker":"[26]"}],"fun_headline_variants":["Circle-valued persistence distances unify via arcs","Annulus arc barcodes tie persistence distances","New isometry bridges interleaving and bottleneck","Circle modules: interleaving equals bottleneck","Arc-based barcodes unify circle persistence distances"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Definition 2.1 defines the morphism $\\Phi_M$ only up to multiplication by a nonzero scalar of the field, and Definition 3.9 uses $\\Phi_M^{2\\delta}$ and $\\Phi_N^{2\\delta}$ in the interleaving equations without fixing those scalars, so the predicate 'M and N are $\\delta$-interleaved' is not precisely defined as written; the isometry theorem inherits this dependence unless a normalization is supplied.","fun_headline_variants_meta":{"raw":{"variants":["Circle-valued persistence distances unify via arcs","Annulus arc barcodes tie persistence distances","New isometry bridges interleaving and bottleneck","Circle modules: interleaving equals bottleneck","Arc-based barcodes unify circle persistence distances"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001451,"raw_usage":{"total_tokens":5794,"prompt_tokens":845,"completion_tokens":4949,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":461,"completion_tokens_details":{"reasoning_tokens":4879}},"tokens_in":461,"tokens_out":4949,"duration_ms":34540,"temperature":1.0,"reasoning_tokens":4879,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:15:10.197137+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix any normalization of $\\Phi_M$ in a homogeneous tube, for example set $\\Phi_M = c\\,\\varphi$ with $c=1$ and $c=2$ on the same quasi-simple $M$, and compute the minimal $\\delta$ for which $M$ is $\\delta$-interleaved with itself under Definition 3.9; the metric axioms require this distance to be zero independent of the scaling, and if the two scalings yield different interleaving predicates or different distances to a fixed module $N$, the isometry theorem fails as stated, with Example 9.1 (the winding-number comparison) as a concrete case to recompute under each scaling.","supporting_citations":[{"cited_title":"Persistence for Circle Valued Maps","cited_arxiv_id":"1104.5646","evidence_quote":"Introduces barcodes and Jordan blocks for circle-valued maps, the invariants whose comparison with the annulus model is worked out in Section 8."},{"cited_title":"Cohen-Steiner, H","cited_arxiv_id":null,"evidence_quote":"Defines the bottleneck distance whose generalisation to arcs and closed curves is Definition 5.12."},{"cited_title":"Hiraoka, Y","cited_arxiv_id":null,"evidence_quote":"Shows how a derived-category equivalence yields an isometry theorem for zigzag persistence, the template this paper adapts to type $\\tilde A$."}],"review_version":1}