{"id":"c0663a84-7d75-4d62-9fd7-3d3d7844b7a3","arxiv_id":"2501.00322","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every bipath persistence module is determined by the barcode of a covering infinite zigzag module, yielding decomposition algorithms and algebraic stability for bipath persistence.","lead":"Bipath persistence modules, one of only three settings where persistence modules always split into interval pieces, are shown to be equivalent to a periodic infinite zigzag module via a covering map. This lets researchers reuse fast zigzag algorithms and stability theorems for bipath persistence, and introduces the fibered arc code as a new invariant for 2D persistence.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central bijection in Theorem 2.3 rests on Lemma 2.2, whose proof is omitted; without verifying that the restricted interval modules are the claimed disjoint periodic zigzag sums, the reduction of bipath to zigzag persistence is not established.","rationale":"The reader's verdict is CONDITIONAL, and our stress-test supports that verdict. The reader's weakest_assumption focuses on the external Aoki-Escolar-Tada theorem that every bipath module is interval-decomposable; that is indeed a load-bearing assumption, but it is a cited external result and the paper's novelty does not rest on proving it. The more pressing internal gap is the unproved Lemma 2.2, which is the key computation behind Theorem 2.3. The entire reduction of bipath persistence to zigzag persistence, as well as the computational reduction in Section 4, depends on knowing the restriction of each interval module and knowing that distinct interval types produce disjoint families of intervals up to the Z-action. The paper states the lemma without proof and gives only a sketch in Theorem 4.1. We checked the covering map on small examples and found no evident contradiction: the apparent conflict in Definition 2.1 at k=0 is resolved by reading the last line as applying to (0,-1) and (0,0), not to (1,0). The formulas in Lemma 2.2 are plausible and consistent with the geometric picture. Nevertheless, the absence of a proof is a genuine gap: if any boundary case is miscomputed, the bijection between the arc code and Z-orbits of B(R(M)) would fail, and the central claim would collapse. The proposed concrete test—an independent enumeration for n=m=4 covering all interval types—would settle whether the lemma is correct. Since this is a fillable gap rather than a demonstrated error, the appropriate verdict remains CONDITIONAL, and our read does not change the reader's verdict. We agree only partially with the reader's weakest_assumption because we identify the omitted proof of Lemma 2.2 as the most load-bearing internal concern, while the reader highlighted the external decomposability theorem; both are relevant, but the internal gap is more directly connected to the paper's central construction.","tokens_in":11764,"tokens_out":23182,"duration_ms":204589,"concrete_test":"Independently verify Lemma 2.2 by direct computation for a small bipath, e.g., n=m=4. For each interval type (full, left with j=0 and with j≠0, right, top, bottom), enumerate the preimage under ζ of the interval, confirm it is exactly the claimed periodic direct sum of ZZ-intervals, and confirm that the internal maps of the restricted module match the interval module structure. Then build a module M that is a direct sum of two intervals of different types, such as a left interval and a right interval, and check that B(R(M)) contains no interval that is a summand of both R(kI) and R(kJ); this verifies the multiplicity-recovery step needed for the bijection in Theorem 2.3. This can be done by hand or with a short computer script enumerating the finite preimages.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2.3 claims that the barcode B(R(M)) of the infinite zigzag restriction is a complete invariant of a bipath module M and that its Z-orbits are in bijection with the arc code B(M). This claim depends entirely on Lemma 2.2, which computes R(kI) for each of the five interval types of a bipath poset. The lemma is stated without proof. A rigorous proof must establish three things: (1) the covering map ζ is a poset map and the preimage of each interval I is exactly the claimed disjoint union of ZZ-intervals; (2) the internal maps of the restricted interval module match those of the direct sum of the corresponding interval modules; (3) for distinct interval types, the resulting intervals in B(R(M)) do not coincide up to the Z-action, so multiplicities can be recovered uniquely. The text asserts the direct-sum preservation as 'easy to check' and the computation in Theorem 4.1 as 'not difficult to see,' but the key bijection in Theorem 2.3 is not proven. If any of these checks fail—for example, if a boundary case such as the j=0 left interval or the j=n right interval yields an interval that overlaps with the image of another type—then the bijection, and hence the claimed completeness of the invariant, would be false. Even granting the external Aoki-Escolar-Tada interval-decomposability theorem, the internal reduction is incomplete without a proof of Lemma 2.2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a poset map ζ from the infinite zigzag poset ZZ to a bipath poset B, and studies the induced restriction functor R from bipath modules to zigzag modules. It claims (Theorem 2.3) that the barcode of R(M) is a complete invariant up to isomorphism and that the arc code of M is recovered as the quotient of this barcode by a free Z-action. It then defines interleaving and bottleneck distances for bipath modules via R, obtaining an isometry theorem (Theorem 3.5) from the known block isometry theorem for zigzag modules. It gives a finite-slice version for computation (Theorem 4.1), and introduces a fibered arc code for R2 persistence modules, with examples comparing it to the fibered barcode and the interval rank invariant.","tokens_in":12055,"tokens_out":14117,"duration_ms":147692,"significance":"If the main theorems hold, the paper's covering construction is a valuable conceptual and computational bridge: bipath persistence becomes a periodic zigzag persistence, which lets known zigzag algorithms and stability results apply directly. The proposed finite-slice reduction to a single finite zigzag is appealing, and the explicit comparison of the fibered arc code with the fibered barcode in Section 5 is useful. The paper is honest about its dependence on the Aoki-Escolar-Tada interval-decomposability theorem and about limitations of the new invariant (Example 5.3). The main reservation is that the key restriction computation Lemma 2.2 is asserted without proof, and the finite-slice uniqueness assertion in Theorem 4.1 is only sketched.","major_comments":[{"comment":"This lemma is the load-bearing step for Theorem 2.3, yet it is stated without proof. Please supply a complete proof that: (a) ζ is a poset map and the preimage of each interval is exactly the claimed disjoint union of ZZ-intervals; (b) the restriction of the interval module kI to each summand has the stated interval support with identity internal maps; and (c) the resulting periodic intervals from different bipath interval types or parameters are pairwise non-isomorphic modulo the Z-action, so that the orbit map in Theorem 2.3 is well defined and injective. Boundary cases (j=0 for left intervals, j=n for right intervals) need explicit treatment. The sentence 'straightforward to verify' in Section 2.1 for the classification of bipath intervals should also be expanded or referenced.","section":"Section 2.2, Lemma 2.2"},{"comment":"The proof reduces to the sentence 'It is not difficult to see...' and the final assertion that each listed zigzag interval is an indecomposable summand of R'(kI) and not of any other R'(kJ). Since the finite slice truncates infinite periodic summands, the identification of which summand survives in ZZ' and the proof of uniqueness require a case-by-case check over all interval types and parameters. Please provide the complete argument; as written, the computational claim is not independently verifiable.","section":"Section 4, Theorem 4.1"},{"comment":"The orbit-replacement argument in the proof does not obviously produce a bijection: if an interval J outside the orbit of I is matched by τ to some translate of τ(I), then after the replacement two source intervals are matched to the same target interval. Since Lemma 3.6 is used to justify the equivalent bottleneck-distance definition on arc codes and the claim that matchings pair intervals of the same type, either give a correct proof (for instance, by matching orbit representatives and using Z-equivariance of the interleaving distance) or present the arc-code matching as a separate definition whose equivalence is proved directly.","section":"Section 3.4, Lemma 3.6"}],"minor_comments":[{"comment":"The global assumption in Section 2.1 is that n,m are greater than 1, but Example 5.2 says n=3, m=1; the displayed poset in that example corresponds to n=3, m=2. Please correct the parameters and ensure the nondegeneracy assumption is consistent.","section":"Section 2.1 / Example 5.2"},{"comment":"The second displayed equality should read E(k[a,b)_ZZ)=k[a,b)_BL, not k[a,b)_ZZ.","section":"Section 3.3, Lemma 3.4"},{"comment":"In the left-interval bullet with j≠0, write µ_M(k[i,j]_◁) instead of µ_M([i,j]_◁) for consistency.","section":"Section 4, Theorem 4.1"},{"comment":"The statement that R preserves direct sums 'since our persistence modules are pointwise finite-dimensional' is misleading: precomposition with ζ preserves direct sums for all persistence modules. Consider rephrasing.","section":"Section 2.2"},{"comment":"The matrices defining M_λ are hard to parse in the present rendering; please typeset them with clear row and column labels.","section":"Section 5, Example 5.2"}],"recommendation":"major_revision","confidential_remarks":"The central construction is elegant and probably correct, but the missing proof of Lemma 2.2 is too large a gap for acceptance in the current form. I expect a case-by-case verification is straightforward and would make the paper acceptable. I do not see circularity or novelty concerns; the relation to Asashiba-Liu is cited. The 'm=1' typo in Example 5.2 and the Lemma 3.6 proof issue should be fixed in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Angel, quick take. This paper does something genuinely useful: it shows that bipath persistence modules can be restricted along a covering map to an infinite zigzag, and that the resulting periodic barcode determines the arc code up to Z-orbits. The reduction is new—prior work used two zigzags, here one suffices—and it lets known zigzag algorithms and stability results apply directly to bipath modules. The isometry theorem (Theorem 3.5) follows cleanly once the covering relation is in place, and the finite-slice theorem in Section 4 is a practical payoff. The fibered arc code examples in Section 5 are a nice touch, showing strictly more discrimination than the fibered barcode. The citation of prior work looks fair, and the main theorems are clearly stated.\n\nWhat is not yet solid: the paper's main structural claim, Theorem 2.3, depends on Lemma 2.2, which states the restriction of each type of interval module. That lemma is given without proof. The accompanying claims that the interval classification is 'straightforward to verify' and certain direct-sum identifications are 'easy to check' are exactly the places where boundary cases can go wrong. In particular, I would want the proof of Lemma 2.2 spelled out: preimage of each interval, internal maps, and confirmation that the resulting ZZ-intervals for distinct bipath interval types do not coincide up to the Z-action. The stress-test note is right that this is load-bearing. Nothing in the text suggests the lemma is false, and the formulas look plausible, so I read this as a fillable gap rather than an error.\n\nMinor things: the proof of Lemma 3.6 hand-waves a bit, and the definition of interleaving distance for bipath modules is by construction rather than derived, which is fine but worth stating. The examples in Section 5 are not accompanied by formal proofs, but they are illustrative. The circularity concern is not warranted—the paper defines the distance via restriction and transfers known stability; that is a definition plus an external theorem, not a derivation that assumes its conclusion.\n\nBottom line: this deserves a serious referee, and I would send it out. The authors need to add the missing proofs, at least for Lemma 2.2 and the interval classification, before the results should be taken as established. If those checks pass, this is a clean, citable reduction.","headline":"Clean covering-map reduction of bipath to zigzag persistence, with a real isometry theorem, but the key lemma is unproved—send it out and ask for the details.","tokens_in":12630,"tokens_out":3050,"would_cite":true,"duration_ms":30998,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N31","16G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that every bipath persistence module is encoded by an infinite periodic zigzag module, making the arc code a complete invariant and giving bipath persistence the same algebraic stability as zigzag persistence.","keywords":["bipath persistence","zigzag persistence","arc code","covering map","algebraic stability","bottleneck distance","interleaving distance","fibered arc code"],"falsifier":"Exhibit two non-isomorphic bipath modules $M$ and $N$ whose restrictions $R(M)$ and $R(N)$ have identical zigzag barcodes; Theorem 2.3 says the barcode of the restriction is a complete invariant, so such a pair cannot exist. Equivalently, finding any pair of bipath modules with $d_I(M,N) \\neq d_B(M,N)$ would refute Theorem 3.5.","tokens_in":11586,"feed_emoji":"🔁","tokens_out":8438,"duration_ms":76372,"temperature":0.7,"pith_summary":"This paper shows that bipath persistence—the third and last setting in which persistence modules always decompose into interval modules—can be reduced to zigzag persistence. The reduction is a covering map from an infinite zigzag poset to the bipath poset: restricting a bipath module along it produces an infinite periodic zigzag module whose barcode, modulo a $\\mathbb{Z}$-shift, is exactly the bipath module's arc code. Because the barcode of that zigzag module is a complete isomorphism invariant, the arc code is complete as well. The paper then defines interleaving and bottleneck distances on bipath modules through the zigzag, proving they coincide and thereby transferring algebraic stability from zigzag to bipath persistence. It also shows that only a finite slice of the zigzag, of size $O(K)$ for maximal chain length $K$, is needed to compute the arc code.","feed_headline":"One covering map turns bipath persistence into zigzag persistence","feed_subtitle":"The arc code becomes a complete invariant, and zigzag stability and O(K) computation carry over.","key_machinery":"The carrying mechanism is the covering map $\\zeta : ZZ \\to B$, which wraps the infinite zigzag poset $ZZ$ around the bipath poset $B$ periodically, and the restriction functor $R : \\mathrm{Vec}^B \\to \\mathrm{Vec}^{ZZ}$ it induces. The map is modeled on the universal covering of the circle and sends each of the two chains of the bipath to alternating segments of the zigzag. Under $R$, every interval module on the bipath becomes a direct sum of zigzag interval modules indexed by $\\mathbb{Z}$, so the barcode of $R(M)$ is periodic and the arc code of $M$ is exactly the orbit space of that barcode under the $\\mathbb{Z}$-shift.","core_discovery":"The central claim is Theorem 2.3: for any bipath module $M$, the barcode $B(R(M))$ of the infinite zigzag module obtained by restricting along the covering map is a complete invariant up to isomorphism, and the arc code $B(M)$ is in bijection with the $\\mathbb{Z}$-orbits of $B(R(M))$. Each interval of the bipath—full, left, right, top, or bottom—lifts to a $\\mathbb{Z}$-periodic family of zigzag intervals, so the infinite zigzag barcode is periodic and its orbit space recovers the arc code. From this, Theorem 3.5 concludes that the interleaving distance $d_I(M,N)$ equals the bottleneck distance $d_B(M,N)$ for all bipath modules, which is algebraic stability: modules that are $\\epsilon$-interleaved have arc codes that are $\\epsilon$-close in bottleneck distance.","pith_inferences":["One extension the authors leave implicit is a general recipe: any finite poset equipped with a periodic zigzag covering would inherit zigzag stability and zigzag algorithms, with bipath posets as the worked-out instance.","Because the arc code is a $\\mathbb{Z}$-quotient of a periodic barcode, one could try to prove stability of the fibered arc code under interleavings of $\\mathbb{R}^2$ modules; the paper does not state such a result.","The finite-slice theorem makes bipath decomposition cost $O(K)$ times the cost of decomposing one zigzag of length about $K$; comparing this with the matrix-algorithm route on random bipath modules would be a direct practical test.","The periodic-zigzag viewpoint connects bipath persistence to persistence modules over finite subsets of the circle, where decompositions into intervals and Jordan cells appear; whether the arc code extends to that setting is left open."],"forward_implications":["To decompose a bipath module, one needs only decompose the finite zigzag slice of size $O(K)$, so any zigzag decomposition algorithm—including fast algorithms for simplex-wise filtrations—can be applied to bipath persistence directly.","Bipath modules inherit the isometry theorem: $d_I(M,N) = d_B(M,N)$, so algebraic stability of zigzag persistence automatically becomes algebraic stability of bipath persistence.","The arc code is a complete isomorphism invariant: two bipath modules are isomorphic if and only if their arc codes coincide.","The fibered arc code of a 2-D persistence module is strictly more discriminating than the fibered barcode: it separates two non-isomorphic modules with equal fibered barcode, though it is not itself complete.","Epsilon-matchings between arc codes can be taken equivariantly with respect to the $\\mathbb{Z}$-action and pair intervals of the same bipath type, giving the bottleneck distance a combinatorial description directly on arc codes."],"supporting_citations":[{"why":"Supplies the classification theorem that every pointwise finite-dimensional module over a bipath poset decomposes into interval modules, the premise for defining arc codes.","marker":"[3]"},{"why":"Introduces bipath modules, their five interval types, and a matrix-based decomposition algorithm that the present zigzag reduction complements.","marker":"[4]"},{"why":"Establishes that pointwise finite-dimensional zigzag modules decompose into interval modules, so the infinite zigzag restriction has a well-defined barcode.","marker":"[12]"},{"why":"Proves the block isometry theorem ($d_I = d_B$ for block-decomposable modules) that is transferred to bipath modules in Theorem 3.5.","marker":"[10]"},{"why":"Develops algebraic stability of zigzag persistence through block extensions, the route by which stability reaches bipath persistence.","marker":"[11]"},{"why":"Provides the fast zigzag decomposition algorithm that the finite-slice computation in Section 4 makes applicable to bipath modules.","marker":"[17]"}],"fun_headline_variants":["Bipath persistence meets zigzag via a covering map","Covering map links bipath and zigzag persistence","Zigzag covering decodes bipath persistence completely","Bipath to zigzag: a covering map gives full invariant","Infinite zigzag barcode from bipath modules"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every pointwise finite-dimensional persistence module over a bipath poset decomposes into interval modules; if that classification ever failed, the arc code and the reduction to a finite zigzag slice would not be defined for arbitrary bipath modules.","fun_headline_variants_meta":{"raw":{"variants":["Bipath persistence meets zigzag via a covering map","Covering map links bipath and zigzag persistence","Zigzag covering decodes bipath persistence completely","Bipath to zigzag: a covering map gives full invariant","Infinite zigzag barcode from bipath modules"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000748,"raw_usage":{"total_tokens":3301,"prompt_tokens":885,"completion_tokens":2416,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":501,"completion_tokens_details":{"reasoning_tokens":2336}},"tokens_in":501,"tokens_out":2416,"duration_ms":17255,"temperature":1.0,"reasoning_tokens":2336,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:53:41.992955+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit two non-isomorphic bipath modules $M$ and $N$ whose restrictions $R(M)$ and $R(N)$ have identical zigzag barcodes; Theorem 2.3 says the barcode of the restriction is a complete invariant, so such a pair cannot exist. Equivalently, finding any pair of bipath modules with $d_I(M,N) \\neq d_B(M,N)$ would refute Theorem 3.5.","supporting_citations":[{"cited_title":"Summand-injectivity of interval covers and monotonicity of interval resolution global dimensions","cited_arxiv_id":"2308.14979","evidence_quote":"Supplies the classification theorem that every pointwise finite-dimensional module over a bipath poset decomposes into interval modules, the premise for defining arc codes."},{"cited_title":"Escolar, and Shunsuke Tada","cited_arxiv_id":null,"evidence_quote":"Introduces bipath modules, their five interval types, and a matrix-based decomposition algorithm that the present zigzag reduction complements."},{"cited_title":"Interval Decomposition of Infinite Zigzag Persistence Modules.Pro- ceedings of the American Mathematical Society, 145(8):3571–3577, January 2017","cited_arxiv_id":null,"evidence_quote":"Establishes that pointwise finite-dimensional zigzag modules decompose into interval modules, so the infinite zigzag restriction has a well-defined barcode."},{"cited_title":"On the Stability of Interval Decomposable Persistence Mod- ules","cited_arxiv_id":null,"evidence_quote":"Proves the block isometry theorem ($d_I = d_B$ for block-decomposable modules) that is transferred to bipath modules in Theorem 3.5."},{"cited_title":"Algebraic Stability of Zigzag Persistence Modules","cited_arxiv_id":null,"evidence_quote":"Develops algebraic stability of zigzag persistence through block extensions, the route by which stability reaches bipath persistence."}],"review_version":1}