{"id":"99ea1d23-ee1d-46b0-b809-79c830852d0a","arxiv_id":"2411.11594","paper_version":6,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"An explicit rank formula computes interval multiplicities for persistence modules over arbitrary finite posets, generalizing the one-parameter persistence formula.","lead":"Mathematicians give a formula that counts how many times an interval module appears inside any multi-parameter persistence module over a finite poset, using only ranks of structure matrices. It turns a hard algebraic question into linear algebra, and makes interval-decomposability checking practical for topological data analysis.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main Theorem 3.27 depends on Proposition 3.13, a projective presentation restated without proof from an unreviewed arXiv preprint; an error there would invalidate the rank formula.","rationale":"The reader's weakest_assumption identifies the same reliance on auxiliary projective presentations. I agree and elevate this to a conditional-acceptance issue because Proposition 3.13 is not proved in the paper and comes from an unreviewed preprint; the main theorem is only as secure as that lemma. The concrete test would settle the concern: if exactness and choice-independence are confirmed, the original ACCEPT stands. No internal inconsistency in the rest of the proof was found, and the examples are consistent with the stated formula.","tokens_in":42349,"tokens_out":33790,"duration_ms":315010,"concrete_test":"Check Proposition 3.13 in the smallest case where sc1(U) has two elements, e.g., the poset of Figure 1: explicitly write the complex 0 -> P_sc1(U) -> P_sc(U) -> V_U -> 0 and verify exactness by computing homology. Then for a randomly chosen finite-dimensional module M over that poset, verify Lemma 2.9 numerically by checking dim Hom(V_U,M) = sum_{a in sc(U)} dim M(a) - rank M(epsilon_1^U). In addition, compute the right-hand side of (3.32) for the same M and I using two different choice maps c,d and two different comparable pairs (j,i); differing values would show the formula is ill-posed and the external presentation is faulty.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is Theorem 3.27, the rank-only formula for d_M(V_I). Its proof reduces, via Lemma 2.9 and Theorem 3.3, to the existence of explicit projective presentations of V_I, tau^{-1}V_I, and the middle term E of the almost split sequence. The presentation of V_U for a connected up-set U is Proposition 3.13, which is restated from Asashiba et al. (2024, Prop. 5.10) with the proof omitted ('We refer the reader...'). The 2D-grid minimality is Lemma 3.30 from Asashiba et al. (2022, Prop. 39), and the minimal presentation of DV_I is Proposition 3.34, also stated without proof. The exact index sets sc1(I), sk1(I), the signs in epsilon1 and pi1, and the form of M(lambda) all come from these auxiliary results. If any of these presentations contained an error, such as a wrong sign, a missing summand in sc1(I), or a failure of exactness, then the block-rank expression in (3.32) would no longer compute d_M(V_I), even though the surrounding Hom-space arguments are sound. Since Proposition 3.13 is drawn from an unreviewed arXiv preprint, this is the least secure link in the proof chain.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper gives an explicit formula for the multiplicity d_M(V_I) of an interval module V_I in any persistence module M over a finite poset P, expressed in terms of ranks of matrices built from the structure maps of M. The central result is Theorem 3.27, abstracted as formula (1.2), which is obtained by combining the Auslander–Reiten formula of Theorem 3.3 with explicit projective presentations of V_I, τ^{-1}V_I, and the middle term E of the almost split sequence. The formula specializes to a simpler form for 2D grids (Theorem 3.36) and is used to introduce the notion of an essential cover ζ:Z→P, under which the multiplicity is preserved by restriction (Theorem 4.15). The final section applies these techniques to zigzag and bipath persistence computations, with worked examples.","tokens_in":42607,"tokens_out":9072,"duration_ms":89954,"significance":"If the main theorem is correct, the paper provides a genuinely useful generalization of the one-parameter persistent Betti number formula: it computes interval multiplicities directly from rank data without decomposing the module, and it identifies which structure maps are essential. The essential-cover results connect the rank formula to zigzag persistence algorithms, which is a valuable practical bridge. The paper is technically detailed: the reduction to Hom-space dimensions, the pushout construction of the almost split sequence, and the worked examples (Example 3.38, Examples 5.4–5.5) are carefully presented. The main caveat is that several load-bearing projective presentations are quoted from earlier papers, including an unreviewed arXiv preprint, rather than proved here.","major_comments":[{"comment":"The projective presentation of V_U for a connected up-set U is the foundation of the whole formula, but it is restated from Asashiba et al. (2024, Proposition 5.10) with the proof omitted, and Proposition 3.34 is likewise stated without proof. The exact index sets sc1(I), sk1(I), the signs inside ε1 and π1, and the exactness of the sequences determine the block-rank expression (3.32); an error in any of these auxiliary statements would change the claimed multiplicity even though the surrounding Hom-space arguments are sound. Since Proposition 3.13 is quoted from an unreviewed arXiv preprint, please either include a self-contained proof (an appendix would suffice) or replace the reference by a peer-reviewed version.","section":"§3.1, Proposition 3.13"},{"comment":"The assertion that formula (3.32) covers the injective case is not proved. The preceding Theorem 3.25 assumes m≥2, while the injective case is handled separately by Theorem 3.20 with the different expression (3.19). Since Theorem 3.27 is stated for all intervals, the paper should include a short verification that for I=↓b the index sets sk(⇓I) and sk1(I) are empty (so that M(π1) disappears) and that the remaining λ-block agrees with M(ε''1) under the choice map. This is a local gap, but it is load-bearing for the full statement of the main theorem.","section":"§3.1.1, Remark 3.26"},{"comment":"After deriving r = s + (rank L(g')_block - rank L(g')_diag), the proof concludes '≥ s' without justification. For an arbitrary matrix [A 0; C B], the difference rank[A 0; C B] - rank A - rank B can be negative, so the displayed implication is not automatic. The step is valid because applying formula (4.38) to M=L identifies this difference with d_L(VI) ≥ 0, but this should be stated explicitly; otherwise the inequality r ≥ s does not follow from the preceding equations.","section":"§4, proof of Theorem 4.15"}],"minor_comments":[{"comment":"There are several typos: 'Auslandr-Reiten' in §1.2, 'examaple' in Notation 3.4, 'Cosider' at the start of Examples 5.4 and 5.5, 'filed' for 'field' in the Introduction, and 'we notice the reader' in Remark 5.8.","section":"Throughout"},{"comment":"There are cross-reference mismatches: Example 5.1 appeals to 'Theorem 4.11' for the essential-cover property, but the relevant statement is Definition 4.11; Example 5.3 says 'adopting Theorem 3.29' where Notation 3.29 is meant; Example 5.5 refers to 'Theorem 4.10' when Lemma 4.9 or Definition 4.11 appears intended.","section":"Examples 5.1, 5.3, 5.5"},{"comment":"Items (2) and (3) of Lemma 3.39 are printed identically; please clarify the intended distinction between the two statements, presumably one is an equality of dimension vectors and the other an equality of total dimensions.","section":"Lemma 3.39"},{"comment":"The remark explains that missing columns in the minimal 2D-grid presentation are eliminated by column operations, but it does not specify the actual column operations. A one-line indication, or a reference to the exact proof in Asashiba et al. (2022), would make the minimal presentation easier to verify.","section":"Remark 3.31"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the paper's reliance on Asashiba et al. (2024, arXiv:2403.08308) for Proposition 3.13, which is not peer-reviewed as far as the reference list indicates. The editor may wish to confirm the status of that preprint and consider whether the manuscript should reproduce the proof of the central projective presentation. The paper also continues the authors' own program with substantial self-citation; this is natural here, but the overlap with the cited arXiv preprint should be checked for duplication requirements."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper gives the first explicit rank formula for interval multiplicities over arbitrary finite posets, and that is the real news. Theorem 3.27 expresses d_M(V_I) as a difference of two ranks built from structure maps of M. It generalizes the classical one-parameter formula and gives you a direct way to get the maximal interval-decomposable summand and test interval-decomposability without full decomposition. The essential-cover theorem (4.15) is a nice second contribution: it tells you which morphisms matter, and in good cases reduces the computation to zigzag persistence. The bipath application with explicit per-type formulas is a useful bonus, and the worked examples actually check out.\n\nThe proofs are detailed and the Hom-space machinery in Lemma 2.9 is clean. The main soft spot is exactly what the stress-test note flags: the central formula depends on Proposition 3.13, a projective presentation of V_U restated without proof from Asashiba et al. 2024, an unreviewed arXiv preprint. Lemma 3.30 and Proposition 3.34 are likewise restated without proof. An error in those index sets or signs would break (3.32). This is not fatal—the 2022 source is published and these are standard Auslander–Reiten computations—but it is a real self-containedness gap. The authors should either give proofs in an appendix or point to a published version of the 2024 preprint. The referee should check those presentations carefully. Remark 3.26, which claims the injective case is covered without proof, is a minor issue.\n\nThis is a paper for both representation theorists and TDA researchers, and it deserves a serious referee. The core claim is likely correct, the formula is genuinely useful, and the limitations are clearly inherited from cited work rather than from sloppy reasoning. I would accept the paper conditional on the auxiliary presentations being proven or properly referenced.\n\nRecommendation: send it to peer review, with referees asked to verify Proposition 3.13 and the related restated lemmas.","headline":"Genuinely new rank formula for interval multiplicities over finite posets, resting partly on the authors' own prior presentations—one from an unreviewed arXiv preprint—but solid enough to warrant serious refereeing.","tokens_in":43168,"tokens_out":1578,"would_cite":true,"duration_ms":18162,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G20","16G70","55N31","62R40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any persistence module over a finite poset, the multiplicity of each interval module is a rank computation.","keywords":["multi-parameter persistence","interval multiplicity","persistence module","finite poset","interval module","rank formula","essential cover","zigzag persistence"],"falsifier":"Take a small finite poset such as a 2D-grid $G_{2,2}$ or $G_{4,2}$, construct a persistence module $M$ by explicit matrices, compute the right-hand side of (3.32) for every interval $I$, and compare with the multiplicities obtained by an independent decomposition of $M$ (for instance by hand or by a separate algorithm). A single mismatch would refute the formula; the paper's Example 3.38 performs this check for one module, and a broader sweep over random modules and all intervals would settle the claim.","tokens_in":42196,"feed_emoji":"📐","tokens_out":4806,"duration_ms":41749,"temperature":0.7,"pith_summary":"This paper establishes an explicit formula that computes how many times a given interval module appears as a direct summand of any persistence module over a finite poset, using only the ranks of matrices assembled from the module's structure maps. This generalizes the classical one-parameter persistence multiplicity formula to arbitrary finite posets, including multi-parameter grids. If correct, it turns interval multiplicity from a decomposition problem into a linear-algebra calculation, and it identifies exactly which maps matter, leading to a reduction technique that can compute multiplicities via zigzag persistence. The formula also yields the maximal interval-decomposable direct summand of a module and a criterion for interval-decomposability.","feed_headline":"Ranks alone count interval summands in persistence modules","feed_subtitle":"A new formula extends the 1-D persistence multiplicity count to any finite poset, no decomposition needed.","key_machinery":"The machinery is Auslander–Reiten theory for the incidence category $k[\\mathbf{P}]$: the paper computes projective presentations of the interval module $V_I$, of the middle term $E$ of the almost split sequence starting at $V_I$, and of $\\tau^{-1}V_I$, then converts Hom-space dimensions into ranks via Lemma 2.9. The assembled morphisms $\\varepsilon_1$ and $\\pi_1$ encode the pre-join and pre-meet combinatorics of $I$, and their block matrix with $\\lambda$ produces the rank expression (3.32). The essential-cover result rests on these explicit morphisms: a poset map $\\zeta$ covers the morphism $g$ if every entry has a preimage, and then $\\operatorname{rank} M(g)$ is preserved under restriction.","core_discovery":"The central claim is Theorem 3.27: for every persistence module $M$ over a finite poset $\\mathbf{P}$ and every interval $I$, the multiplicity $d_M(V_I)$ equals the rank of the block matrix $\\begin{pmatrix} M(\\varepsilon_1) & 0 \\\\ M(\\lambda) & M(\\pi_1) \\end{pmatrix}$ minus $\\operatorname{rank} M(\\varepsilon_1)$ minus $\\operatorname{rank} M(\\pi_1)$, where $\\varepsilon_1$, $\\pi_1$, and $\\lambda$ are built from structure maps of $M$ along the pre-join and pre-meet sets of $I$. This gives the first explicit rank-only formula for interval multiplicities in this generality, and it specializes to the familiar inclusion-exclusion formula in the one-parameter case. It also provides an essential-cover theorem: if an order-preserving map $\\zeta: Z \\to \\mathbf{P}$ covers the matrix data needed for $I$, then the multiplicity is unchanged by restricting $M$ to $Z$, so when $Z$ is a zigzag poset the multiplicity can be read off from zigzag persistence of the filtration.","pith_inferences":["The rank formula likely extends to interval multiplicities in relative or truncated persistence settings, wherever the same projective presentation data can be lifted.","The essential-cover condition suggests a general optimization: search for the smallest poset $Z$ whose cover of the matrix data for $I$ preserves ranks, possibly connecting to existing algorithms that unfold multiparameter modules to zigzag modules.","One could test the formula's stability under perturbations of $M$: since ranks are lower semicontinuous, the multiplicity computed by the formula may be stable under sufficiently small noise in the structure maps, yielding a stability-type statement the paper does not address.","The formula's dependence on choice maps $c$ and $d$ is proven immaterial, but those choices could be exploited to select sparse or well-conditioned matrices for numerical computation."],"forward_implications":["Interval multiplicities can be computed without decomposing $M$, as ranks of matrices.","The maximal interval-decomposable direct summand of $M$ can be read off, and $M$ is interval-decomposable exactly when its dimension vector matches the sum of these interval modules.","The formula shows which morphisms of $\\mathbf{P}$ matter, enabling essential covers to smaller posets; when the smaller poset is zigzag, multiplicities come from zigzag persistence of the filtration without computing all structure maps.","In 2D-grids and bipath posets the formula specializes to practical matrix expressions, bypassing basis changes at global extrema in bipath persistence."],"supporting_citations":[{"why":"Supplies the almost split sequence formula reducing $d_M(L)$ to Hom-space dimensions, the starting point of the proof.","marker":"Asashiba et al. 2017, Theorem 3"},{"why":"Restated as Proposition 3.13, gives the projective presentation of $V_U$ for a connected up-set $U$.","marker":"Asashiba et al. 2024, Proposition 5.10"},{"why":"Restated as Lemma 3.30, gives the minimal projective presentation of interval modules in the 2D-grid case.","marker":"Asashiba et al. 2022, Proposition 39"},{"why":"Provides the pushout construction of almost split sequences used to present the middle term $E$.","marker":"Gabriel 1980, Section 3.6"},{"why":"Restated as Lemma 4.13, shows rank is additive under direct sums, used in the essential-cover theorem.","marker":"Asashiba et al. 2024, Lemma 5.24"}],"fun_headline_variants":["Rank-only formula for interval multiplicities on any poset","Count interval summands without decomposing modules","Persistence multiplicities via ranks, for any finite poset","Multiplicity formula generalized to all finite posets","Zigzag shortcut: ranks reveal interval summands"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The rank formula's correctness depends on the previously established projective presentations of $V_I$, $E$, and $\\tau^{-1}V_I$ being correct; if any of those index sets or signs were wrong, the formula would fail.","fun_headline_variants_meta":{"raw":{"variants":["Rank-only formula for interval multiplicities on any poset","Count interval summands without decomposing modules","Persistence multiplicities via ranks, for any finite poset","Multiplicity formula generalized to all finite posets","Zigzag shortcut: ranks reveal interval summands"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000763,"raw_usage":{"total_tokens":3481,"prompt_tokens":1134,"completion_tokens":2347,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":750,"completion_tokens_details":{"reasoning_tokens":2269}},"tokens_in":750,"tokens_out":2347,"duration_ms":14772,"temperature":1.0,"reasoning_tokens":2269,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T18:21:17.035830+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small finite poset such as a 2D-grid $G_{2,2}$ or $G_{4,2}$, construct a persistence module $M$ by explicit matrices, compute the right-hand side of (3.32) for every interval $I$, and compare with the multiplicities obtained by an independent decomposition of $M$ (for instance by hand or by a separate algorithm). A single mismatch would refute the formula; the paper's Example 3.38 performs this check for one module, and a broader sweep over random modules and all intervals would settle the claim.","supporting_citations":[{"cited_title":"Japan Journal of Industrial and Applied Mathematics 34(2), 489–507 (2017) https://doi.org/10.1007/s13160-017-0247-y","cited_arxiv_id":null,"evidence_quote":"Supplies the almost split sequence formula reducing $d_M(L)$ to Hom-space dimensions, the starting point of the proof."},{"cited_title":"arXiv (2024)","cited_arxiv_id":null,"evidence_quote":"Restated as Proposition 3.13, gives the projective presentation of $V_U$ for a connected up-set $U$."},{"cited_title":"In: Dlab, V., Gabriel, P","cited_arxiv_id":null,"evidence_quote":"Provides the pushout construction of almost split sequences used to present the middle term $E$."},{"cited_title":"arXiv (2024)","cited_arxiv_id":null,"evidence_quote":"Restated as Lemma 4.13, shows rank is additive under direct sums, used in the essential-cover theorem."}],"review_version":1}