{"id":"405028e4-d987-499b-889d-557a7d92a667","arxiv_id":"2505.17858","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A new homological definition of open cobordisms, or tunnels, connecting two subcomplexes, with a matrix reduction algorithm that pairs their birth and death times in a filtration.","lead":"This math paper defines \"tunnels\" between two parts of a shape using homology, and gives an algorithm to compute when each tunnel appears and disappears as the shape is built up. It is aimed at describing voids and channels in atom networks, for example in battery materials or porous glasses.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 2's death-pairing step for cobordisms born in Case F is not fully justified: DPhi stores only image-side columns, so the reduction's ability to recover the true birth cell of a class never born in Ker(iota_A) or Ker(iota_B) is asserted rather than proved.","rationale":"I agree with the reader's identification of the fragile step: the proof of Theorem 4.1 must show that a cokernel class born in Case F, which is not in the image of Phi at birth, can be recovered at its death by reducing the image-side columns stored in DPhi. My reading sharpens this into a spanning/basis concern: DPhi's columns span the image of the domain map, not the ambient codomain, so the recovery claim needs an explicit argument about bases at intermediate filtration values. I do not see a definitive counterexample in the paper, and the surrounding mathematical structure is coherent, so the appropriate verdict remains conditional pending a rigorous proof or implementation-based validation. The reader's conditional verdict already reflects this risk, so no adjustment is needed.","tokens_in":12604,"tokens_out":9903,"duration_ms":114866,"concrete_test":"Write a brute-force checker over Z/2 for all small filtered simplicial complexes (e.g., complexes on up to 6 vertices with fixed disjoint A,B). At each filtration step, store explicit bases of Ker(iota_A), Ker(iota_B), Ker(iota_{A cup B}) and the map Phi, and compute the true interval decomposition of Cok(Phi) by dimension changes and induced maps. Run Algorithm 2 on the same inputs, with search biased toward complexes where a Case F birth is later killed by Case C/G/H. A single mismatch in the paired (birth,death) multiset refutes Theorem 4.1; broad agreement supports the contested spanning claim. The triangulated cylinder of Example 1 with varied insertion orders of top/middle triangles is an analytically checkable sanity case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing assumption in Theorem 4.1 is that the stored columns of DPhi suffice to recover, at death time, a representative of any cokernel class that dies, including classes born in Case F of Table 1. Lines 2-14 build DPhi exclusively from columns of V_A_im and V_B_im (plus absolute cycles), so the span of the columns of DPhi with filtration value at most t is contained in the image of the map H_{*+1}(X_t,A_t) oplus H_{*+1}(X_t,B_t) -> H_{*+1}(X_t,A_t cup B_t). A Case F birth is an element of the codomain that is not in this image at birth. The proof's final assertion ('by adding the columns to the left of tau we can get all equivalent representations') is exactly the missing step: at a later death tau, the dying class is in the image and therefore is a combination of the new tau-columns and older image columns, but nothing in the proof shows that the selected V_A_im/V_B_im columns form a compatible basis of the image at every intermediate filtration value, nor that left-to-right reduction of the new columns yields the lowest 1 at the birth cell of the original Case F class rather than at some later cell. This is the weakest point of the paper; it is not covered by Theorem 3.1, which only classifies when births and deaths occur and does not address pairing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces a homological model for 'open cobordisms' between two disjoint subcomplexes A and B of a filtered cell complex X, defined as elements of the cokernel of a map Φ between kernels of inclusion maps. It proves (Theorem 3.1) a classification of birth/death events in Cok(Φ) in terms of kernel persistence of the inclusions of A, B, and A∪B into X, and presents Algorithm 2 for pairing these birth and death times (Theorem 4.1). The motivating application is detecting tunnels connecting the top and bottom of a box in a point cloud, with birth and death times interpreted as tunnel separation and bottleneck thickness. The paper is primarily theoretical; it does not include an implementation.","tokens_in":12885,"tokens_out":10166,"duration_ms":75004,"significance":"The algebraic setup and the birth/death classification are clean and likely correct. The paper explicitly ships a matrix-reduction algorithm with a claimed correctness theorem; if Theorem 4.1 can be fully proved, the algorithm would be a useful contribution for computing a novel persistence invariant. The connection to kernel persistence is well motivated and the relative-homology formulation is instructive. However, the correctness proof of Algorithm 2 is incomplete at a load-bearing point, so the computational claim should be treated as conditional. The authors are honest about the sensitivity of the construction to the slice thickness ε and about the lack of discussion of representatives, which is positive.","major_comments":[{"comment":"The proof does not justify recovery of Case F births. The columns stored in DΦ are V_A_im and V_B_im columns plus absolute cycles; at filtration value t their span lies in the image of H_{*+1}(X_t,A_t) ⊕ H_{*+1}(X_t,B_t) → H_{*+1}(X_t,A_t∪B_t). A Case F class is born outside this image and enters it only at a later death time. The sentence 'by adding the columns to the left of τ we can get all equivalent representations of the tunnel that dies at time τ' asserts, but does not prove, that the stored columns form a basis of this image at every intermediate filtration step and that left-to-right reduction of the new τ-columns yields a lowest 1 at the true birth cell rather than at a later cell. Since Theorem 3.1 explicitly does not address pairing, this missing argument leaves the correctness of Algorithm 2 for Case F births unproved.","section":"§4.2, Theorem 4.1"},{"comment":"The global row ordering for DΦ is not filtration-compatible. The line orders only the X\\(A∪B) block by filtration value; the A∪B block is not globally ordered by filtration (cells of A and B can interleave in the filtration). The proof of Theorem 4.1 repeatedly invokes 'pushing the lowest 1 as far up as possible' to obtain the earliest born representation, which presupposes that the row order is the filtration order. As written, the reduction on DΦ may therefore select a lowest 1 that does not correspond to the earliest birth time among equivalent representatives, so the pairing rule in Lines 19–20 is not guaranteed to return the true birth–death pair. The algorithm should specify a total filtration-compatible ordering of all rows, or explain why the block ordering suffices.","section":"§4.2, Algorithm 2, Line 1"},{"comment":"Several assertions in the case analysis are made without proof. For example, item 7 claims that a τ-column with τ ∈ A can never be added only in Line 12 because then ∂τ would be trivial and τ would have been added in Line 2; this presumes that a column with lowest 1 in the B-block necessarily corresponds to a cycle in X, a fact that is not established. The phrase 'It is now straightforward to check which cases can occur' is not a substitute for deriving the case analysis from the reduction properties established in Algorithm 1. These gaps, together with the missing Case F argument, make Theorem 4.1 an outline rather than a complete proof.","section":"§4.2, Proof of Theorem 4.1"}],"minor_comments":[{"comment":"There is a typo: 'homlogical' should be 'homological'.","section":"Abstract"},{"comment":"There is a typo: 'graoups' should be 'groups'.","section":"Example 1"},{"comment":"The name 'Delauney' is misspelled; it should be 'Delaunay' in several places.","section":"§2.2 and elsewhere"},{"comment":"The justification that Cok(Φ) is isomorphic to the cokernel of the map in (4) is abbreviated; a short diagram chase would make the quotient argument rigorous and rule out potential issues with the image of j^{A∪B}_*.","section":"§2.1, Equation (4)"},{"comment":"The statement 'there is no class born in Ker(iota_A) and Ker(iota_B)' should be clarified to mean 'no new class born in either', to avoid ambiguity about simultaneous births in both kernels.","section":"Theorem 3.1"},{"comment":"The algorithm is described in prose and pseudocode, but the construction of DΦ from the reduced matrices is not given as a detailed listing; a step-by-step pseudocode block would aid reproducibility and verification.","section":"§4.2"},{"comment":"The paper contains no implementation or experiments. While the theoretical contribution can stand alone, including a small worked example (e.g., the cylinder from Example 1 or a synthetic point cloud) would substantially increase confidence in Algorithm 2 and help readers understand the pairing rule.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely of interest to the computational topology community, but the correctness proof of the main algorithm is not yet complete. The Case F recovery issue and the row-ordering ambiguity are both fixable, but they require substantive additions rather than copy-editing. I would encourage the authors to also consider adding a complexity analysis and at least one explicit example to demonstrate the algorithm's output."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's core object, Cok(Phi), is a sensible homological way to count open cobordisms between disjoint subcomplexes, and Theorem 3.1's case analysis is a clean reduction to kernel persistence. The examples and the discussion of the chemistry motivation are helpful, and the authors are honest about the slice-thickness problem without pretending to solve it. That part of the paper is solid.\n\nThe soft spot is Theorem 4.1. The proof includes phrases like \"It is now straightforward to check\" and then a final assertion that adding columns to the left of tau gives all equivalent representatives, so the reduction finds the earliest birth. That is not demonstrated, and in fact it seems false for Case F. DPhi is built only from columns of V_A_im and V_B_im (plus absolute cycles from zero-columns of R_A_im). A class born in Case F is a birth in Ker(iota_A∪B) with no simultaneous birth in Ker(iota_A) or Ker(iota_B). Therefore, at its birth time there is no column in DPhi representing it. When the class later dies, the algorithm adds new columns from the simultaneous births in Ker(A) and/or Ker(B), but those columns represent the class only at the death time. The reduction can push the lowest 1 down through older columns, but the original birth cell is not present in any column of DPhi, so the lowest 1 cannot point to it. The final paragraph of the proof assumes exactly what needs proof: that the stored columns form a compatible basis covering all classes, including those born in Case F. This is a load-bearing gap, not a minor omission. As stated, Algorithm 2 would either leave Case F classes unpaired or pair them with the wrong birth time.\n\nThere is no implementation or worked small example, which would have helped the authors (and referees) test this. The paper is still worth reading for the definition and the classification, but the algorithmic claim needs major revision. I would send it out for review with a clear request to fix or restrict Theorem 4.1, ideally with a concrete counterexample or a revised algorithm that handles Case F births.","headline":"The cobordism invariant is a good idea, but Algorithm 2's proof skips the one case that matters: classes born in Case F have no column in DPhi, so the death-pairing cannot recover their true birth time.","tokens_in":13444,"tokens_out":5298,"would_cite":true,"duration_ms":42436,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N31"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper defines open cobordisms as classes in a cokernel, proves their births and deaths follow eight kernel-persistence cases, and gives a column-reduction algorithm that pairs them.","keywords":["persistent homology","kernel persistence","open cobordisms","tunnels","bottleneck thickness","alpha complex","column reduction","point cloud"],"falsifier":"Take a filtered triangulated cylinder with A and B the two boundary circles, and arrange the filtration so that a class born directly in $\\mathrm{Ker}(\\iota^{A\\cup B}_*)$ later dies. Run Algorithm 2 and compare the birth time it pairs to the death time with the birth time read off from the defining persistence module $\\mathrm{Cok}(\\Phi)$; a mismatch would refute Theorem 4.1.","tokens_in":12411,"feed_emoji":"🕳️","tokens_out":9510,"duration_ms":69579,"temperature":0.7,"pith_summary":"The paper gives a homological definition of tunnels—open cobordisms—connecting two disjoint parts $A$ and $B$ of a finite cell complex $X$. A tunnel is counted as a class in $\\mathrm{Cok}(\\Phi)$, where $\\Phi$ compares cycles that become trivial inside $A$ or $B$ with cycles that become trivial in their union. When $X$ is filtered, this cokernel is a persistence module, and the paper proves that births and deaths in it occur in exactly eight configurations visible in the kernel persistence of $A$, $B$, and $A\\cup B$. The main result is an algorithm that pairs each birth with its death; in the point-cloud setting, the death time is the bottleneck thickness of the tunnel, the radius of the largest ball that can roll through.","feed_headline":"New algorithm finds tunnels and reports each bottleneck width","feed_subtitle":"For point clouds, it returns when a passage opens and the largest ball that fits through.","key_machinery":"The load-bearing object is the cokernel $\\mathrm{Cok}(\\Phi)$ of the map $\\Phi_k\\colon \\mathrm{Ker}(\\iota^A_k) \\oplus \\mathrm{Ker}(\\iota^B_k) \\to \\mathrm{Ker}(\\iota^{A\\cup B}_k)$; equivalently, by the relative homology long exact sequence, a class is a $(k+1)$-chain with boundary in $A\\cup B$ modulo chains whose boundary lies entirely in $A$ or entirely in $B$. The algorithm works by reordering boundary-matrix rows so that cells of $A$ (or $B$, or $A\\cup B$) come first, then applying standard column reduction so the lowest 1 in a column marks a birth or death. Algorithm 2 assembles the stored relative-cycle columns into a matrix $D_\\Phi$, and double columns in its reduced form mark the deaths.","core_discovery":"On the paper's own terms, the central claim is that open cobordisms between two disjoint subcomplexes are exactly the nonzero classes in $\\mathrm{Cok}(\\Phi)$, with $\\Phi$ the map $\\mathrm{Ker}(\\iota^A_*) \\oplus \\mathrm{Ker}(\\iota^B_*) \\to \\mathrm{Ker}(\\iota^{A\\cup B}_*)$, and that these classes form a persistence module when the complex is filtered. Theorem 3.1 states that a class is born only when a class appears in $\\mathrm{Ker}(\\iota^{A\\cup B}_*)$ without appearing in either kernel for $A$ or $B$, and dies when births occur in the kernels in the combinations called Cases C, G, and H. Algorithm 2 takes birth and death information from the three kernel persistence modules and returns birth–death pairs in $\\mathrm{Cok}(\\Phi)$; Theorem 4.1 asserts this pairing is correct. For a point cloud with $\\alpha$ filtration, the resulting pairs give tunnel birth times and bottleneck thicknesses.","pith_inferences":["One natural extension, left implicit by the paper, is to three or more boundary regions: the pairwise cokernel could be iterated, or a single cokernel defined from all boundary components at once, using the same relative-homology description.","Periodic boundary conditions could be handled by applying the construction after quotienting the box to a torus; the paper notes the torus creates extra homology classes, so the inference is that one would need to quotient those out, perhaps by working relative to the homology of the torus.","The representatives found by Algorithm 2 could be used to define a distance between porous materials, for example by counting tunnels whose death representative touches a given atom or bond.","Empirically, the sensitivity to slab thickness could be studied by running the algorithm for increasing slab size and tracking when the set of tunnels stabilizes; the paper's suggestion to delete slab 2-cells is a concrete preprocessing step to test."],"forward_implications":["For a point cloud in a box, Algorithm 2 turns the alpha filtration into a list of tunnels, each with a birth radius and a death radius equal to the bottleneck thickness.","The same construction applies to any pair of disjoint subcomplexes of any filtered regular cell complex, so it does not rely on Euclidean structure.","Corollary 4.2 provides representatives of each tunnel at birth and just before death, so the cells surrounding the narrowest passage can be inspected directly.","Because the input consists of the kernel persistence modules of three inclusion maps, the algorithm can run on top of existing kernel persistence implementations."],"supporting_citations":[{"why":"It supplies the kernel persistence framework, including the birth and death theory for kernels of inclusion maps that Algorithm 1 and Theorem 3.1 build on.","marker":"[3]"},{"why":"It supplies the standard column reduction algorithm, alpha complexes, and Voronoi/Delaunay duality used throughout the algorithm and in the point-cloud setup.","marker":"[4]"},{"why":"It supplies the regularity condition for cell complexes and the relative homology long exact sequence used to identify kernels with relative cycles.","marker":"[7]"}],"fun_headline_variants":["Kernel persistence reveals tunnels and bottleneck sizes","Algorithm pairs cobordism births and deaths via kernels","New method tracks tunnel openings and widths","Cobordism persistence: find tunnels from kernel maps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The pairing algorithm is correct only if every element of $\\mathrm{Cok}(\\Phi)$ that dies at a cell $\\tau$ can be represented by adding the stored columns lying to the left of $\\tau$, so that column reduction can locate the earliest birth cell.","fun_headline_variants_meta":{"raw":{"variants":["Kernel persistence reveals tunnels and bottleneck sizes","Algorithm pairs cobordism births and deaths via kernels","New method tracks tunnel openings and widths","Cobordism persistence: find tunnels from kernel maps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000525,"raw_usage":{"total_tokens":2453,"prompt_tokens":783,"completion_tokens":1670,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":399,"completion_tokens_details":{"reasoning_tokens":1611}},"tokens_in":399,"tokens_out":1670,"duration_ms":9636,"temperature":1.0,"reasoning_tokens":1611,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:41:23.290703+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a filtered triangulated cylinder with A and B the two boundary circles, and arrange the filtration so that a class born directly in $\\mathrm{Ker}(\\iota^{A\\cup B}_*)$ later dies. Run Algorithm 2 and compare the birth time it pairs to the death time with the birth time read off from the defining persistence module $\\mathrm{Cok}(\\Phi)$; a mismatch would refute Theorem 4.1.","supporting_citations":[{"cited_title":"Cohen-Steiner, H","cited_arxiv_id":null,"evidence_quote":"It supplies the kernel persistence framework, including the birth and death theory for kernels of inclusion maps that Algorithm 1 and Theorem 3.1 build on."},{"cited_title":"Edelsbrunner and J","cited_arxiv_id":null,"evidence_quote":"It supplies the standard column reduction algorithm, alpha complexes, and Voronoi/Delaunay duality used throughout the algorithm and in the point-cloud setup."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the regularity condition for cell complexes and the relative homology long exact sequence used to identify kernels with relative cycles."}],"review_version":1}