{"id":"a4e1d74d-9189-446e-a4e1-9a844eb5280e","arxiv_id":"2507.17994","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces the C-constrained Gromov-Hausdorff distance for chromatic metric pairs and proves that all six diagrams in a six-pack are stable in bottleneck distance with respect to it, up to a factor of 2.","lead":"This paper defines a way to measure distance between metric spaces in which a subset of points is colored, by generalizing the Gromov-Hausdorff distance to respect the coloring. It proves that the six persistence diagrams summarizing colored data, the six-pack, change by at most a fixed factor under this distance, giving a stability guarantee used in topological data analysis.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the six-pack stability proof is internally sound; the flagged issues are non-central typos.","rationale":"The stress-test pass focused on Theorem 5.5, the paper's central claim. I re-derived the key step that was not fully expanded: the claim in Lemma 5.3 that the triangles commute up to contiguity. For a simplex σ ∈ Č^Γ_δ(χ1;A1) with witness a∈A1, set a' = f2(f1(a)). For x∈σ, |d1(x,a') - d2(f1(x),f1(a))| ≤ codis(f1,f2) ≤ 2ε and d2(f1(x),f1(a)) ≤ d1(x,a)+dis f1 ≤ δ+2ε, hence d1(x,a')≤δ+4ε. For y=f2(f1(x)), d1(y,a') ≤ d2(f1(x),f1(a))+dis f2 ≤ δ+4ε. Color containment follows because χ1(σ) is contained in a maximal face τ of Γ and both f1 and f2 are τ-constrained, so χ1(f2(f1(σ)))⊆τ. Thus (f2∘f1)(σ)∪σ is a simplex of Č^Γ_{δ+4ε}(χ1;A1), and the same argument works for Λ using C(Λ). This confirms the 2ε-interleaving construction. The passage from contiguity to commutative homology diagrams via [29, Thm. 12.5 and 12.6] is standard, and the kernel, image, cokernel, and relative interleavings follow by diagram chasing and naturality. Finiteness of X1 and X2 makes the Čech complexes finite, so the persistence modules are finite-dimensional and the algebraic stability theorem applies. I therefore do not see a load-bearing flaw in the main theorem. The two issues raised by the reader are real but peripheral: the displayed inequality in the proof of Proposition 4.12 has the wrong direction (the intended construction gives d^C_H ≤ max{d_H, sup d_Hσ}+ε), and the bracketed question in §5.3 concerns only the alternative proof of Corollary 5.7, which already follows directly from Theorem 5.5 by taking Γ={{0}} and Λ=∅. Neither affects Theorem 5.5. A useful sanity check is to verify the contiguity inequalities explicitly; this would settle the only nontrivial geometric step in the stability proof.","tokens_in":30207,"tokens_out":30721,"duration_ms":332487,"concrete_test":"Independently verify the contiguity step of Lemma 5.3 by writing out, for an arbitrary simplex σ in Č^Γ_δ(χ1;A1) with witness a, the inequalities d1(x, f2(f1(a))) ≤ δ+4ε and d1(f2(f1(x)), f2(f1(a))) ≤ δ+4ε for every x∈σ, together with the maximal-face color containment: if either inequality fails for some valid maps, the interleaving construction in Theorem 5.5 collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After checking the proof of Theorem 5.5 against Lemmas 5.2–5.4, I find no load-bearing flaw in the central stability claim. The 2ε-interleavings are constructed correctly: Lemma 5.2 gives simplicial maps with the required radius shift; Lemma 5.3's triangles are contiguous because for a simplex σ with witness a∈A1, the point f2(f1(a)) is within δ+4ε of every x∈σ and every f2(f1(x)), using dis f1, dis f2, codis(f1,f2) ≤ 2ε; the color constraint follows from containment in a maximal face of Γ or Λ. Applying H_p converts contiguity into commutativity, so kernels, images, cokernels, and relative homology inherit interleavings. Finiteness of X1 and X2 ensures finite-dimensional modules for Theorem 5.1. The two issues the reader notes—the reversed displayed inequality in Proposition 4.12's proof and the bracketed question about rad_A versus Č(X;A) in §5.3—are real but non-central: Proposition 4.12's statement is correct and the proof has an obvious sign error; §5.3 is an alternative proof of Corollary 5.7, which already follows directly from Theorem 5.5 by taking Γ={{0}} and Λ=∅, so an unresolved question there does not undermine the main theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces chromatic metric pairs (A, χ:X→N), C-constrained maps, and a C-constrained Gromov-Hausdorff distance d_C_GH defined by minimizing max{dis f, dis g, codis(f,g)} over C-constrained pairs of maps. It establishes basic properties of this distance, characterizations via C-constrained correspondences and via admissible metrics on disjoint unions, existence of optimal correspondences for C-compact pairs, and stable lower-bound invariants. The central result is Theorem 5.5: for finite X1 and X2 and finite-dimensional simplicial complexes Λ ≤ Γ on N, each of the six persistence modules (domain, codomain, image, kernel, cokernel, relative homology) satisfies dB(Dgm U1, Dgm U2) ≤ 2 d_{C(Γ)∪C(Λ)}_GH((A1, χ1), (A2, χ2)). Corollary 5.7 derives stability for ambient Čech persistence diagrams, and Section 5.3 gives an alternative tripod-based proof under an additional technical assumption.","tokens_in":30415,"tokens_out":12160,"duration_ms":130208,"significance":"The paper supplies a natural and genuinely new distance for colored metric data and proves the first stability theorem for the six-pack with respect to such a distance. The foundational development is careful: the Alexandrov-topology analysis of constraint sets in Section 3.1 is elegant and useful, the computations in Section 4.1 provide sharp worked examples, and the main theorem is a substantial extension of persistence stability to chromatic settings. The proof adapts Chowdhury–Mémoli interleaving techniques and the algebraic stability theorem in a coherent way, and the distance is defined independently of persistence, so there is no circularity. I concur with the stress-test assessment: the flagged issues in Proposition 4.12 and Section 5.3 are real but do not affect the central six-pack stability theorem. The main limitations are the finiteness assumption in Theorem 5.5 and the unresolved question in Section 5.3, both of which are stated explicitly in the manuscript.","major_comments":[],"minor_comments":[{"comment":"The displayed equality 'sup_σ sup_{(y,z)∈R_σ} d(y,z) = sup_{(y,z)∈R} d(y,z)' is false in general, because R also contains the arbitrary correspondence R′ and the supremum over R can be larger. The intended inequality d_C_H(Y,Z) ≤ max{d_H(Y,Z), sup_σ d_H(Y_σ,Z_σ)} can be obtained by choosing R′ and the R_σ with suprema close to the corresponding Hausdorff values, so the statement of the proposition is correct, but the proof as written needs this repair.","section":"§4.2, proof of Proposition 4.12"},{"comment":"The bracketed sentence 'I suspect that even in the general setting the persistence diagrams of (X, rad_A) and Č(X;A) coincide, don’t they?' is an acknowledged open point inside the alternative proof. Since Corollary 5.7 follows directly from Theorem 5.5, this does not threaten the central claim, but the sentence should be resolved, removed, or converted into an explicit open question.","section":"§5.3, bracketed question after Proposition 5.8"},{"comment":"In the proof, after choosing a2, the conclusion 'rad_{A2}(π1(σ)) ≤ rad_{A1}(π1(σ)) + dis R + ε' should refer to rad_{A2}(π2(σ)); the same notational slip occurs in the following displayed line.","section":"§5.3, proof of Proposition 5.8"},{"comment":"The sentence 'we define f_δ ... on the vertex set by setting f_δ(x) = f(x) for every vertex x of Γ' should read 'for every vertex x of Č_δ(X1; A1)' or 'of X1'; as written, it is confusing because Γ has vertices in N and is not the source complex of f_δ.","section":"§5.2, proof of Lemma 5.2"},{"comment":"There are several typos that should be corrected: 'Gromov-Hausdroff' in Section 2, 'C-cconstrained' in the caption of Figure 13, and 'a constraint set N∈C⊆P(N)' in Example 5.6 should read 'C⊆P(N) with N∈C'. None of these affects the mathematics.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"This is a solid contribution. The overlap with the upcoming preprint by Fu, Ver Hoef, Lagoda, Li, Needham, and Weiler is disclosed by the authors and does not affect my assessment. The central stability theorem is sound; the required changes are local proof repairs and exposition fixes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is real. The C-constrained Gromov-Hausdorff distance is a natural generalization of the Kalton-Ostrovskii characterization, and the six-pack stability theorem (Theorem 5.5) is the right validation: finite colored point sets close in this distance have all six persistence diagrams close in bottleneck distance. The proof is standard but correct: Lemma 5.2 turns constrained maps into simplicial maps of Γ-subcomplexes, Lemma 5.3 builds the 2ε-interleavings up to contiguity using distortion and codistortion bounds, and applying H_p plus the algebraic stability theorem gives the result. Finiteness of X_i is exactly what makes the modules finite-dimensional, so no hidden assumption there. This genuinely generalizes the uncolored metric-pair stability from Torras-Casas and Gonzalez-Diaz, and it slots into the Chowdhury-Memoli toolkit as advertised.\n\nWhat the paper also does well: the characterizations (correspondence, admissible metric, zero-distance under compactness) are properly established, and the comparison with the metric-pair distance of Gómez and Che is worked out (they are bi-Lipschitz equivalent). The examples are helpful and the computations check out.\n\nSoft spots, in proportion. Proposition 4.12's proof has a reversed displayed inequality; the statement is true and the sign is an obvious typo. Section 5.3 contains a bracketed author question ('don't they?') about whether the circumradius filtration's persistence diagrams coincide with the ambient Čech diagrams in general. That's an open point in an alternative proof only, since Corollary 5.7 follows directly from Theorem 5.5 (take Γ={{0}}, Λ=∅). So it's a minor blemish, not a gap. The paper is honest about concurrent independent work, and the citation pattern is fair.\n\nWho this is for: people working on chromatic topological data analysis, metric pair stability, or Gromov-Hausdorff-type distances with labels. It's not a paradigm shift, but it's a principled distance with a matching stability guarantee, which is exactly what applied TDA needs.\n\nRecommendation: send it to peer review. The main theorem should survive a serious referee; the typos and the §5.3 question can be handled in revision.","headline":"A correct and genuinely useful stability theorem for chromatic persistence; two minor issues don't threaten the core.","tokens_in":31009,"tokens_out":3234,"would_cite":true,"duration_ms":32757,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C23","55N31"],"pacs":[],"model":"deepseek-v4-flash","headline":"A color-constrained Gromov-Hausdorff distance makes the six-pack of persistence diagrams stable.","keywords":["chromatic metric pairs","C-constrained Gromov-Hausdorff distance","six-pack of persistence diagrams","bottleneck stability","ambient Čech filtration","colored point clouds","Alexandrov topology on colors"],"falsifier":"Take two finite colored point sets in the plane with $\\Lambda = \\{\\{0\\}\\}$ and $\\Gamma = \\{\\{0\\},\\{1\\},\\{0,1\\}\\}$, compute $d^{C}_{GH}$ for $C=\\{\\{0\\},\\{0,1\\}\\}$ by the pair-of-maps definition, and compute the six persistence diagrams on a fine discretization, as in Example 5.6; the theorem is false if any instance gives $d_B$ larger than $2\\,d^{C}_{GH}$.","tokens_in":29963,"feed_emoji":"🎨","tokens_out":8747,"duration_ms":84352,"temperature":0.7,"pith_summary":"This paper defines a distance between chromatic metric pairs — metric spaces carrying a coloring on a distinguished subspace — and proves that a standard invariant called the six-pack of persistence diagrams is stable under this distance. The distance, the C-constrained Gromov-Hausdorff distance, only admits comparison maps that respect chosen color constraints, and reduces to the classical Gromov-Hausdorff distance when colors carry no information. The main theorem states that when two finite colored point sets are ε-close in this distance, with constraints taken from the maximal faces of the two simplicial complexes that define the six-pack, all six persistence diagrams are at most 2ε apart in bottleneck distance. This turns the six-pack into a certificate that can be compared across datasets with labeled subpopulations or components.","feed_headline":"New metric makes six-pack persistence diagrams stable","feed_subtitle":"Color-constrained closeness bounds all six bottleneck diagrams within a factor of two.","key_machinery":"The central object is the $C$-constrained Gromov-Hausdorff distance, defined as half the infimum over pairs $f:A_1\\to A_2$, $g:A_2\\to A_1$ of $\\max\\{\\operatorname{dis} f, \\operatorname{dis} g, \\operatorname{codis}(f,g)\\}$, where all maps are required to be $C$-constrained: for every $\\sigma\\in C$, $f(\\chi_1^{-1}(\\sigma))\\subseteq \\chi_2^{-1}(\\sigma)$. The stability proof is carried by Lemmas 5.2–5.4: a $C(\\Gamma)\\cup C(\\Lambda)$-constrained pair of maps with distortion and codistortion at most $2\\varepsilon$ induces, for every radius $\\delta$, simplicial maps between the $\\Gamma$- and $\\Lambda$-subcomplexes of the ambient Čech filtrations, and these maps form interleavings that commute up to contiguity. Homology converts these into genuine $2\\varepsilon$-interleavings of the six persistence modules, and the algebraic stability theorem turns those into bottleneck-distance bounds on the persistence diagrams.","core_discovery":"The paper's central claim is that the six-pack is stable with respect to the constrained distance: for finite colored subsets $X_1 \\subseteq A_1$ and $X_2 \\subseteq A_2$, with $\\Lambda \\leq \\Gamma$ finite-dimensional simplicial complexes on the color set and $C = C(\\Gamma) \\cup C(\\Lambda)$ the union of their sets of maximal faces, each of the six persistence modules — domain, codomain, image, kernel, cokernel, and relative homology built from the inclusions $\\check{C}^\\Lambda(\\chi;A) \\hookrightarrow \\check{C}^\\Gamma(\\chi;A)$ — satisfies $d_B(\\operatorname{Dgm} U_1, \\operatorname{Dgm} U_2) \\leq 2\\,d^{C}_{GH}((A_1,\\chi_1),(A_2,\\chi_2))$. The constraining family $C$ is not incidental: it is exactly what guarantees that a $C$-constrained map sends $\\Gamma$-colored simplices of the Čech filtration to $\\Gamma$-colored simplices at a controlled radius shift (Lemma 5.2).","pith_inferences":["The same interleaving-by-contiguity skeleton should transfer to other geometric complexes, such as Vietoris-Rips or Dowker filtrations, since the proof only needs the radius-shift and contiguity lemmas, not the spherical geometry of Čech balls.","The paper's bracketed conjecture that the persistence diagrams of $(X,\\operatorname{rad}_A)$ and of the ambient Čech filtration coincide, if true, would remove the circumradius-attainment hypothesis from the tripod-based proof and extend Corollary 5.7 to all finite metric pairs.","For labeled point-cloud matching, this suggests a practical recipe: choose the weakest constraint set compatible with the labels an application cares about, compute $d^{C}_{GH}$, and use the six-pack bottleneck distances as certified lower bounds; Example 5.6 shows such lower bounds can be tight."],"forward_implications":["If two finite colored point sets are close in $d^{C}_{GH}$ with $C=C(\\Gamma)\\cup C(\\Lambda)$, then all six persistence diagrams in the six-pack are close in bottleneck distance, with constant 2.","The ambient Čech persistence diagrams of two finite metric pairs are stable with respect to the induced Gromov-Hausdorff distance for metric pairs (Corollary 5.7).","The new distance inherits classical characterizations: it can be computed via correspondences or common embeddings, and under compactness it is attained and vanishes exactly for constrained isomorphisms (Theorem 4.9, Theorem 4.13, Corollary 4.20).","Weaker constraint sets yield smaller distances, so invariants that ignore some color information can be bounded more tightly by choosing weaker constraints (Remark 4.3(iv), Example 4.4).","The six-pack stability generalizes earlier stability results for metric pairs and for degree-0 image, kernel, and cokernel persistence with Vietoris-Rips filtrations."],"supporting_citations":[{"why":"It supplies the pair-of-maps characterization of classical Gromov-Hausdorff distance that the new constrained definition is modelled on.","marker":"[23]"},{"why":"It supplies the 2ε-interleaving construction and contiguity techniques that Lemma 5.3 adapts to Čech subcomplexes.","marker":"[8]"},{"why":"It introduces chromatic alpha complexes and the six-pack of persistence diagrams whose stability is the paper's target.","marker":"[13]"},{"why":"It provides the algebraic stability theorem that converts persistence-module interleavings into bottleneck-distance bounds on diagrams.","marker":"[5]"},{"why":"It establishes persistence for kernels, images, and cokernels, the modules assembled into the six-pack.","marker":"[11]"},{"why":"Its contiguity theorems justify passing from diagrams that commute up to contiguity to equal induced homology maps.","marker":"[29]"},{"why":"Its tripod distance and stability result give the alternative proof of ambient Čech stability in Section 5.3.","marker":"[27]"},{"why":"It is the previous stability result for metric pairs that Theorem 5.5 and Corollary 5.7 generalize.","marker":"[34]"}],"fun_headline_variants":["Six-pack stable under new chromatic metric","Factor-two stability for color-coded persistence","New Gromov-Hausdorff variant stabilizes six-pack","Chromatic metric keeps six-pack diagrams in check"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole bound rests on the existence, for nearly equal colored point sets, of two maps that respect exactly the chosen color groupings and have both distortion and codistortion below $2\\varepsilon$; if the constraint set is looser or the point sets are infinite, the interleaving construction is not guaranteed and the bottleneck bound can fail.","fun_headline_variants_meta":{"raw":{"variants":["Six-pack stable under new chromatic metric","Factor-two stability for color-coded persistence","New Gromov-Hausdorff variant stabilizes six-pack","Chromatic metric keeps six-pack diagrams in check"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00031,"raw_usage":{"total_tokens":1740,"prompt_tokens":888,"completion_tokens":852,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":504,"completion_tokens_details":{"reasoning_tokens":792}},"tokens_in":504,"tokens_out":852,"duration_ms":8820,"temperature":1.0,"reasoning_tokens":792,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:39:46.041114+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take two finite colored point sets in the plane with $\\Lambda = \\{\\{0\\}\\}$ and $\\Gamma = \\{\\{0\\},\\{1\\},\\{0,1\\}\\}$, compute $d^{C}_{GH}$ for $C=\\{\\{0\\},\\{0,1\\}\\}$ by the pair-of-maps definition, and compute the six persistence diagrams on a fine discretization, as in Example 5.6; the theorem is false if any instance gives $d_B$ larger than $2\\,d^{C}_{GH}$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the pair-of-maps characterization of classical Gromov-Hausdorff distance that the new constrained definition is modelled on."},{"cited_title":"Chowdhury, F","cited_arxiv_id":null,"evidence_quote":"It supplies the 2ε-interleaving construction and contiguity techniques that Lemma 5.3 adapts to Čech subcomplexes."},{"cited_title":"Cultrera di Montesano, O","cited_arxiv_id":null,"evidence_quote":"It introduces chromatic alpha complexes and the six-pack of persistence diagrams whose stability is the paper's target."},{"cited_title":"Chazal, D","cited_arxiv_id":null,"evidence_quote":"It provides the algebraic stability theorem that converts persistence-module interleavings into bottleneck-distance bounds on diagrams."},{"cited_title":"Cohen-Steiner, H","cited_arxiv_id":null,"evidence_quote":"It establishes persistence for kernels, images, and cokernels, the modules assembled into the six-pack."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Its contiguity theorems justify passing from diagrams that commute up to contiguity to equal induced homology maps."},{"cited_title":"A Distance Between Filtered Spaces Via Tripods","cited_arxiv_id":"1704.03965","evidence_quote":"Its tripod distance and stability result give the alternative proof of ambient Čech stability in Section 5.3."},{"cited_title":"Properties and Stability of Persistence Matching Diagrams","cited_arxiv_id":"2409.14954","evidence_quote":"It is the previous stability result for metric pairs that Theorem 5.5 and Corollary 5.7 generalize."}],"review_version":1}