{"id":"5f15b31e-c176-460c-a604-f88d95a7a571","arxiv_id":"2501.09257","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A new cohomological interleaving distance for spaces over BS^1 is shown to equal the homotopy interleaving distances of Blumberg-Lesnick and Lanari-Scoccola, and is computed via barcodes for CP^n and rational homotopy examples.","lead":"This paper defines a cohomology-based interleaving distance for spaces over the classifying space of the circle and proves it matches known homotopy-level interleaving distances. It computes the distance explicitly for complex projective spaces and related examples, linking persistent topology to rational homotopy theory.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the formality lemma is sound for single-parameter dg modules, and the only proof slip is a summation over k≥0 that should read k∈Z.","rationale":"I read the paper as establishing a corrected version of a known phenomenon: over a field, one-parameter persistence dg modules are formal, so homotopy interleaving distances reduce to cohomological interleavings. The reader's weakest assumption identifies exactly the right premise. I checked Lemma 3.5 in detail: F0 is a free graded K[t]-module on cohomology generators; since K[t] is a PID the kernel is free, F1 kills it, and both maps Q→X and Q→H(X) induce isomorphisms on pointwise cohomology. The transfer from Z to R via Proposition 3.7 uses the scaling/restriction lemmas of Lanari-Scoccola; I do not see a hidden model-structure issue. The computations for spaces over BS^1 are extensive, and the barcode arguments are consistent with direct interleaving checks. The only internal inconsistency I found is the summation over k≥0 in the proof of Theorem 3.3; since dCohI is explicitly a supremum over all k and negative degrees are allowed, this line should read k∈Z. This is a minor typo and does not affect the theorem's statement or its applications to singular cohomology, which is nonnegatively graded. The paper also does not overclaim completeness: Remark 6.6 shows dCohI = 0 does not imply isomorphism of spaces, but the paper only asserts equality of distances, which is compatible with that example. I therefore recommend keeping the ACCEPT verdict unchanged.","tokens_in":32516,"tokens_out":33063,"duration_ms":360782,"concrete_test":"Independently re-derive the last paragraph of the proof of Theorem 3.3 with H(Z) = ⊕_{k∈Z} η_k H(Z), and verify for an unbounded persistence dg module with nonzero H^{-1} and zero positive cohomology that the direct-sum δ-interleaving is still constructed from the δ-interleavings of each η_k, so the inequality dHI ≤ dCohI follows across all degrees.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing flaw in the central claim. The equality dHC = dIHC = dHI = dCohI rests on Lemma 3.5, and the hereditary-formality argument via graded K[t]-modules is sound: the two-step free resolution Q gives a genuine zigzag X ← Q → H(X) in Ch(Z,≤)_K, and Proposition 3.7 correctly transfers the Z-indexed statement to R by scaling and restriction. The one internal slip is in the proof of Theorem 3.3, where H(Z) is written as ⊕_{k≥0} η_k H(Z) although dCohI is defined as a supremum over all k∈Z and Ch_K is explicitly not bounded. This is a notational/typing slip rather than a substantive gap: replacing k≥0 by k∈Z makes the direct-sum interleaving argument valid, and the applications to spaces over BS^1 only involve nonnegatively graded singular cohomology.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a cohomology interleaving distance dCohI for persistence dg modules over a field K and proves that, in the single-parameter setting, this distance coincides with the homotopy interleaving distances dHC, dIHC, and dHI introduced by Blumberg–Lesnick and Lanari–Scoccola (Theorem 3.3). The proof relies on a constructive formality lemma for persistence dg modules indexed by Z (Lemma 3.5), together with scaling arguments from Lanari–Scoccola. The paper then specializes to dg K[u]-modules, shows that the relevant distance is the maximum of an even and an odd cohomological bottleneck distance (Theorem 4.7), and applies this to spaces over BS^1 via singular cochains. Concrete computations include complex projective spaces, certain S^1-bundle orbit spaces, and free loop space Borel constructions, with cup-length bounds and a rational homotopy appendix.","tokens_in":32641,"tokens_out":16670,"duration_ms":147772,"significance":"If the main results stand, the paper gives a genuinely computable way to evaluate homotopy interleaving distances in a nontrivial class of persistence dg modules and spaces over BS^1, reducing them to bottleneck distances of cohomology barcodes. The constructive proof of formality for single-parameter persistence dg modules is a useful and clearly explained ingredient, and the authors are explicit that this formality is tied to K[t] being hereditary and fails in the multiparameter setting (Remark 3.6). The computational examples, especially the tetrahedron of distances and the CP^n calculations, are concrete and checkable. There are no fitted parameters or predictions, and the paper relies on standard external results.","major_comments":[{"comment":"Lemma 4.9 is false as stated. Let H^* = K[t]/(t^2) with deg t = 1 and take the filtration F^0 = H^*, F^1 = F^2 = 0. Then H^* is non-negatively graded, dim H^i < infinity, tF^1 = 0 subset F^2, and all stated hypotheses hold. However, E^{0,0} = H^0, E^{0,1} = H^1, and E^{p,q} = 0 for p >= 1, so Tot E has trivial t-action, while t acts nontrivially from degree 0 to degree 1 in H^*. The proof uses the assertion 'F^i H^0 = 0 for i > 0', which does not follow from the stated hypotheses. The lemma becomes true if one adds the natural hypothesis that the filtration is by total degree, i.e. F^i subset H^{>=i}; this extra hypothesis is satisfied in the spectral-sequence applications of Propositions 5.10 and 6.3, but it must be stated and the proof adjusted.","section":"Section 4, Lemma 4.9"},{"comment":"In the proof of Theorem 3.3, the equality H(Z) = direct_sum_{k>=0} eta_k H_*(Z) is incorrect for unbounded persistence dg modules; the direct sum should be over all k in Z, matching the definition of dCohI as a supremum over all integer cohomological degrees. As written, the proof of dHI <= dCohI would not cover objects with nonzero cohomology in negative degrees. This is a local fix, but it should be corrected because Theorem 3.3 is stated for all objects in Ch(R,<=)_K.","section":"Section 3, proof of Theorem 3.3"}],"minor_comments":[{"comment":"In the proof of Lemma 3.5, the generators b_lambda(i)_k should be explicitly chosen as cycles representing the homology classes [b_lambda(i)_k], since the map phi : F_0 -> Ker d is defined by sending each generator to its representative. This is presumably intended but should be stated.","section":"Section 3, Lemma 3.5"},{"comment":"Example 5.7(1) states an assertion for arbitrary field K but the proof uses rational cohomology and the assumption that 1*t^l is nonzero in H^*(LM_hS1; Q). The statement should either be restricted to K = Q or the proof adapted to justify the claim for general K.","section":"Section 5, Example 5.7(1)"},{"comment":"The proof of Proposition 5.13 is terse: the phrase '2(m/2)-trivial' is used without a definition, and the reduction to kernel/cokernel triviality via [4, Corollary 6.6] could be spelled out. This is a clarity issue rather than a mathematical gap.","section":"Section 5, Proposition 5.13"},{"comment":"There are several typographical slips, e.g. 'fincor' in Section 4, 'as a consecuence' in Section 5, and 'more general BV-exact spaces' in Proposition 5.4 where 'more generally, BV-exact spaces' is meant. A careful proofreading pass is recommended.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper's main equality theorem is sound in its intended single-parameter setting, and the applications are valuable. The main obstacle is Lemma 4.9, which is false without an additional filtration hypothesis; since the lemma is used in the spectral-sequence computations, the authors need to correct its statement and proof. The Theorem 3.3 summation issue is minor but should also be fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this paper delivers what it claims. It proves that for persistence dg modules over a field, the cohomology interleaving distance equals the three homotopy-level distances (dHC, dIHC, dHI), and it shows that for spaces over BS^1 this distance can be read off from barcodes in the even and odd cohomology. That is a new and useful bridge: homotopy interleaving distances are abstract, barcodes are computable.\n\nThe main engine is Lemma 3.5, formality of every persistence dg module over a field in the single-parameter poset. The proof via free resolutions over K[t] is constructive and correct; the authors also note in Remark 3.6 that this is a special case of the known fact that derived categories of hereditary abelian categories are formal, and that it fails for multiparameter persistence. I checked the key step in Proposition 3.4 and the transfer to R in Proposition 3.7, and they hold. The transition from dg K[u]-modules to spaces over BS^1 in Theorem 4.7 is also sound: the reduction to the even and odd parts is justified by the 2-periodicity in the persistence direction.\n\nNice touches: the explicit computations for CP^n and the M0/M1 pair, the cup-length bounds, and the appendix separating the two M's by rational toral rank. The examples are not decorative; they demonstrate the triangle inequality and that the distance can be positive even when the underlying graded cohomology is isomorphic.\n\nThe softest spot is in the proof of Theorem 3.3, where H(Z) is written as \\oplus_{k\\ge 0} \\eta_k H(Z). Since Ch_K is unbounded, the sum should be over all k\\in Z. The stress-test note is right: replace k\\ge0 by k\\in Z and the direct-sum interleaving argument goes through. It is a typo-level slip, not a gap in the idea. Second, a few spectral sequence computations in Section 6 (Proposition 6.3 and Lemma 4.9) are compressed; I did not find an error, but they require the reader to supply details. Third, the whole result rests on single-parameter formality, so the equality of distances should not be expected to extend to multiparameter settings without significant new work. The authors are explicit about this limitation.\n\nCitation pattern: self-citation in Proposition 5.4 uses [30], a published paper, for BV-exactness; that is appropriate. Dependence on [6], [8], [31] is standard and the comparison is fair.\n\nWho should read: TDA people who want to compute homotopy interleaving distances for spaces with S^1 actions, and homotopy theorists curious about persistence distances in model categories. Not a breakthrough for a general audience, but a clean, useful contribution to the subfield.\n\nRecommendation: yes, send to a serious referee. The central theorem is provably correct up to minor typographical slips, and the computations are valuable. If I were refereeing, I would ask for the fix in Theorem 3.3 and slightly more detail in the spectral sequence steps, but I would expect acceptance after minor revision.","headline":"A clean, well-written paper proving the equality of four interleaving distances on persistence dg modules and turning that into a computable barcode distance for spaces over BS^1, with only a minor repairable slip in the proof of Theorem 3.3.","tokens_in":33254,"tokens_out":4263,"would_cite":true,"duration_ms":42425,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N31","55U15","55P62","55U35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that, for persistence differential graded modules over a field, the homotopy interleaving distance, the homotopy-commutative interleaving distance, and the interleaving distance in the homotopy category all coincide with…","keywords":["interleaving distance","cohomology interleaving distance","persistence differential graded module","barcode","bottleneck distance","classifying space BS^1","cup-length","Sullivan model"],"falsifier":"Independently compute the homotopy interleaving distance between the two dg $K[u]$-modules coming from the spaces $M_0$ and $M_1$ of Proposition 6.3, directly from the definition of $\\varepsilon$-homotopy interleavings rather than from barcodes. The paper's Theorem 4.7 predicts the value $3$; any other value would falsify the claimed equality.","tokens_in":32276,"feed_emoji":"📏","tokens_out":8809,"duration_ms":82020,"temperature":0.7,"pith_summary":"The paper proves that, for persistence differential graded (dg) modules over a field, three homotopy-level notions of interleaving distance are all equal, and that this common distance is the cohomology interleaving distance obtained by comparing homology barcodes degree by degree. The point is computational: homotopy-level comparisons, which are hard to access directly, become bottleneck distances between ordinary persistence barcodes. The paper applies this to spaces equipped with a map to the classifying space $BS^1$ of the circle, where singular cohomology has a natural module structure over the polynomial ring $K[u]$, so each space over $BS^1$ yields a persistence dg module. This gives a numerical way to compare spaces over $BS^1$ even when no morphism between them exists, with explicit computations for complex projective spaces and for two circle-orbit spaces that are hard to tell apart by rational homotopy invariants.","feed_headline":"Three homotopy interleaving distances collapse to one cohomology distance","feed_subtitle":"For spaces over the circle's classifying space, barcodes from cohomology pin down the distance.","key_machinery":"The load-bearing object is the functor $C$ that converts a dg $K[u]$-module $M$ into a persistence dg module by placing $\\Sigma^{2i}M$ at integer index $i$ and using multiplication by $u$ as the structure map $i\\to i+1$; for spaces over $BS^1$, the singular cochain complex $C^*(X;K)$ becomes a $K[u]$-module via the classifying map. A second component is Lemma 3.5, the formality of single-parameter persistence dg modules, which lets the authors replace any module by its homology without changing homotopy interleavings. The computational engine is then the isometry between interleaving distance and bottleneck distance for barcodes of graded $K[t]$-modules, so $d^0_{CohI}$ and $d^1_{CohI}$ are read off directly from barcodes.","core_discovery":"The central discovery is Theorem 3.3: on the class of persistence dg modules over a field, $d_{HC}=d_{IHC}=d_{HI}=d_{CohI}$. Here $d_{HI}$ is the homotopy interleaving distance, $d_{IHC}$ is the interleaving distance in the homotopy category, and $d_{CohI}$ is the supremum over homological degrees of the ordinary interleaving distances of the homology modules. For dg modules over $K[u]$, the persistence module is obtained by shifting by the action of $u$, and Theorem 4.7 refines the equality to $d_{CohI}=\\max\\{d^0_{CohI},d^1_{CohI}\\}$, where the two superscripts track even and odd cohomology. The proof rests on Lemma 3.5, which says every such persistence dg module is formal: it is quasi-isomorphic to its own cohomology because $K[t]$ is a hereditary ring. The paper also establishes cup-length upper and lower bounds for the distance between spaces over $BS^1$, and works out explicit distances among complex projective spaces and the spaces $M_0$, $M_1$.","pith_inferences":["This suggests a practical route to computing homotopy interleaving distances for finite-type spaces over $BS^1$: compute the cohomology $K[u]$-module structure, read off barcodes, and take a bottleneck distance, with no homotopy-level search required.","The equality is likely special to one parameter: over $K[t_1,\\ldots,t_n]$ for $n\\ge 2$ the formality lemma fails, so the cohomology interleaving distance should be viewed as a lower bound rather than a complete invariant for multiparameter persistence.","Zero-distance equivalence classes over $BS^1$ may serve as a coarse 'persistent shape' invariant for spaces that are otherwise hard to compare, analogous to the role of Gromov–Hausdorff distance in metric geometry.","The cup-length bounds invite refinement: sharper invariants such as rational toral rank could yield better lower bounds, since the examples $M_0,M_1$ separate by toral rank even when cup-length alone does not."],"forward_implications":["The homotopy interleaving distance of persistence dg modules over a field is computable as the bottleneck distance of homology barcodes, degree by degree.","For spaces over $BS^1$, the cohomology interleaving distance is an extended pseudometric and simultaneously controls all three homotopy-level interleaving distances.","Two spaces over $BS^1$ whose cohomology barcodes have distance zero have associated persistence dg modules that are isomorphic in the homotopy category, even if the underlying spaces have different rational homotopy types.","Borel constructions of free loop spaces of formal spaces sit at distance $0$ or $1/2$, depending only on whether their cohomology $K[t]$-modules are isomorphic.","The distance between a space over $BS^1$ and the point is half the cup-length plus one half, and differences of cup-lengths give lower bounds on distances between spaces."],"supporting_citations":[{"why":"Introduces the homotopy interleaving distance whose equality with the cohomology distance is the main result of Theorem 3.3.","marker":"[6]"},{"why":"Introduces the interleaving distance in the homotopy category and the rectification technique used in Proposition 3.7.","marker":"[31]"},{"why":"Supplies the interleaving-distance formalism, the inequalities among the three homotopy distances, and the isometry theorem used for barcode computations.","marker":"[8]"},{"why":"Defines $\\varepsilon$-interleavings and the interleaving distance, the basic objects compared throughout.","marker":"[10]"},{"why":"Gives the interval decomposition of graded $K[t]$-modules that underlies the barcodes.","marker":"[35]"},{"why":"Shows that formality in the derived category is equivalent to the base category being hereditary, which is exactly the single-parameter limitation of Lemma 3.5.","marker":"[27]"},{"why":"Provides the BV-exactness criterion used to compute distances between Borel constructions of free loop spaces.","marker":"[30]"},{"why":"Provides the rational Sullivan models used to construct and compute the examples $M_0$ and $M_1$.","marker":"[18]"}],"fun_headline_variants":["Cohomology interleaving distance equals homotopy interleavings","All interleaving distances coincide on circle-classifying spaces","Cohomology distance matches homotopy for spaces over the circle","Cohomology interleaving distance unifies homotopy interleavings"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The equality rests on the single fact that a one-parameter persistence dg module over a field can be replaced, up to quasi-isomorphism, by its own cohomology, because the polynomial ring $K[t]$ is hereditary; if that formality fails, the distances can diverge.","fun_headline_variants_meta":{"raw":{"variants":["Cohomology interleaving distance equals homotopy interleavings","All interleaving distances coincide on circle-classifying spaces","Cohomology distance matches homotopy for spaces over the circle","Cohomology interleaving distance unifies homotopy interleavings"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001146,"raw_usage":{"total_tokens":4767,"prompt_tokens":971,"completion_tokens":3796,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":587,"completion_tokens_details":{"reasoning_tokens":3715}},"tokens_in":587,"tokens_out":3796,"duration_ms":28660,"temperature":1.0,"reasoning_tokens":3715,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:06:46.695141+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Independently compute the homotopy interleaving distance between the two dg $K[u]$-modules coming from the spaces $M_0$ and $M_1$ of Proposition 6.3, directly from the definition of $\\varepsilon$-homotopy interleavings rather than from barcodes. The paper's Theorem 4.7 predicts the value $3$; any other value would falsify the claimed equality.","supporting_citations":[{"cited_title":"Lanari and L","cited_arxiv_id":null,"evidence_quote":"Introduces the interleaving distance in the homotopy category and the rectification technique used in Proposition 3.7."},{"cited_title":"Bubenik and J.A","cited_arxiv_id":null,"evidence_quote":"Supplies the interleaving-distance formalism, the inequalities among the three homotopy distances, and the isometry theorem used for barcode computations."},{"cited_title":"Chazal, D","cited_arxiv_id":null,"evidence_quote":"Defines $\\varepsilon$-interleavings and the interleaving distance, the basic objects compared throughout."},{"cited_title":"W ebb, Decomposition of graded modules, Proceedings of the American Mathematical Society 94 (1985), 565–571","cited_arxiv_id":null,"evidence_quote":"Gives the interval decomposition of graded $K[t]$-modules that underlies the barcodes."},{"cited_title":"Krause, Homological Theory of Representations, Cam bridge Studies in Advanced Mathe- matics, 195, Cambridge University Press, Cambridge, 2022","cited_arxiv_id":null,"evidence_quote":"Shows that formality in the derived category is equivalent to the base category being hereditary, which is exactly the single-parameter limitation of Lemma 3.5."},{"cited_title":"Kuribayashi, T","cited_arxiv_id":null,"evidence_quote":"Provides the BV-exactness criterion used to compute distances between Borel constructions of free loop spaces."},{"cited_title":"F´ elix, S","cited_arxiv_id":null,"evidence_quote":"Provides the rational Sullivan models used to construct and compute the examples $M_0$ and $M_1$."}],"review_version":1}