{"id":"980278e4-04b0-40a8-a3e9-c6df2c120912","arxiv_id":"2411.16235","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Persistence modules over continuous posets modulo ephemeral modules form a category equivalent to Scott sheaves, and the quotient functor is an isometry for interleaving distances.","lead":"This paper generalizes ephemeral persistence modules from Euclidean spaces to all continuous posets, and proves that the quotient by ephemeral modules is equivalent to sheaves on the Scott topology. It also shows that interleaving distances are preserved by this equivalence, so no metric information is lost in the quotient.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.32 is false as stated: for P=R and M a skyscraper, R^1soc_σ(M)=0 while top_σ(M)=M; the main quotient/isometry argument appears sound but the paper needs a correction.","rationale":"I verified the main categorical mechanism. Proposition 3.4 gives the adjoint triple; the identities j_*j_!=id and j_*j^*=id make j_! and j^* fully faithful, so Theorem 4.16 applies with Ker(j_*)=Eph from Theorem 4.6, yielding Theorem 4.17. The metric part also holds: Lemma 5.9 and Proposition 5.10 give the needed commutation with translations, Theorem 5.12 gives vanishing self-distances for strong translations, and the argument in Theorem 5.13 using j^* (not j_*) to lift interleavings closes the equality. The reader's weakest assumption, continuity of P, is indeed a genuine scope condition rather than a flaw. However, the paper contains a separate, concrete error: Theorem 3.32(i) is false. The skyscraper example is decisive and follows from the paper's own exact sequence in Proposition 3.27. Since the error is in an auxiliary homological statement and not in the derivation of Theorem 4.17 or 5.13, I would not reject the paper; but an unconditional ACCEPT is too strong. The authors should correct Theorem 3.32 (and any statements relying on it, e.g., the displayed consequences in the same subsection) before final acceptance.","tokens_in":95,"tokens_out":58443,"duration_ms":957961,"concrete_test":"Instantiate Theorem 3.32 with P=R and M the skyscraper module k_0 at 0. Compute both sides: top_σ(M)=M (since rad_σ(M)=0), while Proposition 3.27's exact sequence gives R^1soc_σ(M)=0 because M̅=0. If the assertion R^1soc_σ(M)=top_σ(M) fails for this example, the theorem must be corrected or removed.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central equivalence (Theorem 4.17) and isometry (Theorem 5.13) survive scrutiny: the adjoint triple (j^*, j_*, j_!), the identities j_*j_! = id and j_*j^* = id, Ker(j_*)=Eph, and the interleaving arguments under TR1–TR2 check out. The load-bearing defect is a false auxiliary theorem. Theorem 3.32(i) asserts R^1soc_σ(M)=top_σ(M) for all M. Let P=R and let M=k_0 be the skyscraper module at 0. Since no x<0 lies in the support, rad_σ(M)_p=0 for all p, so top_σ(M)=M. Since M is ephemeral, Theorem 4.6 gives j_*M=0, hence the upper closure M̅=0; Proposition 3.27's exact sequence 0→soc_σ(M)→M→M̅→R^1soc_σ(M)→0 then forces R^1soc_σ(M)=0. Thus the claimed equality fails (k_0≠0). The theorem is not used in the proofs of Theorem 4.17 or 5.13, so the main result may be salvageable, but the paper as written contains a demonstrable error and cannot be accepted without revision.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper generalizes the theory of ephemeral persistence modules from real-parameter and cone-ordered settings to arbitrary continuous posets. It defines ephemerality via the way-below relation, proves that ephemeral modules are exactly the kernel of the direct image functor to the Scott topology, and derives an equivalence between the quotient category Fun(P,Mod)/Eph and the category of Scott sheaves, with further equivalences to upper and lower semi-continuous modules. A second main thread defines interleavings for Scott sheaves and proves that the quotient functor is an isometry (Theorem 5.13). The paper is largely self-contained and uses domain-theoretic interpolation systematically.","tokens_in":14,"tokens_out":43218,"duration_ms":526036,"significance":"If the main results stand, they provide a clean domain-theoretic framework for observable persistence modules: the quotient by ephemeral modules is modeled by sheaves on the Scott topology, and the interleaving distance is preserved exactly. The adjoint triple construction and the isometry theorem are nontrivial and potentially useful for multi-parameter persistence. I verified that the central chain Theorem 4.6 -> Theorem 4.16 -> Theorem 4.17, and the interleaving arguments in Propositions 5.10 and 5.12 -> Theorem 5.13, is coherent and does not rely on the flawed auxiliary homological statements discussed below. However, the paper currently contains demonstrable false statements in Section 3.5, so it cannot be accepted without revision.","major_comments":[{"comment":"Theorem 3.32(i), asserting R^1soc_sigma(M) = top_sigma(M), is false as stated. Let P=R and let M be the skyscraper module at 0. Since M(p<=q)=0 whenever p<q, M is ephemeral by Definition 4.1. By Theorem 4.6, j_*M=0; by Lemma 3.8(i), the upper closure \\overline{M} equals j_!j_*M=0. The exact sequence of Proposition 3.27 then forces R^1soc_sigma(M)=0. On the other hand, rad_sigma(M)_0 is the union of images of M(x<=0) for x<0, all of which are zero, so top_sigma(M)_0=M_0=k; hence top_sigma(M)=M. This contradicts the claimed equality. The difficulty is already present in Proposition 3.28(i): for M=k[(-infinity,0)] on R and p=0, we have M_0=0, the lower closure \\underline{M}_0 = colim_{x<0} M_x = k, and top_sigma(M)_0=0, so the asserted exact sequence 0 -> L_1top_sigma(M) -> M -> \\underline{M} -> top_sigma(M) -> 0 becomes 0 -> A -> 0 -> k -> 0 -> 0, which is impossible. The likely correction is that the middle term should be colim_{x<=p} M_x rather than the stalk M_p; this should be worked out carefully and the derived statements adjusted.","section":"3.5"},{"comment":"The false statement of Proposition 3.28(i) also undermines the presentation of Proposition 3.31(ii) and Theorem 3.32(ii), even if the main quotient and isometry theorems are unaffected. Because the exact sequence is used to define/describe L_1top_sigma, the paper should either correct the exact sequence and re-derive Theorem 3.32, or remove Theorem 3.32 and Proposition 3.31(ii) and clearly state what remains true for the Scott-socle and Scott-top. As written, a reader using these results would obtain incorrect conclusions; for example, the skyscraper module above would simultaneously be a counterexample to Theorem 3.32(i). I stress that Theorem 4.17 and Theorem 5.13 do not invoke these results, so the central claims appear salvageable.","section":"3.5"}],"minor_comments":[{"comment":"The notation for j_* and j^* is nearly indistinguishable in the text (e.g., in Proposition 3.4, Lemma 3.8, and Theorem 4.6). Please use unambiguous superscript and subscript stars in the final version, since the adjunction identities depend on which is which.","section":"Throughout"},{"comment":"The proof says 'As in the proof of Proposition 3.27' but the dual argument is not literally the same: the tensor-product description of top_sigma involves colimits over down-sets, not limits over up-sets. Once the exact sequence is corrected, the proof should spell out the colimit computation.","section":"3.5"}],"recommendation":"major_revision","confidential_remarks":"The central equivalence and isometry results have been stress-tested and appear sound; the errors are confined to Section 3.5 and the auxiliary socle/top statements. The paper should not be rejected, but it cannot be accepted while Proposition 3.28(i) and Theorem 3.32(i) are false as stated. I recommend asking the authors to correct the exact sequence and the derived results, and to verify whether Proposition 3.31(ii) and Proposition 4.14(iv) remain valid under the correction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main takeaway: the paper's principal results are worth engaging with, but there is a concrete error in an auxiliary theorem that needs fixing before publication.\n\nThe genuinely new content is the extension of Chazal et al. and Berkouk-Petit from R and R^n to arbitrary continuous posets: the quotient category Fun(P)/Eph is shown equivalent to Scott sheaves, and the quotient functor is an isometry for superlinear translations satisfying TR1-TR2. The adjoint triple (j^*, j_*, j_!) around the identity map from Alexandrov to Scott topology is cleanly developed, and the proofs are detailed and mostly self-contained. I checked the key adjunction identities and the ephemeral characterization (Theorem 4.6); they hold. The homological side, with Scott-socle and Scott-top and their derived functors, is original even if it now appears to contain a mistake.\n\nThe stress-test is right: Theorem 3.32(i) fails. Over P=R, take the skyscraper module k_0 at 0. It is ephemeral, so j_*M=0 and hence the upper closure \\overline{M}=j_!j^*M=0. The exact sequence of Proposition 3.27 then forces R^1soc_σ(M)=0. But rad_σ(M)=0 and top_σ(M)=M≠0. So R^1soc_σ(M)≠top_σ(M). The dual formula (ii) likely breaks too. The good news is that Theorem 3.32 is not used in the main quotient equivalence (Theorem 4.17) or the isometry theorem (Theorem 5.13); those arguments rely only on the adjoint triple, Theorem 4.6, and the interleaving lemmas. So the main results are likely salvageable, but the paper as submitted cannot be accepted verbatim.\n\nOther soft spots are minor: the definition of ephemeral via ≪ differs from the strict-order version when the poset has a bottom element, but that is acknowledged. The dependence on continuity of P is standard and well-motivated.\n\nFor whom: anyone working in multiparameter persistence or sheaf-theoretic TDA will want to know these results once the error is corrected. The paper deserves a serious referee: the central argument is substantial and mostly correct. My recommendation: send it to review, but require the authors to fix Theorem 3.32 and check the surrounding homological statements carefully.","headline":"Solid generalization of ephemeral modules to continuous posets, but Theorem 3.32 is false as stated; the main quotient and isometry theorems look sound.","tokens_in":27703,"tokens_out":2945,"would_cite":false,"duration_ms":25503,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N31","06B35","18F20","18E35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Over any continuous poset, quotienting persistence modules by the ephemeral modules yields exactly the sheaves on the Scott topology, and the quotient functor preserves interleaving distances.","keywords":["ephemeral modules","persistence modules","Scott sheaves","continuous posets","interleaving distance","semi-continuous modules","way-below relation","quotient categories"],"falsifier":"Find a continuous poset $P$ and a superlinear family of translations satisfying TR1–TR3 with two persistence modules $M,N$ such that $d_a(M,N)\\ne d_\\sigma(j_*M,j_*N)$; that would disprove the isometry Theorem 5.13. Concretely, the easiest place to look is a module with nonzero interleaving distance to zero that is not ephemeral, since Corollary 5.14 asserts the two conditions coincide.","tokens_in":26647,"feed_emoji":"🕸️","tokens_out":6206,"duration_ms":51954,"temperature":0.7,"pith_summary":"The paper extends the notion of an ephemeral persistence module, whose internal maps vanish along the way-below relation, from the real line to any continuous poset. Its central result is that, after quotienting out the ephemeral modules, the remaining category is exactly the category of sheaves on the Scott topology of the poset. This identifies the observable part of a persistence module with sheaf-theoretic data, and the paper further shows that the interleaving distance is preserved by this quotient. The upshot is a single framework in which persistence modules, Scott sheaves, and upper or lower semi-continuous modules are equivalent descriptions of the same objects.","feed_headline":"Quotient by ephemeral modules is the Scott sheaf category","feed_subtitle":"On continuous posets, the quotient functor is an isometry, so interleaving distances survive passing to sheaves.","key_machinery":"The adjoint triple $(j^*, j_*, j_!)$ linking Alexandrov sheaves (which are the same as persistence modules) to Scott sheaves, via the identity map $j\\colon P^a\\to P^\\sigma$. On a continuous poset, the interpolation property of the way-below relation makes $j_*j^*=\\mathrm{id}$ and $j_*j_!=\\mathrm{id}$, so $j_*$ becomes an exact localization whose kernel is the ephemeral modules. The endofunctors $M\\mapsto \\overline{M}$ (limits over elements way above $p$) and $M\\mapsto \\underline{M}$ (colimits over elements way below $p$) form the unit and counit of this adjunction and connect the sheaf picture to upper and lower semi-continuity.","core_discovery":"For a continuous poset $P$, the category of persistence modules over $P$ has a Serre subcategory $\\mathrm{Eph}$ of ephemeral modules, defined by $M(p\\le q)=0$ whenever $p$ is way below $q$. The main theorem (Theorem 4.17) states that the quotient category $\\mathrm{Fun}(P,\\mathrm{Mod})/\\mathrm{Eph}$ is equivalent to the category of sheaves on the Scott topology, and also equivalent to the full subcategories of upper and lower semi-continuous modules. Theorem 5.13 sharpens this to a metric statement: the quotient functor $j_*$ is an isometry, so $d_a(M,N)=d_\\sigma(j_*M,j_*N)$, and Corollary 5.14 characterizes ephemerality as having zero interleaving distance to the zero module.","pith_inferences":["The continuity assumption is likely necessary in a strong sense: without the interpolation property, the identities $j_*j^*=\\mathrm{id}$ and $j_*j_!=\\mathrm{id}$ may fail, so the quotient-sheaf equivalence would have to take a different form for non-continuous posets.","The isometry theorem suggests a transfer principle for stability: any stability bound proved for sheaf interleavings automatically applies to persistence modules, and conversely, because the two distances coincide.","The equivalence with semi-continuous modules may connect to symplectic topology, where semi-continuous persistence modules are already used, giving a concrete sheaf-theoretic handle on those modules.","A natural testable extension is to replace the Scott topology by other topologies generated by a different approximating relation; the meager-set characterization of ephemerality suggests that any topology with a basis of the same form would yield an analogous quotient."],"forward_implications":["Ephemeral modules form a bilocalizing Serre subcategory, so the quotient category has both a section and a cosection, and the Scott-socle and Scott-top are the torsion and torsion-free parts.","Scott sheaves, upper semi-continuous modules, and lower semi-continuous modules are three equivalent concrete models for the observable quotient of persistence modules.","Because the quotient functor is an isometry, the interleaving distance between persistence modules equals the interleaving distance between their Scott sheaves, so no metric information is lost when passing to the quotient.","A module is ephemeral exactly when its interleaving distance to the zero module vanishes (under conditions TR1–TR3), so the quotient removes precisely the modules that are indistinguishable from zero by interleavings.","Over $\\mathbb{R}^n$, the Scott-socle and Scott-top of boundary-type indicator modules recover standard topological boundaries, giving sheaf-theoretic formulas for socle and top."],"supporting_citations":[{"why":"Introduces observable modules, the quotient-category viewpoint that this paper generalizes to continuous posets.","marker":"[9]"},{"why":"Defines ephemeral persistence modules over $\\mathbb{R}^n$ via $j_*M=0$, the definition the paper adopts and extends.","marker":"[3]"},{"why":"Establishes the equivalence between persistence modules and sheaves on the Alexandrov topology.","marker":"[10]"},{"why":"Supplies the general theory of interleaving distances used throughout Section 5.","marker":"[5]"},{"why":"Provides the domain-theoretic foundations, including continuous posets, the way-below relation, and the Scott topology.","marker":"[14]"},{"why":"Theorem 4.16 is the quotient-category criterion that turns the adjoint triple into the main equivalence.","marker":"[7]"},{"why":"Gives the basis of the Scott topology and the interpolation property used in Lemma 2.2 and throughout.","marker":"[13]"},{"why":"The gamma-sheaves over $\\mathbb{R}^n$ that Scott sheaves generalize.","marker":"[19]"},{"why":"Classifies indecomposable injective persistence modules, used for Proposition 4.20.","marker":"[16]"}],"fun_headline_variants":["Ephemeral quotient yields Scott sheaves","Modding by ephemeral modules gives Scott sheaves","Interleaving distance preserved in sheaf quotient","Persistence mod ephemeral: Scott sheaf category","Quotient functor is an isometry to Scott sheaves"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument rests on the poset $P$ being continuous, meaning every element is the directed supremum of elements way below it, which yields the interpolation property used in nearly every proof; if $P$ is not continuous, the key identities $j_*j^*=\\mathrm{id}$ and $j_*j_!=\\mathrm{id}$ may fail and the main equivalence is not established.","fun_headline_variants_meta":{"raw":{"variants":["Ephemeral quotient yields Scott sheaves","Modding by ephemeral modules gives Scott sheaves","Interleaving distance preserved in sheaf quotient","Persistence mod ephemeral: Scott sheaf category","Quotient functor is an isometry to Scott sheaves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000126,"raw_usage":{"total_tokens":1212,"prompt_tokens":769,"completion_tokens":443,"prompt_tokens_details":{"cached_tokens":768},"prompt_cache_hit_tokens":768,"prompt_cache_miss_tokens":1,"completion_tokens_details":{"reasoning_tokens":367}},"tokens_in":1,"tokens_out":443,"duration_ms":25896,"temperature":1.0,"reasoning_tokens":367,"cache_read_input_tokens":768,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:24:42.515386+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a continuous poset $P$ and a superlinear family of translations satisfying TR1–TR3 with two persistence modules $M,N$ such that $d_a(M,N)\\ne d_\\sigma(j_*M,j_*N)$; that would disprove the isometry Theorem 5.13. Concretely, the easiest place to look is a module with nonzero interleaving distance to zero that is not ephemeral, since Corollary 5.14 asserts the two conditions coincide.","supporting_citations":[{"cited_title":"18 (2016), no","cited_arxiv_id":null,"evidence_quote":"Introduces observable modules, the quotient-category viewpoint that this paper generalizes to continuous posets."},{"cited_title":"Thesis (Ph.D.)–Univers ity of Pennsylvania","cited_arxiv_id":null,"evidence_quote":"Establishes the equivalence between persistence modules and sheaves on the Alexandrov topology."},{"cited_title":"22, Cambridge University Press, Cambridge, 2013","cited_arxiv_id":null,"evidence_quote":"Provides the domain-theoretic foundations, including continuous posets, the way-below relation, and the Scott topology."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Theorem 4.16 is the quotient-category criterion that turns the adjoint triple into the main equivalence."},{"cited_title":"Gierz, K","cited_arxiv_id":null,"evidence_quote":"Gives the basis of the Scott topology and the interpolation property used in Lemma 2.2 and throughout."},{"cited_title":"2 92, Springer-Verlag, Berlin, 1994","cited_arxiv_id":null,"evidence_quote":"The gamma-sheaves over $\\mathbb{R}^n$ that Scott sheaves generalize."},{"cited_title":", Manuscripta mathematica 44 (1983), 45-50","cited_arxiv_id":null,"evidence_quote":"Classifies indecomposable injective persistence modules, used for Proposition 4.20."}],"review_version":1}