{"id":"f9182432-850b-4b66-9b37-1f5fa60cb5dd","arxiv_id":"2606.29135","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Closed manifolds with positive scalar curvature and an Ingham-decay fundamental group have vanishing simplicial volume.","lead":"This paper proves Gromov's conjecture that positive scalar curvature forces the simplicial volume to vanish, for manifolds whose fundamental group satisfies a new, much weaker form of rapid decay called Ingham decay. The proof pairs the authors' quantitative index theory with Ingham's Fourier-decay theorem.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Prop 4.2's key bound (4.11)–(4.12) drops the closed-walk tuple count |B|^m, which is exponential for hyperbolic groups and cannot be absorbed by Ingham decay; the estimate is invalid as printed and needs the cyclic-walk collapse repair.","rationale":"The reader's weakest_assumption singles out exactly the step (4.11)→(4.12) in Proposition 4.2, and my independent check confirms that this is the most load-bearing point. If (4.11)→(4.12) is false as written, the key estimate (4.4) does not follow, and the theorem's proof collapses for the very groups (hyperbolic, property RD) that the paper claims as applications. The reader and I also share the judgment that this is not a fundamental obstruction: the natural cyclic-walk collapse replaces the erroneous tuple sum by (m+1)! a_ε^{*(m+1)}(e), for which the displayed bound in (4.12) is valid by  ⟨a^{*m}, a⟩ ≤ ‖a^{*m}‖_{ℓ_2}‖a‖_{ℓ_2} ≤ ‖a‖_{C*}^m ‖a‖_{ℓ_2}. Thus the appropriate verdict is CONDITIONAL, exactly as the reader stated. I see no need to move to REJECT, because the gap is localized and repairable; but until the repair is written, the central claim is not fully supported. Secondary issues (the abstract's Ricci-curvature application is absent from the body; Theorem 3.1 is imported from the authors' unverified preprint [43]) reinforce the conditional verdict but are less decisive than the counting gap.","tokens_in":19201,"tokens_out":10147,"duration_ms":89116,"concrete_test":"Re-derive Proposition 4.2 for Γ = F_2 (or any hyperbolic group with exponential growth), m = 1, and a_ε the indicator of the word ball of radius R = Cε + C. Compute the printed left side of (4.11), the sum over closed walks of length 2, and compare it with the right side of (4.12). The ratio should grow like |B_R| if the printed step is false. Then compute the corrected expression 2(a_ε ∗ a_ε)(e) and verify it is bounded by 2‖a_ε‖_{C*}‖a_ε‖_{ℓ_2}; finally check that with this corrected bound the Ingham-decay argument in §4.4 still makes the right-hand side of (4.4) tend to zero.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The estimate that drives Theorems 1.1 and 1.2 is the step from (4.11) to (4.12) in Proposition 4.2. What precedes (4.11) gives, for each permutation σ, a sum over tuples (γ_0, …, γ_{m+1}) with γ_0 = γ_{m+1} = e of products ∏ a_ε(γ_{σ(j)}^{-1}γ_{σ(j+1)}). Summing over the Γ-variables with the product constraint should produce the single convolution value a_ε^{*(m+1)}(e), up to the (m+1)! permutation factor. The displayed (4.11), however, replaces this by a sum over the same tuples of |(a_ε ∗ ⋯ ∗ a_ε)(γ_{σ(0)})|. Taken literally, for each σ the sum over all closed walks has roughly |B_{Cε+C}|^m terms, and the convolution value at γ_{σ(0)} does not collapse the walk count. For exponential-growth groups, such as the hyperbolic groups the theorem explicitly covers, this introduces a factor e^{cε}. The remaining Ingham decay factor e^{CΦ(Cε+C)} is subexponential by (1.7), so it cannot absorb e^{cε}; hence (4.4) and (4.5) would not tend to zero. The argument does admit a natural repair: replace (4.11) by the exact cyclic-walk collapse (m+1)! a_ε^{*(m+1)}(e), then bound this by (m+1)!‖a_ε‖_{C*}^m‖a_ε‖_{ℓ_2}. That would restore the printed right-hand side of (4.12). But as written the proof of the central estimate has a genuine gap, and without the repair the main vanishing theorem is unsubstantiated for exponential-growth fundamental groups.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to prove Gromov's conjecture that a closed oriented manifold admitting a positive scalar curvature metric has vanishing simplicial volume, under the additional assumption that the fundamental group satisfies an Ingham decay property (Theorems 1.1 and 1.2). The proof uses a quantitative index theorem from the authors' preprint [43] to represent the fundamental class (and, more generally, Poincaré duals of the A-hat class) by cyclic traces of a finite-propagation idempotent. The central estimate, Proposition 4.2, bounds simplicial norms by a factor e^{CΦ(Cε+C)} times an operator norm; under positive scalar curvature, Ingham's theorem on Fourier decay is used to make this bound tend to zero as ε→∞. Section 5 extends the method to characteristic numbers paired with cohomology classes of Ingham growth. The abstract also promises a geometric application involving a universal negative Ricci lower bound, but no such result appears in the body.","tokens_in":19488,"tokens_out":25116,"duration_ms":222144,"significance":"If the main estimate is correct, this is a substantive advance: it proves Gromov's simplicial volume vanishing conjecture for a wide class of fundamental groups (in particular hyperbolic groups via property RD), and strengthens the conclusion to all Poincaré duals of the A-hat class. The paper is well structured, the Ingham lemma is proved essentially self-containedly, and the overall architecture — quantitative index theory plus Fourier decay — is convincing and likely repairable. It should be credited for making the group-theoretic hypothesis explicit and for avoiding heavy K-homology machinery. However, as printed, the proof of the key estimate in Proposition 4.2 is invalid, and the abstract advertises a Ricci-curvature application that is not proved in the text. These are load-bearing gaps that must be fixed before the paper can be accepted.","major_comments":[{"comment":"The chain of inequalities used to prove the central estimate (4.4) is not correct as written. Combining (4.6)-(4.9) gives, for each σ, the sum over γ0=e, γ1,...,γm of ∏_{j=0}^m aε(γ_{σ(j)}^{-1}γ_{σ(j+1)}), with γ_{σ(m+1)}=γ_{σ(0)}. This is not |(aε*...*aε)(γ_{σ(0)})|; in fact the sum in (4.11) is infinite for infinite Γ because the summand depends only on one of the free γ variables. Even on the charitable reading, bounding the tuple sum by (m+1)!∥aε*...*aε∥_{ℓ2} drops the number of closed walks of length m+1 with steps in supp(aε). For hyperbolic Γ this count is roughly |B_{Cε+C}|^m, exponential in ε, and the subexponential factor e^{CΦ(Cε+C)} cannot absorb it. The repair is standard: for each σ the tuple sum collapses exactly to aε^{*(m+1)}(e), so the σ-sum is (m+1)! aε^{*(m+1)}(e), and this is at most (m+1)!∥aε∥_{C*}^m∥aε∥_2, recovering the right-hand side of (4.12). This cyclic-walk","section":"Section 4.2, Eqs. (4.6)-(4.12)"},{"comment":"The abstract promises a geometric application: in every dimension there exists a universal negative constant such that any closed oriented manifold with scalar curvature bounded below by a normalized positive constant and Ricci curvature bounded below by the universal constant has vanishing simplicial volume. No such statement is proved or even stated in the body; Section 5 concerns characteristic numbers and cocycles of Ingham growth, not Ricci lower bounds. This advertised result must either be stated and proved in the paper or removed from the abstract.","section":"Abstract, last sentence"},{"comment":"The indexing conventions need to be made precise to support the repair of (4.11). The sum in (3.19) is over γ0=γm+1=e and γ1,...,γm, but the chain δ(γ0,...,γm) has only m+1 entries and the trace product uses σ(m+1)=σ(0); the role of γ_{m+1}=e is unclear. In the cyclic collapse one must identify exactly which γ_i are free and which is constrained by γ0=e. The authors should parametrize the sum by the cyclic increments h_j=γ_{σ(j)}^{-1}γ_{σ(j+1)} and prove the bijection with {h_0...h_m=e}; without this, the claimed equality S_σ=aε^{*(m+1)}(e) is not formally justified.","section":"Section 3.3 and Section 4.2, indexing in Theorem 3.2 and Proposition 4.2"}],"minor_comments":[{"comment":"The Fourier transform notation is inconsistent: in (4.18)-(4.21) the paper writes ψ_n where it means the Fourier transform, and in (4.22)-(4.24) qψ should be широко смотреть на рукопись.","section":"Section 4.3, Eqs. (4.18)-(4.21)"},{"comment":"The term 'logarithmic decay rate function' is misleading for the sublinear Φ used in the Ingham condition; property RD is the logarithmic case Φ(i)=C log i + C. Consider renaming it 'Ingham decay rate function'.","section":"Section 4.2, terminology"},{"comment":"The sentence 'we can choose the decay rate Ψ so that Ψ(cε) dominates CΦ(Cε+C)' should be justified: the chosen Ψ must also satisfy the Ingham conditions (4.16). This is true, but a short construction (e.g., Ψ(λ)=C'Φ(C''λ+C''') with adjusted constants) should be written out.","section":"Section 4.4, after Eq. (4.32)"},{"comment":"The claim that the preceding proof applies 'with only one modification' is too terse. After the repair of Proposition 4.2, one must check that the cocycle factor α(γ_{σ(k)},...,γ_{σ(0)}) is absorbed into the weighted convolution argument. This is probably routine, but it should be written out, or Proposition 5.1 should be moved to an appendix.","section":"Section 5, Proposition 5.1"},{"comment":"The paper relies on [43, Theorem 3.3] for the quantitative index theorem, which is load-bearing and is only quoted. Since [43] is a preprint, the authors should either include a proof sketch or state clearly that the main result depends on an external preprint.","section":"Section 3.3, Theorem 3.1"},{"comment":"The identity ∑γ ∥Iε,γ,e∥_HS^2 = ∥∑γ Iε,γ,e∥_HS^2 uses the fact that the operators Iε,γ,e have orthogonal ranges on the distinct fundamental domains γF; this should be stated explicitly.","section":"Section 4.2, Eq. (4.13)"}],"recommendation":"major_revision","confidential_remarks":"Dear Editor, the main theorems are likely correct and the gap in Proposition 4.2 appears repairable by a standard cyclic-walk collapse; I therefore recommend major revision rather than rejection. Please note that the key quantitative index theorem is quoted from the authors' own preprint [43]; the final version should either prove it or clearly flag it as an external dependency. The abstract's Ricci-curvature application seems to be a leftover from another version and is not supported by the body; it must be fixed. I saw no evidence of circularity or fitted constants: the Ingham decay hypothesis is an external group-theoretic condition and the conclusion is not forced by normalization."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the bottom line: if Theorem 1.1 is right, it is a major step on Gromov's simplicial volume conjecture, and the Ingham decay property is a genuinely new weakening of RD. The paper is worth taking seriously. But the proof as written has a real gap in Proposition 4.2 that needs fixing, and the abstract promises an application that is not in the body.\n\nThe new idea is good. The Ingham decay property (1.7)-(1.8) sits naturally between RD and the unconditional linear bound, and the summability condition Σ Φ(i)/i² < ∞ is exactly what Ingham's theorem needs to produce Fourier decay that can absorb the C*-norm growth. The extension to all A-hat classes is a nice strengthening, and the quantitative index theorem from [43] is a plausible engine. If the main theorem survives refereeing, it covers all rapid-decay groups, including hyperbolic groups.\n\nNow the soft spots. The step (4.11)->(4.12) in Proposition 4.2 is, as written, not justified. The sum over tuples in (4.11) has roughly |B_{Cε}|^m terms for each σ, and the displayed inequality drops that factor. For exponential-growth groups (which the theorem claims to cover), that factor is e^{cε}, and the Ingham decay cannot absorb it. The natural repair is to replace (4.11) by the exact cyclic-collapse: the sum over γ₁,...,γ_m of the product should give (m+1)! aε^{*(m+1)}(e), and then the stated (m+1)! ||aε||_{C*}^m ||aε||_{ℓ2} bound follows from the standard convolution estimate. That repair is straightforward and does not change the architecture, but as printed the proof of the central estimate is incomplete.\n\nTwo more things. The abstract claims a geometric application (universal negative Ricci constant + normalized PSC force vanishing simplicial volume in every dimension), but no such theorem appears in the body. Either include it or drop it from the abstract. And the load-bearing Theorem 3.1 is inherited verbatim from the authors' own preprint [43]; the current paper cannot be fully evaluated until [43] is either published or independently checkable. That is not by itself a flaw, but it is a dependency the referee needs to see.\n\nOverall: the strategy is coherent, the result is precisely stated, and the flaw is a gap with a plausible repair, not a demonstration that the theorem is false. I would send this to a serious referee.","headline":"Strong new result if the Proposition 4.2 gap is repaired; the Ingham decay idea is real, and the proof has a fixable counting error.","tokens_in":20171,"tokens_out":4763,"would_cite":true,"duration_ms":39282,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C21","53C23","19K56","58J20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under an Ingham decay condition on the fundamental group, positive scalar curvature forces the simplicial volume — and every Â-cap simplicial norm — of a spin-universal-cover manifold to vanish.","keywords":["simplicial volume","positive scalar curvature","Ingham decay property","fundamental group","index theory","A-hat genus","cap product","scalar curvature geometry"],"falsifier":"A direct counterexample — a closed oriented manifold with spin universal cover, Ingham-decay fundamental group, positive scalar curvature, and nonzero simplicial volume — would refute the theorem. Short of that, the proof's fate is settled by checking the omitted counting factor in the step (4.11)→(4.12): for a group with exponential word growth, the number of admissible tuples (γ₁,…,γ_m) with |γ_i|_w ≤ Cε+C is exponential in ε, so the displayed bound (m+1)!∥aε∥^m_{C*_r} ∥aε∥_{ℓ²} cannot hold unless a cyclic-walk collapse is intended; verifying this for one exponential-growth group would decid","tokens_in":18887,"feed_emoji":"📐","tokens_out":14444,"duration_ms":137271,"temperature":0.7,"pith_summary":"Gromov's conjecture says that any closed oriented manifold admitting a metric of positive scalar curvature must have zero simplicial volume — the infimal number of simplices needed to represent the fundamental class. The paper proves this conjecture for every closed oriented manifold whose universal cover is spin and whose fundamental group satisfies the Ingham decay property, a subexponential bound on the reduced C*-algebra norm of finitely supported functions that is strictly weaker than the classical rapid-decay (RD) property. In fact, it proves more: the simplicial norm of every cap product Â[k](TM) ∩ [M] vanishes for all 0 ≤ k ≤ m, so not just the fundamental class but every Â-class Poincaré dual is topologically cheap. The argument works directly in singular homology, using a quantitative index theorem to write the fundamental class as cyclic traces of a finite-propagation idempotent built from the Dirac operator; positive scalar curvature opens a spectral gap, and Ingham's theorem supplies a compactly supported Schwartz function whose Fourier transform decays subexponentially with a prescribed rate that dominates the group-theoretic weight. A byproduct stated in the abstract is a geometric application: in every dimension, a normalized positive scalar curvature lower bound together with a universal negative Ricci curvature lower bound forces vanishing simplicial volume.","feed_headline":"Positive scalar curvature forces simplicial volume to vanish","feed_subtitle":"For spin-universal-cover manifolds with Ingham-decay fundamental group, every Â-cap simplicial norm vanishes.","key_machinery":"The load-bearing construction is the finite-propagation 'difference idempotent' I_ε, a 2×2 operator-valued idempotent built from smooth functions of the Dirac operator with Fourier support in a ball of radius about ε/m; its pointwise cyclic traces Ch^ε_k give representatives of Â[m-k](TM) ∩ [M] in ε-homology. A quantitative index theorem (stated as Theorem 3.1 and spelled out in Theorem 3.2) expresses the image f_*[M] in group homology as an explicit sum of cyclic L²-traces of I_ε on translates of a fundamental domain. Gromov's mapping theorem lets the simplicial norm be computed in group homology. On the group side, the Ingham decay property (1.7)-(1.8) bounds the reduced C*-norm of finitel","core_discovery":"The central claim is Theorem 1.1 and its generalization Theorem 1.2: for a closed oriented manifold M whose universal covering is spin, if M admits a metric of positive scalar curvature and π₁(M) satisfies the Ingham decay property, then ||[M]||_{ℓ¹} = 0 and, more generally, ||Â[k](TM) ∩ [M]||_{ℓ¹} = 0 for every 0 ≤ k ≤ m. The proof identifies the fundamental class (and each Â-cap product) with a cycle in ε-homology whose simplicial norm is controlled by an operator bound; positive scalar curvature gives the Dirac operator a spectral gap away from zero, and the Ingham decay property is exactly the hypothesis that makes the finite-propagation representatives ε-small after the chosen functiona","pith_inferences":["The same index-theoretic template should work with any decay condition that is dominated by a subexponential Fourier-decay function; the Ingham condition is sufficient, not necessary, so other group invariants (e.g., a weighted ℓ² estimate tied to the growth function) could substitute without changing the proof's architecture.","Because the estimates are written before taking the limit ε → ∞, the argument yields explicit, dimension-dependent bounds on ||Â[k](TM) ∩ [M]||_{ℓ¹} in terms of the scalar-curvature lower bound and the group's decay function; this points toward a quantitative strengthening of the conjecture.","The theorem effectively upgrades the Lichnerowicz-type obstruction from a single characteristic number (the Â-genus) to an entire family of simplicial norms paired with group cohomology classes; testing these new invariants on known positive-scalar-curvature manifolds could reveal topological constraints not visible through the Â-genus alone."],"forward_implications":["Gromov's conjecture is settled for all closed oriented manifolds with spin universal cover and Ingham-decay fundamental group; since property RD implies Ingham decay, this covers hyperbolic groups and many other groups for which the conjecture was previously open.","The stronger conclusion means the Poincaré duals of the Â-class have zero simplicial norm, not just the fundamental class — a new quantitative topological obstruction: positive scalar curvature makes every Â-cap product ℓ¹-cheap.","The characteristic-number vanishing says ⟨Â(TM) ∪ f*α, [M]⟩ = 0 for every group cocycle of Ingham growth, a direct extension of the Lichnerowicz vanishing theorem beyond the Â-genus.","The abstract's geometric application gives a uniform dimension-dependent Ricci lower bound under which a normalized positive scalar curvature bound forces simplicial volume to vanish — a macroscopic analogue of the microscopic conjecture.","Because the proof operates directly in singular homology, it supplies explicit estimates on the simplicial norm before sending ε to infinity, so the method quantifies how fast the norm can be driven to zero in terms of the curvature lower bounds and the group's decay function."],"fun_headline_variants":["Positive scalar curvature and π₁ decay force simplicial volume to zero","Gromov's simplicial volume conjecture proved under Ingham decay","Spin cover, positive scalar curvature, π₁ decay: simplicial volume zero","Simplicial volume vanishes for spin manifolds with PSC and π₁ decay"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing step is the inequality (4.11)→(4.12), which bounds the ℓ¹-sum over tuples of word length O(ε) by (m+1)! times the ℓ² norm of the convolution power; this step does not count how many such tuples exist, and if their number grows exponentially in ε the Ingham-decay factor would need to be subexponential in a way the printed bound does not provide.","fun_headline_variants_meta":{"raw":{"variants":["Positive scalar curvature and π₁ decay force simplicial volume to zero","Gromov's simplicial volume conjecture proved under Ingham decay","Spin cover, positive scalar curvature, π₁ decay: simplicial volume zero","Simplicial volume vanishes for spin manifolds with PSC and π₁ decay"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00066,"raw_usage":{"total_tokens":2801,"prompt_tokens":640,"completion_tokens":2161,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":384,"completion_tokens_details":{"reasoning_tokens":2077}},"tokens_in":384,"tokens_out":2161,"duration_ms":14658,"temperature":1.0,"reasoning_tokens":2077,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T09:46:48.411506+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct counterexample — a closed oriented manifold with spin universal cover, Ingham-decay fundamental group, positive scalar curvature, and nonzero simplicial volume — would refute the theorem. Short of that, the proof's fate is settled by checking the omitted counting factor in the step (4.11)→(4.12): for a group with exponential word growth, the number of admissible tuples (γ₁,…,γ_m) with |γ_i|_w ≤ Cε+C is exponential in ε, so the displayed bound (m+1)!∥aε∥^m_{C*_r} ∥aε∥_{ℓ²} cannot hold unless a cyclic-walk collapse is intended; verifying this for one exponential-growth group would decid","supporting_citations":[],"review_version":3}