{"id":"2e31c66c-96a9-4c83-8c6e-21495e8c0a0b","arxiv_id":"2607.08621","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"For closed oriented manifolds X with dim X ≥ 5 and χ(X)=0, the Rosenberg S¹-stability conjecture holds: X × S¹ admits a PSC metric if and only if X does.","lead":"The paper proves that for closed oriented manifolds X of dimension at least 5 with zero Euler characteristic, X admits a positive scalar curvature (PSC) metric if and only if X × S¹ does. This resolves a 2006 conjecture by Rosenberg under the topological condition χ(X)=0, linking the vanishing of Euler characteristic to the existence of PSC metrics on product manifolds.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"Theorem 2.1's diffeomorphism construction may not yield a separating hypersurface, and the pointwise angle verification has a potential gap in the Jacobian computation.","rationale":"The reader correctly identifies Theorem 1.3 as a load-bearing external dependency from an unpublished preprint, and this is a legitimate concern. However, I identify an additional internal concern in Theorem 2.1 that is arguably more critical: the pointwise verification of the angle condition relies on special local coordinates where the Jacobian takes a particularly simple form (14), and the argument that this suffices at every point of {ξ̃=0} is not fully justified. The computation g(∂_ξ̃, ∂_ξ̃)·g^{-1}(dξ̃, dξ̃) = 1/g^{ξξ} + 1 uses the specific Jacobian structure from (15), which depends on the O(|(x,ξ)|^2) terms being irrelevant. At the point Q itself this is fine (the Jacobian is evaluated exactly at the center), but the claim that this works at every point of the hypersurface requires that suitable coordinates can be chosen at each point independently, which the paper asserts but does not fully verify. The separating property in Theorem 3.1 is also not fully established. These concerns, combined with the external dependency on Theorem 1.3, support maintaining the CONDITIONAL verdict. The core idea of connecting χ(X)=0 to the angle condition via a nowhere-vanishing vector field is sound and interesting, but the execution has gaps that need closing.","tokens_in":11198,"tokens_out":880,"duration_ms":1439420,"concrete_test":"Independently re-derive the Jacobian J_F at a general point Q ∈ {ξ̃=0} without using the special coordinates where U(Q)=∂_1, by computing dF_Q directly from the flow definition (7). Then verify that g(∂_ξ̃, ∂_ξ̃)·g^{-1}(dξ̃, dξ̃) < 2 holds at points where the O(|(x,ξ)|^2) correction terms in (14) are non-negligible. If the product exceeds 2 at any such point, the angle condition fails and Theorem 2.1 does not hold.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The reader correctly identifies Theorem 1.3 as an external dependency, but the more critical concern is internal to Theorem 2.1 itself. The proof constructs a diffeomorphism F via the flow of a vector field U and then verifies the angle condition (5) pointwise using special local coordinates. The key computation at equation (15) uses the Jacobian J_F and J_{F^{-1}}, which are derived from the local expression (14): F(x,ξ) = (x_1+ξ, x_2,...,x_n, ξ) + O(|(x,ξ)|^2). From this, the authors compute g(∂_ξ̃, ∂_ξ̃)(Q)·g^{-1}(dξ̃, dξ̃)(Q) = 1/g^{ξξ} + 1, and since g^{ξξ} > 1, conclude this is < 2. However, the local expression (14) is only first-order accurate at Q, while the computation of g(∂_ξ̃, ∂_ξ̃) involves the metric g evaluated on vectors derived from the Jacobian. The O(|(x,ξ)|^2) terms in (14) could contribute first-order corrections to the Jacobian at nearby points, and the argument that verification at Q suffices for all points on {ξ̃=0} needs more justification. Additionally, the claim in Theorem 3.1 that {ξ̃=0} separates ∂Y_{a,-} and ∂Y_{a,+} relies on the limiting behavior of h_q(t), but the separating property requires more than just existence of a unique t — it requires that {ξ̃=0} is properly embedded and disconnects the cylinder, which is not explicitly verified.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This paper addresses the 2006 Rosenberg S^1-stability conjecture and the 1994 Rosenberg-Stolz conjecture for X × ℝ. The main result (Theorem 1.1) states that for a closed, oriented manifold X with dim X ≥ 5 and χ(X) = 0, the S^1-stability conjecture holds: X × S^1 admits a positive scalar curvature (PSC) metric if and only if X does. The nontrivial direction proceeds by lifting a PSC metric from X × S^1 to X × ℝ, constructing a diffeomorphism F of X × ℝ (Theorem 2.1) using a nowhere-vanishing vector field on X (guaranteed by χ(X) = 0) so that an angle condition ∠_g(ν_g, ∂_ξ̃) ∈ [0, π/4) holds on the hypersurface {ξ̃ = 0}, then invoking a prior result (Theorem 1.3, from [16]) to obtain a PSC metric on that hypersurface, and finally applying Proposition 3.1 (Råde's result [11]) to conclude X admits PSC. A T^n-stability generalization (Theorem 1.2) follows by induction.","tokens_in":11447,"tokens_out":1762,"duration_ms":182720,"significance":"The Rosenberg S^1-stability conjecture is a well-known open problem in scalar curvature geometry. The paper's approach of connecting the topological condition χ(X) = 0 to a geometric angle condition on the metric is novel and dimension-independent (for dim ≥ 5). The key technical contribution is Theorem 2.1, which is self-contained and provides an explicit diffeomorphism construction. The overall argument depends on Theorem 1.3 from the author's companion preprint [16] (arXiv:2509.24016), which is not independently verified here; this is a significant external dependency that should be transparently acknowledged. The results are falsifiable in the sense that the angle condition is checkable for specific metrics.","major_comments":[{"comment":"§2, proof of Theorem 2.1, around Eq. (14)–(15): The pointwise verification of the angle condition (5) uses the local expression F(x,ξ) = (x_1+ξ, x_2,...,x_n, ξ) + O(|(x,ξ)|^2), from which the Jacobians J_F and J_{F^{-1}} at Q are computed. The computation of g(∂_ξ̃, ∂_ξ̃)(Q)·g^{-1}(dξ̃, dξ̃)(Q) = 1/g^{ξξ} + 1 then uses these Jacobians together with the metric components at Q. While the first-order Jacobian at the center point Q is correct, the claim that this pointwise computation at each Q (with Q-dependent coordinates) suffices to establish (5) on all of {ξ̃ = 0} needs more explicit justification. Specifically, the O(|(x,ξ)|^2) terms in (14) do not affect the Jacobian at the center Q itself, but the argument should state clearly that the computation is performed exactly at Q (where the coordinates are centered) and that the choice of coordinates varies smoothly with Q. The current phra","section":null},{"comment":"§3, proof of Theorem 3.1: The claim that {ξ̃ = 0} separates ∂Y_{a,-} and ∂Y_{a,+} in Y = X × [-a, a]_ξ for sufficiently large a is asserted to follow from the limiting behavior of h_q(t). The monotonicity and surjectivity of h_q(t) established in Theorem 2.1 show that {ξ̃ = 0} is a properly embedded hypersurface diffeomorphic to X, but the separating property in the compact cylinder Y requires that {ξ̃ = 0} disconnects Y into two components each containing one boundary face. This should be verified more explicitly: for instance, by showing that the projection π_R restricted to {ξ̃ = 0} is bounded (so {ξ̃ = 0} lies in some Y for large a) and that the two sides of {ξ̃ = 0} in X × ℝ connect to the two ends, which then implies separation in Y for large a. The current argument is plausible but incomplete as stated.","section":null},{"comment":"Theorem 1.3 (cited from [16], arXiv:2509.24016) is a load-bearing external result: the entire passage from the angle condition to PSC on the hypersurface depends on it. Since [16] is an unpublished preprint by the same author, the paper should either (a) include a self-contained proof sketch of Theorem 1.3 sufficient for a reader to verify the key analytic steps, or (b) clearly state that the main results are conditional on the acceptance of [16]. As it stands, a reader cannot verify the main theorems without accessing and verifying a separate preprint.","section":null}],"minor_comments":[{"comment":"Abstract and throughout: 'smallest eigenvalue of g' should be clarified as 'smallest eigenvalue of the metric tensor g' (i.e., the uniform lower bound on g(v,v) for unit v in some background metric), to avoid ambiguity.","section":null},{"comment":"§1, Theorem 1.2 statement: 'X admits a PSC metric if and only if X × T^n, n ≥ 1' is missing a verb — should read 'if and only if X × T^n admits a PSC metric.'","section":null},{"comment":"§2, line below Eq. (3): 'Clearly X_0 = {ξ = 0}' — the notation X_0 is introduced here but the subscript 0 is used both for the hypersurface and for the point P = 0. Consider using X_P or Σ_P for clarity.","section":null},{"comment":"§2, proof of Theorem 2.1: The scaling argument 'we may thus assume that g_{ξξ} > 1, √g^{ξξ} ≤ 1 − ζ uniformly' should explain more explicitly how the one-time scaling achieves both conditions simultaneously (scaling the ξ-direction vs. scaling the whole metric).","section":null},{"comment":"§2, Eq. (9) and surrounding: The notation max_{W ∈ Γ(TX), ||W||_g = 1} g²(∂_ξ/||∂_ξ||_g, W) uses g² to denote the square of the inner product; this is nonstandard and could be confused with the second metric power. Consider writing [g(·,·)]².","section":null},{"comment":"§3, proof of Corollary 3.1: 'χ(M) = 0' should be 'χ(X) = 0' (M is not defined in this context).","section":null},{"comment":"References: [16] (arXiv:2509.24016) and [14] (arXiv:2412.12479) are cited as load-bearing but are unpublished preprints. This is acceptable but should be noted for the reader.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper's novelty is real but heavily front-loaded into the companion preprint [16]. The core new result here (Theorem 2.1) is a geometric construction that is largely checkable, but the analytic step (Theorem 1.3) is the actual mechanism producing PSC. I recommend minor revision with the request that the author either sketch the proof of Theorem 1.3 or clearly flag the dependency. The stress-test concern about the Jacobian computation is valid but addressable: the computation is pointwise at the center of adapted coordinates, and the O(|(x,ξ)|^2) terms genuinely do not contribute at the center point — but this needs to be stated more carefully. The separation concern is more substantive and should be addressed."},"author_rebuttal":null,"desk_editor":{"model":"glm-5.2","letter":"The paper proves that if X is closed, oriented, dim X ≥ 5, and χ(X) = 0, then X × S¹ admits a PSC metric iff X does, resolving the Rosenberg S¹-stability conjecture for this class. The key new idea is Theorem 2.1: χ(X) = 0 gives a nowhere-vanishing vector field, which is used to construct a diffeomorphism F of X × ℝ such that the angle condition ∠_g(ν, ∂_ξ̃) ∈ [0, π/4) holds on {ξ̃ = 0}. This bridges a topological condition to a geometric one without curvature assumptions, which is a clean and genuinely new connection. The T^n generalization follows by a straightforward recursion argument. The construction of F via the flow of a normalized vector field U, the monotonicity argument for h_q(t), and the pointwise Jacobian computation are the core technical content and appear largely correct on a careful read. The stress-test concern about the O(|(x,ξ)|²) terms in the local expression (14) affecting the Jacobian does not land: the computation is done pointwise at Q where the coordinates are centered, so the first-order expression is exact at that point, and the argument is repeated at each point of {ξ̃ = 0}. That part holds up. The real soft spots are two. First, the entire conclusion depends on Theorem 1.3 from arXiv:2509.24016 (same author, unpublished), which converts the angle condition into a PSC metric on the hypersurface via conformal geometry and elliptic PDE. If that theorem has gaps, this paper's argument collapses. Second, the claim in Theorem 3.1 that {ξ̃ = 0} separates the two boundary components of the cylinder X × [−a, a] is asserted but not properly justified. The monotonicity of h_q(t) gives existence and uniqueness of a level set, but separating requires that {ξ̃ = 0} is properly embedded and disconnects the cylinder — this is not explicitly verified and needs an argument. This is a gap, though likely fillable. The paper is for differential geometers working on scalar curvature, especially those familiar with the Rosenberg-Stolz program and Gromov's μ-bubble techniques. It deserves a serious referee who can check Theorem 1.3 in the companion paper and verify the separating-hypersurface claim. Recommend conditional acceptance pending resolution of these two issues.","headline":"New proof that χ(X)=0 implies Rosenberg S¹-stability for dim X≥5, but the argument rests on an unpublished companion paper and has a gap in the separating-hypersurface claim.","tokens_in":12212,"tokens_out":608,"would_cite":true,"duration_ms":121079,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Zero Euler characteristic unlocks scalar curvature stability","keywords":[],"falsifier":"A counterexample would be a closed, oriented manifold X with dim X ≥ 5 and χ(X) = 0 such that X admits no positive scalar curvature metric but X × S¹ does. Alternatively, if Theorem 1.3 fails — meaning there exists a uniformly PSC metric of bounded curvature on some X × ℝ satisfying the angle condition but for which no conformal deformation yields positive scalar curvature on the hypersurface — then the main argument collapses.","tokens_in":11392,"feed_emoji":"⭕","tokens_out":1345,"duration_ms":94099,"temperature":0.7,"pith_summary":"The paper proves that for a closed, oriented manifold X of dimension at least 5 with zero Euler characteristic, the product X × S¹ admits a metric of everywhere positive scalar curvature if and only if X itself does. This confirms a 2006 conjecture of Rosenberg under the topological hypothesis χ(X) = 0, and extends the result to higher-dimensional tori X × Tⁿ. The argument also establishes the analogous Rosenberg–Stolz conjecture for X × ℝ under additional geometric regularity on the metric. The central mechanism connects the purely topological condition χ(X) = 0 — equivalently, the existence of a nowhere-vanishing vector field on X — to a geometric angle condition on the noncompact cylinder X × ℝ. Specifically, the vanishing Euler characteristic lets one construct a global diffeomorphism of X × ℝ that reorients the coordinate direction so that the unit normal to a distinguished hypersurface makes an angle strictly less than π/4 with the R-direction. Once this angle condition is met, a prior result of the author (Theorem 1.3, from arXiv:2509.24016) produces a conformally related metric whose restriction to that hypersurface has positive scalar curvature. A topological result of Råde then propagates the positive scalar curvature from the separating hypersurface back to X itself, completing the nontrivial direction of the equivalence.","feed_headline":"Zero Euler characteristic unlocks scalar curvature stability","feed_subtitle":"A 2006 conjecture on when X × S¹ has positive scalar curvature iff X does is proved for all closed manifolds with χ = 0 and dimension at 5.","key_machinery":"The nowhere-vanishing vector field on X (equivalent to χ(X) = 0); the Gram–Schmidt orthogonalization of that vector field against ∂_ξ to produce a complete unit vector field U orthogonal to ∂_ξ; the global flow of U defining the diffeomorphism F; the angle condition ∠_g(ν_g, ∂_ξ) ∈ [0, π/4) and its equivalence to the algebraic bound 1 ≤ g(∂_ξ, ∂_ξ) · g⁻¹(dξ, dξ) < 2; Theorem 1.3 (the author's prior result) converting the angle condition to positive scalar curvature on a hypersurface via conformal deformation; and Proposition 3.1 (Råde's topological result) transferring positive scalar curvature from a separating hypersurface in X × [−a, a] to X itself when dim X ≥ 5.","core_discovery":"The key discovery is that χ(X) = 0 is sufficient to construct a global diffeomorphism F of X × ℝ such that, on the hypersurface {π_R ∘ F⁻¹ = 0}, the angle between the metric unit normal and the reoriented R-direction lies in [0, π/4). This converts a topological obstruction — the absence of a nowhere-zero vector field would force χ(X) ≠ 0 — into a geometric angle condition that is strong enough to guarantee positive scalar curvature on the hypersurface via conformal deformation. The bridge between topology and geometry is the nowhere-vanishing vector field guaranteed by χ(X) = 0, which is used to build a complete flow whose time-reparameterization defines the diffeomorphism F.","pith_inferences":["If the angle condition could be verified under a weaker topological hypothesis than χ(X) = 0 — for instance, under a Rosenberg index condition or an enlargeability condition — the same machinery would yield stability results for a strictly larger class of manifolds.","The dimensional threshold dim X ≥ 5 is inherited from Proposition 3.1 (Råde's result), which itself relies on the existence of smooth minimal hypersurfaces. Recent advances in generic regularity for minimizing hypersurfaces in higher dimensions could potentially push this threshold upward.","The dependence on Theorem 1.3 means the conformal deformation step is the least verified link in the chain; if the elliptic PDE analysis underlying that theorem extends to metrics with weaker curvature bounds, the bounded-curvature hypothesis on X × ℝ could potentially be relaxed."],"forward_implications":["For all odd-dimensional closed oriented manifolds X with dim X ≥ 5, the S¹-stability conjecture holds unconditionally, since χ(X) = 0 automatically for odd-dimensional closed manifolds.","The Rosenberg–Stolz conjecture for X × ℝ is confirmed for the class of complete, bounded-curvature, uniformly PSC metrics whose smallest eigenvalue is bounded below by a positive constant, whenever χ(X) = 0 and dim X ≥ 5.","The Tⁿ-stability theorem (Corollary 3.2) shows that positive scalar curvature on X is equivalent to positive scalar curvature on X × Tⁿ for any n ≥ 1, under the same hypotheses.","The angle-condition framework may extend to other product manifolds X × M where a suitable coordinate direction and flow can be controlled, potentially broadening the class of manifolds for which stability-type conjectures can be verified."],"fun_headline_variants":["χ(X) = 0 proves Rosenberg's S¹-stability conjecture","Zero Euler characteristic confirms Rosenberg S¹-stability","Rosenberg's S¹-stability holds when χ(X) = 0","Nowhere-zero vector fields from χ = 0 yield Rosenberg stability","χ = 0 bridges topology and geometry in Rosenberg's conjecture"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The argument depends on Theorem 1.3, a result from the author's own unpublished preprint, which states that a geometric angle condition on a hypersurface of X × ℝ is sufficient to produce a conformally deformed metric with positive scalar curvature on that hypersurface. If that theorem has gaps in its elliptic PDE or conformal geometry analysis, the entire chain from χ(X) = 0 to positive scalar curvature on X breaks, since the angle condition verified in this paper only has购买","fun_headline_variants_meta":{"raw":{"variants":["χ(X) = 0 proves Rosenberg's S¹-stability conjecture","Zero Euler characteristic confirms Rosenberg S¹-stability","Rosenberg's S¹-stability holds when χ(X) = 0","Nowhere-zero vector fields from χ = 0 yield Rosenberg stability","χ = 0 bridges topology and geometry in Rosenberg's conjecture","Zero Euler characteristic proves two 2006 Rosenberg conjectures","χ(X) = 0 suffices for S¹-stability in dimensions 5 and up","Vanishing Euler characteristic proves Rosenberg-Stolz for X × ℝ","χ = 0 forces the angle condition needed for Rosenberg stability","Rosenberg conjecture settled for closed manifolds with χ = 0"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":1777,"prompt_tokens":500,"completion_tokens":1277,"prompt_tokens_details":null},"tokens_in":500,"tokens_out":1277,"duration_ms":15120,"temperature":1.0,"reasoning_tokens":1018,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T04:16:17.785889+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"A counterexample would be a closed, oriented manifold X with dim X ≥ 5 and χ(X) = 0 such that X admits no positive scalar curvature metric but X × S¹ does. Alternatively, if Theorem 1.3 fails — meaning there exists a uniformly PSC metric of bounded curvature on some X × ℝ satisfying the angle condition but for which no conformal deformation yields positive scalar curvature on the hypersurface — then the main argument collapses.","supporting_citations":[],"review_version":1}