{"id":"bc6dd7e5-f7df-407c-b476-b8865911dfae","arxiv_id":"2505.17263","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"There exist two sequences of closed 4-manifolds with nonnegative Ricci curvature, bounded diameter and volume, which converge to the same Gromov-Hausdorff limit yet have fundamental groups Z/2Z and trivial.","lead":"Mathematicians constructed two sequences of curved four-dimensional spaces that have different global loop structure but converge to the same limiting shape. This settles an open question about whether such loop structure can be read off from the limit alone.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central construction rests on unverified metric-existence theorems: Theorem 5 is delegated by analogy, and Theorem 2's statement disagrees with its proof's warping function, so the claimed nonnegative-Ricci metrics are not established.","rationale":"The reader's weakest assumption identified the same bottleneck: the existence of the metrics described in Theorems 2 and 5 is asserted through sketches and external lemmas rather than fully demonstrated. My pass sharpens this: Theorem 5 is not proved at all, and Theorem 2 as written contains an internal mismatch between the stated metric and the warping function used in its proof. Because the main theorem cites these theorems directly, the central claim is conditional on repairing this mismatch and completing the omitted verification. This does not amount to evidence that the construction is impossible; the tools cited are standard, and the topological reasoning about fundamental groups is plausible. It does mean the manuscript should not be accepted without the missing derivation. The verdict therefore remains CONDITIONAL, matching the reader's assessment.","tokens_in":7845,"tokens_out":41264,"duration_ms":324600,"concrete_test":"Write out the proof of Theorem 5 by following the proof of Theorem 2 verbatim, specifying the exact psi, the Ricci computation via Lemma 3, and the smooth matching at r = d/2 and r = pi - d/2. Independently recompute the metric in Theorem 2 from its proof: if the resulting psi does not equal the stated c(sin(r) - sin(9d/20) + d) on (d, pi-d), or if the two copies of the cap do not glue with a C^2 metric of nonnegative Ricci, then the main construction lacks a verified input. As a secondary check, rescale the final metrics by 1/pi and re-verify the diameter, volume, and GH limits.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main theorem depends entirely on Theorems 2 and 5 producing closed 4-manifolds with nonnegative Ricci curvature, the stated fundamental groups, and a common Gromov-Hausdorff limit. The load-bearing gap is the existence of these metrics. Theorem 5 is not proved, only asserted to be 'exactly as the proof of Theorem 2'. Theorem 2's proof is a sketch: it invokes an Anderson perturbation, an Otsu perturbation to a smooth concave 1-Lipschitz function, and one-line claims that the gluing remains simply connected and the free involution extends. More concretely, the statement of Theorem 2 gives ds^2 = dr^2 + c^2(sin(r) - sin(9d/20) + d)^2 ds_3^2 for r in (d, pi-d), whereas the proof constructs a function psi with a c/2 factor in the middle segment and only on (d/2, pi-d/2); these formulas are not compatible as written, and the claimed interval leaves a transition region unexplained. If the gluing does not preserve nonnegative Ricci through the caps, or if the free involution does not extend to the smoothed metric, the quotient M_i need not have pi_1 = Z/2 and the two sequences need not share a limit. The diameter bound in the main theorem also requires an unstated rescaling, since Theorem 2 has diameter at least pi.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to construct two sequences of closed 4-dimensional manifolds, (M_i) and (N_i), with non-negative Ricci curvature, diameter bounded above by 1, volume bounded below by a positive constant v, and d_GH(M_i,N_i)→0, such that π_1(M_i)=Z/2Z and π_1(N_i)=1. If correct, this gives a negative answer to Pan's question of whether the fundamental group is determined by a non-collapsed Ricci limit. The construction combines an Eguchi–Hanson-type metric with a free involution quotient for M_i, and a desingularization of C^2/µ_4 for N_i, with the common limit being a spherical suspension of a rescaled S^3/µ_4. The main argument is presented as Theorems 2 and 5, which are asserted to produce the needed warped-product metrics, followed by a one-sentence proof of the main theorem.","tokens_in":8039,"tokens_out":5707,"duration_ms":48072,"significance":"The result, if fully established, would settle a natural open question in the stability theory of fundamental groups under non-collapsed Ricci limits. The paper is concise and the overall strategy is plausible, combining known tools (Eguchi–Hanson metrics, Anderson's gluing, Otsu's perturbation lemma, and a doubly warped product desingularization) in a explicit way. However, the current manuscript is best read as a well-motivated construction outline: the two key existence theorems (Theorems 2 and 5) are only sketched, and the proof of the main theorem omits the rescaling or conformal modification needed for the stated diameter bound. With the missing details supplied, the construction would be a valuable contribution; at present, the central claim is not yet verified to the standard expected for a journal publication.","major_comments":[{"comment":"The statement of Theorem 2 and its proof are not compatible as written. The theorem asserts, for r in (d, π-d), the metric ds^2 = dr^2 + c^2(sin(r) - sin(9d/20) + d)^2 ds_3^2, but the proof constructs a function ψ on (d/2, π-d/2) with middle branch c/2 sin(r) - c/2 sin(9d/10) + cd and endpoint branches c(r + d/10) and c(π - r + d/10). The interval, the factor c/2, and the constant sin(9d/10) all differ from the theorem's displayed formula. Since this explicit warping is the basis for the claimed convergence to the spherical suspension, the metric that is actually constructed is not the one whose asymptotic form is used later.","section":"Theorem 2"},{"comment":"The proof of Theorem 5 is delegated to the sentence 'The proof of Theorem 5 is exactly as the proof of Theorem 2,' but Theorem 5 concerns S^3/µ_4 and does not involve the free involution quotient that is central to Theorem 2. No verification is given that the analog of the ψ-gluing preserves non-negative Ricci curvature for the doubly warped product of Theorem 4, nor that the completion is simply connected in the absence of the quotient. Because Theorem 5 is load-bearing for the sequence (N_i), this is a genuine gap rather than a harmless repetition.","section":"Theorem 5"},{"comment":"The verification of conditions (2) and (3) in Lemma 8 is asserted with 'provided that c is small enough' and the displayed lower bounds for r ∈ (3/4, 5/4) are not derived. In particular, the expressions such as (2ρ/φ)[(n/2)^3/(n+c/2)^3 - nc] and 4 - 2 - (n+c/2)/(n/2) nc - (n+c/2)c∥φ∥∞ - c^2 are introduced without justification, and the notation φ is used both for the mollification kernel and for the warping function, making the argument difficult to follow. Since the non-negative Ricci curvature of the metric in Theorem 4 rests on these inequalities, a complete proof of Lemma 8 is required.","section":"Lemma 8"},{"comment":"The proof of the main theorem is a single sentence that does not address the stated diameter bound diam(M_i), diam(N_i) ≤ 1. The metrics in Theorems 2 and 5 have a radial variable r ∈ (0, π), so their diameters are comparable to π, not bounded by 1, unless an explicit rescaling or conformal modification is made. The overview mentions a conformal change and gluing 'cf. [1]', but Section 1 does not specify the rescaling, the resulting volume lower bound v, or why the convergence to the spherical suspension survives the modification. These are necessary steps for the theorem as stated.","section":"Section 1"}],"minor_comments":[{"comment":"The symbol ds_3 is used for the round metric on S^3 and also for the induced metric on quotients S^3/µ_2 and S^3/µ_4; this should be clarified, since the quotient metrics are not literally the same as the round metric on S^3.","section":"Notation"},{"comment":"The notation φ is used for the mollification kernel and also for the warping function in the same paragraph; using different symbols (e.g., η for the kernel) would remove avoidable confusion.","section":"Lemma 8"},{"comment":"The sentence 'Notice first that ˆρ = ρ on (0, 3/4) ∪ (5/4, ∞)' is missing the hat on one side; the intended meaning is that the mollified function agrees with the piecewise linear one away from the transition region.","section":"Lemma 8"},{"comment":"The one-line claim that the glued space remains simply connected by the Seifert–van Kampen theorem and that the free involution extends naturally needs a more detailed justification, especially because the gluing regions are not explicitly described topologically.","section":"Proof of Theorem 2"},{"comment":"The paper repeatedly invokes [8, Lemma 1.5] for the existence of smooth concave 1-Lipschitz perturbations; since this lemma is central to both main constructions, it would be helpful to state its exact content or at least the precise conditions under which it applies.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a natural question and the proposed construction is plausible, but the current version is more of an extended sketch than a complete proof. The main issues are technical gaps in Theorems 2 and 5 and in Lemma 8, plus the omitted diameter rescaling in Section 1. These appear fixable within the scope of the paper, so I recommend major revision rather than rejection. I saw no evidence of problematic citation behavior; the references are standard and the reliance on external results is legitimate, provided the cited lemmas are stated precisely."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this note answers Pan's question in the negative with a clean construction: two non-collapsed sequences of closed 4-manifolds with nonnegative Ricci curvature, uniformly bounded diameter and volume, same GH limit, but fundamental groups Z/2Z and {1}. That is a real advance over Otsu's earlier example, which only gives one sequence. Second, the construction is credible but the proofs of the two key metric-existence theorems are sketches, and one of them contains a concrete mismatch between its statement and proof.\n\nWhat is new and good: the two-sequence counterexample is genuinely new; the strategy of combining the Eguchi-Hanson resolution with the quotient by a free involution and a desingularized C2/µ4 is elegant and the high-level picture is clear. The paper is honest about relying on known tools (Anderson's gluing, Otsu's concave 1-Lipschitz perturbations, Zhou's doubly warped products). The main theorem follows directly if Theorems 2 and 5 hold, and there is no circularity or data-fitting.\n\nThe soft spots are in those theorems. Theorem 5 is not proved; it is declared 'exactly as Theorem 2', but the two constructions are structurally different (warped products over Hopf fibrations vs a perturbed Eguchi-Hanson metric), so that analogy does not carry the Ricci-check. Theorem 2's proof has a specific mismatch: the statement claims ds^2 = dr^2 + c^2(sin r − sin(9d/20)+d)^2 ds_3^2 on (d, π−d), while the proof builds a warping function ψ with a c/2 factor, using 9d/10 and intervals (d/2, π−d/2). Either the statement has a typo or the construction differs from what is claimed; a referee will need this sorted out. The gluing step, the simple-connectivity after gluing, and the extension of the free involution are each asserted in a line or two, not demonstrated. Lemma 8's inequalities are plausible but the 'c small enough' is not quantified, and Theorem 5's Ricci verification is absent. The diameter bound in the main theorem needs a rescaling that is not stated, though that part is harmless.\n\nNet: the idea is good and the answer is probably correct, but the proof is not referee-ready as is. The paper deserves a serious referee who can fill or fix these gaps; it should not be desk-rejected. I would recommend asking for complete proofs of Theorems 2 and 5 before publication.","headline":"A genuinely new two-sequence counterexample to Pan's stability question, but the metric-existence proofs are too sketched to verify the main theorem as written.","tokens_in":8641,"tokens_out":3929,"would_cite":true,"duration_ms":23981,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C23","53C20","53C21"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two sequences of closed 4-manifolds with identical Gromov–Hausdorff limit have different fundamental groups.","keywords":["nonnegative Ricci curvature","Gromov–Hausdorff limit","fundamental group stability","Eguchi–Hanson space","spherical suspension","Ricci limit spaces","desingularization of quotient singularity","4-manifolds"],"falsifier":"Compute the Ricci curvature of the metric $ds^{2}$_{M',c,d} of Theorem 2 and the metric $ds^{2}$_{N,c,d} of Theorem 5 on the gluing intervals (d/2, d) and (π−d, π−d/2) using Lemma 3 and Lemma 7; if any Ricci component is negative for all small c and d, the main theorem collapses. A concrete check would evaluate the three nonnegativity conditions of Lemma 7 at the seam points where the profiles switch between the original and the perturbed forms.","tokens_in":7546,"feed_emoji":"📐","tokens_out":4643,"duration_ms":37734,"temperature":0.7,"pith_summary":"The paper aims to settle a question posed in [9]: whether the fundamental group of a closed manifold is determined, for nearby metrics, by a non-collapsed Gromov–Hausdorff limit. It constructs two sequences of closed 4-dimensional manifolds with non-negative Ricci curvature, diameter at most 1, and volume bounded below by a positive constant, such that the two sequences converge to the same Gromov–Hausdorff limit, yet one manifold in each pair has fundamental group Z/2Z and the other is simply connected. If the construction is correct, the fundamental group is not invariant under Gromov–Hausdorff convergence even in the non-collapsed, uniformly bounded case.","feed_headline":"Same Ricci limit, different fundamental groups","feed_subtitle":"A pair of closed nonnegatively curved 4-manifolds converge to one space yet have fundamental groups Z/2Z and trivial.","key_machinery":"The construction rests on two models. The first is the Eguchi–Hanson space, a complete simply connected 4-manifold asymptotic to the cone over $RP^{3}$ = $S^{3}$/µ_2; a free involution on it, conjugated by a mapping of $R^{4}$, produces a manifold asymptotic to the cone over $S^{3}$/µ_4 with fundamental group Z/2Z. The second is a Berger-sphere type doubly warped product metric, used for the desingularization of the orbifold $C^{2}$/µ_4, which gives the simply connected model. The main mechanism is a gluing and rescaling procedure: the warped products are modified with concave 1-Lipschitz profiles (using a perturbation lemma and a gluing step from the literature) to make them extend smoothly over a compact piece with the spherical suspension as Gromov–Hausdorff limit, while preserving non-negative Ricci curvature through the curvature formulas in Lemma 3 and Lemma 7.","core_discovery":"The central claim is that there exist sequences (M_i) and (N_i) of closed 4-manifolds with d_GH(M_i, N_i) → 0 and uniform bounds on curvature, diameter, and volume, with π1(M_i) = Z/2Z and π1(N_i) = 1. Both sequences converge to the same singular space, the spherical suspension of a scaled quotient sphere c $S^{3}$/µ_4 (the sphere quotiented by the diagonal action of the fourth roots of unity). This gives a negative answer to both versions of the question posed in [9].","pith_inferences":["Editorial inference: the method likely extends to produce limits where the fundamental groups differ by other finite abelian groups, by replacing the quotients µ_2 and µ_4 with other group actions, although the paper does not claim this.","Editorial inference: a direct numerical or symbolic check of the three nonnegativity conditions in Lemma 7 at the gluing seams would convert the sketchy proof of Theorem 5 into a fully verified construction; this is the most natural testable follow-up.","Editorial inference: if the perturbative gluing steps in the paper fail, a plausible rescue would be to use explicit hyper-Kähler ALE metrics instead of the mollified warped products, but that alternative is not explored in the paper."],"forward_implications":["The fundamental group is not determined by the Gromov–Hausdorff limit of a non-collapsed sequence of closed manifolds with uniform Ricci lower bound, diameter bound, and volume lower bound.","Both forms of the question in [9] — an ε-stability of fundamental groups and the determination of π1 from the limit — have a negative answer.","The example lives in dimension 4 and produces a singular common limit that is a spherical suspension, showing that such instability can appear with a particularly simple limit space.","It highlights the gap between the known surjective homomorphism from π1(M_i) to the fundamental group of the limit and the absence of injectivity or uniqueness for such homomorphisms."],"supporting_citations":[{"why":"Provides the Eguchi–Hanson gravitational instanton metric, the starting point for the simply connected model M'.","marker":"[5]"},{"why":"Supplies the gluing and perturbation technique used to modify the Eguchi–Hanson metric while preserving non-negative Ricci curvature.","marker":"[1]"},{"why":"Gives the lemma about concave 1-Lipschitz perturbations that is used to smooth the warped profiles while keeping non-negative Ricci curvature.","marker":"[8]"},{"why":"Exhibits a smooth non-negatively curved metric on the desingularization of C^2/µ_4, used for the simply connected model N.","marker":"[7]"},{"why":"Poses the question that the main theorem answers negatively, giving the paper its purpose.","marker":"[9]"},{"why":"Provides the standard Berger-sphere metric formulas and the exercise about smoothness on quotients S^3/µ_n used in the construction for N.","marker":"[12]"},{"why":"Supplies the doubly warped product metric and the Ricci-curvature conditions (Lemma 7 here) for the desingularization that yields N.","marker":"[17]"}],"fun_headline_variants":["Fundamental groups diverge despite identical Ricci limits","Ricci limit fails to determine fundamental group","Closed 4-manifolds share limit, not fundamental group","Fundamental group unstable under non-collapsed Ricci limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the metrics described in Theorems 2 and 5 actually exist, are smooth, have non-negative Ricci curvature, and admit the stated quotient behavior with fundamental groups Z/2Z and trivial, especially after the gluing step.","fun_headline_variants_meta":{"raw":{"variants":["Fundamental groups diverge despite identical Ricci limits","Ricci limit fails to determine fundamental group","Closed 4-manifolds share limit, not fundamental group","Fundamental group unstable under non-collapsed Ricci limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001065,"raw_usage":{"total_tokens":4361,"prompt_tokens":737,"completion_tokens":3624,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":353,"completion_tokens_details":{"reasoning_tokens":3562}},"tokens_in":353,"tokens_out":3624,"duration_ms":21823,"temperature":1.0,"reasoning_tokens":3562,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:50:13.227639+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Ricci curvature of the metric $ds^{2}$_{M',c,d} of Theorem 2 and the metric $ds^{2}$_{N,c,d} of Theorem 5 on the gluing intervals (d/2, d) and (π−d, π−d/2) using Lemma 3 and Lemma 7; if any Ricci component is negative for all small c and d, the main theorem collapses. A concrete check would evaluate the three nonnegativity conditions of Lemma 7 at the seam points where the profiles switch between the original and the perturbed forms.","supporting_citations":[{"cited_title":"Eguchi and A","cited_arxiv_id":null,"evidence_quote":"Provides the Eguchi–Hanson gravitational instanton metric, the starting point for the simply connected model M'."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the gluing and perturbation technique used to modify the Eguchi–Hanson metric while preserving non-negative Ricci curvature."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the lemma about concave 1-Lipschitz perturbations that is used to smooth the warped profiles while keeping non-negative Ricci curvature."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Poses the question that the main theorem answers negatively, giving the paper its purpose."},{"cited_title":"Petersen","cited_arxiv_id":null,"evidence_quote":"Provides the standard Berger-sphere metric formulas and the exercise about smoothness on quotients S^3/µ_n used in the construction for N."},{"cited_title":"A family of $4$-manifolds with nonnegative Ricci curvature and prescribed asymptotic cone","cited_arxiv_id":"2406.02279","evidence_quote":"Supplies the doubly warped product metric and the Ricci-curvature conditions (Lemma 7 here) for the desingularization that yields N."}],"review_version":1}