{"id":"efe0d098-2209-4e92-9b38-8d73f55a152b","arxiv_id":"2501.01113","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper asserts width and S1-stability PSC results for simply connected 4-manifolds up to homeomorphism, but the proof rests on a false disk-filling lemma.","lead":"The authors claim Gromov's band-width inequality and Rosenberg's S1-stability conjecture hold for simply connected smooth 4-manifolds when 'admits positive scalar curvature' is replaced by 'admits it on some smooth structure up to homeomorphism'. A key lemma used in the proof appears false, so the main theorems are not established as written.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 23 is false: a nullhomotopic embedded circle in a 4-manifold need not bound a smoothly embedded disk, and this lemma is load-bearing in Lemma 24 and Proposition 27 for Theorems A, B, and C.","rationale":"The reader's verdict identifies a genuine false statement at the core of the proof. Lemma 23 is not a harmless technicality: it is invoked in Lemma 24 to produce the embedded disk that allows a 1-handle to be cancelled by a 2-handle in a 5-dimensional cobordism. Without Lemma 24, Proposition 27 cannot rule out 1-handles, and the concluding step that M#k(S2×S2) is obtained from the PSC hypersurface by codimension-at-least-3 surgeries is unsupported. Since Theorems A, B, and C all rely on Proposition 27 (directly or via Theorem 28/32), the main results are unproved as written. The paper's overall strategy is plausible and the authors may be able to repair the argument, perhaps by replacing Lemma 23 with a statement appropriate to the specific cobordism or by using a different handle-trading technique in dimension 5. But the current manuscript does not establish its central claims, so the reader's REJECT verdict is appropriate.","tokens_in":15755,"tokens_out":6859,"duration_ms":70001,"concrete_test":"Test Lemma 23 directly with the trefoil knot K in the equatorial S3 of S4. Compute the Arf invariant of the trefoil: it is 1, so K is not slice. Since π1(S4)=0, K is nullhomotopic; if it bounded an embedded disk in S4, standard surgery on its intersection with S3 would produce a slice disk in B4, contradicting the Arf invariant. This single counterexample disproves Lemma 23. To test the paper's specific use, reconstruct the loop α ∪ beta' in Lemma 24 as the trefoil in a suitable ∂+V and check whether an embedded disk exists; the absence of such a disk invalidates the handle-trading step.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central surgery step rests on Lemma 23, which claims that every embedded nullhomotopic loop in a 4-manifold bounds a smoothly embedded disk. The proof attempts to remove self-intersections of an immersed disk by Whitney finger moves, but finger moves only rearrange intersections; they do not eliminate them. In dimension 4, removing double points requires Whitney disks whose existence is obstructed. The claim is false: let K be a non-slice knot, e.g. the trefoil, embedded in an equatorial S3 inside S4. Since S4 is simply connected, K is nullhomotopic. If K bounded a smoothly embedded disk in S4, then by restricting to one of the two 4-balls bounded by S3 and surgering the closed intersection curves, K would bound a smooth disk in B4, i.e. K would be slice. The trefoil is not slice, so Lemma 23 is contradicted. Lemma 24 uses Lemma 23 to construct an embedded disk bounding the loop alpha ∪ beta' in ∂+V, which is needed to create a cancelling 2-handle/3-handle pair and trade away 1-handles. Proposition 27 then depends on Lemma 24 to remove 1-handles before the 2-handle argument. Since Lemma 23 fails, Lemma 24 and Proposition 27 are not established, and the proofs of Theorems A, B, and C collapse as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper addresses Gromov's band width inequality and Rosenberg's S1-stability conjecture for closed simply connected smooth 4-manifolds, proving them 'up to homeomorphism' (Theorems A and B), together with a more general Theorem C for arbitrary 4-manifolds under a stabilization hypothesis. The strategy is standard in broad outline: use µ-bubble descent to produce a separating PSC hypersurface in a long band, and then use surgery theory in a 5-dimensional cobordism to transfer PSC from the hypersurface to M, possibly after stabilization by S^2×S^2 summands. The surgery part is carried out through Lemma 26, Lemma 24, and Proposition 27; the latter is the central technical result. The paper also contains an extension of the Gromov-Lawson-Stolz classification to dimension 4 up to homeomorphism (Observation 30) and an example involving a non-simply connected 4-manifold (Example 11).","tokens_in":15986,"tokens_out":6855,"duration_ms":59994,"significance":"If the main results were correct, they would establish the expected up-to-homeomorphism analogues of two central conjectures in 4-dimensional positive scalar curvature geometry and would support the picture that failure of S1-stability in dimension 4 is due to exotic smooth structures. The µ-bubble descent argument in Section 2 is a careful and standard exposition, and the introductory example showing PSC on V5×S1 via an exotic copy and the s-cobordism theorem is illuminating. However, because the central surgery step relies on a false 4-dimensional unknotting lemma, the paper does not provide a valid proof of its main theorems.","major_comments":[{"comment":"Lemma 23 is false as stated. A nullhomotopic embedded circle in a 4-manifold need not bound a smoothly embedded disk. For instance, take a non-slice knot K in an equatorial S^3 ⊂ S^4. Since S^4 is simply connected, K is nullhomotopic, but if K bounded a smoothly embedded disk in S^4, then, cutting S^4 along the equatorial S^3 and surgering the closed curves of intersection, K would bound a smooth disk in B^4, i.e., would be slice; the trefoil is a counterexample. The proof's use of Whitney finger moves cannot remove double points; in dimension 4 the Whitney trick is obstructed, and finger moves only rearrange intersections. This lemma is load-bearing for the paper's surgery argument.","section":"§3.2, Lemma 23"},{"comment":"Lemma 24 is not established because it depends directly on Lemma 23. In the proof, the existence of an embedded disk in ∂+V bounded by α∪β' is exactly the false statement of Lemma 23. Without such a disk, the cancelling 2-handle/3-handle pair cannot be produced, and the conclusion that all 1-handles can be traded for 3-handles fails. Consequently, Lemma 24 cannot be used to eliminate 1-handles from the handle decomposition in the subsequent arguments.","section":"§3.2, Lemma 24"},{"comment":"Proposition 27 is the central surgery step and it relies on Lemma 24 to choose a handle decomposition of V without 1-handles. Since Lemma 24 is invalid, the proof of Proposition 27 collapses. In particular, the claimed construction of the level set diffeomorphic to M#k(S^2×S^2) and the transfer of PSC via codimension-≥3 surgeries are not justified. The subsequent results that depend on Proposition 27—Theorem 28, Corollary 29, Proposition 31, Theorem 32, and Theorem 33 (Theorems A, B, and C in the introduction)—are therefore unproven.","section":"§3.4, Proposition 27"}],"minor_comments":[{"comment":"The statement labeled Observation 9 in the introduction and Observation 30 in Section 4.2 is identical; this duplication with different labels is confusing and should be consolidated.","section":"§1.2 and §4.2"},{"comment":"The phrase 'Since i induces an isomorphism on π1, we may choose a handle decomposition of (V;M,Σ) without 1-handles, by Lemma 24' presupposes the false Lemma 24; the proof should be revised to justify handle cancellation independently or with a correct statement.","section":"§3.4, proof of Proposition 27"},{"comment":"The proof says 'due to Proposition 27, the band X does not admit any PSC hypersurface that separates the faces'; this is only valid if Proposition 27 holds, so the statement is conditional on the invalid proposition.","section":"§4.1, Theorem 28 proof"},{"comment":"There is a typo in the line 'Cauchy-Schwarzh2/n≤|A|2': it should read 'h^2/n ≤ |A|^2'. The mathematical content is unaffected.","section":"§2.3, Proposition 18 proof"}],"recommendation":"reject","confidential_remarks":"The false Lemma 23 is a fundamental obstruction: it asserts a version of the Whitney trick in dimension 4, which is known to fail. The counterexample is global (knot slicing), so the error cannot be repaired by a local modification of the proof. Since Proposition 27 and all main theorems depend on Lemma 24, which in turn depends on Lemma 23, the central claim of the paper is not established. The µ-bubble section and the examples have independent value, but as submitted the paper is not correct."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: the paper asks a genuinely good question, and the high-level strategy is believable, but Lemma 23 is false and it is load-bearing. The proofs of Theorems A, B, and C do not go through as written.\n\nWhat is new: the statement that Gromov's width inequality and Rosenberg's S1-stability should hold for simply connected 4-manifolds up to homeomorphism is a natural and useful reformulation. Observation 30, extending the Gromov-Lawson-Stolz classification to dimension 4 in the homeomorphism sense, is correct and is a direct consequence of Freedman's classification plus the known surgery theorem. The µ-bubble descent section is standard, carefully written, and accurate. The idea of applying Wall's normal surgery to the 5-dimensional cobordism is the right shape.\n\nThe soft spot is not minor. Lemma 23 claims every embedded nullhomotopic loop in a 4-manifold bounds a smoothly embedded disk. That is false: take a non-slice knot in an equatorial S3 inside S4. The loop is nullhomotopic in S4, but if it bounded a smooth disk, removing a small ball disjoint from the disk would make the knot slice, which it is not. The proof of Lemma 23 uses Whitney finger moves to 'push self-intersections out of the boundary,' but finger moves do not eliminate intersections; they rearrange them, and in dimension 4 the needed Whitney disks are obstructed. Since Lemma 24 uses Lemma 23 to build an embedded disk for the 1-handle trading, and Proposition 27 depends on Lemma 24 to remove 1-handles, the entire surgery step collapses. The rest of the paper cannot compensate.\n\nIt is worth saying the issue might be repairable: the loop constructed in Lemma 24 is quite special, and in that particular 5-dimensional cobordism one might prove the required disk exists without asserting the general false lemma. But as it stands, the central derivation is invalid.\n\nWho should read it: anyone working on scalar curvature in dimension 4 will find the formulation and the observations useful, and the flaw is a good cautionary example about 4-manifold topology. A referee could reasonably recommend major revision. I would not desk-reject it: the question is important and the approach is largely sound, but I would not accept the main theorems until the handle-trading lemma is fixed.\n\nMy recommendation: send to a knowledgeable referee, prepare for a negative first report.","headline":"Good question and a correct homeomorphism-invariant reformulation, but the surgery proof hinges on a false lemma, so the main theorems are not established as written.","tokens_in":16562,"tokens_out":4655,"would_cite":false,"duration_ms":42507,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C21","57R65","57K40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Simply connected 4-manifolds obey the scalar-curvature width and $S^1$-stability theorems once \"PSC\" is read up to homeomorphism.","keywords":["positive scalar curvature","4-manifolds","band width inequality","S1-stability","exotic smooth structures","μ-bubbles","surgery theory","Seiberg-Witten invariants"],"falsifier":"Test Lemma 23 on the loop in $S^4$ given by a non-slice knot in an equatorial $S^3$: the lemma predicts a smooth embedded disk, while the non-slice property predicts none, so this single example decides whether the surgery step in Proposition 27 is valid.","tokens_in":15471,"feed_emoji":"📏","tokens_out":12303,"duration_ms":103358,"temperature":0.7,"pith_summary":"Four-manifolds are the only dimension where the standard positive-scalar-curvature (PSC) conjectures fail: Seiberg–Witten invariants produce simply connected 4-manifolds that are not PSC even though their product with a circle is PSC. This paper argues that the failure is entirely due to exotic smooth structures. The main theorems state that Gromov's band width inequality and Rosenberg's $S^1$-stability become true for simply connected smooth 4-manifolds if the conclusions are read up to homeomorphism: a band $M^4\\times[-1,1]$ with scalar curvature $\\ge 20\\kappa^2$ has width at most $2\\pi/(5\\kappa)$, and $M\\times S^1$ is PSC exactly when some smooth structure on $M$ is PSC. If correct, this extends the Gromov–Lawson–Stolz classification to dimension 4 in the homeomorphism sense and pins the known counterexamples on exotic structures rather than on topology.","feed_headline":"Width and S1-stability hold for 4-manifolds up to homeomorphism","feed_subtitle":"Known counterexamples come from exotic smooth structures, not from topology.","key_machinery":"The load-bearing object is the separating $\\mu$-bubble: a minimizer of Gromov's modified area functional with a carefully chosen weight function $h$ that blows up on the faces of the band. Its second variation converts the scalar curvature lower bound on the band into a positive lower bound for $\\lambda_1(-\\Delta+R/2)$ on a hypersurface $\\Sigma$, which by conformal deformation makes $\\Sigma$ PSC. The second half of the machinery is 5-dimensional surgery theory: a cobordism from $M$ to a PSC $\\Sigma$ is simplified by normal-map surgery and 1-handle trading until $M\\#k(S^2\\times S^2)$ is obtained from $\\Sigma$ by surgeries of codimension $\\ge 3$, which preserve PSC by the Gromov–Lawson–Schoen–Yau surgery theorem.","core_discovery":"The central claim is Theorem A: if $M^4$ is a closed simply connected smooth 4-manifold that is not PSC up to homeomorphism, then every metric on the band $M^4\\times[-1,1]$ with scalar curvature $R_g\\ge 20\\kappa^2$ has width at most $2\\pi/(5\\kappa)$. From this the paper derives Theorem B: $M$ is PSC up to homeomorphism if and only if $M\\times S^1$ is PSC. The proof passes through the more general Theorem C, which replaces the simple-connectedness hypothesis by the assumption that no stabilization $M\\#k(S^2\\times S^2)$ is PSC, and through an up-to-homeomorphism version of the Gromov–Lawson–Stolz classification in dimension 4. The mechanism is that a band longer than the bound would contain a separating PSC hypersurface, a 5-dimensional surgery argument would then make some $M\\#k(S^2\\times S^2)$ PSC, and in the simply connected case that forces $M$ itself to be PSC up to homeomorphism.","pith_inferences":["If the results hold, the known Seiberg–Witten counterexamples to $S^1$-stability are best understood as statements about smooth structures: the homeomorphism type still satisfies the PSC dichotomy, and the failure must be encoded in the smooth structure.","A natural next question is whether the stabilization number $k$ in Theorem C can be bounded by a quantity such as the minimal $b_2$ contribution needed to kill Seiberg–Witten invariants; if so, it would give a quantitative measure of how exotic a 4-manifold is.","The proof relies on Lemma 23, which states that every embedded nullhomotopic circle in a 4-manifold bounds a smoothly embedded disk; testing this lemma on a non-slice knot in an equatorial $S^3\\subset S^4$ would show whether the handle-trading step in Proposition 27 needs a different argument.","The same up-to-homeomorphism relaxation may revive other dimension-4 PSC statements currently obstructed by Seiberg–Witten invariants, such as classification up to stabilization or width estimates for other bands."],"forward_implications":["Every closed simply connected 4-manifold is either PSC up to homeomorphism or satisfies the sharp width bound with constant $2\\pi/(5\\kappa)$.","The Gromov–Lawson–Stolz dichotomy extends to dimension 4: non-spin simply connected 4-manifolds are PSC up to homeomorphism, while spin ones are PSC up to homeomorphism exactly when $\\hat A(M)=0$.","For $M=K3$, the band $M\\times[-1,1]$ obeys the width inequality, resolving that specific case of Gromov's conjecture.","For any closed 4-manifold $M$, if $M\\times S^1$ is PSC, then some stabilization $M\\#k(S^2\\times S^2)$ is PSC.","If $M\\times S^1$ is PSC and $M$ is simply connected, then $M$ is homeomorphic to a PSC manifold, so the known counterexamples to $S^1$-stability are all exotic-structure effects."],"supporting_citations":[{"why":"States the $S^1$-stability conjecture and gives the $V_5$ counterexample that motivates the paper.","marker":"[Ros07]"},{"why":"Outlines the $\\mu$-bubble version of the band width argument and the descent to a PSC separating hypersurface, which Section 2 follows.","marker":"[Gro23]"},{"why":"Completes Gromov's width argument in dimensions 5 and 6 and supplies the surgery strategy that the paper adapts to dimension 4.","marker":"[Räd23]"},{"why":"Provides the existence and second-variation theory for separating $\\mu$-bubbles used in the descent argument.","marker":"[Zhu21]"},{"why":"Gives the surgery theorem preserving PSC under codimension $\\ge 3$ surgeries and the classification of non-spin simply connected PSC manifolds.","marker":"[GL80]"},{"why":"Supplies the Schoen–Yau surgery theorem for PSC and the conformal-descent circle of ideas behind the $\\mu$-bubble argument.","marker":"[SY79]"},{"why":"Classifies spin simply connected PSC manifolds in dimensions $\\ge5$ via $\\hat\\alpha$, the analogue used in Observation 30.","marker":"[Sto92]"},{"why":"Freedman's homeomorphism classification of simply connected 4-manifolds is used to identify homeomorphism types in the up-to-homeomorphism statements.","marker":"[Fre82]"},{"why":"Wall's h-cobordism result for simply connected 4-manifolds is used to pass between exotic smooth structures and to prove the forward direction of $S^1$-stability.","marker":"[Wal64]"},{"why":"Lichnerowicz's spin obstruction, $\\hat A\\neq0$ implies no PSC metric, is the spine of the spin case in Observation 30.","marker":"[Lic63]"}],"fun_headline_variants":["Up to homeomorphism, width and S^1-stability hold in 4D","Homeomorphism defeats exotic counterexamples in 4D","Ignore smooth structure: width and S^1-stability hold in 4D","S^1 stability and band width: true up to homeomorphism in 4D","Topological 4-manifolds obey width and S^1-stability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Lemma 23, that every embedded nullhomotopic circle in a 4-manifold bounds a smoothly embedded disk, which is used to trade 1-handles for 3-handles; if that lemma fails for a non-slice knot in an equatorial $S^3\\subset S^4$, the surgery step in Proposition 27 breaks.","fun_headline_variants_meta":{"raw":{"variants":["Up to homeomorphism, width and S^1-stability hold in 4D","Homeomorphism defeats exotic counterexamples in 4D","Ignore smooth structure: width and S^1-stability hold in 4D","S^1 stability and band width: true up to homeomorphism in 4D","Topological 4-manifolds obey width and S^1-stability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000915,"raw_usage":{"total_tokens":3877,"prompt_tokens":840,"completion_tokens":3037,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":456,"completion_tokens_details":{"reasoning_tokens":2946}},"tokens_in":456,"tokens_out":3037,"duration_ms":22729,"temperature":1.0,"reasoning_tokens":2946,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:38:05.666398+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test Lemma 23 on the loop in $S^4$ given by a non-slice knot in an equatorial $S^3$: the lemma predicts a smooth embedded disk, while the non-slice property predicts none, so this single example decides whether the surgery step in Proposition 27 is valid.","supporting_citations":[],"review_version":1}