{"id":"e4af8a63-cf30-467e-a57d-8fa3cfbd4f83","arxiv_id":"2608.04855","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A smooth sphere metric and a curve-shrinking flow are built so that different sequences of times pull the flow to different closed geodesics, refuting the Grayson-Gage uniqueness conjecture.","lead":"The authors construct a metric on the 2-sphere and an immortal curve-shrinking flow whose late-time limit is not a single geodesic but an entire circle of distinct geodesics. This settles, in the negative, a uniqueness question raised by Grayson and Gage.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The construction hinges on the unverified applicability of the AMN Nash-Moser theorem to the antipodally projected operators; if the reduced right-inverse identity (2.25) fails, the almost-geodesic family with exact length profile does not exist.","rationale":"The reader's weakest assumption identifies the same point. The manuscript's proof of Proposition 2.8 is the foundation of the whole construction, and its validity depends on an external Nash-Moser theorem whose hypotheses are not restated. The internal argument for the symmetry reduction is coherent, and the paper is careful about its dependencies; no internal contradiction or obvious gap was found. Because the concern is about verification of a black box rather than a demonstrated error, the appropriate verdict remains CONDITIONAL, so the reader's verdict is unchanged. The secondary use of Grayson's uniform derivative decay is also external, but it only affects the final C-infinity upgrade in Theorem 5.1, not the existence of the non-unique family; hence it is less load-bearing. If the AMN applicability is confirmed, the central claim appears to go through.","tokens_in":25595,"tokens_out":24333,"duration_ms":259972,"concrete_test":"Independently re-derive the right-inverse identity (2.25) for the projected operators, verifying that Pi_Y^+ Q^+{Lambda^+(z,tau),h} = Pi_Y^+ Q^+{Lambda(z),h} uses only (2.24), and then check the tame estimates of [AMN25, Corollary 8.9] pass through the projections without derivative loss. A specific check: compute the operator norm bounds for V^+ and Q^+ on the finite-dimensional Fourier truncations of X^+ and Y^+, with the antipodal projection included, and confirm the constants are uniform as the truncation degree grows. If any estimate loses a power of the smoothing parameter, or if the identity (2.25) only holds on a neighborhood that the Nash-Moser iteration leaves, Proposition 2.8 is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim relies on Proposition 2.8, which produces the almost-geodesic family with exact length profile L_epsilon + epsilon W and Lambda_2 = 0. That proposition applies the Nash-Moser inverse function theorem of [AMN25, Theorem 5.1, Corollary 5.2] to the reduced map Lambda^+(z,tau)=Lambda(z)-tau(W,0) on the antipodally symmetric tame subspaces X^+ x R and Y^+. The algebraic steps are provided: equivariance Lambda(T_X z)=T_Y Lambda(z), the identity (2.23), and the cancellation Q^+{(b,0),h}=0, yielding the quadratic right-inverse identity (2.25). What is load-bearing and not fully checked in the manuscript is that the projected operators V^+ = Pi_X^+ V and Q^+ = Pi_Y^+ Q satisfy all tameness and derivative-gain hypotheses assumed by [AMN25, Theorem 5.1]. The paper asserts smooth tameness of the unprojected V,Q via [AMN25, Corollary 8.9] and then composes with the tame projections; this is plausible, but the projections act on the second factor through the antipodal map and could alter the grading constants or the required order of smoothing in the Nash-Moser iteration. Since Sections 3-5 and the conclusion of non-unique geodesic limits are built entirely on this family, a failure of this external hypothesis would invalidate the main theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a smooth antipodally symmetric Riemannian metric g on S^2, a one-parameter family of distinct simple closed geodesics all of the same length, and an immortal embedded curve-shortening flow whose C^infinity omega-limit set is exactly that family. The proof has three parts: an almost-geodesic family with a prescribed length profile is built using the Ambrozio-Marques-Neves Nash-Moser inverse function theorem (Proposition 2.8); a potential W is chosen so that its negative gradient flow spirals into a prescribed curve Z (Section 3); and a spectral-gap/bootstrap analysis of the graph equation (4.13) shows that the stable Fourier modes decay while the center modes follow the spiraling gradient flow (Proposition 4.34). The final section upgrades finite-regularity convergence to smooth convergence via Grayson's derivative decay and proves the main theorem.","tokens_in":25869,"tokens_out":12349,"duration_ms":142384,"significance":"If valid, the result settles the Grayson-Gage uniqueness question in the negative, in a strong form: non-uniqueness occurs for metrics arbitrarily C^infinity-close to the round metric, and the entire omega-limit set is a circle of equal-length geodesics. The construction is detailed and the spectral-gap estimate (4.29)-(4.30) together with the bootstrap in Proposition 4.34 are explicit and checkable. The main risk is the applicability of the AMN Nash-Moser theorem to the antipodally projected operators, on which the existence of the almost-geodesic family rests. The argument does not assume its conclusion; the length profile W is chosen independently and the metric is built around it.","major_comments":[{"comment":"The entire construction of the almost-geodesic family {Gamma_{epsilon,v}} with exact length profile L_epsilon + epsilon W(v) is obtained by applying [AMN25, Theorem 5.1 and Corollary 5.2] to the reduced map Lambda^+(z,tau)=Lambda(z)-tau(W,0) on the projected tame spaces X^+ x R and Y^+. The manuscript verifies the equivariance identity (2.20), the quadratic right-inverse identity (2.23), and the cancellation (2.24), and it asserts that the projected maps V^+ and Q^+ are smooth and tame because they are compositions of smooth tame maps with the tame projections (2.19). What is not shown is that the projected spaces X^+, Y^+ X^+ x R are tame spaces of the exact type required by [AMN25, Section 5.1], and that the tame estimates, derivative-gain orders, and smoothing operators for V and Q survive composition with Pi_X^+ and Pi_Y^+. Since Sections 3 through 5 and the main theorem rest entirely on Proposition 2.8, a failure of any of these hypotheses would invalidate the result. Please add a complete verification of every hypothesis of the AMN theorem as applied to the reduced map, or state and prove a variant that accepts the quadratic right-inverse identity in the projected categories.","section":"Section 2, Proposition 2.8, equations (2.23)-(2.25)"},{"comment":"The upgrade from omega_{C^{m-1}}(gamma) to omega_{C^infinity}(gamma) rests on the invocation of [Gra89, Theorem 7.2] and on the claim that the decay of |nabla^r_s kappa| gives uniform bounds for every derivative of constant-speed parametrizations of gamma(t). The paper does not state the exact hypotheses of [Gra89, Theorem 7.2] or explain in detail how the decay of curvature derivatives along the geometric flow yields smooth convergence of unparametrized curves in the topology of E(S^2). This step is load-bearing because it converts the finite-regularity identification (5.3) into the C^infinity statement (5.5). Please provide the precise argument, including the role of the uniform length bounds and the diagonal Arzela-Ascoli application.","section":"Section 5, proof of Theorem 5.1, equations (5.3)-(5.5)"}],"minor_comments":[{"comment":"The domain of Theta is written as 's > s_0 - 2' in (3.2) but as 's >= s_0 - 2' elsewhere in the proof; please unify the notation.","section":"Lemma 3.1, equation (3.2)"},{"comment":"The verification of (3.8) for the function B(s) at k=0 is only summarized; the one-line computation using the decay of alpha(s) = exp(-e^{s/2}) would make the proof easier to follow.","section":"Lemma 3.1, estimate (3.8)"},{"comment":"The identity C(-Psi circ bA) = C(Psi) is stated without derivation; since this is the first nontrivial use of the center-mode extraction, a short explanation would improve readability.","section":"Section 2, equation (2.17)"},{"comment":"The word 'Grayson' is split across lines in the title; please fix the line break in the journal version.","section":"Title page"},{"comment":"The phrase 'by the C^1-stability of embeddings on compact sets' is acceptable but could be replaced by an explicit uniform inverse-function-theorem statement as epsilon varies; the current wording is slightly informal.","section":"Lemma 4.6, proof"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a serious construction and, modulo the AMN verification, the main theorem is plausible. The paper's only external self-citation ([LZ26]) is contextual and does not affect the proof. The decisive issue is whether the projected Nash-Moser setup satisfies all hypotheses of [AMN25]; if the authors can supply a complete verification, the paper should be acceptable. If the verification reveals a genuine obstruction, the main theorem would not follow from the presented argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my take. This is the first paper I know that actually constructs an immortal simple closed curve shortening flow with a non-unique geodesic limit, and it does it in a strong form: for every C-infinity neighborhood of the round metric on S^2, the C-infinity omega-limit set is exactly an entire circle of distinct simple closed geodesics all having the same length. If the construction holds, it settles a 35-year-old question in the negative. The proof is genuinely structured: an almost-geodesic family with prescribed length profile from the AMN Nash-Moser theory, a spiral gradient flow, and a center-stable decomposition with an explicit spectral gap coming from the antipodal symmetry. The spectral gap estimate in Lemma 4.28 is concrete, the bootstrap in Proposition 4.34 is the standard \"shrink epsilon, enlarge S\" routine, and the paper is honest about what it needs from outside.\n\nThe reader's main concern is fair. The entire almost-geodesic family rests on applying the AMN Nash-Moser theorem to the reduced map on the symmetric tame subspaces. The paper supplies the key algebraic identities, especially the quadratic right-inverse identity (2.25) and the cancellation Q^+((b,0),h)=0. But it does not verify in detail that the projected operators V^+ and Q^+ satisfy the tameness estimates with the exact grading constants and smoothing orders required by the Nash-Moser iteration. Composing tame maps with projections is plausible, but because the projections act through the antipodal map on the second factor, a referee should check that the derivative-gain and smoothing conditions survive. I would not call that a fatal flaw; I would call it the necessary referee step. The second external input, Grayson's uniform derivative decay [Gra89, Theorem 7.2], is used only to upgrade C^{m-1} limits to C-infinity limits, which is a much lighter dependency.\n\nOne thing I liked: Remark 5.6 directly addresses Gage's monotone quantity G(g,t). The constructed flow eventually intersects each limiting geodesic in exactly two points, so Gage's round-sphere argument really does not generalize, and the remark explains why. The citation pattern looks clean; the only author self-citation is contextual.\n\nBottom line: this deserves a serious referee. The right verdict is conditional: verify the AMN applicability, especially the symmetry reduction, and check the exact statement of Grayson's theorem. I would want to see that referee report before relying on the construction, but I would happily cite this as the negative answer once those black boxes are confirmed.","headline":"A credible construction paper that plausibly answers Grayson-Gage negatively; the load-bearing Nash-Moser black box needs a referee check before full certification.","tokens_in":26420,"tokens_out":2138,"would_cite":true,"duration_ms":24543,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C44","53C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"An immortal curve shortening flow on a sphere can have an entire one-parameter family of geodesics as its complete set of subsequential limits, answering Grayson and Gage's uniqueness question in the negative.","keywords":["curve shortening flow","geodesic limits","non-uniqueness","Grayson–Gage question","Nash–Moser inverse function theorem","almost geodesics","omega-limit set","Fourier mode analysis"],"falsifier":"A direct numerical check: integrate the two-dimensional center-mode system (3.21) for the potential of Lemma 3.1 with a perturbation $e(t)$ of relative size $\\eta$ as in (3.18), and test whether the tube $C_S$ is forward-invariant with the constants of Lemma 3.17; a single trajectory starting in $C_{\\mathrm{in}}$ that exits through a lateral boundary $z=\\pm h(s)/5$ would break the trapping argument. For the full geometric claim, run the curve shortening flow on the constructed metric starting from $\\gamma_0$ and look for any subsequential limit curve at positive distance from the family $\\{\\Gamma(z)\\}$; finding one would contradict the asserted equality $\\omega_{C^\\infty}(\\gamma)=\\{\\Gamma(z)\\}$.","tokens_in":25362,"feed_emoji":"🌀","tokens_out":19119,"duration_ms":163959,"temperature":0.7,"pith_summary":"Curve shortening flow moves a curve on a surface in the direction of its curvature vector, and Grayson and Gage proved that a simple closed curve that flows for all time must accumulate, along some subsequence of times, on a closed geodesic; they asked whether that geodesic is forced to be unique. This paper answers the question in the negative. It constructs a smooth metric on the two-sphere, chosen from an arbitrarily small $C^\\infty$ neighborhood of the round metric, and an immortal simple curve shortening flow whose subsequential limits are exactly a prescribed one-parameter family of distinct simple closed geodesics, all of the same length. The result shows that a smooth ambient metric does not by itself force a single asymptotic limit for this gradient-type flow, and that the Grayson–Gage convergence theorem cannot be upgraded to full convergence.","feed_headline":"Curve shortening flow can refuse to settle on a single geodesic","feed_subtitle":"A curve-shortening flow that never collapses can accumulate on a whole circle of geodesics instead of one.","key_machinery":"The construction is carried by three mechanisms working in sequence. First, a Nash–Moser inverse function theorem from [AMN25] is applied to the length functional on a space of graphs over great circles, producing, for any prescribed mean-zero function $W$ on $\\mathbb{RP}^2$ and any small $\\epsilon>0$, a family of almost-geodesics $\\Gamma_{\\epsilon,v}$ whose lengths in the conformal metric $g_\\epsilon=e^{2\\rho_\\epsilon}g_{\\mathrm{rd}}$ are exactly $L_\\epsilon+\\epsilon W(v)$; the component of curvature tangent to the family equals $-\\nabla_{q_\\epsilon}(\\epsilon W)$, and the transverse component is smaller by an additional factor of $\\epsilon$ (Lemma 2.33). Second, a potential $W$ is engineered (Lemma 3.1) so that negative gradient trajectories of $\\epsilon W$ with respect to metrics close to the round one spiral indefinitely around a prescribed critical circle $Z\\subset\\mathbb{RP}^2$, with speed decaying to zero while its logarithmic time-derivative stays arbitrarily small (Lemma 3.17). Third, every nearby curve is written as a graph $f=\\psi_\\epsilon(p)+u$ over a reference curve: the first two odd Fourier modes $p$ form a slow center variable that follows the spiral, while the remaining odd modes $u$ — the lowest of which is the third Fourier mode, with eigenvalue $-8$ for the round linearized operator $\\partial_x^2+1$ — decay much faster (Lemma 4.28), so the full flow tracks the spiral and accumulates on exactly the geodesics indexed by $Z$. Grayson's uniform derivative decay ([Gra89, Theorem 7.2]) then promotes the finite-regularity limits to $C^\\infty$ limits.","core_discovery":"The paper's central discovery is that the $\\omega$-limit set of an immortal curve shortening flow on a closed surface can be a whole circle of geodesics rather than a single one. Theorem 5.1 states that for every $C^\\infty$ neighborhood of the round metric on $\\mathbb{S}^2$ there exist an antipodally invariant metric $g$ in that neighborhood, an embedded initial curve $\\gamma_0$, and an embedded circle $Z\\subset\\mathbb{RP}^2$ with a smooth injective map $\\Gamma$ from $Z$ into the space of smooth unparametrized embedded curves, such that every $\\Gamma(z)$ is a simple closed $g$-geodesic, all the $\\Gamma(z)$ have equal length, the curve shortening flow $\\gamma(t)$ starting from $\\gamma_0$ exists and remains embedded for all $t\\geq 0$, and the $C^\\infty$ $\\omega$-limit set of $\\gamma$ is exactly $\\{\\Gamma(z):z\\in Z\\}$. In particular, for every $z$ there is a sequence of times along which the flow converges smoothly to $\\Gamma(z)$, and no curve outside the family is a subsequential limit. Because the metric can be taken arbitrarily close to the round metric, the non-uniqueness is a smooth phenomenon rather than a consequence of extreme geometry.","pith_inferences":["The center-mode/stable-mode template is general: any flow admitting a finite-dimensional family of almost-stationary solutions with a spectral gap after removing finitely many modes, plus a gradient spiral in those modes, could in principle exhibit a circle-valued $\\omega$-limit set — a pattern that may transfer to higher-order curve flows or to mean curvature flow near families of minimal hypersu","If the expected uniqueness for real-analytic metrics is correct, then the metrics constructed here cannot be chosen real-analytic; the smoothing machinery would be genuinely necessary rather than a convenience.","A quantitative signature of the construction is that length should approach the common geodesic length at a subexponential rate while the curve's position keeps rotating through the family — a rate pattern one could look for numerically as a diagnostic of non-uniqueness."],"forward_implications":["Grayson and Gage's convergence theorem is sharp: subsequential convergence to a closed geodesic is the best general statement available, and convergence to a unique geodesic cannot be asserted for arbitrary smooth metrics.","Non-uniqueness occurs in every $C^\\infty$ neighborhood of the round metric, so positive curvature and near-round geometry do not restore uniqueness.","The limiting geodesics all have equal length and are parametrized by an embedded circle, so the $\\omega$-limit set is uncountably large yet rigidly structured.","The monotonicity of Gage's set $G(g,t)$ of geodesics meeting the flow in at least four points cannot detect the phenomenon: in this example the evolving curve eventually meets each limiting geodesic in exactly two points, so $G(g,t)$ eventually contains none of them (Remark 5.6).","The construction runs inside antipodally invariant metrics, so even a large symmetry group of the ambient metric does not force a unique limit, in contrast with the round metric, where every immortal flow converges to a unique great circle."],"supporting_citations":[{"why":"Supplies the Nash–Moser inverse function theorem and companion estimates (Theorems 5.1 and 7.6, Corollaries 5.2 and 8.9) that produce the almost-geodesic family with the prescribed length profile in Proposition 2.8.","marker":"[AMN25]"},{"why":"Establishes subsequential convergence of immortal simple closed curve shortening flows to closed geodesics, poses the uniqueness question, and provides the uniform derivative decay (Theorem 7.2) used to upgrade limits to smooth convergence.","marker":"[Gra89]"},{"why":"Proves the contrasting uniqueness result for the round metric, poses the same question as Question (6), and supplies the monotonicity of the geodesic-intersection set G(g,t) discussed in the introduction.","marker":"[Gag90]"},{"why":"Angenent's zero-counting principle is used in Remark 5.6 to justify that the evolving curve eventually intersects each limiting geodesic in exactly two points.","marker":"[Ang91]"}],"fun_headline_variants":["Curve flow traces a full circle of geodesics","Immortal curve flow accumulates on a family of geodesics","Geodesic limit under curve flow is not unique","Curve shortening flow limit can be a whole circle","Answer to Grayson-Gage: geodesic limit not unique"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything depends on the Nash–Moser inverse function theorem from [AMN25] still producing the prescribed nearly-geodesic family after the problem is restricted to antipodally symmetric data; if that restriction breaks the theorem's key technical identity, the family of almost-geodesics on which the whole construction rests does not exist.","fun_headline_variants_meta":{"raw":{"variants":["Curve flow traces a full circle of geodesics","Immortal curve flow accumulates on a family of geodesics","Geodesic limit under curve flow is not unique","Curve shortening flow limit can be a whole circle","Answer to Grayson-Gage: geodesic limit not unique"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000899,"raw_usage":{"total_tokens":3842,"prompt_tokens":883,"completion_tokens":2959,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":499,"completion_tokens_details":{"reasoning_tokens":2880}},"tokens_in":499,"tokens_out":2959,"duration_ms":22712,"temperature":1.0,"reasoning_tokens":2880,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:51:08.411878+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct numerical check: integrate the two-dimensional center-mode system (3.21) for the potential of Lemma 3.1 with a perturbation $e(t)$ of relative size $\\eta$ as in (3.18), and test whether the tube $C_S$ is forward-invariant with the constants of Lemma 3.17; a single trajectory starting in $C_{\\mathrm{in}}$ that exits through a lateral boundary $z=\\pm h(s)/5$ would break the trapping argument. For the full geometric claim, run the curve shortening flow on the constructed metric starting from $\\gamma_0$ and look for any subsequential limit curve at positive distance from the family $\\{\\Gamma(z)\\}$; finding one would contradict the asserted equality $\\omega_{C^\\infty}(\\gamma)=\\{\\Gamma(z)\\}$.","supporting_citations":[],"review_version":1}