{"id":"d9345a39-3b45-45da-8c9d-4e3de4a8e739","arxiv_id":"2509.26515","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every dimension n≥3, there exists an embedded, rotationally symmetric, non-convex ancient mean curvature flow that looks like two parallel pancakes joined by a neck and lies in a slab.","lead":"This paper constructs a new class of ancient solutions to mean curvature flow: closed, non-convex hypersurfaces shaped like two stacked pancakes joined by a neck, moving inside a slab for all past times. It is the first compact non-convex ancient example in a slab, and shows that previous convexity-based classification results do not extend without convexity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The asserted uniform entropy bound λ(M^i_t)≤5 in Lemma 3.3 is unproved and is the gatekeeper for Brakke compactness; without it the ancient limit B_t is not constructed.","rationale":"The reader identified the same load-bearing assumption as I did: the uniform entropy bound in Lemma 3.3. This is the precise point where the proof first depends on a quantitative estimate that is asserted but not shown. It is genuinely load-bearing because without it Brakke compactness cannot be applied, and there is no ancient flow B_t to which the later regularity and compactness arguments could apply. The paper's own phrasing 'One can check' marks the spot; in a rigorous construction, such a check must be supplied. I do not think this invalidates the central claim—the bound is plausible and likely provable by a comparison argument—but it is exactly the kind of gap that warrants a conditional verdict rather than acceptance. The regularity lemma (3.4) is also sketchy, but it is less crisply testable and may admit fixes; the entropy bound is the cleaner, more elementary missing step. Therefore I agree with the reader's conditional verdict and recommend no change.","tokens_in":14738,"tokens_out":29635,"duration_ms":268183,"concrete_test":"Carry out the omitted 'one can check' in Lemma 3.3. For each initial profile Γ^m_s (two copies of the ancient pancake at time s joined by the circular arc f_m), prove λ(Σ^m_s)≤5 by decomposing the Gaussian area integral into the two pancake regions and the neck region, using the known entropy bound for the ancient pancake from [3] (or a crude comparison with two parallel planes) and an explicit bound for the catenoidal neck (e.g., by comparison with a catenoid of entropy ≤2). Show the estimate for all m∈(1/2,1) and s≤−100. As a numerical cross-check, sample n=3, s=−10^{100}, m=0.6 and evaluate the Gaussian surface area integral by quadrature; if any sample exceeds 5, the asserted bound is false and Lemma 3.3 collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The entire construction of the weak ancient solution B_t depends on extracting a subsequential limit of the old-but-not-ancient flows M^i_t via Brakke compactness. The only quantitative control offered for this step is the sentence in Lemma 3.3: “One can check that we have the (crude) uniform entropy bound λ(M^i_t)≤5.” No derivation or calculation is given. This matters because Brakke compactness requires uniform local area bounds on parabolic cylinders; a uniform entropy bound is the standard way to obtain them. The bound is not a trivial consequence of the construction: each initial surface Σ_{s_i}^{m_i} consists of two large pancake components (each expected to contribute entropy near 2) joined by a catenoidal neck, and the entropy of the neck at its own scale must also be controlled. The paper explicitly flags the missing verification by writing “One can check,” but the check is not supplied. Since the authors themselves locate the load-bearing step here, this is not an artifact of the reader's reading. Without this estimate, B_t is never shown to exist, and all subsequent lemmas (3.4, 3.5) and Theorem 1.1 are unmoored. The regularity lemma is also only sketched, but the entropy bound is the more elementary and unavoidable gap: it is the first point where the proof relies on an unverified quantitative input.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for every n≥3, an ancient solution of mean curvature flow in R^{n+1} that is O(n)×O(1)-invariant, embedded, non-convex, closed, contained in a slab, and whose profile curve has two strict local maxima and one strict local minimum. The construction proceeds by building a sequence of rotationally symmetric 'old-but-not-ancient' flows from two copies of the Bourni–Langford–Tinaglia ancient pancake joined by a catenoidal neck, extracting a weak ancient Brakke limit via compactness, then upgrading the limit to a smooth ancient flow through a sequence of regularity claims involving Sturmian intersection arguments and barrier constructions.","tokens_in":15087,"tokens_out":2663,"duration_ms":24190,"significance":"If correct, Theorem 1.1 provides the first compact non-convex ancient solution of MCF trapped in a slab, thereby showing that the convexity hypothesis in Bourni–Langford–Tinaglia's Theorem 1.2 is necessary for n≥3. The paper's ambitious strategy—using Brakke compactness to obtain a weak ancient limit and then proving regularity—is a natural and potentially influential approach. The authors are explicit about the limitations of their method, including the obstruction in n=2 and the heuristic nature of parts of the regularity argument. The paper also credits and builds on the non-fattening results for rotationally symmetric flows, which is appropriate.","major_comments":[{"comment":"The proof asserts 'One can check that we have the (crude) uniform entropy bound λ(M^i_t)≤5' with no derivation. This bound is the only quantitative input for Brakke compactness, and without it the existence of the ancient Brakke flow B_t is not established. The bound is not obvious: each initial surface Σ_{s_i}^{m_i} consists of two pancake components (each contributing entropy near 2) joined by a catenoidal neck, whose entropy at its own scale must be controlled. Please provide a proof or a precise reference for this estimate; otherwise Lemma 3.3 and all subsequent claims are unsupported.","section":"§3.1, Lemma 3.3"},{"comment":"The proof that the singular set of B_t avoids the axis of rotation relies on the assertion that the partial profile functions f_m have 'uniformly controlled geometry' (a uniform interval of definition, uniform C^1 bounds, and uniformly bounded minima). These are not proved, and they are load-bearing: they justify the existence of a fixed catenoid W whose profile intersects each approximating profile in exactly two points. Without a rigorous verification of these three conditions, the Sturmian intersection argument for excluding axis singularities fails.","section":"§3.2, Lemma 3.4, Claim 1"},{"comment":"The regularity upgrade via 'wide but thin pancakes' and 'thin grim reaper' barriers is described largely in qualitative terms ('must quickly push in', 'would lead to a contradiction', 'one can see'). These claims are the core of the proof that the weak limit B_t is smooth; they require precise barrier constructions and quantitative estimates. As written, the argument is a sketch rather than a proof. This is a central gap because the final theorem requires a smooth ancient flow, not merely a Brakke flow.","section":"§3.2, Lemma 3.4, Claims 3–4"},{"comment":"The final step invokes Theorem 2.4 from [1] to conclude that the flow starting from a smooth slice B_{t_i} is non-fattening and agrees with B_t, but the applicability to the weak limit is not fully justified. In particular, the graphicality of B_{t_i} is asserted from Theorem 2.1 and the graphicality of the approximating flows; this inference for a weak limit needs a proof. The phrase 'as long as m>0' also needs to be tied to the evolution of the neck parameter m(t) and to the fact that B_t has no neck singularity after t_i.","section":"§3.3, Proof of Theorem 1.1"}],"minor_comments":[{"comment":"There are several typographical and formatting issues: 'particulary' in the acknowledgements, 'Its easy' should be 'It is easy', and the reference to 'nvent. Math' is missing 'I' in 'Invent.'.","section":"General"},{"comment":"The captions for Figures 1 and 2 contain ambiguous notation such as 'a = 2.05, h = 1.3t = -19.5'; please clarify the parameters being displayed.","section":"Captions"},{"comment":"The definition of Brakke flow is standard but the statement of Brakke compactness is informal; since the paper depends heavily on this theorem, a precise statement with the exact hypotheses (including the entropy bound) would help the reader.","section":"§2.1"},{"comment":"Several references are to preprints or in-press items; please update where possible, especially [2], [8], [14], and [16].","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper's core idea is promising and the result would be significant, but the missing entropy bound in Lemma 3.3 is a hard obstruction: without it the weak limit B_t is not constructed. The regularity proof in Lemma 3.4 is also too sketchy for a journal proof. I recommend major revision rather than rejection, as these gaps appear fixable with additional work, but the authors must supply the missing estimates and a complete regularity argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: this paper constructs a genuinely new ancient solution to mean curvature flow — a stack of two ancient pancakes joined by a neck — which is closed, non-convex, and trapped in a slab. If correct, it's the first compact non-convex example in a slab and shows the convexity hypothesis in Bourni–Langford–Tinaglia's classification is needed. The construction is interesting and the continuity argument is used well.\n\nWhat's new: the object itself. The authors take two ancient pancakes, join them with a catenoidal neck, and then use a continuity method close to Altschuler–Angenent–Giga to produce a sequence of 'old-but-not-ancient' flows. They then pass to a Brakke limit and upgrade to smoothness. The geometric picture is clear and the paper is honest about what is not proved.\n\nThe soft spots are real. The first and most serious is in Lemma 3.3: the uniform entropy bound λ(M^i_t) ≤ 5 is asserted with 'One can check' and no derivation. This is not a throwaway estimate — it is exactly what makes Brakke compactness applicable. The pancakes each have entropy near 2, and the neck needs separate control, so 5 is plausible, but it is not automatic. Without this bound the ancient limit B_t is not constructed. The authors themselves flag it, so this is not a misreading.\n\nThe second soft spot is Lemma 3.4, the regularity upgrade. The proof is long and has several claims (about singular sets, blowdown cones, tips, barriers) that are only sketched. Some steps, for example the entropy comparison in Lemma 3.5 Claim 5, are terse enough that I'd want more detail. These are downstream of the entropy bound, but they still need to be written out.\n\nThe citation pattern looks fine, the paper is built on established tools, and there are no fitted parameters or invented entities. The authors are open about limitations (backwards asymptotics, the n=2 case).\n\nMy recommendation: this deserves a serious referee. The construction is novel and significant, and the gaps are addressable rather than fatal. If the entropy bound can be justified, the main theorem likely stands. Don't desk reject; send it out, but make sure the referee asks for the entropy estimate and a fuller regularity argument.","headline":"A novel and significant construction whose main theorem is plausible, but the key uniform entropy bound in Lemma 3.3 is asserted without proof, so the current version is incomplete.","tokens_in":15519,"tokens_out":2664,"would_cite":false,"duration_ms":21620,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every n≥3 there is a closed, non-convex ancient mean curvature flow trapped in a slab, shaped like two pancakes joined by a neck.","keywords":["ancient solution","mean curvature flow","non-convex","slab","rotational symmetry","pancake","catenoid","Brakke flow"],"falsifier":"Compute the entropy of the approximating flows M^i_t on a fixed parabolic cylinder; if the bound ever exceeds 5 for some i, the compactness step fails. More directly, if the catenoid used in Claim 1 of Lemma 3.4 intersects some approximating profile more than twice, the singularity-free argument collapses.","tokens_in":14659,"feed_emoji":"🥞","tokens_out":4097,"duration_ms":34944,"temperature":0.7,"pith_summary":"The paper constructs the first compact, embedded ancient solution to mean curvature flow that is non-convex and confined to a slab. Ancient solutions exist for all negative times and serve as models for singularity formation; previously, the only slab-trapped compact examples were convex 'ancient pancakes.' The new solution looks like two such pancakes stacked and joined by a neck, and its existence shows that convexity is not a hidden requirement for a compact ancient flow to stay inside a slab. The construction works in every dimension n≥3.","feed_headline":"Non-convex pancake stack flows inside a slab for all time","feed_subtitle":"The closed ancient solution exists for every n≥3 and shows convexity is not needed for slab confinement.","key_machinery":"The construction is carried out on the profile curve in a plane, where rotational symmetry reduces mean curvature flow to a forced curve-shortening flow. The initial profiles are two copies of the ancient pancake profile joined by a catenoidal neck, parametrized by the neck width m. A continuity argument selects a neck width that sits at the boundary between eventual neck-pinch and quick convexification, producing a family of 'old-but-not-ancient' flows whose Brakke limit is the ancient solution. The Sturmian theorem on intersection numbers is the main tool that controls singularities and upgrades the weak limit to a smooth flow.","core_discovery":"For every n≥3 there exists an ancient solution M^n_t ⊂ R^{n+1}, t∈(−∞,0], that is rotationally symmetric, embedded, closed, non-convex, and contained in a slab; its profile curve has two strict local maxima (the pancake tips) and one strict local minimum (the neck). The flow is obtained as a limit of long-lived smooth rotationally symmetric flows, and the paper proves the weak limit is actually smooth by combining Sturmian intersection-counting with catenoid and pancake barriers.","pith_inferences":["One could test the continuity argument numerically: evolve the two-pancake profile for large negative times and track the neck width; the paper's proof predicts the neck remains bounded and no pinch occurs before time 0.","The failure at n=2 is tied to the catenoid being entire; a similar construction using a different entire self-shrinker might produce an analogous stack in R^3 with different topology.","If the asserted entropy bound in Lemma 3.3 could be verified with an explicit constant, the compactness step would be placed on firmer ground and might extend to k>2 pancakes without new ideas.","The halfspace-convex-hull scenario, if realized from an infinite stack, would give an embedded counterpart to the known immersed example and sharpen the Chini–Møller trichotomy."],"forward_implications":["The convexity assumption in the earlier classification of compact slab-trapped ancient flows is genuinely necessary for n≥3.","At every time the profile has exactly two tips and one neck, so the flow is a stack of two pancakes, not a single pancake.","The flow is rotationally and reflection symmetric, and stays inside a fixed slab for all t≤0.","The resulting ancient flow is smoothly embedded for all t≤0, not merely a weak Brakke flow.","The construction suggests that similar stacks of finitely many pancakes can be built, and possibly an infinite stack whose convex hull is a halfspace."],"fun_headline_variants":["Two pancakes, one neck: ancient slab flow for n≥3","Ancient non-convex pancake stack stays slab-bound","Closed slab-bound ancient flow from two pancakes and a neck","Pancake stack with neck persists in slab for all n≥3","Non-convex ancient slab flow built from a pancake stack"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The asserted but unproved uniform entropy bound λ(M^i_t) ≤ 5 in Lemma 3.3 is load-bearing: without it, Brakke compactness cannot produce the ancient limit flow B_t.","fun_headline_variants_meta":{"raw":{"variants":["Two pancakes, one neck: ancient slab flow for n≥3","Ancient non-convex pancake stack stays slab-bound","Closed slab-bound ancient flow from two pancakes and a neck","Pancake stack with neck persists in slab for all n≥3","Non-convex ancient slab flow built from a pancake stack"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001343,"raw_usage":{"total_tokens":5179,"prompt_tokens":514,"completion_tokens":4665,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":258,"completion_tokens_details":{"reasoning_tokens":4593}},"tokens_in":258,"tokens_out":4665,"duration_ms":28834,"temperature":1.0,"reasoning_tokens":4593,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T13:30:24.190008+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the entropy of the approximating flows M^i_t on a fixed parabolic cylinder; if the bound ever exceeds 5 for some i, the compactness step fails. More directly, if the catenoid used in Claim 1 of Lemma 3.4 intersects some approximating profile more than twice, the singularity-free argument collapses.","supporting_citations":[],"review_version":1}