{"id":"276be076-c209-4e81-bf59-6ce8fe75f8b6","arxiv_id":"2412.00561","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The stabilized symplectic embedding capacity of the round four-ball is computed exactly: a Fibonacci staircase below tau^4, then 3a/(a+1).","lead":"This paper solves the stabilized symplectic embedding problem for four-dimensional ellipsoids into the round ball, giving an exact piecewise formula with a Fibonacci staircase below the golden-ratio accumulation point and a rational tail above it. It proves the necessary lower bounds by constructing new families of singular algebraic curves, linked to scattering diagrams.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 6.3.2, which identifies scattering-diagram rays with the ellipsoid staircase, is stated without proof and is load-bearing for Theorems B and F.","rationale":"The paper solves a major problem. The chain is: Theorem B/F (curve existence via scattering) -> Theorem 2.0.1 (curve obstruction) -> Theorem E/A (embedding function). The reader flagged Theorem 2.0.1 as a black box; that is an external citation and can in principle be checked. But Lemma 6.3.2 is internal: the proof of the main theorems literally says 'combining Corollary 6.3.1 and Lemma 6.3.2', and no proof of the lemma appears in the full text. The lemma's content is the exact dictionary that matches the algebraic curves to the staircase points; a small error in the inequalities defining the dense region would break the density argument and thus the lower bound for the stabilized embedding function. The test is a finite verification of at most four cases, so the concern is settleable. Verdict remains CONDITIONAL, pending a proof or a computational verification of Lemma 6.3.2.","tokens_in":42665,"tokens_out":24351,"duration_ms":226599,"concrete_test":"Supply the missing proof of Lemma 6.3.2 by direct computation. For each J=2 entry in Table 4.3.1 (CP^2, CP^1 x CP^1, Bl_3 CP^2, Bl_4 CP^2), substitute W_X, compute nu from (6.1.1), and verify the equivalence xi_- < slope(phi(W_X(p,q))) < xi_+ iff p/q (or q/p) > a_X^acc, where a_X^acc is the fixed point of S_X. Also verify that the discrete rays from [GP10, Thm. 5] are exactly the images of outer-corner pairs. If a single counterexample emerges, the lemma is false and Theorems B/F lack a key step.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 6.3 introduces Lemma 6.3.2 as the bridge between the algebraic curve construction and the symplectic embedding staircase. For each rigid del Pezzo surface with J=2 strands, it asserts that a ray in the basic scattering diagram is discrete exactly when p/q (or q/p) is an outer corner of the infinite staircase c_X, and that W_X(p,q) lies in the dense region exactly when p/q (or q/p) > a_X^acc. The proof of Theorem B and Theorem F then simply cites this lemma. But the full text contains no proof of Lemma 6.3.2: after its statement the paper moves directly to Corollary 6.3.3 and the proofs of the main theorems. This is not a matter of an external black box; it is an omitted proof inside the paper of a nontrivial identification involving the explicit W_X maps of Table 4.3.1, the change-of-lattice factors nu, and the roots xi in (6.2.1). If the identification is incorrect for any of the J=2 surfaces, for example if the boundary of the dense region corresponds to a value different from the staircase accumulation point, then the constructed well-placed curves may not give the needed lower bounds c_{X x R^{2N}}(a) >= a/(a+1) on a dense subset of [a_X^acc, infinity), and Theorem A/E would not follow. The lemma is therefore the most load-bearing internal step of the paper.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper computes the stabilized symplectic embedding function c_{B^4 \\times \\mathbb{R}^{2N}}(a) for ellipsoids into the round ball, proving an explicit piecewise formula: an infinite Fibonacci staircase for a below the accumulation point \\tau^4, followed by the rational tail 3a/(a+1) for a \\ge \\tau^4. The proof proceeds by converting symplectic embedding obstructions into statements about existence of (p,q)-well-placed rational algebraic curves, then using a correspondence with scattering diagrams, the change-of-lattice trick, and positivity results for basic scattering diagrams from Gross--Pandharipande and Gr\\\"afnitz--Luo. The paper also proves analogous results for del Pezzo surfaces and gives new families of sesquicuspidal plane curves, including a resolution of the minimal-degree problem in many cases.","tokens_in":42967,"tokens_out":8407,"duration_ms":84694,"significance":"If the proof is completed, the paper fully solves the stabilized ellipsoid embedding problem for the round ball, a central open problem in quantitative symplectic geometry after McDuff--Schlenk. The explicit formula is parameter-free and exhibits a sharp phase transition from the unstabilized staircase to a rational folding tail. Conceptually, the paper builds a bridge from singular algebraic curve theory and scattering diagrams to symplectic nonsqueezing, and it gives a large new supply of low-degree cuspidal rational plane curves. The exposition is largely careful and the main reductions are clearly laid out; the paper is explicit that it avoids unproved symplectic field theory virtual techniques. The main reservations are the unproved internal Lemma 6.3.2 and the heavy reliance on the imported obstruction theorem from [McS23].","major_comments":[{"comment":"Lemma 6.3.2 is stated without proof, and it is the hinge for Theorem B and for the J=2 cases of Theorem F(a), and hence, through Section 3 and Section 2, for Theorem A. The lemma identifies, under the bijection W_X, the discrete rays of S(D^{\\ell_1,\\ell_2}_{m_1,m_2})_{min} with outer corners of the infinite staircase c_X|_{[1,a_X^{acc}]} and the dense region with (a_X^{acc},\\infty). After the statement the text moves directly to Corollary 6.3.3 and the proofs of Theorem B and Theorem F, with no further argument. If this identification is incorrect at any boundary value, or if a non-outer fraction maps to a discrete ray, the dense lower bounds c_{X\\times\\mathbb{R}^{2N}}(a)\\ge a/(a+1) would not follow, and Theorems A and E would fail. Please supply a complete proof of both bullets, or reduce them explicitly to published results such as [GP10, Thm. 5] together with the staircase classification in [MS24]/[Cri+25], including the congruence conditions stated in Corollary 6.3.3.","section":"Section 6.3, Lemma 6.3.2"},{"comment":"All symplectic embedding obstructions in the paper pass through Theorem 2.0.1, imported from [McS23] (Cor. 2.7.2, Cor. 2.3.8, Thm. D, Thm. E), which is described only by a short sketch involving moduli of J-holomorphic curves. This is the unique bridge from existence of algebraic curves to the lower bounds c_{X\\times\\mathbb{R}^{2N}}\\ge a/(a+1). The manuscript should state the publication status of [McS23] and either give a complete proof of Theorem 2.0.1 or a precise reference to a published version with all hypotheses verified. In particular, the hypotheses of Corollary 2.0.2 (monotonicity of X or N\\le 1, semipositivity, index-zero condition) should be checked explicitly for the unimonotone del Pezzo surfaces used in Theorem E(b), and the perturbation step from algebraic sesquicuspidal curves to symplectic sesquicuspidal curves in the proof of Theorem E should be justified in the presence of auxiliary singularities.","section":"Section 2, Theorem 2.0.1"}],"minor_comments":[{"comment":"The monotonicity inequalities in the proof of Corollary D appear reversed. Since U\\subset B^4(3), one has c_{U\\times\\mathbb{R}^{2N}}\\ge c_{B^4(3)\\times\\mathbb{R}^{2N}}, and since X\\subset U, one has c_{U\\times\\mathbb{R}^{2N}}\\le c_{X\\times\\mathbb{R}^{2N}}. The displayed chain of inequalities should be corrected so that the two directions give the claimed equality.","section":"Corollary D"},{"comment":"The sentence 'Inspecting Table 4.2.1' refers to the wrong table; the relevant data appears in Table 4.3.1.","section":"Corollary 6.3.3"},{"comment":"There are several typographical errors: 'especicially' in Remark 1.0.9, 'geoemtry' and 'resuls' in Remark 1.0.10, 'Cantour' in Remark 5.2.10, 'necesarrily' in Remark 3.0.6, and 'if and only of one if' in Corollary 6.3.1.","section":"Various"},{"comment":"The sentence 'an inspection of their argument shows that we can take \\kappa=1' would benefit from a precise pointer to the relevant part of [GP10], since the sharp 'if and only if' in Theorem B depends on nonvanishing for the primitive ray, not merely for some positive multiple of it.","section":"Section 6.2, Theorem 6.2.1"}],"recommendation":"major_revision","confidential_remarks":"The central formula is impressive and the reduction is mostly transparent, but the refereeing hinges on two items: the unproved Lemma 6.3.2 and the black-box status of [McS23]. The first is an omission inside the paper and should be fixable; the second requires either a published reference or a substantial proof sketch. I recommend major revision rather than rejection, with the expectation that the authors can supply the missing proof of Lemma 6.3.2 and clarify the status of [McS23]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this paper solves a central open problem in quantitative symplectic geometry — the stabilized ellipsoid embedding function for B^4 is now known for all a, and the answer is as clean as conjectured: Fibonacci staircase up to tau^4, then 3a/(a+1). The novelty and significance are real, and the scattering-diagram bridge to singular algebraic curves is clearly the right framework.\n\nWhat is new: Theorem A's lower bound 3a/(a+1) on the whole tail (previously only isolated values), the iff characterization of minimal-degree (p,q) cuspidal curves in Theorem B, the general correspondence Theorem G (well-placed curves iff nonzero scattering coefficient), and the density results in Theorem F. There are no free parameters; the staircase constants come from quiver moduli and prior honest results.\n\nSoft spots: two. First, the paper's internal bridge Lemma 6.3.2 (discrete rays vs outer corners, dense region vs (a_acc,∞)) is stated and used as the linchpin for Theorems B and F, but I could not find a proof anywhere in §6.3; it is not deferred, it is simply missing. This is the load-bearing step that the stress-test identifies, and the concern is valid. The statement is plausible and probably follows from GP10's classification plus the known staircase numerology, but as written it is an assertion, not a proved lemma. The second soft spot is the black-box obstruction theorem 2.0.1 imported from [McS23]; if that theorem has hidden assumptions, the symplectic consequences don't go through. The authors acknowledge the dependency and sketch the argument, so this is a normal (if heavy) external dependence rather than an internal error. Both soft spots are fixable in principle; the first definitely needs a proof in any final version.\n\nCitation pattern: heavy self-citation, but to papers that actually prove the needed tools (e.g. staircase stability). Not a flaw by itself.\n\nAudience: symplectic topologists, algebraic geometers interested in cuspidal curves and scattering diagrams. Deserves a serious referee. Recommendation: send to peer review; the referee should treat Lemma 6.3.2 as the main thing to verify.","headline":"A major result: the stabilized ellipsoid embedding function for the ball is now known exactly, via a new scattering-diagram bridge to singular curves, but one internal lemma is asserted without proof and needs a referee's attention.","tokens_in":43481,"tokens_out":2526,"would_cite":true,"duration_ms":28252,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D35","14H50","14J26","14N35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Exact formula for stabilized ellipsoid embeddings into the round ball: Fibonacci staircase below τ⁴, then the rational fold 3a/(a+1).","keywords":["symplectic nonsqueezing","stabilized ellipsoid embeddings","embedding capacity","Fibonacci staircase","sesquicuspidal curves","scattering diagrams","minimal degree plane curves","del Pezzo surfaces"],"falsifier":"Evaluate the stabilized capacity at any $a>\\tau^4$ by an independent method, for example by computing the relevant higher symplectic capacities at $a=7$ or $a=8$, and compare with $3a/(a+1)$; any value below $3a/(a+1)$ would refute Theorem A. On the algebraic side, run the completion of the scattering diagram $D^{3,3}_{e_1,e_2}$ to sufficiently high order in $t$ and check that the coefficient at the relevant lattice point is nonzero for every coprime $p,q$ with $p+q$ divisible by $3$ and $p/q>\\tau^4$; a single zero coefficient, or a failure of the predicted Fibonacci condition below $\\tau^4$, would refute Theorem B and with it the curve construction behind the embedding theorem.","tokens_in":2268,"feed_emoji":"📐","tokens_out":2841,"duration_ms":123194,"temperature":0.7,"pith_summary":"The paper claims to settle the stabilized symplectic ellipsoid embedding problem for the round four-ball: after increasing the ambient dimension by an arbitrary amount, the sharp embedding capacity is an explicit function that repeats the known four-dimensional Fibonacci staircase up to the accumulation point $\\tau^4=(7+3\\sqrt{5})/2$ and then becomes the rational fold $3a/(a+1)$. The route is indirect: stabilized ellipsoid embeddings are obstructed by rational algebraic curves in the complex projective plane that carry a prescribed $(p,q)$ cusp and only node-like auxiliary singularities, so the embedding result is deduced from a purely algebro-geometric existence theorem for such curves. The existence theorem is proved via scattering diagrams, converting each candidate curve into a term in a completed diagram, with the needed terms shown to be nonzero using positivity results for basic scattering diagrams. If correct, this gives the first complete solution of Problem 1.0.2 for the round ball and explains the phase transition between the staircase and the simple folding tail as a transition from unicuspidal to sesquicuspidal obstructions.","feed_headline":"Ellipsoid squeezing solved: Fibonacci stairs then a rational fold","feed_subtitle":"The round ball's stabilized embedding capacity is now explicit for every ratio and stabilization.","key_machinery":"The load-bearing bridge is a bijection between $(p,q)$-well-placed rational curves in a uninodal Looijenga pair and curves in a toric model that meet one distinguished toric divisor with contact order one, together with the theorem that existence of such curves is detected by nonvanishing of a coefficient in the minimal scattering diagram $S(D_T)_{\\min}$. A scattering diagram is a collection of rays in the plane labeled by power series; the completion algorithm adds outgoing rays until the monodromy around every loop is trivial. For rigid del Pezzo surfaces the paper exhibits toric models whose diagrams are basic scattering diagrams with two or three initial rays, and a change-of-lattice reduction turns the relevant two-ray cases into the standard diagrams $D^{\\ell_1,\\ell_2}_{e_1,e_2}$. Known positivity results for these standard diagrams then supply the required nonzero coefficients, yielding the curve existence theorems and hence the embedding obstructions.","core_discovery":"On the paper's own terms, the central discovery is Theorem A: for every $N\\ge 1$, the stabilized ellipsoid embedding function of the round ball is $c_{B^4\\times\\mathbb{R}^{2N}}(a) = \\frac{1}{\\sqrt{\\alpha_k}}\\,a$ on each interval $[\\alpha_k,\\beta_k]$, is $\\sqrt{\\alpha_{k+1}}$ on each interval $[\\beta_k,\\alpha_{k+1}]$, and is $3a/(a+1)$ for all $a\\ge\\tau^4$, where the numbers $\\alpha_k,\\beta_k$ are built from Fibonacci numbers and accumulate at $\\tau^4$. The paper derives this from Theorem B, a complete answer to the minimal-degree problem for rational plane curves with a $(p,q)$ cusp when $p+q$ is divisible by $3$: such a curve of degree $(p+q)/3$ exists exactly for the Fibonacci pairs $(\\mathrm{Fib}_{k+4},\\mathrm{Fib}_k)$ with $k$ odd, or when $p/q>\\tau^4$. Because the symplectic obstruction mechanism turns these curves into lower bounds and known folding constructions give matching upper bounds, the two theorems together determine the capacity. The same framework is extended to del Pezzo surfaces, yielding a complete description of their stabilized embedding functions in the rigid cases and a universal rational lower bound in the non-rigid cases.","pith_inferences":["Editorial inference: if the scattering-diagram bridge is as robust as the paper suggests, the same coefficient-nonvanishing criterion should compute stabilized capacities for other monotone targets beyond del Pezzo surfaces, where the rational tail $a/(a+1)$ would be governed by the same folding construction.","Editorial inference: the byproduct equality $c_{B^4\\times\\mathbb{R}^{2N}} = c_{\\mathbb{CP}^2\\times\\mathbb{R}^{2N}}$ suggests that in the stabilized regime the capacity depends mainly on the symplectic area class and the minimal cusp ratios available, not on finer features of the four-dimensional target.","Editorial inference: the conjectural exact count of well-placed curves carrying the tail obstructions, if proved, would give a quantitative refinement of Theorem A and a sharp check on the scattering coefficients at $t=1$.","Editorial inference: the refined multi-variable scattering coefficients described in the paper should detect the auxiliary singularity counts and homology classes of well-placed curves, potentially linking the stabilized embedding problem to the subtle combinatorial phenomena seen in the classification of unicuspidal curves."],"forward_implications":["The stabilized embedding capacity of the round four-ball is now known for every $a\\ge 1$ and every $N\\ge 1$, replacing the previous piecemeal lower bounds with one explicit formula.","Hind's folding embeddings are sharp in the whole tail region $a\\ge\\tau^4$, so no stronger stabilized obstruction can exist beyond the accumulation point.","For plane curves with a prescribed $(p,q)$ cusp and $p+q$ divisible by $3$, the minimal degree is $(p+q)/3$ except in finitely many cases of low singularity excess, and the curves realizing the minimum are rational and can be chosen well-placed with respect to any nodal cubic.","For rigid del Pezzo surfaces, the same framework computes the stabilized capacity as the unstabilized staircase up to the accumulation point and $a/(a+1)$ beyond; for non-rigid del Pezzo surfaces it proves the lower bound $a/(a+1)$ for all $a$.","The ratio set $S_X$ of cusp types realized by rational curves is dense beyond the accumulation point for rigid del Pezzo surfaces and dense in $[1,\\infty)$ for non-rigid ones, giving an abundance of algebraic obstructions."],"supporting_citations":[{"why":"It supplies Theorem 2.0.1, the black-box obstruction: index-zero sesquicuspidal rational symplectic curves forbid stabilized embeddings, and every symplectic lower bound in this paper flows from it.","marker":"[McS23]"},{"why":"It provides the theorem equating scattering-diagram coefficients with relative Gromov–Witten counts, which is the core of the bridge from curves to scattering diagrams.","marker":"[GPS10]"},{"why":"It analyzes basic scattering diagrams with equal labels and proves the positivity in the dense region for the equal-label case needed for the complex projective plane.","marker":"[GP10]"},{"why":"It supplies the change-of-lattice trick and the positive-factorization result used to reduce toric-model diagrams to the standard form and to prove coefficient positivity.","marker":"[Gro+18]"},{"why":"It extends scattering positivity to unequal labels and the dense region, handling the remaining product case $\\mathbb{CP}^1\\times\\mathbb{CP}^1$.","marker":"[GL23]"},{"why":"It gives the explicit folding embeddings that provide the upper bound $3a/(a+1)$, the matching half of Theorem A.","marker":"[Hin15]"},{"why":"It provides the ECH-based obstructions at outer staircase corners which, together with scaling, establish the staircase on $[1,\\tau^4]$.","marker":"[CH18]"},{"why":"It introduces the well-placed condition and prior outer-corner curves, and states the stability of staircases used for the low-accumulation regime.","marker":"[MS24]"},{"why":"It justifies passing from compact ellipsoid embeddings with large extra dimensions to the noncompact stabilization $E(a_1,a_2)\\times\\mathbb{R}^{2N}$ used in the central problem.","marker":"[PV15]"}],"fun_headline_variants":["Ellipsoid squeezing fully solved: Fibonacci stairs, then rational fold","Symplectic nonsqueezing cracked by cuspidal curves and scattering","Exact ellipsoid-ball capacity via Fibonacci staircase and folding","From Fibonacci to rational: ellipsoid embedding in all stabilizations","Scattering diagrams and cuspidal curves settle ellipsoid embedding"],"cache_read_input_tokens":45568,"weakest_assumption_plain":"The paper assumes as a black box that a singular rational sphere in a symplectic four-manifold, with one cusp whose parameters satisfy the rigidity condition $p+q=c_1([C])$ and only node-like other singularities, always obstructs stabilized ellipsoid embeddings in the stated way; the present paper does not reprove the compactness and transversality behind that implication.","fun_headline_variants_meta":{"raw":{"variants":["Ellipsoid squeezing fully solved: Fibonacci stairs, then rational fold","Symplectic nonsqueezing cracked by cuspidal curves and scattering","Exact ellipsoid-ball capacity via Fibonacci staircase and folding","From Fibonacci to rational: ellipsoid embedding in all stabilizations","Scattering diagrams and cuspidal curves settle ellipsoid embedding"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000981,"raw_usage":{"total_tokens":4192,"prompt_tokens":1001,"completion_tokens":3191,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":617,"completion_tokens_details":{"reasoning_tokens":3098}},"tokens_in":617,"tokens_out":3191,"duration_ms":24087,"temperature":1.0,"reasoning_tokens":3098,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:16:24.524366+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the stabilized capacity at any $a>\\tau^4$ by an independent method, for example by computing the relevant higher symplectic capacities at $a=7$ or $a=8$, and compare with $3a/(a+1)$; any value below $3a/(a+1)$ would refute Theorem A. On the algebraic side, run the completion of the scattering diagram $D^{3,3}_{e_1,e_2}$ to sufficiently high order in $t$ and check that the coefficient at the relevant lattice point is nonzero for every coprime $p,q$ with $p+q$ divisible by $3$ and $p/q>\\tau^4$; a single zero coefficient, or a failure of the predicted Fibonacci condition below $\\tau^4$, would refute Theorem B and with it the curve construction behind the embedding theorem.","supporting_citations":[],"review_version":1}