{"id":"7af308fe-4a12-47c8-9946-86689e928fa8","arxiv_id":"2412.11103","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"An integer-valued count of immersed minimal tori in closed Riemannian manifolds of dimension at least 8 is introduced and asserted to be invariant under metric perturbations.","lead":"The paper defines a weighted count of minimal tori in a Riemannian manifold and claims the count is unchanged under perturbations of the metric. It imports recent transversality tools from symplectic geometry and geodesic counting to a new setting, but the current version has internal inconsistencies that undermine the claim.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.3 proves only linear jet-rank growth, whereas Wendl's condition (Definition 3.6) and Theorem 3.2 require l^n growth with n=2 for a torus; the infinite-codimension step in Proposition 4.2 is therefore unsupported.","rationale":"The reader's weakest_assumption identifies exactly the mismatch between the linear rank bound in Proposition 3.3 and the l^n growth required by Wendl's condition for a 2-dimensional surface. This is the most load-bearing concern because it sits at the base of the super-rigidity genericity theorem (Theorem 1.1), which in turn is required for the counting function's invariance (Theorem 1.2). Without a correct verification of Wendl's condition, Theorem 3.2—the source of the infinite-codimension conclusion about Petri-condition failures—cannot be applied, and the wall-crossing analysis of Section 5 has no transversality foundation. The issue is concrete and quantitative: the paper's own Proposition 3.3 gives a lower bound of k/2, a full power of l short of the required l^2. The claim that setting c_0=1/2 and l_0=5d+3 satisfies Definition 3.6 is only consistent if the exponent in that definition is 1, but for a torus it must be 2. I also considered other weaknesses noted by the reader, including the index formula in Proposition 4.1 and the dimension hypotheses; however, the Wendl-condition mismatch is the first point where the argument's central machinery fails, and it alone invalidates the proof as written. The paper does contain substantial original calculations and a serious attempt to adapt Wendl's methods, so the issue is not one of fraud or carelessness across the board; it is a specific technical gap that a strengthened rank estimate or a corrected application of the transversality theorem might repair. The proposed concrete test would distinguish between a repairable underestimation of the rank and a fundamental failure of the transversality approach. Since the concern lands on the same assumption the reader highlighted, I agree with the reader's weakest_assumption and do not recommend changing the verdict.","tokens_in":34946,"tokens_out":4403,"duration_ms":42101,"concrete_test":"Compute exactly the rank of L^{≤l}_{Δ,B} for the Laplacian on R^2 using the explicit right inverses R̂ and R̂† given in §3.3, for a minimal non-zero element B in ker(ω_Δ), e.g. the constant element B = 1⊗1 (degree d=0) or a degree-1 element. Evaluate the rank for l = 1,...,50. If the rank grows like O(l), then Proposition 3.3's linear bound is optimal, Wendl's condition fails, and Theorem 3.2 cannot be invoked. If the rank grows like Ω(l^2), then a stronger estimate is provable and the paper's bound is merely suboptimal; in that case, check whether a repair of Proposition 3.3 would restore the argument. Independently, verify the exact exponent in Doan–Walpuski Definition 1.6.15 to confirm it equals the dimension of the surface (2) rather than being an adjustable parameter.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central count-invariance result (Theorem 1.2) rests on Theorem 1.1, whose proof passes through Proposition 4.2. Proposition 4.2 invokes Theorem 3.2 (Doan–Walpuski Theorem 1.6.17) to conclude that the locus where the ∞-jet Petri condition fails has infinite codimension. That theorem is applicable only if the symbol satisfies Wendl's condition as in Definition 3.6: for each homogeneous B in ker(ω_D), there must exist right inverses with rk L^{≤l}_{σ,B} ≥ c_{d,ρ} l^n for all l ≥ l_{d,ρ}, where n is the dimension of the manifold on which the operator acts. Here the operator is the Jacobi operator on a Riemann surface (the torus), so n = 2; the required growth is quadratic in the jet order. Proposition 3.3 establishes only rk L^{≤k}_{Δ,B} ≥ k/2 for k ≥ 10d+6, i.e. linear growth. The text then asserts that setting c_0(ρ,d)=1/2 and l_0=5d+3 satisfies Definition 3.6, but that would only be correct if the exponent in Definition 3.6 were 1. Indeed, Definition 3.6 never fixes the value of n, and the subsequent application assumes the exponent matches the surface dimension. A linear lower bound is not enough to force the Brill–Noether strata to have infinite codimension, so the conclusion of Theorem 3.2 cannot be applied. Consequently, the super-rigidity genericity theorem and the counting invariance built on it lack a necessary ingredient as written. This is a genuine gap in the argument, not merely a disagreement with consensus: the cited transversality theorem has a precise quantitative hypothesis that the supplied computation does not meet.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a count function n(g, U) for minimal immersed tori in a Riemannian manifold of dimension at least 8 and claims that this count is invariant under continuous deformations of the metric. The proof strategy transplants the transversality and super-rigidity framework of Wendl and Doan–Walpuski from J-holomorphic curves to the Jacobi operator of minimal surfaces. After reviewing White's moduli-space theory, the paper introduces Petri and flexibility conditions, attempts to verify Wendl's condition for the Jacobi operator, computes indices of twisted Jacobi operators over orbifold Riemann surfaces, and then constructs local models near critical points and weak limit points to prove the invariance of the count. The main theorems are Theorem 1.1 (generic super-rigidity of metrics) and Theorem 1.2 (invariance of the count).","tokens_in":35432,"tokens_out":8123,"duration_ms":71612,"significance":"If the main results were correct, they would provide a new invariant counting minimal tori in high-dimensional Riemannian manifolds, generalizing recent counting results for closed geodesics and adapting powerful transversality machinery from symplectic geometry to minimal surface theory. The paper also gives an explicit weight system for multiple covers, which is a useful concrete structure. However, the central technical step—the verification of Wendl's condition for the Jacobi operator—is not carried out correctly, and the gap is load-bearing for both main theorems. The paper is partly a translation of known results rather than an independent breakthrough, but the attempted adaptation is nontrivial and could be valuable if the missing estimate were supplied.","major_comments":[{"comment":"The verification of Wendl's condition is insufficient. Definition 3.6 is stated with an exponent n that is never defined in the manuscript; in the cited source (Doan–Walpuski, Definition 1.6.15, and Wendl's theorem), the exponent n is the dimension of the manifold on which the operator acts. Here the operator is the Jacobi operator on a torus, so n = 2, and Wendl's condition requires rk L^{≤l}_{σ,B} ≥ c_{d,ρ} l^2 for large l. Proposition 3.3 proves only the linear bound rk L^{≤k}_{Δ,B} ≥ k/2 for k ≥ 10d+6. The sentence 'Therefore it is enough to set c0(ρ, d) = 1/2 and l0(ρ, d) = 5d + 3 in Definition 3.6' is therefore incorrect: for n = 2, the linear function (1/2)l cannot dominate any positive multiple of l^2 for all large l. Consequently, Theorem 3.2 cannot be applied to conclude that the failure locus of the ∞-jet Petri condition has infinite codimension. This gap undermines Proposition 4.2 and, in turn, the proofs of Theorems 1.1 and 1.2.","section":"Section 3.3, Proposition 3.3; Definition 3.6; Theorem 3.2"},{"comment":"The statement of Proposition 4.1 omits the factor 1/2 that its own proof derives. The displayed formula in the statement reads index(J^V_{g,v;ϖ}) = - rk_C N_{v,ϖ} Σ_x dim(V/V_{ρx}), while the proof concludes 'index(J^V_{g,v;ϖ}) = - 1/2 rk_C N_{v,ϖ} Σ_x dim(V/V_{ρx}).' The subsequent codimension estimates in Proposition 4.2 use the version with the factor 1/2, including the formula index_{K_i} J^{V_i}_{g,v;ϖ} = -((n-2)/2) Σ_x dim(...). The Proposition as stated is false by a factor of 2 and the inconsistency should be corrected so that the statement matches the proof and the later usage.","section":"Section 4.1, Proposition 4.1 and its proof"}],"minor_comments":[{"comment":"The abstract at the top of the full text states 'dim M ≥ 8', while an earlier version and the introduction's discussion of Theorem 1.1 use 'dim M ≥ 6'. The paper should state the precise dimensional hypotheses consistently in the abstract, the introduction, and the theorems.","section":"Abstract and Introduction"},{"comment":"The exponent n in the definition of Wendl's condition is never defined. It should be explicitly identified as the dimension of the underlying manifold of the elliptic operator, to avoid the ambiguity that currently allows the later substitution n = 1.","section":"Definition 3.6"},{"comment":"In the display of weight relations after setting n2 = n4 = n8 = 0, the expression 'n2_{+3} = n − 3' appears to contain a typo; it should presumably be 'n2_{+3} = n2 − 3'. Please check and correct all such formulas.","section":"Section 5.2.3"},{"comment":"The local model for a weak limit point is asserted to be the zero set of g(ϵ, r) = r((ϵ − t)f(ϵ, r) − r^2 h(r)), with f(t,0) and h(0) nonzero. This is plausible but the derivation is not given in the manuscript; a more explicit derivation or a precise reference to the corresponding argument in [5] would help the reader verify the model.","section":"Section 5.2.2, Proposition 5.4"},{"comment":"The definition of the count function involves a free choice of normalization constants n2, n4, n8, which are later set to zero. The paper should explicitly acknowledge that the numerical value of the count depends on this choice, even though the invariance property holds for any consistent choice satisfying the derived relations.","section":"Section 5.1 and Definition 5.1"}],"recommendation":"reject","confidential_remarks":"The paper has a coherent overall strategy and demonstrates familiarity with the relevant literature, but the central technical estimate needed to apply Wendl's transversality theorem is not established. Proposition 3.3 gives only linear growth in the jet order, while the required Wendl condition for an operator on a torus is quadratic. This is not a mere typographical issue: it is a genuine gap in the proof of the infinite-codimension statement, which is the backbone of the super-rigidity genericity theorem and of the counting invariance. Without a new argument proving the quadratic rank bound (or an alternative transversality theorem), the main claims are unsupported. I would encourage the authors to pursue the missing estimate, but as written the manuscript is not suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely new: it attempts the first perturbation-invariant integer count of minimal tori in high-dimensional Riemannian manifolds, importing the Taubes/Eftekhary counting framework and Wendl's transversality technology. The organizational debt to Eftekhary and Doan–Walpuski is explicit and mostly honest, and the expository appendices on orbifolds and twisted operators are useful. If the main theorems held, this would be a real contribution.\n\nThat said, the central argument has load-bearing gaps. The most serious is the Wendl-condition verification. Proposition 3.3 proves rank growth that is linear in the jet order, roughly k/2, but Definition 3.6 and the cited Theorem 3.2 require growth like c·l^n with n equal to the dimension of the underlying surface, i.e. n=2 for a torus. The paper never fixes the exponent in Definition 3.6, and the application to the torus requires quadratic growth. A linear bound does not imply the infinite-codimension conclusion for the Brill–Noether strata, so the genericity of super-rigidity and the invariance count built on it are not supported as written. This is not a cosmetic mismatch; it is exactly the quantitative hypothesis the cited theorem needs.\n\nThere are also internal inconsistencies. Proposition 4.1 states an index formula without the factor 1/2 that its own proof derives, and the subsequent codimension estimates in Proposition 4.2 use the version with 1/2. The counting weights are fixed by setting n2=n4=n8=0 with no real justification, and the dimension hypotheses drift: the abstract and Theorem 1.2 say dim M≥8, the introduction says dim M≥6, and Section 5 says dim M≥7. These may be fixable, but they add real friction.\n\nWho is this for? Someone working in geometric counting or minimal surface theory who wants to see whether the Taubes/Eftekhary strategy can transfer to minimal submanifolds. The paper deserves a serious referee in principle, because the question is good and the author is engaging the right sources. But as it stands, the main theorems are not proven. I would send it back for a careful revision addressing the Wendl-condition exponent, the index formula factor, and the normalization, then look again. I would not cite it in its current form.","headline":"A serious and ambitious adaptation of Taubes–Eftekhary counting to minimal tori, but the main transversality input is quantitatively mismatched and the paper is not yet reliable as written.","tokens_in":35931,"tokens_out":2019,"would_cite":false,"duration_ms":21119,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53A10","58E12","58D27"],"pacs":[],"model":"deepseek-v4-flash","headline":"For closed manifolds of dimension at least eight, the paper defines an integer counting minimal immersed tori and proves the count does not change when the metric is perturbed.","keywords":["minimal tori","super-rigid metrics","Jacobi operator","transversality","metric perturbations","counting function","wall-crossing","Riemannian geometry"],"falsifier":"For the model symbol $\\Delta = \\partial_1^2 + \\partial_2^2$, take a minimal-rank homogeneous element $B \\in \\ker \\hat\\omega_\\Delta$ and compute $\\operatorname{rk} L^{\\le l}_{\\Delta,B}$ numerically for $l$ up to a few hundred; if the growth is $O(l)$ rather than $O(l^2)$, the hypothesis of the cited transversality theorem (which needs exponent $n=2$) is not satisfied, and the conclusion that Petri-condition failures form an infinite-codimension locus collapses.","tokens_in":34742,"feed_emoji":"🍩","tokens_out":10764,"duration_ms":90561,"temperature":0.7,"pith_summary":"This paper claims that minimal immersed tori in a closed Riemannian manifold of dimension at least eight can be counted by an integer that does not change when the metric is perturbed. The count is not a naive cardinality: each torus receives a weight determined by its covering degree and its type under a $\\mathbb{Z}_2$-index classification, and embedded tori under a generic super-rigid metric contribute $\\pm 1$. The paper argues that generic metrics are super-rigid, meaning every embedded minimal torus and all its multiple covers have trivial Jacobi kernel, and that failures of super-rigidity occur only along a codimension-one wall. Crossing a wall or a birth-death point changes which tori exist, but the weighted sum is shown to be unchanged, which makes the count well-defined for arbitrary metrics as well. If true, this gives a higher-dimensional analogue of counting closed geodesics.","feed_headline":"A metric-invariant count of minimal tori exists for dim ≥ 8","feed_subtitle":"Embedded tori contribute ±1; multiply covered ones get weights, so the total survives metric perturbations.","key_machinery":"The argument is carried by the Jacobi operator $J_{g,u}$, the linearization of the mean-curvature functional; along an immersed torus that factors as a covering $\\pi$ followed by an embedding $v$, the operator satisfies $J_{g,v\\circ\\pi} \\cong \\pi^*J_{g,v}$, which is equivalent to $J_{g,v}$ twisted by a flat local system. The paper verifies a technical symbol-level growth condition for this operator, proves a flexibility property by constructing metric variations that realize any symmetric endomorphism of the normal bundle as the derivative of $J$, and derives an orbifold index formula showing that twisted Jacobi operators on multiple covers have index at most $-(n-2)$. Local models near critical points and weak-limit points (double covers) reduce the invariance of the count to a finite list of type-change diagrams, from which the weight table $n^d_{\\pm k}$ is forced.","core_discovery":"The paper's central claim is that the moduli space of minimal immersed tori carries a deformation-invariant integer count. Precisely, for a closed manifold $M$ of dimension at least eight and a compact open set $U^g$ of $g$-minimal torus maps up to reparameterization, the paper defines $n(g,U^g) \\in \\mathbb{Z}$ using signs attached to the determinant of the Jacobi operator twisted by the three nonzero elements of $H^1(T,\\mathbb{Z}_2)$. The defining properties are: when $g$ is super-rigid and the torus is embedded, the contribution is $\\pm 1$; and for any path of metrics whose associated one-parameter moduli space is generic, the total count at the two endpoints is equal. The paper also claims that super-rigid metrics form a comeager set for $\\dim M \\ge 6$, with the non-super-rigid locus a codimension-one wall, so the local wall-crossing terms can be computed and cancel.","pith_inferences":["A deformation-invariant count of this kind could be regarded as a Riemannian analogue of enumerative curve counts, but the paper does not claim invariance under changing the smooth structure or complex structure.","The dimension gap between the genericity result ($\\dim M \\ge 6$) and the torus count ($\\dim M \\ge 8$) suggests that the boundary case $\\dim M = 7$ might require new local models for higher-degree branching; the paper leaves this case unexamined.","One could try to compute the invariant on explicit examples, such as flat tori or products of a circle with a sphere, where minimal tori are known, to see whether the weight table reproduces a known integer across different metrics.","If the growth-rate hypothesis is the weak point, the failure would show up in the codimension estimate for non-super-rigid metrics rather than in the local weight relations, so the counting function might be salvageable under a stronger transversality result."],"forward_implications":["A metric-invariant count of minimal tori exists whenever $\\dim M \\ge 8$, and super-rigid metrics are generic in dimensions at least 6.","For super-rigid metrics, embedded tori contribute $\\pm 1$, while higher-degree covers contribute through the weight table $n^d_{\\pm k}$, with degrees 2 and 4 governed by explicit recurrence relations.","The count extends to non-generic metrics by perturbing a given metric to a nearby super-rigid metric; the paper defines this extension and claims it is independent of the chosen perturbation.","Any path of metrics between two endpoints can be chosen so that only finitely many critical points and weak-limit points occur, making the invariant computable by summing local wall-crossing corrections."],"supporting_citations":[{"why":"Supplies the equivariant Brill-Noether transversality theorem, the Petri and flexibility conditions, and the symbol-growth framework that the paper adapts to the Jacobi operator.","marker":"[3]"},{"why":"Provides the counting-function construction and the wall-crossing local models for closed geodesics that the paper rewrites for minimal tori.","marker":"[5]"},{"why":"Supplies the classification of tori by determinant signs and the weight relations used to define the count.","marker":"[12]"},{"why":"Establishes the super-rigidity genericity result in the J-holomorphic setting whose method is imported to prove Theorem 1.1.","marker":"[13]"},{"why":"Proves that the space of embedded minimal surfaces for varying metrics is a Banach manifold, giving the implicit-function-theorem framework for the moduli space.","marker":"[15]"},{"why":"Proves that generic metrics are bumpy, providing the base genericity result that the super-rigidity theorem extends.","marker":"[16]"}],"fun_headline_variants":["Metric-perturbation invariant count of minimal tori for dim ≥ 8","Counting minimal tori survives metric changes in high dimensions","A deformation-invariant integer counts minimal tori in dim 8+","Invariant minimal torus count defined for manifolds of dim ≥ 8"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the technical growth-rate condition verified in the paper is strong enough for the transversality theorem it invokes; the paper proves only linear growth in the jet order, while the invoked theorem appears to require quadratic growth, and if that quadratic rate is genuinely necessary the genericity theorem and the count's invariance both fail.","fun_headline_variants_meta":{"raw":{"variants":["Metric-perturbation invariant count of minimal tori for dim ≥ 8","Counting minimal tori survives metric changes in high dimensions","A deformation-invariant integer counts minimal tori in dim 8+","Invariant minimal torus count defined for manifolds of dim ≥ 8"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000259,"raw_usage":{"total_tokens":1504,"prompt_tokens":779,"completion_tokens":725,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":395,"completion_tokens_details":{"reasoning_tokens":647}},"tokens_in":395,"tokens_out":725,"duration_ms":6517,"temperature":1.0,"reasoning_tokens":647,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:19:21.786585+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the model symbol $\\Delta = \\partial_1^2 + \\partial_2^2$, take a minimal-rank homogeneous element $B \\in \\ker \\hat\\omega_\\Delta$ and compute $\\operatorname{rk} L^{\\le l}_{\\Delta,B}$ numerically for $l$ up to a few hundred; if the growth is $O(l)$ rather than $O(l^2)$, the hypothesis of the cited transversality theorem (which needs exponent $n=2$) is not satisfied, and the conclusion that Petri-condition failures form an infinite-codimension locus collapses.","supporting_citations":[{"cited_title":"and Walpuski, T., Equivariant Brill-Noether theory for elliptic op- erators and super-rigidity of J-holomorphic maps, J","cited_arxiv_id":null,"evidence_quote":"Supplies the equivariant Brill-Noether transversality theorem, the Petri and flexibility conditions, and the symbol-growth framework that the paper adapts to the Jacobi operator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the counting-function construction and the wall-crossing local models for closed geodesics that the paper rewrites for minimal tori."},{"cited_title":"H., Counting pseudo-holomorphic submanifolds in dimension 4, J","cited_arxiv_id":null,"evidence_quote":"Supplies the classification of tori by determinant signs and the weight relations used to define the count."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the super-rigidity genericity result in the J-holomorphic setting whose method is imported to prove Theorem 1.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves that the space of embedded minimal surfaces for varying metrics is a Banach manifold, giving the implicit-function-theorem framework for the moduli space."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves that generic metrics are bumpy, providing the base genericity result that the super-rigidity theorem extends."}],"review_version":1}