{"id":"7be535d9-5504-44bf-ac34-5384e270edd0","arxiv_id":"2505.07228","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For mirrors of toric weak Fano manifolds, a Lagrangian section is Hamiltonian isotopic to a special Lagrangian if and only if certain phase/slope inequalities hold, proving a toric form of Thomas-Yau.","lead":"This paper proves a toric version of the Thomas-Yau conjecture: a Lagrangian section of a mirror Calabi-Yau fibration can be moved by Hamiltonian isotopy to a special Lagrangian exactly when a slope inequality in the Fukaya-Seidel category holds. The result links symplectic geometry, algebraic stability, and a PDE criterion, giving a new tool for detecting special Lagrangians.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The if-and-only-if in Theorem 1.2 depends on uniform asymptotic control in the complete-intersection induction (§5.2.2–5.2.3) that is only sketched and not verified.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing issue: the proof must choose k >> kV >> 1 uniformly over all toric subvarieties and convert non-strict leading-order inequalities into strict phase inequalities, with the complete intersection induction only sketched. My stress-test agrees and sharpens the point: the induction in §5.2.2–5.2.3 must also control all remainder terms and the cone orientations so that the Γ(SV) period has the correct phase and sign for every codimension. The divisor case is explicit and convincing, and the cited external theorems (Chu-Li-Takahashi, Iritani, Fang) are not in question. The concern is therefore about the completeness of the reduction inside this paper, not about its framework or conclusions. Since the reader already returned CONDITIONAL and this concern supports that verdict, no change is needed. The proposed test—writing out the r = 3 induction with explicit bounds—would settle whether the gap is merely expository or affects correctness.","tokens_in":34879,"tokens_out":21870,"duration_ms":200675,"concrete_test":"Complete the induction for the first non-divisor case, V = V1 ∩ V2 ∩ V3 with n = 3, by writing out the object SV from the iterated cone construction and the full expansion for ∫_V e^{-iω0} ch(L) with every remainder explicitly bounded in the regime k, k̂ >> k1, k2, k3 >> 1. Verify (a) the leading term carries the exact sign (-1)^{codim V} and factor (2π i)^{-n} k^r k̂^{-n+1} (∏ k_i) ∫_{Γ(SV)} e^{-W}; and (b) each remainder is O(k^{-1}) + O(k_i^{-1}) relative to that leading term, uniformly in the subvariety. If either fails, Theorem 2.5 is not established for codimension-3 toric subvarieties and the iff in Theorem 1.2 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim reduces dHYM positivity, via Iritani's Gamma theorem, to the phase inequalities (1.3). The divisor case in §5.2.1 is written out and is plausible: equation (5.7) isolates the subleading term and the genericity assumption of Remark 2.6(v) upgrades a non-strict leading-order inequality to a strict one. The general toric subvariety, however, is handled by an induction over complete intersections that is only sketched. The displayed formulas (5.11), (5.12), and the final display of §5.2.3 involve iterated cones whose K-theory classes are differences of two objects; the sign, the factors k_i^{-1}, and the hierarchy k, k̂ >> k1, ..., kr >> 1 are asserted but the remainder terms are not shown to be uniformly dominated by the displayed leading term. The equivalence in Theorem 2.5 and hence the 'if and only if' in Corollary 2.9/Theorem 1.2 requires that, for every toric subvariety V, the phase of the Γ(SV) period reproduce the phase of ∫_V e^{-iω0} ch(L) up to o(1) with a uniform choice of constants. If for some high-codimension V the error terms are not uniformly small, the converse step—deriving the leading-order non-strict inequality and then strictness by genericity—can fail, and the claimed equivalence would not follow. The paper explicitly marks the divisor case as the one worked out and the complete intersection case as induction, so this is a genuine gap rather than a matter of disagreement with existing results.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a toric version of the Thomas-Yau conjecture. For a toric weak Fano manifold X with a fixed torus-invariant Kähler form, a SYZ Lagrangian section mirror to a high power L^k of a line bundle is claimed to be Hamiltonian isotopic to a supercritical special Lagrangian section if and only if certain phase inequalities hold for periods attached to all toric subvarieties. The proof reduces this statement to the supercritical dHYM Nakai-Moishezon criterion of Chu-Li-Takahashi, then uses Iritani's toric Gamma theorem and toric homological mirror symmetry in the sense of Fang-Liu-Treumann-Zaslow to translate dHYM positivity into phase inequalities for periods. Applications are given to Bridgeland stability for SYZ sections on toric weak del Pezzo surfaces, to the unstable case via weak dHYM solutions, to a higher-rank analogue, and to a weaker statement for general projective toric manifolds.","tokens_in":35148,"tokens_out":3787,"duration_ms":43636,"significance":"If the main theorem is established, it is a substantial result: it gives a concrete, numerically checkable criterion for existence of special Lagrangian representatives in a natural toric setting, and it makes contact with several strands of current research: the dHYM Nakai-Moishezon criterion, the Gamma conjecture, homological mirror symmetry, and Joyce-Li stability conjectures. A notable strength of the paper is its clean reduction: the argument contains no fitted parameters or ad-hoc numerical inputs, and the main claim is reduced to independently established external theorems. The surface application is also valuable, and the paper is careful to flag where it relies on the Arcara-Miles conjecture. The principal risk is not circularity but the quantitative asymptotic control in the complete-intersection induction; if that gap is filled, the main theorem appears credible and significant.","major_comments":[{"comment":"The proof of Theorem 2.5 for general toric submanifolds is not complete. The divisor case in §5.2.1 is written out and gives a plausible leading-order comparison, but the induction for complete intersections is only sketched. In the displays leading to (5.11) and (5.12), and in the final display of §5.2.3, remainder terms are asserted to be dominated by the leading term, but no uniform bound is supplied. More specifically, the proof must show that after dividing by the displayed prefactors, the error terms are o(1) uniformly over all toric submanifolds V and over the choices k, \\hat k >> k_1, ..., k_r >> 1. This uniformity is load-bearing: the if-and-only-if statement of Theorem 2.5 and hence of Corollary 2.9/Theorem 1.2 requires that a leading-order non-strict inequality can be upgraded to a strict phase inequality by the genericity of [ω0], and that this happens for every toric submanifold simultaneously. Without a completed estimate, the converse direction of the main theorem is not established.","section":"§5.2.2–5.2.3"},{"comment":"The objects appearing in the complete-intersection step are K-theory differences of two complexes, and the period integral is a difference of two Gamma-cycle integrals. The phase of a difference of two complex numbers is not determined by the phases of the individual terms unless one has a precise bound on their relative sizes. In (5.11), the right-hand side is a difference of two oscillatory integrals, and in (5.12) this is replaced by the integral over a cone object. The paper states that the natural choice h = k_2 gives the displayed leading term, but it does not show that the difference of the two Γ-integrals is approximated by the leading term with a sign that is uniform in all parameters. This is not a cosmetic issue: the claimed equivalence of dHYM positivity with the phase inequalities for higher-codimension toric subvarieties depends on exactly this approximation.","section":"§5.2.2, Eq. (5.11)–(5.12)"},{"comment":"Even in the divisor case, the conversion from dHYM positivity to the phase inequalities uses an approximate equality rather than an exact one. The proof correctly identifies the leading-order term and uses genericity to rule out equality, but the strictness argument should be stated more carefully: from the inequality (5.9) one obtains only a non-strict inequality of leading-order phases, and then genericity of [ω0] is used to exclude equality. This step is valid only if the error terms after division by the prefactor are uniformly small for all divisors and all line bundles in a bounded K-theory class. The paper asserts this uniformity but does not give the required bounds; a precise statement of the error estimates would strengthen the proof considerably.","section":"§5.2.1, Eq. (5.7) and Remark 5.5"}],"minor_comments":[{"comment":"In the display after the sentence beginning \"is equivalent to\", the right-hand side is written as an integral without \"arg\"; it should read \"arg ∫ Γ(L^{⊗k\\tilde k}) ...\".","section":"§5.2.2"},{"comment":"There is a typo in \"in the uion of finitely many proper analytic subvarieties\"; it should be \"union\".","section":"§5.2.2"},{"comment":"\"Hamiltionian\" should be \"Hamiltonian\".","section":"Introduction, Theorem 1.9"},{"comment":"The display referenced as \"the expansion (5.2.2)\" is not numbered; the reference is confusing and should be fixed.","section":"§5.2.2"},{"comment":"The derivation of the phase inequalities assumes that the relevant periods do not vanish and that arg is chosen continuously; it would be helpful to state this convention explicitly when first introducing the inequalities (1.3).","section":"§3.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about its conditional statements, and the main concern is not circularity or reliance on the target result. The referee's central objection is the incompleteness of the uniform asymptotic analysis for complete intersections in §5.2.2–5.2.3. This is a fixable but genuine gap in the proof of the main if-and-only-if statement. If the author supplies a rigorous induction with explicit uniform remainder estimates, the paper would likely be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First: the paper is worth taking seriously. Stoppa proves a new toric form of Thomas-Yau: for SYZ Lagrangian sections mirror to line bundles on toric weak Fano manifolds, being Hamiltonian isotopic to a supercritical special Lagrangian section is equivalent to a family of phase inequalities on mirror objects LV[−codim V]→L. The reduction runs through Iritani's Gamma theorem and toric HMS, landing on the Chu-Li-Takahashi Nakai-Moishezon criterion for dHYM. That is a real bridge, not a repackaging: nobody had made the equivalence between dHYM positivity and Fukaya-Seidel phase inequalities before. The surface application to Bridgeland stability (Theorem 1.9) and the unstable-case construction (Theorem 1.12) are also substantive.\n\nThe paper is honest about what is conditional. The surface converse relies on the Arcara-Miles conjecture, known for Bl_p P2 and Bl_{p,q} P2; the author says so. Remark 2.11 correctly notes that a direct comparison with Li's inequality is not immediate because the SYZ representative is noncompact, which slightly qualifies the abstract's 'agrees with Li'. That is a minor overstatement, not a defect in the mathematics.\n\nThe soft spot is the one the stress-test note identifies. §5.2.1 gives a full asymptotic argument for toric divisors. §5.2.2–5.2.3 handle complete intersections by induction, with formulas like (5.11) and (5.12) for iterated cones, but the error terms are not shown to be uniformly dominated by the displayed leading term. The equivalence in Theorem 2.5 needs, for every toric subvariety V, the Gamma-theoretic period of Γ(SV) to track the period of ∫_V e^{-iω0}ch(L) up to o(1) with a uniform choice of k >> kV >> 1. If some high-codimension error is not uniformly small, the converse—upgrading a non-strict leading-order inequality using genericity—could fail. The paper explicitly says the divisor case is worked out and the rest is induction, so this is a genuine gap in the written proof, not a hidden circularity or a disagreement about known theorems. My guess is the gap is fillable; the structure is clear and the divisor case shows the method. But it is exactly the kind of estimate a referee should ask to see in full.\n\nWho is it for: anyone working on SYZ, special Lagrangians, dHYM, or Fukaya-Seidel stability. It deserves a serious referee. I would send it out, with the request that the complete-intersection induction be written out carefully.","headline":"A substantial new reduction of toric Thomas-Yau to dHYM positivity; the proof is complete for divisors and sketched for complete intersections, which is the one real gap.","tokens_in":35723,"tokens_out":2529,"would_cite":true,"duration_ms":23904,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J32","14J45","53C38","53D37","53D12"],"pacs":[],"model":"deepseek-v4-flash","headline":"For toric mirrors of weak Fano manifolds, a SYZ Lagrangian section is Hamiltonian isotopic to a supercritical special Lagrangian exactly when the phase inequalities for all toric subvarieties hold.","keywords":["toric weak Fano","SYZ transform","special Lagrangian","Thomas-Yau conjecture","deformed Hermitian Yang-Mills","Nakai-Moishezon criterion","Bridgeland stability","Fukaya-Seidel category"],"falsifier":"Compute both sides of the phase inequality (1.3) to next-to-leading order in $k$ for a toric weak del Pezzo surface, for example the blow-up of $\\mathbb{P}^2$ at one point with the Kähler class of Remark 1.11, and compare the sign of the phase difference with the sign of the corresponding dHYM Nakai-Moishezon integral (2.3). The central claim predicts exact agreement of signs for all sufficiently large $k$; a single toric curve $V$ where the signs disagree for arbitrarily large $k$ would refute the theorem. A second check is to put $[\\omega_0]$ on one of the analytic subvarieties excluded in Remark 2.6(v) and verify that the strict inequalities fail exactly when a toric subvariety is strictly semi-stabilising in (5.10).","tokens_in":34614,"feed_emoji":"🌀","tokens_out":15594,"duration_ms":140325,"temperature":0.7,"pith_summary":"The paper establishes a toric form of the Thomas-Yau conjecture: on the mirror of a toric weak Fano manifold, a SYZ Lagrangian section is Hamiltonian isotopic to a supercritical special Lagrangian section precisely when a list of phase inequalities holds, one for every toric subvariety. Those inequalities compare the argument of the period of the section with the arguments of periods of mirror objects $L_V$ attached to subvarieties through morphisms $L_V[-\\mathrm{codim}\\,V]\\to L$. The proof reduces the statement to the Nakai-Moishezon criterion for supercritical deformed Hermitian Yang-Mills connections via the SYZ transform, the toric Gamma-class theorem, and toric homological mirror symmetry. As an application, on mirrors of toric weak del Pezzo surfaces, a naturally defined Bridgeland stable Lagrangian section is Hamiltonian isotopic to a special Lagrangian section, with partial converses; unstable sections produce weak special Lagrangian solutions at the minimal phase. The result converts a PDE existence question into a cohomological slope condition, exactly as the Thomas-Yau programme predicted.","feed_headline":"Special Lagrangians exist exactly when stability phases align","feed_subtitle":"A toric proof ties Thomas-Yau to a Nakai-Moishezon criterion for deformed Hermitian Yang-Mills.","key_machinery":"The central mechanism is the Gamma-class integral transform $\\Gamma:K_0(X)\\to H_n(Y_{q_k},\\{\\mathrm{Re}(W(k\\omega_0))\\gg 0\\};\\mathbb{Z})$ supplied by the toric Gamma-class theorem, combined with the SYZ transform. It converts integrals of $\\mathrm{ch}(E)$ against $e^{-ik\\omega_0}$ into periods $\\int_{\\Gamma(E)} e^{-W(k\\omega_0)}\\Omega_0$ up to controlled powers of $k$; a scaling analysis as $k\\to\\infty$ isolates the contribution of a toric subvariety $V$ by looking at $L^{\\otimes k}(k_V V)$ and taking the cone in the exact sequence $0\\to L^{\\otimes k}\\to L^{\\otimes k}(k_V V)\\to S_V\\to 0$. The resulting objects $S_V$ satisfy $S_V[-\\mathrm{codim}\\,V]\\to L^{\\otimes k}$ and are used to rewrite the dHYM Nakai-Moishezon inequalities as the phase inequalities for periods.","core_discovery":"On the paper's own terms, the central claim is Theorem 1.2 (Corollary 2.9): for $X$ a toric weak Fano manifold with fixed torus-invariant Kähler form $\\omega_0$, $L$ a holomorphic line bundle with $L^\\vee$ ample, and $k$ sufficiently large, the SYZ Lagrangian section $L=L(L^{\\otimes k},h)$ in the mirror to $(X,k\\omega_0)$ is Hamiltonian isotopic to a supercritical special SYZ Lagrangian section if and only if, for every toric subvariety $V$, $$\\arg\\left((-1)^{\\mathrm{codim}\\,V}\\int_{[L_V]} $e^{{-W(k\\omega_0)}}$\\Omega_0\\right) < \\arg \\int_{[L]} $e^{{-W(k\\omega_0)}}$\\Omega_0,$$ where $L_V$ are mirror objects with morphisms $L_V[-\\mathrm{codim}\\,V]\\to L$. When $X$ is a toric weak del Pezzo surface, the supercritical assumption can be dropped. The proof shows this phase condition is equivalent, through the Gamma-class mirror transform, to the Nakai-Moishezon criterion for a supercritical deformed Hermitian Yang-Mills connection on $L^\\vee$.","pith_inferences":["A plausible extension the paper does not pursue: the same leading-order comparison should work for any compact toric manifold with a Gamma-class theorem, with the missing ingredient being a categorical interpretation of the complex cycles in Theorem 2.20.","The strictness in the phase inequalities is probably not removable: equality should correspond to semistable-but-not-stable objects, so non-generic Kähler classes should produce sections that are semistable but not special, a toric analogue of wall-crossing.","A quantitative check: the gap between the two arguments in (1.3) has a definite size at order $O(k^{-1})$; computing it on a del Pezzo surface would give a numerical Thomas-Yau threshold and could be compared with known dHYM wall-chamber decompositions.","For higher rank, the two phase inequalities with a B-field suggest that special Lagrangian multi-sections should be governed by the same slope conditions, with the sub-bundle and curve-restriction morphisms as destabilisers; the example of Section 8.2 is a test case where dHYM solvability is not yet known."],"forward_implications":["On toric weak del Pezzo surfaces, a naturally defined Bridgeland stable Lagrangian section $L[1]$ is Hamiltonian isotopic to a shifted special Lagrangian section; the converse holds on the blow-ups of $\\mathbb{P}^2$ at one or two points, and conditionally on the stated destabilisation conjecture in general.","When the phase inequalities fail, the Lagrangian is Bridgeland unstable and there is a morphism to a weak solution of the special Lagrangian equation with minimal phase, providing a toric counterpart of the conjectured unstable case.","For an arbitrary projective toric manifold, a weaker analogue holds in which the phase inequalities involve complex cycles with strictly nested support, though these cycles are not yet known to come from exact triangles.","For rank-two bundles on toric weak del Pezzo surfaces, higher-rank dHYM positivity with a B-field implies the same kind of phase inequalities for periods, and violations force Bridgeland instability of the mirror object.","The common rescaling invariance of the dHYM equation and the uniform choice of $k$ mean the theorem applies simultaneously to all sufficiently high tensor powers of $L$, with $k$ uniform over bounded sets in K-theory."],"supporting_citations":[{"why":"Supplies the Nakai-Moishezon criterion: supercritical dHYM solvability is equivalent to the integral positivity condition (2.3).","marker":"[8]"},{"why":"Provides the toric Gamma-class isomorphism used to convert Chern character integrals into oscillatory periods.","marker":"[21]"},{"why":"Shows compatibility of the Gamma-class isomorphism with the SYZ transform and identifies integration cycles with Lagrangian sections.","marker":"[18]"},{"why":"Establishes toric homological mirror symmetry identifying the bounded derived category with the Fukaya-Seidel category.","marker":"[19]"},{"why":"Gives the local correspondence equating the special Lagrangian condition for the SYZ section with the dHYM equation on the dual bundle.","marker":"[29]"},{"why":"Gives the phase-inequality formulation of the Thomas-Yau programme for exact triangles that the theorem realizes.","marker":"[30]"},{"why":"Supplies the Bridgeland-stability/dHYM dictionary on surfaces used for the del Pezzo converse and for the unstable case.","marker":"[11]"}],"fun_headline_variants":["Stability phase condition yields special Lagrangian isotopy","Toric Thomas-Yau solved via Nakai-Moishezon criterion","Special Lagrangians exist exactly under stability phases","Mirror symmetry proves Thomas-Yau for toric fibrations","Stable objects imply special Lagrangian sections in toric mirrors"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that one large rescaling constant can be chosen to work simultaneously for all toric subvarieties and that, for generic Kähler classes, the approximate comparison of leading-order quantities is always strict; if these fail, the claimed equivalence breaks.","fun_headline_variants_meta":{"raw":{"variants":["Stability phase condition yields special Lagrangian isotopy","Toric Thomas-Yau solved via Nakai-Moishezon criterion","Special Lagrangians exist exactly under stability phases","Mirror symmetry proves Thomas-Yau for toric fibrations","Stable objects imply special Lagrangian sections in toric mirrors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000317,"raw_usage":{"total_tokens":1877,"prompt_tokens":1113,"completion_tokens":764,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":729,"completion_tokens_details":{"reasoning_tokens":683}},"tokens_in":729,"tokens_out":764,"duration_ms":7415,"temperature":1.0,"reasoning_tokens":683,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:22:07.634239+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute both sides of the phase inequality (1.3) to next-to-leading order in $k$ for a toric weak del Pezzo surface, for example the blow-up of $\\mathbb{P}^2$ at one point with the Kähler class of Remark 1.11, and compare the sign of the phase difference with the sign of the corresponding dHYM Nakai-Moishezon integral (2.3). The central claim predicts exact agreement of signs for all sufficiently large $k$; a single toric curve $V$ where the signs disagree for arbitrarily large $k$ would refute the theorem. A second check is to put $[\\omega_0]$ on one of the analytic subvarieties excluded in Remark 2.6(v) and verify that the strict inequalities fail exactly when a toric subvariety is strictly semi-stabilising in (5.10).","supporting_citations":[{"cited_title":"Chu, M.-C","cited_arxiv_id":null,"evidence_quote":"Supplies the Nakai-Moishezon criterion: supercritical dHYM solvability is equivalent to the integral positivity condition (2.3)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the toric Gamma-class isomorphism used to convert Chern character integrals into oscillatory periods."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows compatibility of the Gamma-class isomorphism with the SYZ transform and identifies integration cycles with Lagrangian sections."},{"cited_title":"Fang, C.-C","cited_arxiv_id":null,"evidence_quote":"Establishes toric homological mirror symmetry identifying the bounded derived category with the Fukaya-Seidel category."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the local correspondence equating the special Lagrangian condition for the SYZ section with the dHYM equation on the dual bundle."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the phase-inequality formulation of the Thomas-Yau programme for exact triangles that the theorem realizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Bridgeland-stability/dHYM dictionary on surfaces used for the del Pezzo converse and for the unstable case."}],"review_version":1}