{"id":"18cec612-7bf5-4b3e-a81c-5fbd99613d97","arxiv_id":"2411.16543","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For polarized tropical affine tori, the fibered Lagrangian cobordism group carries a parallelotope filtration that vanishes after n+1 steps, and the associated symplectic Fourier transform mirrors Mukai's transform.","lead":"A new theorem shows that the Lagrangian cobordism group of a symplectic torus built from a polarized tropical affine torus admits a geometric filtration that ends after finitely many steps, even though the group itself is infinite-dimensional. The result gives a symplectic analogue of Bloch's filtration on Chow groups of abelian varieties and comes with a Fourier transform between Fukaya categories of dual tori.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem A's proof passes through a cobordism group with immersed generators; the promised return to the embedded group is asserted but not established.","rationale":"I read the paper in good faith and agree with the reader's assessment. The main theorem is novel and the geometric strategy is plausible: reinterpret Pontryagin products as fiberwise addition, approximate flat sections by tropical rational functions, use surgery cobordisms to pass to tropical lifts, and then use transversality to make iterated fiberwise sums empty. However, the key reduction from the immersed cobordism group back to the embedded group is exactly where the proof is weakest. Lemma 3.10 is explicit that its relations hold only with immersed ends; Lemma 3.17 inherits this and still concludes 'possibly immersed' representatives; and Theorem 3.18 then treats vanishing in the immersed group as if it were vanishing in the original group. The paper flags this passage but does not prove it, and the cited Proposition 3.15 controls only transversality of tropical hypersurfaces, not the embeddedness of the composed cobordism. This is a genuine gap in the central argument, so the reader's CONDITIONAL verdict is appropriate. I do not see an alternative load-bearing flaw: the Fourier-transform part of the paper is independent of Theorem A, and Assumption 5.6 is explicitly flagged and confined to the Chow-group comparison. The concrete test above would settle whether the asserted embedded cobordism can actually be constructed.","tokens_in":29534,"tokens_out":10102,"duration_ms":109115,"concrete_test":"Fix n = 2 and take a polarized tropical affine 2-torus (e.g. the family in Example 1.13). For a generator L = (F_1^+ - F_1^-) ⊗ (F_2^+ - F_2^-) ⊗ (F_3^+ - F_3^-), write out the iterated addition correspondence L_⊗^{(3)} applied to the product of the three surgery cobordisms supplied by Proposition 2.10. Verify explicitly that the projection of this composition to X × C is an embedded Lagrangian with all ends either empty or matching the original embedded fibers; in particular check transversality of all fiber products and clean self-intersections. If the projected object has immersed ends or is only an immersed cobordism, then Theorem 3.18 is not proved as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem A (Theorem 3.18) reduces an element of Cob_fib(X)^{⋆(n+1)} to a fiberwise sum of n+1 differences of flat sections and then invokes Lemma 3.17. But Lemma 3.17 is proved only in a cobordism group whose generators may be immersed Lagrangians. Lemma 3.10 explicitly states that its relation (14) is valid 'in a cobordism group of immersed Lagrangians ... although not in our original cobordism group', and Remark 3.11 only makes the individual lifts L_{φ±} embedded, not their fiberwise products. The paragraph before Lemma 3.10 asserts that after fiberwise summing n+1 of these cobordisms one gets a valid relation in the original group, but no formal construction or proof is provided. Proposition 3.15, cited there, only arranges transversality of tropical hypersurfaces, not the embedding of the projected iterated composition. Consequently the final step of Theorem 3.18—'lives over tropical subvarieties of codimension at least n+1, hence vanishes'—proves vanishing of the image of L in the immersed group. Since the natural map from the embedded cobordism group to the immersed cobordism group can have kernel, this does not by itself imply L = 0 in Cob_fib(X). The missing piece is either an explicit embedded cobordism realizing the asserted relation, or a proof that the map from Cob_fib(X) to the immersed group is injective on the relevant filtration.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the fibered Lagrangian cobordism group Cob_fib(X) of a symplectic torus X = X(B) associated to a polarized tropical affine torus B. It defines a decreasing parallelotope filtration by powers of the ideal generated by differences F_p - F_q under the Pontryagin product, and claims Theorem A: this filtration terminates after n+1 steps. The proof strategy is to exchange Pontryagin products of fibers for fiberwise addition of flat sections of the dual fibration, replace flat sections by tropical Lagrangian sections using a polarization, and then use Hicks's surgery cobordisms to convert differences of sections into lifts of tropical hypersurfaces. Transversality of the tropical hypersurfaces is then used to conclude that n+1-fold fiberwise additions vanish. The second half constructs a symplectic Fourier transform on Fukaya categories induced by the graded symplectomorphism (q,p) -> (-p,q), proves that under Abouzaid's homological mirror symmetry it corresponds to Mukai's Fourier-Mukai transform, and connects the filtration to the Bloch filtration on Chow groups of abelian varieties.","tokens_in":29846,"tokens_out":8934,"duration_ms":92152,"significance":"If Theorem A is established, it is a genuine structural result: previous work of Sheridan-Smith and the author showed that these cobordism groups are infinite-dimensional, so a natural finite geometric filtration with understood first graded pieces is new and interesting. The Fourier transform theorem is elegant, explicit, and independent of the main gap in Section 3; identifying the symplectic Fourier transform with a graded symplectomorphism is a clean geometric statement that should be of independent value. The paper is also unusually honest about its conditional inputs: Remark 1.11 and Assumption 5.6 explicitly flag statements that have not appeared in the literature, and Remark 1.6 states that the Fourier transform material is not needed for Theorem A. The central proof, however, currently passes through a cobordism group with immersed generators without a rigorous return to the embedded group, so Theorem A is not yet established as written.","major_comments":[{"comment":"The proof of the main theorem proves a statement in the wrong cobordism group at a load-bearing step. Lemma 3.10 concludes Eq. (14) only in \"a cobordism group of immersed Lagrangians modulo immersed unobstructed cobordisms (although not in our original cobordism group)\"; Lemma 3.17 then states that its representatives are \"possibly immersed tropical Lagrangians\". The paragraph immediately before Lemma 3.10 says that after fiberwise summing n+1 of these cobordisms one obtains a valid relation in the original group, but no construction or proof of this assertion is given. Proposition 3.15 proves only transversality of the tropical hypersurfaces, not that the iterated fiberwise compositions are embedded or that a projected cobordism can be chosen embedded. Remark 3.11 ensures only that each L_{φ±} is individually embedded. Therefore the final step of Theorem 3.18—\"lives over tropical subvarieties of codimension at least n+1, hence vanishes\"—establishes vanishing of the image of L in the immersed cobordism group. Since the map from Cob_fib(X), whose generators are embedded by Definition 2.8, to the immersed cobordism group is not shown to be injective, Theorem A does not follow as written. The missing ingredient is either an explicit embedded cobordism realizing the asserted n+1-fold fiberwise sum, or a proof of injectivity of the embedded-to-immersed map on the filtration under consideration.","section":"§3.3 (Theorem 3.18), §3.1 (Lemma 3.10), §2.3 (Definition 2.8)"},{"comment":"The paper's advertised connection to the Bloch filtration is conditional rather than a proved theorem. Proposition 5.8 uses the homomorphism Cob(X) → K0(Fuk(X)) whose existence Remark 1.11 says \"has not yet appeared in the literature\", and Assumption 5.6 states an unobstructed-immersed-cobordism-to-cone-decomposition principle that the paper also says has not appeared. Neither input is needed for Theorem A or Theorem B, and the paper is transparent about this, but the abstract and §1.4 present the mirror statement to Bloch's theorem as one of the main outcomes. This should be clearly labelled as conditional, or the missing functor and assumption should be proved.","section":"§1.4, Remark 1.11; §5.1, Assumption 5.6; Proposition 5.8"}],"minor_comments":[{"comment":"The statement reads \"A tropical affine torus B is z if and only if...\"; the missing word is presumably \"polarized\".","section":"Lemma 3.4"},{"comment":"In Equation (19), the right-hand side \"a\" should presumably be the skyscraper sheaf O_a; as written it is not of the same type as the right-hand side of Equation (20).","section":"§4.1, Equations (19)–(20)"},{"comment":"The phrase \"δ± satisfying Equation (12)\" appears to be a cross-reference error: the admissibility condition on δ is Equation (10), while Equation (12) is the quasi-periodicity of f_{k,δ}.","section":"Corollary 3.8"},{"comment":"The algebraic Fourier transform, the symplectic Fourier transform, and individual Lagrangian fibers are all denoted F. This is manageable in context but could be confusing; a distinct notation for the algebraic versus the symplectic functor would improve readability.","section":"§1.3, Definition 4.6 and §1.2, Definition 1.4"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern is valid and should be the central focus of the revision. The paper is well organized and honest about its conditional assumptions, and the Fourier transform section appears sound; if the embedded-versus-immersed gap can be closed, the paper would be a strong contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper has two real results. Theorem B, the symplectic Fourier transform, is clean: the map ι is a graded symplectomorphism, the grading computation is explicit, and the identification with Mukai's transform via Abouzaid's HMS is a nice piece of work. That half deserves to be published. Theorem A, the finite parallelotope filtration, is the headline and it's a good idea—the switch from Pontryagin product on fibers to fiberwise addition on the dual fibration is elegant, and the use of tropical hypersurfaces is genuinely new. The first two graded pieces being H^0(B) and Alb(B) is a real structural observation.\n\nThe soft spot is exactly where the stress-test puts it. Lemma 3.10 proves a cobordism relation in a group where immersed Lagrangians are allowed as generators. The paper then uses Lemma 3.17 and Theorem 3.18 to conclude vanishing in the original embedded cobordism group. The paragraph after Lemma 3.10 says that after fiberwise summing n+1 such cobordisms one gets a valid relation in the embedded group, but that is asserted, not proved. Proposition 3.15 arranges transversality of the tropical hypersurfaces; it does not construct an embedded cobordism or prove that the map from the embedded group to the immersed group is injective on the relevant subgroup. Since that map can have kernel, vanishing in the immersed group does not automatically imply vanishing in Cob_fib(X). This is a load-bearing gap, not a cosmetic one.\n\nA couple of smaller points. Assumption 5.6 is flagged, and the paper is honest that it is only needed for the Chow-group comparison, so I wouldn't call that a flaw, just a limitation. The references are fair: Sheridan-Smith, Hicks, Abouzaid, Subotic are the right names, and the author says where each tool comes from. No circularity: the filtration theorem isn't assumed anywhere.\n\nWho should read this? Symplectic topologists and anyone working on mirror symmetry for tori. The Fourier transform section is a good read now. Theorem A is worth serious referee time, but only with a request to close the immersed-to-embedded gap: either an explicit embedded cobordism realizing the relation after taking n+1 fiberwise sums, or a proof that the relevant map from Cob_fib(X) to the immersed group is injective. I'd accept it for review, with the clear expectation of a major revision.","headline":"A substantial paper whose Fourier transform half is solid, but whose main filtration theorem currently rests on an unproved immersed-to-embedded passage.","tokens_in":30317,"tokens_out":3757,"would_cite":true,"duration_ms":39671,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D12","53D37","14C15","14T05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For symplectic tori built from polarized tropical affine tori, the fibered Lagrangian cobordism group carries a finite geometric filtration of length n+1.","keywords":["Lagrangian cobordism","symplectic tori","tropical affine tori","parallelotope filtration","Pontryagin product","Fourier transform","homological mirror symmetry","Chow groups"],"falsifier":"For $n=1$, Theorem A asserts that $(F_a-F_b)\\star(F_c-F_d)=0$ in the fibered Lagrangian cobordism group of the $2$-dimensional symplectic torus. Writing down the corresponding embedded Lagrangian in $X(B)\\times\\mathbb{C}$, or showing by a direct topological obstruction that no embedded Lagrangian with those ends exists, would settle whether the immersed-to-embedded reduction is valid.","tokens_in":29289,"feed_emoji":"🔁","tokens_out":16686,"duration_ms":152018,"temperature":0.7,"pith_summary":"This paper studies the group generated by Lagrangian torus fibers of a symplectic torus modulo Lagrangian cobordism, the equivalence relation generated by Lagrangian submanifolds of $X\\times\\mathbb{C}$ with prescribed cylindrical ends. For symplectic tori built from a polarized tropical affine $n$-torus, it proves that the parallelotope filtration, defined by Pontryagin products of differences of fibers, terminates after $n+1$ steps. Equivalently, any $n+1$-fold Pontryagin product of fiber differences is Lagrangian cobordant to zero. Earlier work had shown these cobordism groups are infinite-dimensional, so the result establishes that infinite generation can coexist with a finite, geometrically meaningful filtration. The paper also constructs a Fourier transform between the Fukaya categories of dual tori and shows it is mirror to the algebraic Fourier transform, connecting the filtration to a classical vanishing statement for zero-cycles on abelian varieties.","feed_headline":"Polarized tori: n+1 fiber differences vanish in cobordism","feed_subtitle":"A geometric filtration on the fibered Lagrangian cobordism group terminates after n+1 steps.","key_machinery":"The central object is the parallelotope filtration $F^i\\mathrm{Cob}_{\\mathrm{fib}}(X)=\\mathrm{Cob}_{\\mathrm{fib}}(X)_{\\mathrm{hom}}^{\\star i}$, where $\\mathrm{Cob}_{\\mathrm{fib}}(X)_{\\mathrm{hom}}$ is generated by differences of Lagrangian torus fibers and $\\star$ is the Pontryagin product induced by the group law on the base. The argument is carried by three mechanisms: the duality between the two Lagrangian torus fibrations on $X(B)$, which turns the Pontryagin product into fiberwise addition; the use of polarizations to write every constant section, up to Hamiltonian isotopy, as the graph of a tropical rational function; and Lagrangian surgery along tropical hypersurfaces, which converts differences of sections into lifts of tropical subvarieties. A transversality statement for the tropical hypersurfaces $V(f_{k,\\delta})$ ensures that an $(n+1)$-fold fiberwise sum is empty. In the second half, the key object is the graded symplectomorphism $\\iota(q,p)=(-p,q)$ between $X(B)$ and $X(B^\\vee)$, which induces the symplectic Fourier transform and is proved to be the mirror of the algebraic Fourier transform.","core_discovery":"The central claim, Theorem A, is that for a polarized tropical affine torus $B$ of dimension $n$, with $X(B)$ the associated symplectic torus, the parallelotope filtration satisfies $F^{n+1}\\mathrm{Cob}_{\\mathrm{fib}}(X(B))=0$. Here $F^i$ is the subgroup generated by $i$-fold Pontryagin products of elements $F_p-F_q$, where $F_p$ and $F_q$ are Lagrangian torus fibers. The proof reinterprets fibers of the fibration $X(B)\\to B$ as constant sections of the dual fibration $X(B)\\to F$, so the Pontryagin product becomes fiberwise addition. In the polarized case, constant sections are Hamiltonian isotopic to graphs of tropical rational functions, and differences of such sections are cobordant to lifts of tropical hypersurfaces of codimension at least one. Choosing the tropical functions generically makes the hypersurfaces transverse, so a fiberwise sum of $n+1$ of them is empty; the paper concludes that the corresponding element of the fibered Lagrangian cobordism group vanishes. A second theorem identifies the symplectic Fourier transform between Fukaya categories of dual tori with the functor induced by the graded symplectomorphism $(q,p)\\mapsto(-p,q)$, and shows that under homological mirror symmetry it matches the algebraic Fourier transform; this is then used to present the filtration as the mirror of the classical Pontryagin-power filtration on zero-cycles of an abelian variety.","pith_inferences":["The paper does not determine sharpness of the filtration length; since $F^1/F^2$ is nonzero, at least the first step carries information, but whether lower graded pieces can be nonzero for $n>1$ is left open. Computing explicit examples for $2$- and $3$-dimensional polarized tori would test this.","The main unresolved technical step is the passage from immersed intermediate Lagrangians to embedded cobordisms after summing $n+1$ of them; if that passage fails, the immersed version of Theorem A would stand while the embedded statement would not. This could be settled by writing down explicit regularizations for small $n$.","Because the symplectic Fourier transform is a symplectomorphism, it should commute with all Lagrangian cobordism invariants, not just the objects of the Fukaya category; a concrete consequence would be a coordinate-swap relation for Lagrangian Floer cohomology groups of dual tori, up to the grading shift computed in the paper.","The transversality machinery for tropical hypersurfaces is not obviously restricted to tori; it may extend to other affine manifolds with Lagrangian torus fibrations, yielding finite filtrations on cobordism groups of a broader class of mirror-symmetric spaces if the embeddedness gap can be closed."],"forward_implications":["The subgroup generated by differences of Lagrangian fibers is nilpotent under the Pontryagin product, with nilpotency index at most $n+1$, so high powers of fiber differences carry no Lagrangian-cobordism information.","The first two graded pieces are explicit: $F^0/F^1\\cong H^0(B;\\mathbb{Z})$ and $F^1/F^2\\cong \\mathrm{Alb}(B)$, the Albanese torus of the base, giving computable invariants of the otherwise infinite-dimensional group.","The symplectic Fourier transform between dual tori is induced by an honest graded symplectomorphism, not by a kernel object, so Lagrangian submanifold invariants can be transported between dual tori by a coordinate change.","Under homological mirror symmetry the filtration corresponds to the classical Pontryagin-power filtration on zero-cycles of the dual abelian variety, placing the symplectic cobordism result inside the theory of algebraic cycles.","The polarization hypothesis is essential: for non-polarizable bases the filtration can be unbounded rather than terminating, by previously known results on Lagrangian cobordism of torus fibers."],"supporting_citations":[{"why":"provides the Lagrangian surgery cobordism that turns the difference of the zero-section and the graph of a tropical polynomial into a lift of a tropical hypersurface.","marker":"[Hic20]"},{"why":"supplies the polarization-to-metric dictionary, the regularity result for the tropical functions $f_{k,\\delta}$, and the map from $\\mathrm{Cob}_{\\mathrm{fib}}(X)_{\\mathrm{hom}}$ to $\\mathrm{Alb}(B)$ used for the low graded pieces.","marker":"[SS21]"},{"why":"supplies the tropical intersection theory used to ensure that transverse intersections of tropical hypersurfaces remain tropical subvarieties.","marker":"[Mik06]"},{"why":"gives the homological mirror symmetry equivalence used to compare the symplectic Fourier transform with the algebraic Fourier transform and to relate the filtration to Chow groups.","marker":"[Abo21]"},{"why":"introduces the algebraic Fourier transform on derived categories of dual abelian varieties that the symplectic construction is mirror to.","marker":"[Muk81]"},{"why":"provides the uniqueness criterion for integral Fourier equivalences by their action on skyscraper sheaves, used to pin down the symplectic Fourier transform.","marker":"[Huy06]"},{"why":"identifies the rigid-analytic dual of the torus $Y(B)$ with $Y(B^\\vee)$, which matches dual abelian varieties with dual tropical tori.","marker":"[Fos+18]"},{"why":"defines polarizations of tropical affine tori, the class of bases for which the main filtration theorem is stated.","marker":"[MZ08]"}],"fun_headline_variants":["Filtration on torus cobordism group truncated at n+1","Fourier transform bridges dual symplectic tori","Lagrangian cobordism filtration terminates after n+1 layers","Mirror to Bloch filtration: torus cobordism group filtered","Polarized tori: cobordism filtration ends in finite depth"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the vanishing in the embedded cobordism group relies on the assertion that, although the intermediate cobordisms have immersed ends, fiberwise summing $n+1$ of them produces a valid relation in the original group; the paper states this but does not supply a formal embedding or gluing argument.","fun_headline_variants_meta":{"raw":{"variants":["Filtration on torus cobordism group truncated at n+1","Fourier transform bridges dual symplectic tori","Lagrangian cobordism filtration terminates after n+1 layers","Mirror to Bloch filtration: torus cobordism group filtered","Polarized tori: cobordism filtration ends in finite depth"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000225,"raw_usage":{"total_tokens":1486,"prompt_tokens":987,"completion_tokens":499,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":603,"completion_tokens_details":{"reasoning_tokens":409}},"tokens_in":603,"tokens_out":499,"duration_ms":5895,"temperature":1.0,"reasoning_tokens":409,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:00:24.202836+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $n=1$, Theorem A asserts that $(F_a-F_b)\\star(F_c-F_d)=0$ in the fibered Lagrangian cobordism group of the $2$-dimensional symplectic torus. Writing down the corresponding embedded Lagrangian in $X(B)\\times\\mathbb{C}$, or showing by a direct topological obstruction that no embedded Lagrangian with those ends exists, would settle whether the immersed-to-embedded reduction is valid.","supporting_citations":[],"review_version":1}