{"id":"4851d184-cfa6-4159-9777-7b471a4428d6","arxiv_id":"2505.00910","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves that smooth hypersurfaces in complex projective space of degree a > 2n+1 have a Fukaya category equivalent to a category of equivariant matrix factorizations.","lead":"This paper claims to prove homological mirror symmetry for every smooth high-degree projective hypersurface: the symplectic category built from Lagrangian submanifolds matches an algebraic category of matrix factorizations. A specialist would read it for a new bridge from wrapped Fukaya categories to the Fukaya category of the boundary at infinity.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof asserts fullness of the correspondence functor Phi without proof ('defined using the Yoneda embedding'); this surjectivity is the load-bearing step, and no theorem in [Fuk] is cited for it.","rationale":"Reader's weakest assumption identifies the same step: the functor Phi and its fullness. I agree that this is the load-bearing gap. The existence of Phi is plausibly obtained from [Fuk] with routine modifications since C is compact, so I do not rest the concern there; the explicit assertion 'Phi is full since it is defined using the Yoneda embedding' is the vulnerable point. Fullness of a correspondence functor is a geometric property of quilted counts, not a formal consequence of Yoneda, and the paper gives no proof or citation for it. The dimension equality (1.22) would only complete essential surjectivity after full faithfulness of Psi is established; it cannot supply the missing surjectivity on Hom spaces. Since the reader already conditioned the verdict on this concern, my stress-test does not move the verdict: it remains CONDITIONAL until the fullness claim is either proved or replaced by a direct computation. I see no basis for rejection: the overall strategy is coherent and the remaining steps (the graded scalar argument, the mirror computation of morphisms, and the Hochschild dimension count) are checkable. A concrete test would be to verify full faithfulness on the finite split-generating set.","tokens_in":6056,"tokens_out":16596,"duration_ms":182767,"concrete_test":"Check whether [Fuk] contains a theorem asserting that an unobstructed compact Lagrangian correspondence induces a full A-infinity functor in a Liouville setting. If it does not, perform the following computational check on the split-generators: fix the distinguished basis {L_i} of Lefschetz thimbles in U and {V_i} of vanishing cycles in X. Compute the induced map on morphisms Phi: Hom^*_{W(U)}(L_i,L_j) -> Hom^*_{F(X)}(V_i,V_j). The domain is computed in the paper as a free k[s^a]-module of rank 1 or 2; compute the target independently from Seidel's long exact sequence for the vanishing cycles in the degree-a hypersurface X. Surjectivity of Phi on this finite generating set is exactly the fullness claim; if any morphism is not hit, the proof of Theorem 1.1 fails at this step.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The theorem stands on the unproved fullness of the A-infinity functor Phi: W(U) -> F(X) in (1.8). The paper says: 'The functor Phi is full since it is defined using the Yoneda embedding.' That is not a mathematical argument in this context. Fullness of a Lagrangian-correspondence functor is the statement that the quilted counts HF*(L,K) -> HF*(C o L, C o K) are surjective; Yoneda's lemma only gives fullness for an embedding of a category into its own category of modules, not for a geometric functor between Fukaya categories. No theorem in [Fuk] is identified that proves fullness for the Liouville correspondence C, and the paper cites [Fuk] only for the existence of the functor. If Phi is not full, the subsequent localization argument collapses: the map from S(U) tensor_{k[s]} k[s]/(s^a - lambda) equivalent to mf([A^{n+2}/G], tilde w) to F(X) would not be known to be an equivalence, and the computation of Phi(s_L^a) as a scalar cannot repair a missing surjectivity on Hom spaces. This is not an internal contradiction, but it is the point where the proof most needs a supporting theorem or computation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves (Theorem 1.1) homological mirror symmetry for every smooth degree-a hypersurface X ⊂ CP^{n+1} with a > 2n+1, identifying the balanced, Z/2-graded, idempotent-complete Fukaya category F(X) with the equivariant matrix-factorization category mf([A^{n+2}/G], w − x_1⋯x_{n+2}), where w = Σ x_i^a and G is the finite group in (1.3). The strategy is to choose a smooth degree-a hypersurface Y ⊃ X and study the affine complement U = Y∖X, a Liouville manifold whose contact boundary C is an S^1-bundle over X. The author argues that C is an unobstructed Lagrangian correspondence in U × X: the Maslov/dimension bound (1.7) forces the obstruction m0(1) to vanish, so by [Fuk] the correspondence induces an A∞ functor Φ: W(U) → F(X) (with a one-sentence Liouville generalization). Since compact Lagrangians in U are disjoint from C, Φ descends to the stable Fukaya category S(U) = W(U)/F(U). Mirror-side inputs [LU, Thm 4.1] and [LU, §5.1] identify W(U) and S(U) with equivariant matrix-factorization categories, in which the thimbles L_i have explicitly computed Hom spaces (free modules over k[s^{±a}] of rank one or two). A grading argument gives Φ(s_L^a) = λ·id_V with λ ∈ k^*, so Φ descends to S(U) ⊗_{k[s^a]} k[s^a]/(s^a − λ) ≅ mf([Spec S/G], w̃).","tokens_in":6287,"tokens_out":34265,"duration_ms":331728,"significance":"If the proof can be completed, this is a major result: homological mirror symmetry for all smooth projective hypersurfaces of sufficiently high degree, obtained by a uniform argument. The mechanism — passing from the wrapped Fukaya category of an affine complement to the Fukaya category of its contact boundary at infinity via a Lagrangian correspondence — is attractive and likely reusable beyond this example. The paper includes several concrete and checkable computations that deserve explicit credit: the Maslov/dimension bound (1.7), the explicit Hom computation in S(U) on the mirror side, and the Hochschild dimension count (1.28)–(1.35), which matches equivariant matrix-factorization Hochschild cohomology against Griffiths' primitive-cohomology count and is independent of the construction of Φ. The dependence on [LU, Thm 4.1] and [LU, §5.1] is not, on its face, circular: those statements concern Brieskorn–Pham Milnor fibers and do not by themselves imply the target equivalence.","major_comments":[{"comment":"The statement 'The functor Φ is full since it is defined using the Yoneda embedding' is the load-bearing step of the paper and is not a valid argument. The Yoneda embedding is full and faithful as an embedding of a category into its category of right modules, but Φ is a geometric A∞ functor between Fukaya categories induced by the Lagrangian correspondence C; its fullness is the statement that the quilted/disk counts induce surjections Hom_W(U)(L,K) → Hom_F(X)(C∘L, C∘K), and this does not follow from the Yoneda lemma. No theorem in [Fuk] is identified that proves fullness of a correspondence functor in this setting, and the papers [WW10, MWW18, LL13] are cited only for the existence of the underlying structures. If Φ (or its descent Ψ) is not full, the map from S(U) ⊗_{k[s^a]} k[s^a]/(s^a − λ) ≅ mf([Spec S/G], w̃) to F(X) is not known to be full, the 'full and faithful' conclusion collapses, and the Hochschild dimension check at the end cannot repair a missing surjectivity on Hom spaces. The author should either prove fullness for the relevant pairs of thimbles (for instance, by computing the maps HF(L_i,L_j) → HF(V_i,V_j) explicitly in the coordinates of (1.16) and the identification (1.13)–(1.14)) or provide a precise citation to a theorem establishing fullness of correspondence functors in the Liouville setting.","section":"§1, p. 3, sentence after the Hom computation"},{"comment":"The existence of Φ: W(U) → F(X) is cited to [Fuk], an unpublished preprint developed in the compact setting, and the extension to the Liouville setting is dispatched with the sentence 'the generalization to the Liouville setting is straightforward, since C is compact.' This is a substantial black box: the source is the wrapped Fukaya category of a Liouville manifold, whose objects are noncompact Lagrangians and whose A∞ structure involves Hamiltonian functions growing at infinity, and the compatibility of the correspondence construction with that wrapped structure — including the asserted vanishing on the compact subcategory F(U) used in (1.9) — is not spelled out. Since [Gaoa] and [Gaob] are also preprints, the reader cannot verify this step from the published record. The paper should state precisely which theorem in which reference constructs a filtered A∞ functor from a wrapped Fukaya category to another Fukaya category from a compact Lagrangian correspondence, or should provide the argument.","section":"§1, Eq. (1.8)"},{"comment":"The passage from the scalar computation Φ(s_L^a) = λ·id_V to the conclusion that 'Φ induces a full and faithful functor' from the localized category is not written out and is incomplete as it stands. Fullness of the descended functor inherits from the unproved fullness of Φ (see the first major comment), while faithfulness requires showing that the rank-one (respectively rank-two) free generators of Hom_S(U)(L_i,L_j) computed on the mirror side do not map to zero or to proportional classes in HF(V_i,V_j); this injectivity is not demonstrated. In addition, the independence of λ from L is justified by the sentence 'any pair of vanishing cycles can be connected by a chain of vanishing cycles with non-trivial Floer cohomologies between them', which is only a sketch: the connecting morphisms need to be specified with their degrees, and the naturality argument requires the corresponding source-side morphisms in S(U) to exist and to map to nonzero classes. Please provide a complete proof of the full-and-faithfulness of the descended functor.","section":"§1, final paragraph (p. 3)"},{"comment":"Several load-bearing inputs are cited to the author's companion works [LU, Theorem 4.1], [LU, Section 5.1], and [LU22, Theorem 6.11], which are arXiv preprints. In particular, the quasi-equivalence (1.13), the identification S(U) ≅ W(U) ⊗_{k[s^a]} k[s^{±a}], and the mirror statement about the monodromy natural transformation are the bridge between the geometric stable Fukaya category and the matrix-factorization categories used throughout the rest of the proof, and I cannot verify these inputs from the present text. This is a verifiability concern about load-bearing inputs, not an accusation of circularity: the quoted statements do not by themselves imply the target equivalence, and the final Hochschild check in (1.22) is a separate computation. The author should either state the exact theorems being used, with all hypotheses and grading conventions, or indicate precisely where these results have appeared in refereed form.","section":"§1, Eqs. (1.13)–(1.14), (1.19), (1.22)"}],"minor_comments":[{"comment":"The inequality 2 − μ(β) ≥ 2 + 2(a − n − 2) is not readable without the Maslov index convention: with the standard convention μ(β) is nonnegative, so the displayed inequality appears to force a negative lower bound on μ(β). Please specify the convention used for the Maslov index of disks with boundary on C and rewrite (1.7) so that the sign convention is explicit.","section":"§1, Eq. (1.7)"},{"comment":"The word 'full' should be 'cohomologically full' throughout, with the relevant Hom complexes and degrees made explicit; the geometric meaning in terms of surjectivity of the induced maps on Floer cohomology should be stated.","section":"§1, p. 3"},{"comment":"Please specify the character χ and the grading conventions used in (1.28)–(1.29), and state explicitly why the invariant part of the Jacobi ring is spanned by the classes (x_1⋯x_{n+2})^i for i = 0,…,n; the factors a in the relations a x_i^{a−1} − ∏_{l≠i} x_l require a normalization that is not discussed in the text.","section":"§1, Eq. (1.28)–(1.29)"},{"comment":"The grading degree t to which each summand in (1.28) contributes is only implicit (through the condition t − dim N_γ = 2); the author should state explicitly that the nontrivial γ-summands described after (1.32) lie in HH^2, since the reader otherwise has to reconstruct this to compare with the Poincaré-polynomial content of (1.22).","section":"§1, Eq. (1.28)"},{"comment":"The phrase 'of general type' is never justified in the text; it follows from K_X = O_X(a − n − 2), which is ample under (1.1), and this should be stated for the reader's convenience.","section":"Title and abstract"},{"comment":"The numbering and hypotheses of [Gan, Theorem 3] and of the results cited from [San21] should be checked against the current arXiv versions, and the hypotheses used in applying [Gan, Theorem 3] (such as the relevant smoothness, properness, and the meaning of dim QH^*(X)) should be stated in the text.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"To the editor: this is a very short note (five pages) supporting a major claim, and its load-bearing inputs are (i) an unpublished preprint of Fukaya [Fuk] with an asserted but unproved Liouville generalization, and (ii) several results from the author's own companion preprints ([LU, Thm 4.1; §5.1] and [LU22, Thm 6.11]). I recommend asking the author to disclose the relationship between this paper and [LU] and to confirm whether the companion results are available in refereed form or will be stated in sufficient detail in the present paper. The one-sentence fullness justification cannot carry the main theorem, and the paper would benefit from either an expanded proof or a precise theorem statement drawn from [Fuk]. The idea is promising and the closed-string dimension computations are solid; the recommendation of major revision reflects proof gaps rather than suspected falsity. On fit: the result is of high interest to the symplectic and algebraic-geometry communities, and the brevity of the note is acceptable for a letters-style journal only if the companion works are readily accessible to referees."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Hi — quick take on Ueda's arXiv:2505.00910. The claim is big: HMS for all smooth projective hypersurfaces of degree a > 2n+1, via a Lagrangian correspondence from the wrapped Fukaya category of an affine complement to the Fukaya category of the boundary. If the proof holds, this is a major new chunk of general-type HMS. The overall strategy is coherent and the paper is honest about its dependencies. The Hochschild dimension count at the end is explicit and the Jacobi ring comparison is a nice piece of work.\n\nThe trouble is a load-bearing assertion that doesn't hold up. The author writes that the correspondence functor Φ is full \"since it is defined using the Yoneda embedding.\" That isn't an argument. Yoneda gives fullness for an embedding of a category into its own presheaf category, not for an A-infinity functor between Fukaya categories induced by a Lagrangian correspondence. Fullness of such a functor means the quilted maps Hom(L,K) → Hom(Φ(L),Φ(K)) are surjective, and no theorem in [Fuk] or elsewhere is cited for that. If Φ isn't full, the localization that identifies the stable category S(U) with the matrix factorization side collapses, and the essential surjectivity step can't be rescued by the HH dimension count because the functor isn't known to be full faithful to begin with.\n\nThere are secondary soft spots: the core mirror equivalence (1.13) comes from a companion preprint [LU] by the same author, and the extension of Fukaya's correspondence machinery to the Liouville setting is waved at as \"straightforward.\" These could be acceptable to specialists, but they're not demonstrated here.\n\nMy verdict: this deserves a serious referee, but not acceptance in the current form. The gap in the fullness proof is structural, not cosmetic. If a referee can extract a proof of that point, the paper becomes a major result. As it stands, I'd treat it as an interesting proof sketch rather than an established theorem. I wouldn't cite it as a proven result yet.","headline":"A plausible but underproved HMS theorem for high-degree hypersurfaces; the fullness of the Lagrangian-correspondence functor is asserted without proof.","tokens_in":6814,"tokens_out":6809,"would_cite":false,"duration_ms":71123,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D37","14J33","14J70","53D12"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that every smooth projective hypersurface of degree $a>2n+1$ satisfies homological mirror symmetry, via a functor from the wrapped Fukaya category of an affine hypersurface to the Fukaya category of its boundary at…","keywords":["homological mirror symmetry","Fukaya category","wrapped Fukaya category","matrix factorizations","Lagrangian correspondence","projective hypersurfaces","stable Fukaya category","Hochschild cohomology"],"falsifier":"Compute both sides of the dimension identity (1.22) for a concrete pair, say $n=2$, $a=6$: the Hochschild cohomology of $\\mathrm{mf}([\\mathbb{A}^4/G], x_1^6+\\cdots+x_4^6 - x_1x_2x_3x_4)$ and the quantum cohomology of a smooth sextic surface in $\\mathbb{CP}^3$. Any mismatch would refute the essential-surjectivity step and therefore Theorem 1.1.","tokens_in":5847,"feed_emoji":"🪞","tokens_out":13434,"duration_ms":114804,"temperature":0.7,"pith_summary":"The paper aims to prove homological mirror symmetry for every smooth projective hypersurface of sufficiently high degree: if $X\\subset\\mathbb{CP}^{n+1}$ is smooth of degree $a>2n+1$, then its Fukaya category (the symplectic invariant built from Lagrangian submanifolds and their intersection theory) is equivalent to an algebraic category of equivariant matrix factorizations. The strategy is geometric: an affine hypersurface $U$ whose boundary at infinity is $X$ supplies a Lagrangian correspondence $C\\subset U\\times X$, and the paper shows this correspondence induces an equivalence between the stable wrapped Fukaya category of $U$ and the Fukaya category of $X$. If correct, this is a uniform mirror-symmetry statement covering every smooth projective hypersurface of general type beyond a dimension-dependent degree bound, and it makes the geometry at infinity the engine of the proof.","feed_headline":"Mirror symmetry proven for all high-degree hypersurfaces","feed_subtitle":"A boundary-at-infinity functor equates symplectic and algebraic sides for every smooth hypersurface of degree a > 2n+1.","key_machinery":"The central object carrying the argument is the Lagrangian correspondence $C\\subset U\\times X$ obtained from the contact boundary of the Liouville domain whose completion is $U$. The key identity that makes it work is the obstruction formula (1.5)--(1.7): every pseudoholomorphic disk class $\\beta$ with boundary on $C$ has Maslov index $\\mu(\\beta)\\ge 2(a-n-2)$, so the virtual evaluation $(\\mathrm{ev}_\\beta)_!^{\\mathrm{virt}}(1)$ has degree $2-\\mu(\\beta)$ at least $2n+2$, exceeding $\\dim C$; hence $m_0(1)=0$ and $C$ is unobstructed. Once unobstructed, $C$ defines an $A_\\infty$ functor $\\Phi\\colon W(U)\\to\\mathcal{F}(X)$ via a Yoneda-style construction, and the paper attributes fullness to that Yoneda-embedding definition. The quotient by compact objects isolates the boundary contribution to $\\Phi$, turning the mirror equivalence into a statement about a stable category.","core_discovery":"Theorem 1.1 asserts that for a smooth degree-$a$ hypersurface $X\\subset\\mathbb{CP}^{n+1}$ with $a>2n+1$, there is an equivalence $$\\mathcal{F}(X)\\simeq \\mathrm{mf}\\left([\\mathbb{A}^{n+2}/G], w-x_1\\cdots x_{n+2}\\right),$$ where $w=x_1^a+\\cdots+x_{n+2}^a$, $G$ is the diagonal symmetry group preserving the superpotential, and $\\mathrm{mf}$ is the idempotent-complete category of equivariant matrix factorizations. The proof constructs the affine hypersurface $U=Y\\setminus X$ for a smooth degree-$a$ $Y\\subset\\mathbb{CP}^{n+2}$ containing $X$ as a hyperplane section, so $X$ is the boundary at infinity of the Liouville completion of $U$. The contact boundary $C$ of that Liouville domain forms a Lagrangian correspondence $C\\to U\\times X$; the degree hypothesis forces the disk-counting obstruction $m_0(C)$ to vanish, so $C$ induces an $A_\\infty$ functor $\\Phi\\colon W(U)\\to\\mathcal{F}(X)$. Because compact Lagrangians in $U$ are disjoint from $C$, this functor descends to the stable quotient $S(U)=W(U)/\\mathcal{F}(U)$. The paper proves $\\Phi$ is full and faithful on that quotient using explicit matrix-factorization morphism computations, and then uses a Hochschild cohomology dimension count to conclude essential surjectivity, completing the equivalence.","pith_inferences":["The explicit threshold $a>2n+1$ is likely not sharp: the proof really uses the Maslov-index bound (1.7) and the non-congruence of Floer cohomology degrees $0$ and $n$ modulo $2(a-n-2)$, so any degree range preserving those inequalities should admit the same equivalence.","The same contact-boundary mechanism should extend to other families of projective varieties that are boundaries at infinity of Liouville domains, for example complete intersections, whenever the corresponding disk obstruction vanishes by a dimension count.","A categorical consequence of the argument is a general template: the Fukaya category of a closed symplectic manifold can be recovered from the wrapped Fukaya category of its affine complement after localizing away compact objects, which may offer a uniform route from Landau–Ginzburg models to closed mirror symmetry."],"forward_implications":["For every smooth hypersurface $X\\subset\\mathbb{CP}^{n+1}$ of degree $a>2n+1$, homological mirror symmetry holds: $\\mathcal{F}(X)$ is equivalent to the equivariant matrix factorization category with superpotential $x_1^a+\\cdots+x_{n+2}^a-x_1\\cdots x_{n+2}$.","The stable wrapped Fukaya category $W(U)/\\mathcal{F}(U)$ of the affine Milnor fiber $U$ is equivalent to the Fukaya category of its boundary $X$, so the non-compact geometry of $U$ fully determines the closed Fukaya category of $X$.","The dimension identity $\\dim HH^\\ast = \\dim QH^\\ast$ for these hypersurfaces follows from the Jacobian-ring computation (1.28)--(1.35), identifying the quantum cohomology of $X$ with primitive cohomology as in the classical Griffiths description.","The degree bound is explicit: for a fixed dimension $n$, all degrees $a\\ge 2n+2$ are covered, so the proof supplies infinitely many new mirror equivalences in every dimension."],"supporting_citations":[{"why":"Supplies the construction of an $A_\\infty$ functor from an unobstructed Lagrangian correspondence, the main tool producing $\\Phi\\colon W(U)\\to\\mathcal{F}(X)$.","marker":"[Fuk]"},{"why":"Provides the theorem that equality of Hochschild cohomology dimensions forces essential surjectivity, used to complete the equivalence.","marker":"[Gan]"},{"why":"Gives the quasi-equivalence between the wrapped Fukaya category of the Milnor fiber and equivariant matrix factorizations, the mirror-side model underlying the proof.","marker":"[LU]"},{"why":"Computes the monodromy autoequivalence and the Hochschild cohomology class of the natural transformation $s$, governing the action on the stable category.","marker":"[LU22]"},{"why":"Supplies the Hom-space computations among Lefschetz thimbles in matrix factorizations, used to establish fullness of the functor.","marker":"[Dyc11]"},{"why":"Provides the Hochschild cohomology decomposition for equivariant matrix factorizations used in the dimension count (1.28).","marker":"[BFK14]"},{"why":"Gives the formal-completion equivalence for matrix factorizations, used to pass to the local singularity category in (1.20).","marker":"[Orl11]"}],"fun_headline_variants":["Boundary functor proves mirror symmetry for high-degree hypersurfaces","Fukaya categories equivalent for high-degree projective hypersurfaces","Mirror symmetry via wrapped Fukaya to boundary Fukaya functor","High-degree hypersurfaces: mirror symmetry proof from boundary"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole proof leans on the assumption that the contact-boundary correspondence over $X$ can be turned into a fully faithful functor between the two symplectic invariants; this transformation is cited from another work, with the Liouville generalization said to be straightforward and fullness asserted rather than shown.","fun_headline_variants_meta":{"raw":{"variants":["Boundary functor proves mirror symmetry for high-degree hypersurfaces","Fukaya categories equivalent for high-degree projective hypersurfaces","Mirror symmetry via wrapped Fukaya to boundary Fukaya functor","High-degree hypersurfaces: mirror symmetry proof from boundary"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000683,"raw_usage":{"total_tokens":3076,"prompt_tokens":898,"completion_tokens":2178,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":514,"completion_tokens_details":{"reasoning_tokens":2106}},"tokens_in":514,"tokens_out":2178,"duration_ms":17484,"temperature":1.0,"reasoning_tokens":2106,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:32:29.011199+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute both sides of the dimension identity (1.22) for a concrete pair, say $n=2$, $a=6$: the Hochschild cohomology of $\\mathrm{mf}([\\mathbb{A}^4/G], x_1^6+\\cdots+x_4^6 - x_1x_2x_3x_4)$ and the quantum cohomology of a smooth sextic surface in $\\mathbb{CP}^3$. Any mismatch would refute the essential-surjectivity step and therefore Theorem 1.1.","supporting_citations":[],"review_version":1}