{"id":"bd49dc35-4072-433d-9985-24464e46a343","arxiv_id":"2412.04437","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new classification of bounding cochains on normed matrix factorizations yields algebraic invariants matching open Gromov-Witten invariants of RP^1 and RP^3.","lead":"The authors introduce normed matrix factorization categories, a mirror-symmetry framework with non-Archimedean norms, and classify their bounding cochains for spherical objects. They use this to define numerical invariants that, for RP^1 and in low degrees for RP^3, agree with open Gromov-Witten and Welschinger counts.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 11 overstates the codomain: bounding cochains only hit the positive-valuation subspace F^0(R_s)_{1-n}, not all of (R_s)_{1-n}; for n=3, T^{-1/4} is a concrete element of the stated codomain that cannot be in the image.","rationale":"The paper's headline claim is Theorem 11. Reading Sections 5.2-5.4 carefully, every existence and injectivity statement is made relative to F^0R_s, the positive-valuation subspace. The proof never addresses elements of degree 1-n with negative valuation, and a direct valuation argument shows such elements cannot be attained: b has positive valuation, c has positive valuation, the canonical y_b of Corollary 4.19 is expressed entirely through positive-valuation products with rational coefficients, and the normed property of τ forces positive valuation on τ(gexp(b)). For n = 3, T^{-1/4} is a concrete element of (R_s)_{-2} with valuation -1/4, so it is a counterexample to surjectivity as stated. This is an internal inconsistency rather than a clash with external consensus, and it is directly checkable. The fix is simple: state the codomain as F^0(R_s)_{1-n}. The numerical invariants and the construction of point-like bounding cochains use r = s ∈ F^0, so the applications of Theorem 11 survive essentially unchanged. The reader's flagged concerns about the residue estimates in Theorem 15 and the missing n = 3 code are legitimate, but the codomain mismatch is more immediately verifiable and strikes at the central theorem's statement, which is why I focus on it here.","tokens_in":71820,"tokens_out":24517,"duration_ms":238104,"concrete_test":"Compute the valuation of the canonical gexp(b) from Corollary 4.19 for any b with ν_{M,s}(b) > 0: every summand has positive valuation, so ν_R(τ(gexp(b))) > 0 by the normed property. Then attempt Proposition 5.4 with r = T^{-1/4} for n = 3: the base case b(1) = r h0 has negative valuation and violates the bounding-cochain requirement ν(b) > 0, confirming that the proof cannot produce this element and the stated codomain is unreachable.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 5 proves a weaker statement than Theorem 11. Proposition 5.4 only constructs bounding cochains for r ∈ F^0R_s with |r| = 1-n, and Proposition 5.19 only compares bounding cochains whose common value r lies in F^0R_s. The stated codomain (R_s)_{1-n} is strictly larger: for n = 3, T^{-1/4} ∈ R_3 has degree 2(n+1)(-1/4) = -2 = 1-n and valuation -1/4 < 0. Meanwhile every bounding cochain b satisfies ν_{M,s}(b) > 0; exp(b) and the canonical y_b of Corollary 4.19 are built as sums of products of b and c (with ν(c) > 0) with rational coefficients, hence have positive valuation. Since τ is normed, ν_R(τ(gexp(b))) ≥ ν(gexp(b)) > 0, so T^{-1/4} cannot lie in the image. Thus the claimed bijection to (R_s)_{1-n} is false; the correct target is F^0(R_s)_{1-n}. The point-like construction τ(gexp(b)) = s and Definition 13 are unaffected because s ∈ F^0R_s, but the central classification theorem as stated requires correction.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a normed DG category of matrix factorizations over a Novikov field, defines a notion of bounding cochain for spherical normed Calabi-Yau objects, and proves a classification theorem (Theorem 11) asserting that gauge-equivalence classes of bounding cochains are in bijection with the degree-(1-n) part of R_s. The numerical invariants extracted from a point-like bounding cochain are then matched, for the simplex Delta_n with n odd, against open Gromov-Witten invariants: exactly for n=1 (Theorem 20) and by computer calculation for n=3 in low degrees (Section 1.3.3). The construction passes through a toric valued Landau-Ginzburg model, the Dirac factorization M_n, proofs of sphericity and of the normed property of Shklyarov's infinity-trace, and a detailed inductive construction of bounding cochains.","tokens_in":1795,"tokens_out":2258,"duration_ms":109962,"significance":"If the classification theorem is corrected as described below, this is a substantial piece of mirror-symmetry infrastructure: it gives an algebraic setting in which one can talk about point-like bounding cochains and numerical invariants for objects that are invisible in the characteristic-zero Fukaya category, and the n=1 computation is a complete, checkable match with the open Gromov-Witten invariants of RP^1 in CP^1. The toric construction of valued Landau-Ginzburg models and the Dirac factorization are natural and likely to be reused. However, the paper's central theorem as stated has a false codomain, and the advertised n=3 evidence is not reproducible from the text.","major_comments":[{"comment":"Theorem 11 states that the map from gauge-equivalence classes of bounding cochains to the full degree-(1-n) part of R_s is bijective, but the proof only treats values in the positive-valuation subspace F^0 R_s. Proposition 5.4 constructs bounding cochains only for r in F^0 R_s with |r| = 1-n, and Proposition 5.19 only compares bounding cochains with a common value r in F^0 R_s. The stated codomain is strictly larger: for n=3, the element T^{-1/4} has degree 2(n+1)(-1/4) = -2 = 1-n and valuation -1/4 < 0. Meanwhile every bounding cochain b satisfies nu_{M,s}(b) > 0, and since tau is normed, nu_R(tau(gexp(b))) >= nu(gexp(b)) > 0. Hence T^{-1/4} lies in the stated codomain but not in the image. The correct target is F^0 (R_s)_{1-n} (indeed the image has positive valuation). The classification theorem, the surrounding statements such as Definition 13's point-like bounding cochain, and Propositions 5.4 and 5.19 need to be restated consistently with this corrected target. The point-like value s lies in F^0 R_s, so the numerical invariants themselves are not destroyed, but Theorem 11 as written is false.","section":"Section 1.2.4, Theorem 11; Section 5.2, Proposition 5.4; Section 5.4, Proposition 5.19"},{"comment":"The n=3 agreement with open Gromov-Witten-Welschinger invariants is a central advertised claim, but it rests entirely on computer calculations for which no code, scripts, or raw residue data are provided. The text reports only final assembled values, e.g. N_{1/2,2} = -2 and N_{3/2,6} = 2 in Equation (9), after mentioning 345,912 residues appearing at m=7. This is not independently verifiable. In addition, the sign of the first value is not discussed in relation to the claimed coincidence with Welschinger invariants, and it differs from the n=1 value N_{1/2,2} = 2 of Corollary 23 without explanation. Please supply a reproducible computation, including code or detailed residue tables, and state explicitly, with signs, the OGW/Welschinger values being matched.","section":"Section 1.3.3 and Section 1.2.7, Theorem 20"},{"comment":"The proof that Theta is normed reduces to residue estimates whose boundary cases need justification. In Corollary 6.10, the assertion that the assumption sum_j m_j >= (n+1)(n+l-1)-n implies 'not all m_j are negative' is not immediate from the inequality alone, and the subsequent identification of the sum over critical points of Res_x(g_vec m) with -Res_infinity(g_vec m) requires ruling out residues at coordinate hyperplanes z_j=0, which can appear when some m_j<0. Since Theorem 15 and hence Theorem 16 rely on this normedness, please either prove these two facts explicitly or reformulate the argument so that the pole structure at the coordinate hyperplanes is controlled.","section":"Section 6.3.3, Lemmas 6.9-6.12 and Corollary 6.10"}],"minor_comments":[{"comment":"The sentence 'In Section 6.5 we prove Proposition 5.4 which yields the numerical invariants of M_n for n=1' refers to the wrong statement; it should refer to Proposition 22 (or Corollary 23), not Proposition 5.4.","section":"Section 1.4, Outline"},{"comment":"Several congruences in the inductive step are written modulo F^{E_l} End(M)_s when the quantity is a scalar in R_s; they should be modulo F^{E_l} R_s, e.g. the line containing tau(gexp(b_(l))) = r.","section":"Section 5.2, proof of Proposition 5.4"},{"comment":"The reference to 'Lemma 4.2' for the statement that Theta is an infinity-trace should be to Theorem 4.2.","section":"Section 6.3, first paragraph"},{"comment":"Equation (9) reads N_{1/2,2} = -2, N_{3/2,6} = 2, N_{d,k} = 0, d+k <= 7; since 1/2+2 and 3/2+6 are not greater than 7, the last clause should be 'for all other (d,k) with d+k <= 7'.","section":"Section 1.3.3, Equation (9)"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know up front. The paper has a genuinely new idea — normed matrix factorization categories, a Dirac factorization for toric polytopes, and a classification of bounding cochains for spherical normed Calabi-Yau objects — and the n=1 computation matching the open Gromov-Witten invariants of RP^1 ⊂ CP^1 is a complete, explicit proof. But Theorem 11, the main classification theorem, is not correct as stated. The proof only reaches the positive-valuation subspace F^0(R_s)_{1-n}, not all of (R_s)_{1-n}. A bounding cochain b has ν(b)>0, and the normed trace inequality forces ν_R(τ(gexp(b))) ≥ 0, so for n=3 an element like T^{-1/4} s lies in the stated codomain but cannot be in the image. This is a genuine flaw in the statement, not a cosmetic one; the correct target should be F^0(R_s)_{1-n}. The good news is that the point-like construction uses r = s, which has positive valuation, so Definition 13 and the invariants are unaffected.\n\nWhat is actually new and solid: the normed matrix factorization category with the normed ∞-trace; the Dirac factorization for Delzant polytopes with a combinatorial relative spin condition; the proof that the Dirac factorization is spherical for odd n (Theorem 18); the proof that Shklyarov's trace is normed for the n-simplex (Theorem 15); and the complete n=1 calculation producing N_{1/2,2}=2 with all other invariants vanishing. The n=1 argument is a real proof of principle.\n\nThe soft spots, in proportion. The Theorem 11 overstatement is the most serious mathematical issue; it needs a corrected statement and a quick check that nothing downstream depends on the false bijection. The n=3 computer verification is presented as evidence but no code or raw residue data are shipped; the paper lists the intermediate values, but a referee cannot independently verify 345,912 residues by hand. This should be fixed before publication — even a short script would help. Conjectures 14, 17, 19 are clearly labeled as conjectures, and the paper does not oversell them. The residue estimates in Section 6.3.3 are intricate and load-bearing; I did not find an error, but they are exactly the kind of place where a sign mistake can hide.\n\nWho this is for: symplectic topologists working on open Gromov-Witten and Welschinger invariants, and algebraists interested in matrix factorizations with non-Archimedean norms. The framework is new enough to deserve serious referee time, and the n=1 result is publishable, but the paper needs a substantive revision before it is ready.\n\nMy recommendation: send it to peer review. A competent referee should be able to verify the corrected Theorem 11 and check the n=1 proof; the n=3 reproducibility issue should be resolved as a condition of acceptance.","headline":"New framework and a clean n=1 proof, but Theorem 11 misstates its own target: the bijection lands in F^0(R_s)_{1-n}, not all of (R_s)_{1-n}.","tokens_in":72658,"tokens_out":6384,"would_cite":true,"duration_ms":62512,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D37","14J33"],"pacs":[],"model":"deepseek-v4-flash","headline":"For spherical normed Calabi-Yau objects, bounding cochains are classified by one trace value, making the new numerical invariants well defined.","keywords":["mirror symmetry","open Gromov-Witten invariants","Welschinger invariants","matrix factorization","non-Archimedean norm","Calabi-Yau structure","bounding cochains","toric geometry"],"falsifier":"Recompute the $n=3$ residues with an independent implementation: the values of Section 1.3.3 must come out as $\\Theta_2(b^{\\otimes 2}/2)\\equiv s^5T/8$, $\\Theta_3(b^{\\otimes 3}/3)\\equiv s^5T/24$, $\\Theta_4(b^{\\otimes 4}/4)\\equiv -3s^5T/10$, $\\Theta_5(b^{\\otimes 5}/5)\\equiv s^5T/6$ modulo $F^6$, and the corresponding three $\\Theta$-contributions from $y_{b_{(5)}}$ must be $s^5T/8$, $-11s^5T/30$, $4s^5T/15$; then $\\alpha_5=s^5T^{1/2}z_1z_2z_3/120$ follows, and the invariants must be $N_{1/2,2}=-2$, $N_{3/2,6}=2$, with all others zero for $d+k\\leq 7$. Any deviation would falsify the normed-trace proof or the advertised matching with real enumerative invariants.","tokens_in":71575,"feed_emoji":"🔢","tokens_out":15655,"duration_ms":127449,"temperature":0.7,"pith_summary":"This paper introduces a category of matrix factorizations in which objects and morphism spaces carry non-Archimedean valuations, and uses it to build an algebraic mirror for a real Lagrangian sphere inside a Fano toric manifold. The central claim is that for a spherical normed Calabi-Yau object $M$, the assignment $\\varrho([b])=\\tau(\\operatorname{gexp}(b))$ bijects gauge-equivalence classes of bounding cochains with the degree $1-n$ part of the base Novikov field; this classification makes the numerical invariants $N_{d,k}$ extracted from a point-like bounding cochain independent of all choices. For the $n$-simplex with $n$ odd, the Dirac factorization is shown to be a spherical normed Calabi-Yau object, mirroring $\\mathbb{R}P^n \\subset \\mathbb{C}P^n$. For $n=1$ the invariants are computed exactly and coincide with the open Gromov-Witten invariants of $\\mathbb{R}P^1 \\subset \\mathbb{C}P^1$; for $n=3$ low-degree computations agree with the real enumerative invariants of $\\mathbb{R}P^3 \\subset \\mathbb{C}P^3$. If the construction works, it provides the missing algebraic side of mirror symmetry for real Lagrangians even when the Lagrangian is zero in the ordinary characteristic-zero Lagrangian category.","feed_headline":"Normed matrix factorizations reproduce real sphere invariants","feed_subtitle":"A classifying map for bounding cochains makes these invariants canonical, matching disk counts for RP^1 and RP^3.","key_machinery":"The load-bearing object is the normed matrix factorization category $\\operatorname{MF}(W,w)$: a DG category whose objects are $\\mathbb{Z}$-graded free $S$-modules $M$ with an endomorphism $D_M$ squaring to the superpotential shift $W-w$, together with a family of valuations, and whose morphism complexes carry the induced operator norm. The normed $\\infty$-trace $\\Theta$, a chain-level Calabi-Yau structure defined by residue and supertrace formulas on the cyclic complex, is what upgrades the category from algebraic to enumerative: it supplies the map $\\tau(\\operatorname{gexp}(b))$ that classifies bounding cochains and defines the invariants. A point-like bounding cochain is a Maurer-Cartan solution $b\\in \\operatorname{End}(M)_s$ normalised by $\\tau(\\operatorname{gexp}(b))=s$; Theorem 11 says its gauge class is unique, so the coefficients $N_{d,k}$ of its potential are canonical. The Dirac matrix factorization $M_{\\triangle}$ is the Clifford-algebra quotient $\\operatorname{Cl}(\\triangle)/\\langle e_0\\cdots e_n - T^{1/2}\\rangle$, constructed from the polytope $\\triangle$ together with a combinatorial relative spin structure; it is the proposed algebraic counterpart of the real locus of the associated toric manifold.","core_discovery":"The paper's main result is Theorem 11: if $M$ is a spherical object that is normed Calabi-Yau, the map $\\varrho: x_M(\\operatorname{End}(M)_s)/{\\sim} \\to (R_s)_{1-n}$, $[b] \\mapsto \\tau(\\operatorname{gexp}(b))$, is bijective. In words, the only invariant of a bounding cochain up to gauge equivalence is the single element $\\tau(\\operatorname{gexp}(b))$, obtained by applying the normed $\\infty$-trace to the cyclic-chain element $\\operatorname{gexp}(b)$ formed from the exponential of $b$ after subtracting a canonical correction. Because a point-like bounding cochain is defined by the normalisation $\\tau(\\operatorname{gexp}(b))=s$, the potential $c=\\sum_{k,d} N_{d,k+1} \\frac{s^k}{k!}T^d$ of that cochain yields well-defined numerical invariants $N_{d,k}$ (Definition 13). On the toric side, the paper constructs a valued Landau-Ginzburg model for every Delzant polytope and, under a combinatorial relative spin condition, a Dirac factorization; for the odd $n$-simplex this object is spherical, with $H^*(\\operatorname{End}(M_{\\triangle_n})_0) \\cong H^*(\\mathbb{R}P^n;(R_{\\triangle_n})_0)$, and the $\\infty$-trace $\\Theta$ is normed. For $\\triangle_1$, the invariants are $N_{1/2,2}=2$ and all others zero, matching the open Gromov-Witten invariants of $\\mathbb{R}P^1 \\subset \\mathbb{C}P^1$; for $\\triangle_3$, low-degree computer calculations give $N_{1/2,2}=-2$, $N_{3/2,6}=2$ with all other $N_{d,k}$ vanishing for $d+k \\leq 7$, matching the real enumerative invariants of $\\mathbb{R}P^3 \\subset \\mathbb{C}P^3$ up to a factor of 2.","pith_inferences":["Editorial inference: the classification mechanism should produce point-like bounding cochains and numerical invariants for other Fano toric pairs with spherical real loci, since the proof of Theorem 11 uses only sphericity and the normed Calabi-Yau property; testing Conjecture 19 on the next odd simplex $n=5$ would separate the general method from the specific residue computations.","Editorial inference: the six residue values listed for the $n=3$ computation (for example $\\Theta_4(b^{\\otimes 4}/4)\\equiv -3s^5T/10$ modulo $F^6$) are checkable by an independent implementation, and a mismatch would localise the error to the canonical choice of $y_b$ or to the residue bounds, not to the classification theorem.","Editorial inference: because the normed structure is valued in $\\frac{1}{n+1}\\mathbb{Z}$, the invariants may admit a combinatorial reading in terms of polytope data that the paper does not pursue.","Editorial inference: the exceptional nontrivial background class chosen for $\\triangle_1$, whereas higher simplices use the tautological class, suggests that the sign conventions of the invariants are tied to the combinatorial relative spin structure; varying that structure is a concrete test of the factor of 2 appearing in the $\\mathbb{R}P^3$ comparison."],"forward_implications":["For every spherical normed Calabi-Yau object, bounding cochains up to gauge equivalence are parameterised by the base field: the map $[b] \\mapsto \\tau(\\operatorname{gexp}(b))$ is a bijection, so the numerical invariants of Definition 13 do not depend on any choice.","For each odd $n$, the Dirac factorization of the $n$-simplex is a spherical normed Calabi-Yau object with $H^*(\\operatorname{End}(M_{\\triangle_n})_0) \\cong H^*(\\mathbb{R}P^n;(R_{\\triangle_n})_0)$, realising the mirror of $\\mathbb{R}P^n \\subset \\mathbb{C}P^n$ at the cohomological level.","For $n=1$, the invariants are $N_{1/2,2}=2$ and all others zero, exactly the open Gromov-Witten invariants of $\\mathbb{R}P^1 \\subset \\mathbb{C}P^1$.","For $n=3$, the low-degree invariants $N_{1/2,2}=-2$, $N_{3/2,6}=2$, and $N_{d,k}=0$ for $d+k\\leq 7$ coincide with the real enumerative invariants of $\\mathbb{R}P^3\\subset\\mathbb{C}P^3$ up to a factor of 2.","These statements hold over a Novikov ring, so the invariants remain non-trivial even though $\\mathbb{R}P^n$ is the zero object in the ordinary characteristic-zero Lagrangian category."],"supporting_citations":[{"why":"Supplies the $\\infty$-trace $\\Theta$ whose residue and supertrace formula defines the normed Calabi-Yau structure used throughout.","marker":"[20]"},{"why":"Defines point-like bounding cochains and open Gromov-Witten invariants of rational cohomology spheres; Theorem 11 is its algebraic counterpart and Theorem 20 compares against it.","marker":"[22]"},{"why":"Introduces bounding cochains, the Maurer-Cartan obstruction theory and the relative spin condition that the normed category adapts.","marker":"[6]"},{"why":"The Delzant polytope/toric-manifold correspondence provides the combinatorial input for the toric Landau-Ginzburg construction.","marker":"[4]"},{"why":"Establishes the matrix factorization category as the algebraic side of homological mirror symmetry, which the paper refines to a normed setting.","marker":"[18]"},{"why":"Gives the Kapustin-Li trace formula that $\\Theta$ extends with higher-order residue corrections.","marker":"[11]"},{"why":"Shows point-like bounding cochains are universal for the open Gromov-Witten invariants, justifying the algebraic point-like normalisation.","marker":"[23]"},{"why":"Defines the real enumerative invariants in dimension three to which the $n=3$ low-degree invariants are compared.","marker":"[26]"}],"fun_headline_variants":["Normed factorizations reproduce real sphere invariants","Mirror symmetry test: invariants match RP^1 and RP^3","New invariants from normed categories mirror real projective spaces","Spherical bounding cochains yield canonical numerical invariants","Normed matrix factorizations count curves on real loci"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction of point-like bounding cochains and of the invariants $N_{d,k}$ depends on the intricate residue estimates (Lemmas 6.9-6.12) proving that the $\\infty$-trace $\\Theta$ on the projective-space Landau-Ginzburg model is normed; if any sign or convergence check is wrong, the normed Calabi-Yau structure, the sphere-cohomology computation, and the numerical invariants all collapse, and the $n=3$ computer verification is a further load-bearing premise with no code or raw data supplied.","fun_headline_variants_meta":{"raw":{"variants":["Normed factorizations reproduce real sphere invariants","Mirror symmetry test: invariants match RP^1 and RP^3","New invariants from normed categories mirror real projective spaces","Spherical bounding cochains yield canonical numerical invariants","Normed matrix factorizations count curves on real loci"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001215,"raw_usage":{"total_tokens":5190,"prompt_tokens":1327,"completion_tokens":3863,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":943,"completion_tokens_details":{"reasoning_tokens":3782}},"tokens_in":943,"tokens_out":3863,"duration_ms":28500,"temperature":1.0,"reasoning_tokens":3782,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:23:51.184884+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the $n=3$ residues with an independent implementation: the values of Section 1.3.3 must come out as $\\Theta_2(b^{\\otimes 2}/2)\\equiv s^5T/8$, $\\Theta_3(b^{\\otimes 3}/3)\\equiv s^5T/24$, $\\Theta_4(b^{\\otimes 4}/4)\\equiv -3s^5T/10$, $\\Theta_5(b^{\\otimes 5}/5)\\equiv s^5T/6$ modulo $F^6$, and the corresponding three $\\Theta$-contributions from $y_{b_{(5)}}$ must be $s^5T/8$, $-11s^5T/30$, $4s^5T/15$; then $\\alpha_5=s^5T^{1/2}z_1z_2z_3/120$ follows, and the invariants must be $N_{1/2,2}=-2$, $N_{3/2,6}=2$, with all others zero for $d+k\\leq 7$. Any deviation would falsify the normed-trace proof or the advertised matching with real enumerative invariants.","supporting_citations":[{"cited_title":"Shklyarov, Calabi-Yau structures on categories of matrix factorizations , J","cited_arxiv_id":null,"evidence_quote":"Supplies the $\\infty$-trace $\\Theta$ whose residue and supertrace formula defines the normed Calabi-Yau structure used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines point-like bounding cochains and open Gromov-Witten invariants of rational cohomology spheres; Theorem 11 is its algebraic counterpart and Theorem 20 compares against it."},{"cited_title":"Delzant, Hamiltoniens p´ eriodiques et images convexes de l’application moment , Bull","cited_arxiv_id":null,"evidence_quote":"The Delzant polytope/toric-manifold correspondence provides the combinatorial input for the toric Landau-Ginzburg construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the matrix factorization category as the algebraic side of homological mirror symmetry, which the paper refines to a normed setting."},{"cited_title":"Welschinger, Spinor states of real rational curves in real algebraic convex 3-manifolds and enu- merative invariants, Duke Math","cited_arxiv_id":null,"evidence_quote":"Defines the real enumerative invariants in dimension three to which the $n=3$ low-degree invariants are compared."}],"review_version":1}