{"id":"1de36a31-88c8-4a70-ae27-9715ed23fab5","arxiv_id":"2412.09724","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The paper gives explicit formulas for Kalck-Karmazyn algebras and their flat deformations, including Kawamata's matrix order, by computing endomorphism algebras of an immersed Lagrangian with a bounding cochain via mirror symmetry.","lead":"The authors use homological mirror symmetry to compute explicit deformations of Kalck-Karmazyn algebras, which arise from cyclic quotient surface singularities. Their method identifies an immersed Lagrangian in a punctured torus whose endomorphism algebra gives the deformed algebra, yielding an explicit formula for Kawamata's matrix order.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (10) is asserted, not proved: the mirror-symmetry bridge from [16] is extended to the once-punctured torus with one divisor and a nodal genus-one family, and a failure there would invalidate the bounding-cochain computations and Theorem 1.10.","rationale":"The reader's weakest_assumption identifies the same load-bearing premise: the quasi-equivalence (10) is imported from the authors' prior work without proof, and the entire bounding-cochain computation in Sections 3 and 4 depends on it. I agree that this is the most serious gap, because Theorem 1.10's matrix formula is a statement about an algebraic order R = End(F|E), but the proof computes the mirror side first and then identifies the two objects via (10) and Lemma 4.7. If (10) fails, the computation could still be interesting as a Fukaya-categorical construction, but it would not prove Kawamata's matrix order formula. The concern does not amount to a demonstrated error: the explicit examples in Section 1.12–1.19 and Proposition 1.11 provide nontrivial consistency checks, and the r ≤ 32 verification of Conjecture 1.9 is credible supporting evidence if the A∞ table of Theorem 3.4 is correct. Still, the proof of (10) is a theorem-sized missing piece, not a routine remark. The reader's CONDITIONAL verdict is therefore appropriate, and I would not change it: the gap is addressable, but until it is filled, the central claim is conditional on an unproved mirror-symmetry bridge.","tokens_in":23690,"tokens_out":7377,"duration_ms":83835,"concrete_test":"Establish (10) for the smallest Wahl cases by direct comparison with algebraic deformation theory: for (n,q) = (2,1), (3,1), and (3,2), compute the algebra End(F|E) over k[t] purely algebraically from the construction in [24] (or from Kawamata's matrix order [13]) and compare its k[t]-basis and multiplication table with the algebra R_b produced by Lemma 4.1 and Corollary 3.5 with s = t^n. If the two algebras are isomorphic as k[t]-orders in these cases, the mirror bridge (10) is validated in the range used; any mismatch identifies the precise t2=0 step in [16] where the extension fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (10) states a quasi-equivalence F(T1,{s}) ≃ Perf(E) over the base B of the family of nodal genus-one curves obtained from E2 by smoothing the black node and keeping the orange node. The paper invokes [16, Theorem A], which treats a compact torus relative to two marked points {s,o} and a two-parameter Tate family over Z[[t1,t2]]; the one-puncture/one-node statement is obtained by setting t2=0 and replacing the compact torus by T1 = T\\{o}. This specialization is not a formal consequence of [16]: it requires checking that the relative A∞ structure remains defined when one marked point becomes a puncture, that the t2=0 reduction of the Tate family is exactly E → Spec B, and that the quasi-equivalence survives this reduction. No such check is supplied. Since the deformed endomorphism algebra R_b in Lemma 4.1, the identification of K_b with F|E in Lemma 4.7, and hence every matrix entry in Theorem 1.10 are computed inside F(T1,{s}), a failure of (10) would sever the link between the bounding-cochain algebra and the algebraic deformation R = End(F|E). This is a load-bearing premise independent of the internal A∞ bookkeeping in Section 3.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a mirror-symmetry approach to deformations of Kalck–Karmazyn algebras associated to cyclic quotient surface singularities. It constructs an immersed Kawamata Lagrangian K_{r,a} in the two-punctured torus mirror to the Kawamata vector bundle, computes its endomorphism A∞-algebra (Theorem 3.4), and uses bounding cochains in the relative Fukaya category F(T1,{s}) to describe the locus Def^0_{F_E/E} and the flat deformations of the Kalck–Karmazyn algebra. For Wahl singularities, an explicit bounding cochain is shown to yield a flat deformation R_b, which is then embedded into Mat_n(k[t]) via an action on Hom(K_b,H); this yields the explicit matrix order in Theorem 1.10 and its extension to P^1 in Proposition 1.11. The paper also gives a new multiplication table for R_{r,a} and reports computational verification of Conjecture 1.9 for r≤32.","tokens_in":24006,"tokens_out":5642,"duration_ms":58922,"significance":"If correct, this is a substantial advance: it gives the first explicit computation of Kawamata's matrix order for all Wahl singularities, illustrates a general method for using homological mirror symmetry to compute categorical absorption algebras, and makes concrete predictions about deformation spaces that can be checked computationally. The authors are careful to separate proven statements from conjectures, and the paper contains explicit worked examples and a computer implementation for the deformation computations. However, the central argument currently rests on an unproved extension of a mirror-symmetry theorem and on A∞ computations that are only carried out in full for r=2, so the significance is conditional pending those gaps.","major_comments":[{"comment":"The quasi-equivalence F(T1,{s}) ≃ Perf(E) over B is asserted with the comment that the techniques from [16, Theorem A] apply directly. This is not a formal consequence of [16], because [16] treats a compact torus relative to two marked points and a two-parameter Tate family; obtaining the once-punctured relative category and the t2=0 family requires checking that the relative A∞ operations remain defined after removing one marked point and that the quasi-equivalence is compatible with the reduction. Since every bounding-cochain computation in Sections 3 and 4, and hence the identification of R_b with End(F|E) in Lemma 4.7, takes place in F(T1,{s}), the statement needs either a proof or a precise reference that covers this specialization. This is the main load-bearing gap.","section":"§3, Eq. (10)"},{"comment":"The A∞ algebra of K_{r,a} is stated for all r and a, but the proof computes only the case r=2 and says the general case is similar; Corollary 3.5 then summarizes all differential and product contributions in Figure 14. This is a problem because the matrix D and the subscheme Def^0_{F_E/E} are defined from those contributions, and Lemma 4.1 later relies on cancellations among them. Please provide the general computation, at least as a reproducible computer-verified derivation with explicit sign conventions, or state precisely which contributions are being asserted and prove or verify them. A proof-by-figure for arbitrary r is not sufficient for a paper whose central output is explicit formulas.","section":"§3, Theorem 3.4 and Corollary 3.5"},{"comment":"The bounding cochain locus is introduced as ad hoc, and the proof that it lies in Def^0 uses a geometric cancellation of pairs of rectangles. The argument contains several steps justified by 'obviously' and 'straightforward' and by Figures 16–17; in particular the reduction to x,y≤n and the verification of Equation (15) are not fully formal about all possible signs and about the exceptional cases where [⋅]=0 and an extra n^2 is added. Since this lemma controls flatness of R_b and all matrix entries of Theorem 1.10, it needs a complete sign-checked proof or an accompanying verified computation.","section":"§4, Lemma 4.1 and Eq. (15)"}],"minor_comments":[{"comment":"The paragraph begins 'Legtus analyze the vanishing locus'; this should be 'Let us analyze'.","section":"§3, before Corollary 3.7"},{"comment":"The sentence 'In Theorem 3, we found the mirror Lagrangian' should refer to Theorem 2.4 rather than an unnamed 'Theorem 3'.","section":"§3, first paragraph"},{"comment":"The displayed presentation of the deformed algebra ends with a trailing '+' and an unresolved line break; please fix the typesetting of the relations.","section":"§3, Example 3.9"},{"comment":"The text writes 'Kalck–Karamazyn algebra'; the name should be 'Kalck–Karmazyn'.","section":"§3, Example 3.10"},{"comment":"The computer code is described as 'available upon request'; for reproducibility of the r≤32 verification and the examples in 3.11 and 3.12, the code should be archived in a public repository with a persistent identifier.","section":"§3, Reference [19]"},{"comment":"The notation in Figure 14 is not fully explained in the text; in particular the colors, dots, and the replacement of s by s(1−t0) in Corollary 3.5 deserve a short gloss so the reader can extract the matrix D without reconstructing the whole A∞ computation.","section":"§3, Figure 14"}],"recommendation":"major_revision","confidential_remarks":"The paper's dependence on the authors' own prior theorems [16] and [17] is legitimate, since those are published results; the main issue is not novelty or scope but the incompleteness of the proof of Equation (10) and of the general A∞ computation. A revised version that supplies the missing quasi-equivalence argument and a complete or computer-verified computation of Theorem 3.4 would make the paper suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know before you read this: the main value is a concrete, likely-correct computation—an explicit multiplication table for Kalck–Karmazyn algebras (Corollary 1.3) and a closed-form matrix order for the Q-Gorenstein smoothing of Wahl singularities (Theorem 1.10)—plus a method that actually produces these algebras. The soft spot is that the crucial mirror-symmetry bridge for a relative Fukaya category, equation (10), is stated and not proved. If it fails, the bridge between the bounding-cochain algebra and the algebraic deformation is severed. The rest of the paper is a mixture of careful algebraic geometry and symplectic bookkeeping that, as far as I can tell, holds together.\n\nWhat is genuinely new: the multiplication table, the matrix order embedding, and the computed deformation spaces Def^0_{FE/E} in Section 3 are not in the literature. The idea of using an immersed Lagrangian with a bounding cochain in a relative Fukaya category to see flat deformations of endomorphism algebras is good, and the figures and examples make the constructions concrete. The authors also state a conjecture (1.9) and verify it for r≤32; that is honest evidence.\n\nWhere I would push back: equation (10) is not a direct corollary of [16, Theorem A]. The cited theorem is for a compact torus relative to two points; here you have a once-punctured torus with one compactification divisor, and the family E is obtained by setting t2=0. The authors say \"techniques apply directly,\" but that is a real check: the relative A∞ structure with a puncture, the reduction of the Tate family, and the quasi-equivalence surviving that reduction. The stress-test note is on target. This is load-bearing, not cosmetic.\n\nTwo smaller issues: the A∞ computation of Theorem 3.4 is only detailed for r=2, with the general case summarized as similar. A referee will want more. The computer code in [19] is \"available upon request,\" not public; given the computational claims, public code would help. Lemma 4.1's bounding cochain is ad hoc, and the proof that it lies in Def^0 is a long cancellation argument, not easy reading.\n\nWho is this for? People who care about categorical absorption, noncommutative resolutions, and mirror symmetry for punctured tori. If the bridge in (10) is supplied, the results are significant. As it stands, this is a conditional paper: the main formulas are probably right, but the proof has a gap that needs filling.\n\nMy recommendation: send it to a serious referee. It deserves referee time, not a desk rejection. The referee should focus on (10) and the general A∞ computation. If those hold up, this will be a useful paper.","headline":"Fresh, concrete computations of Kalck–Karmazyn deformations and Kawamata's matrix order via mirror symmetry, but the load-bearing mirror-symmetry bridge (10) is asserted rather than proved.","tokens_in":24539,"tokens_out":2592,"would_cite":true,"duration_ms":25686,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D37","14J33","14B05","16G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Mirror symmetry yields an explicit formula for Kawamata's matrix order.","keywords":["mirror symmetry","Fukaya category","Kalck-Karmazyn algebra","Wahl singularity","matrix order","cyclic quotient singularity","bounding cochain","Q-Gorenstein smoothing"],"falsifier":"Compute the endomorphism algebra of the Kawamata Lagrangian K_{$n^{2}$,nq-1} with the bounding cochain of Lemma 4.1 for a case not worked out in the paper (for example n=6, q=1) purely from the holomorphic-polygon counts of Section 3, and compare the resulting k[t]-algebra with the matrix order built independently from Kawamata's algebraic construction; any mismatch in the multiplication table or in flatness over k[t] would falsify Theorem 1.10.","tokens_in":23485,"feed_emoji":"🪞","tokens_out":6948,"duration_ms":63201,"temperature":0.7,"pith_summary":"The paper aims to compute explicitly the one-parameter flat families of algebras that absorb the singularities of threefold smoothings of cyclic quotient singularities, especially the matrix order attached by Kawamata to a Q-Gorenstein smoothing of a Wahl singularity. The authors argue that homological mirror symmetry makes this computation tractable: the mirror of the Kawamata vector bundle is an immersed Lagrangian in the two-punctured torus, and equipping it with a bounding cochain in the relative Fukaya category produces an endomorphism algebra isomorphic to the desired flat family. If the argument is correct, the matrix order is given by an explicit embedding into Mat_n(k[t]) with a concrete formula, and the Kalck-Karmazyn algebra itself has a simple multiplication table indexed by a Young diagram. The value is that a family of algebras previously accessible only through abstract birational geometry can now be written down and studied by hand or by computer.","feed_headline":"Mirror symmetry writes Kawamata's matrix order explicitly","feed_subtitle":"A Lagrangian with a bounding cochain in the punctured torus computes the flat algebra family that absorbs the singularity.","key_machinery":"The load-bearing object is the Kawamata Lagrangian K_{r,a}, the immersed exact Lagrangian in the two-punctured torus obtained as the mirror of the Kawamata vector bundle under homological mirror symmetry for the cycle of rational curves E_2. Deformations are studied by reinterpreting one puncture as a compactification divisor and working in the relative Fukaya category F(T1,{s}); a bounding cochain b = Σ t_i \\bar{w}_i deforms the A∞-structure, and the flat family of algebras is the deformed endomorphism algebra ($hom^{0}$(K,K), m^b_2). The computation is carried by the hidden A∞-algebra of the immersed Lagrangian together with visible holomorphic polygons, which encode the rectangle combinatorics of the Young diagram.","core_discovery":"The central discovery is an explicit correspondence between algebraic deformations and symplectic data: the endomorphism algebra of the Kawamata Lagrangian K_{$n^{2}$,nq-1} endowed with the specific bounding cochain b of Lemma 4.1 in the relative Fukaya category F(T1,{s}) is isomorphic to the matrix order R_{$n^{2}$,nq-1} that absorbs the Q-Gorenstein smoothing of the Wahl singularity 1/$n^{2}$(1,nq-1). This yields Theorem 1.10, an explicit k[t]-basis {w_i}_{i∈Z_{$n^{2}$}} and an embedding R ↪ Mat_n(k[t]) with matrix entries given by a closed formula. Alongside, the paper proves that the Kalck-Karmazyn algebra R_{r,a} has basis w_i indexed by Z_r with product w_j w_i = w_{j+i} exactly when a rectangle condition in a Young diagram is met, and that the flat deformation locus $Def^{0}$_{F_E/E} is cut out by the entries of an explicitly computed skew-symmetric matrix D.","pith_inferences":["The approach suggests a general recipe: whenever a categorical absorption is defined by an endomorphism algebra of a vector bundle on a genus-one fibration, the mirror Lagrangian with a bounding cochain should yield an explicit presentation of the flat family; testing this on other cyclic quotient singularities with multiple smoothing components would show how far the method extends.","The rectangle and Young-diagram rule for the multiplication table hints that Kalck-Karmazyn algebras form a combinatorial family that might be studied without reference to singularities, possibly connecting to lattice-path or cluster-algebra structures.","The unproved quasi-equivalence (10) is the main technical risk to the mirror-symmetry identification; if it is established, the same framework would give explicit deformations for all components of the versal deformation space, not just the Q-Gorenstein one.","A direct check of the formula of Theorem 1.10 for a new pair (n,q) by independent algebraic construction would provide strong evidence for the mirror-symmetry identification, and a mismatch would locate the failure precisely."],"forward_implications":["For every Wahl singularity 1/n^2(1,nq-1), the matrix order absorbing its Q-Gorenstein smoothing can be written down explicitly from Theorem 1.10, with an embedding into Mat_n(k[t]) ready for computer implementation.","The Kalck-Karmazyn algebra R_{r,a} acquires a closed multiplication rule (basis w_i, product w_j w_i = w_{j+i} when the rectangle condition holds), parametrized entirely by the inverse b of a modulo r.","The subscheme Def^0_{F_E/E} deforming the restriction of the Kawamata bundle while preserving the dimension of its endomorphism algebra is the vanishing locus of an explicit skew-symmetric matrix D, so flat deformations of R_{r,a} are computable in examples.","If Conjecture 1.9 holds, all deformations of R_{r,a} that are captured by bundle deformations come from deformations of the ambient surface W, matching the irreducible components of the Kollár-Shepherd-Barron correspondence.","The order of Theorem 1.10 extends to P^1 with the fiber at infinity also isomorphic to R_{n^2,nq-1}, giving a full compactification of the deformation."],"supporting_citations":[{"why":"Supplies the homological mirror symmetry quasi-equivalence for genus-one curves with marked points and the relative Fukaya category formalism on which the identification (10) is based.","marker":"[16]"},{"why":"Establishes that a Q-Gorenstein smoothing of a Wahl singularity gives rise to a one-parameter flat degeneration to a matrix algebra, the object this paper makes explicit.","marker":"[13]"},{"why":"Constructs the flat family R = End(F) over deformations of the surface and shows general fibers are hereditary algebras, providing the algebraic deformation context.","marker":"[24]"},{"why":"Proves existence and properties of the Kawamata vector bundle F, including the vanishing of Ext^k(F,F) used in Lemma 2.2.","marker":"[11]"},{"why":"Gives the construction of F via universal extensions and the earlier presentation of the Kalck-Karmazyn algebra R_{r,a}.","marker":"[9]"},{"why":"Provides the bounding cochain and Maurer-Cartan formalism used to deform the endomorphism algebra of the Lagrangian.","marker":"[5]"},{"why":"Defines the maximal iterated extension construction that produces the Kawamata vector bundle.","marker":"[12]"},{"why":"Introduces the torsion-free sheaf H whose n-fold sum splits the Kawamata bundle on general fibers, used in the proof of Lemma 4.4.","marker":"[6]"}],"fun_headline_variants":["Mirror symmetry makes Kawamata's matrix order explicit","Lagrangian with bounding cochain codes flat algebra family","Kawamata matrix order via a punctured-torus Lagrangian","Explicit matrix order from a bounding cochain in a torus","Mirror symmetry pinpoints Kawamata's matrix order"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the claim, stated without proof in the paper, that the relative Fukaya category of the once-punctured torus with one compactification divisor is equivalent to the perfect derived category of the family of nodal genus-one curves; if this quasi-equivalence fails, the bounding-cochain computations would not describe the intended algebraic deformations.","fun_headline_variants_meta":{"raw":{"variants":["Mirror symmetry makes Kawamata's matrix order explicit","Lagrangian with bounding cochain codes flat algebra family","Kawamata matrix order via a punctured-torus Lagrangian","Explicit matrix order from a bounding cochain in a torus","Mirror symmetry pinpoints Kawamata's matrix order"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000198,"raw_usage":{"total_tokens":1357,"prompt_tokens":921,"completion_tokens":436,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":537,"completion_tokens_details":{"reasoning_tokens":352}},"tokens_in":537,"tokens_out":436,"duration_ms":4061,"temperature":1.0,"reasoning_tokens":352,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T16:48:00.211477+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the endomorphism algebra of the Kawamata Lagrangian K_{$n^{2}$,nq-1} with the bounding cochain of Lemma 4.1 for a case not worked out in the paper (for example n=6, q=1) purely from the holomorphic-polygon counts of Section 3, and compare the resulting k[t]-algebra with the matrix order built independently from Kawamata's algebraic construction; any mismatch in the multiplication table or in flatness over k[t] would falsify Theorem 1.10.","supporting_citations":[{"cited_title":"Lekili, A","cited_arxiv_id":null,"evidence_quote":"Supplies the homological mirror symmetry quasi-equivalence for genus-one curves with marked points and the relative Fukaya category formalism on which the identification (10) is based."},{"cited_title":"Kawamata, Semi-orthogonal decomposition and smoothing, J","cited_arxiv_id":null,"evidence_quote":"Establishes that a Q-Gorenstein smoothing of a Wahl singularity gives rise to a one-parameter flat degeneration to a matrix algebra, the object this paper makes explicit."},{"cited_title":"Karmazyn, A","cited_arxiv_id":null,"evidence_quote":"Proves existence and properties of the Kawamata vector bundle F, including the vanishing of Ext^k(F,F) used in Lemma 2.2."},{"cited_title":"Fukaya, Y.-G","cited_arxiv_id":null,"evidence_quote":"Provides the bounding cochain and Maurer-Cartan formalism used to deform the endomorphism algebra of the Lagrangian."},{"cited_title":"Kawamata, On multi-pointed non-commutative deformations and Calabi–Yau threefolds, Compositio Mathematica, 154 (2018), 1815–1842","cited_arxiv_id":null,"evidence_quote":"Defines the maximal iterated extension construction that produces the Kawamata vector bundle."},{"cited_title":"Hacking,Exceptional bundles associated to degenerations of surfaces, Duke Math","cited_arxiv_id":null,"evidence_quote":"Introduces the torsion-free sheaf H whose n-fold sum splits the Kawamata bundle on general fibers, used in the proof of Lemma 4.4."}],"review_version":1}