{"id":"0440d8ff-34b1-44f9-a80e-15ad5d93cc81","arxiv_id":"2608.06696","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For coprime positive a,b, the derived category of the product of a type-A path algebra with the rational Dyck poset is equivalent to the derived category of the full path lattice, proving the CLR conjecture.","lead":"This mathematics paper proves a 2018 conjecture by Chapoton, Ladkani, and Rognerud: for coprime numbers a and b, certain algebraic objects built from Dyck paths and from higher Auslander algebras are derived equivalent. The proof introduces a new tilting technique that works for general finite linear categories, not just simple posets.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 6.6 depends on Xing's unproved identifications B0 ≅ eA'e and B0^{(a+b)} ≃ A^a_{b+1}; the paper's own Section 8 checks stop before this junction.","rationale":"The paper's genuinely new machinery—the relative interval-tilting theorem, its iteration to staircases, and the Dyck-corner equivalence Corollary 5.8—is presented in substantial detail, and the Section 8 finite checks (Cartan determinants, Coxeter polynomials, multiplication tables for (3,4) and (4,5)) support that part. I found no internal inconsistency in the new constructions. The remaining gap is exactly where the new result is spliced into the known chain: the identification of the staircase corner with Xing's B0 and the replicated-algebra equivalence B0^{(a+b)} ≃ A^a_{b+1} are cited from an unrefereed preprint, and Lemma 6.5 is comparatively sketchy. Theorem 6.6 cannot be derived from Corollary 5.8 alone without these inputs, so the conditional verdict is appropriate. The reader's weakest-assumption analysis locates the same dependency, which is why I agree with it. If the suggested independent check of the Xing junction succeeds, the conditional verdict can be upgraded.","tokens_in":33363,"tokens_out":13144,"duration_ms":134583,"concrete_test":"For the smallest coprime cases (a,b) = (2,3), (3,4), (4,5), independently implement Xing's construction: compute the Dyck-indexed projective summand P in A^a_{b+1}, form the Serre-orbit sum T = ⊕_{i=0}^{a+b-1} S^iP, verify the orthogonality conditions in [14, Proposition 4.17 and Corollary 4.21], and check that End(T)^op is the replicated algebra B0^{(a+b)} derived equivalent to A^a_{b+1}; in addition, enumerate all ordered triples in os_a(a+b) to confirm that the reversal in Lemma 6.5 preserves every zero composite.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim Theorem 6.6 is a chain: Corollary 5.8 gives D^b(kDyck^{above}_{a,b}) ≃ D^b((B^st_{a,b})^op), and (45)-(46) and (49) are standard tensor, replication, and lattice steps. The load-bearing middle junction is (47): Xing's B0^{(a+b)} ≃der A^a_{b+1}, cited from [14, Theorem 4.5 and Proposition 4.25], together with the identification B0 ≅ B^st_{a,b} cited from [14, Proposition 4.33] in Proposition 5.5. These external statements are not reproved in the paper. The genuinely new Dyck-corner equivalence in Corollary 5.8 is not sufficient for Theorem 6.6: without the Xing junction, the chain terminates at B0^op and does not reach kL_{a,b}. Lemma 6.5, used to pass from (B0^{(a+b)})^op to A^a_{b+1}, is also only a one-paragraph sketch; its reversal map should preserve the zero relations of the higher Auslander category, but this is asserted rather than demonstrated. The finite checks in Section 8 validate the new staircase and Dyck-corner constructions; they do not independently verify the Xing junction or the self-oppositeness lemma.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a relative interval-tilting theorem for finite k-linear categories with arbitrary finite-dimensional Hom spaces, non-semisimple diagonal endomorphism algebras, and forced-zero composites. It iterates this theorem to establish derived equivalences between incidence algebras of finite coordinate staircases and idempotent corners of higher Auslander algebras of type A. It then identifies rational Dyck posets with such staircases, proves that the resulting corner is Xing's algebra B0 in the coprime case, and combines the new Dyck-corner equivalence with results of Ladkani, Xing, and Gottesman to prove the Chapoton--Ladkani--Rognerud conjecture for coprime positive integers. The paper also gives Fukaya-categorical interpretations of the staircase equivalences and includes explicit finite checks for the (3,4) and (4,5) examples.","tokens_in":33622,"tokens_out":10654,"duration_ms":104632,"significance":"If the external inputs are correct, Theorem 6.6 resolves the CLR conjecture. The paper's own contribution is substantial: the relative tilting theorem genuinely extends the CLR interval-tilting mechanism to categories with higher-dimensional Hom spaces and zero composites, the iterated staircase theorem is explicit and does not require coprimality, and the non-coprime replicated statements are new. The paper is also commendably explicit about the division between new results and cited ones: Section 8 provides reproducible finite checks (Cartan determinants, Coxeter polynomials, and complete composable-pair counts) that are consistent with the staircase and Dyck-corner constructions. The main weakness is that the final CLR proof depends at two load-bearing points on results from the recent preprint [14] that are cited but not proved, and on a one-paragraph self-oppositeness lemma; these need to be addressed before the central claim can be regarded as fully established.","major_comments":[{"comment":"The proof of Theorem 6.6 is a chain whose final junction passes through two results from the preprint [14] that are not proved or even stated in this paper: the identification B0 ≅ eA'e in Proposition 5.5 (citing [14, Proposition 4.33]) and the derived equivalence B0^{(a+b)} ≃ A^a_{b+1} used in the proof of Theorem 6.6 (citing [14, Theorem 4.5 and Proposition 4.25]). The new Dyck-corner equivalence of Corollary 5.8 alone does not reach kL_{a,b}; without the Xing junction the chain stops at B0^op. Because Theorem 6.6 is the central claim, this dependence is load-bearing. The finite checks in Section 8 validate the staircase and Dyck-corner constructions but do not independently verify the Xing junction. Please either include proofs or detailed convention-matching derivations of these two external results, or explicitly state that the main theorem is conditional on [14] and give the precise statements and version used.","section":"Section 6.d, proof of Theorem 6.6"},{"comment":"The proof of self-oppositeness of A^a_{b+1} is a one-paragraph sketch. It asserts that σ(z) = (a+b+1-z_a, ..., a+b+1-z_1) sends arrows to arrows in the opposite direction and preserves the zero relations, but it does not verify the strict interlacing inequalities under σ, nor does it check the half-square zero-product rule for two composable arrows whose outer pair fails to interlace. Since this lemma is used in the proof of Theorem 6.6 to identify (A^a_{b+1})^op with A^a_{b+1}, it is load-bearing. Please provide a complete proof, for example by explicitly verifying that the interlacing inequalities are reversed appropriately under σ and by checking the composite rule (8) under the reversal.","section":"Lemma 6.5"},{"comment":"The identification of the staircase corner with Xing's B0 depends on matching the vertex labeling of rational Dyck paths with the indexing used in [14, Proposition 4.33]. The paper correctly tracks the above/below convention and takes opposites in Corollary 5.8, but because this identification is cited rather than proved, a mismatch in coordinate order or in the direction of the path order would propagate through the proof of Theorem 6.6 and invalidate the conclusion. Please include an explicit translation between the coordinates β(x) used here and Xing's vertex set for B0, or state precisely which statement in [14] is being used and why it applies verbatim to the conventions of this paper.","section":"Proposition 5.5 / Corollary 5.8"}],"minor_comments":[{"comment":"The phrase 'By Theorem 7.4' should read 'By Corollary 7.4'.","section":"Section 7.e"},{"comment":"The acknowledgments contain a nonstandard poetic passage beginning 'For many years, the fundamental axiom of my mathematical creed...'; this is inappropriate for a research paper and should be removed.","section":"Acknowledgments"},{"comment":"The sentence 'The second family of inequalities is empty when j = 0' is unclear; it should say that the interlacing family of inequalities is empty when j = 0.","section":"Definition 4.1"},{"comment":"The symbol C_j is used both for the category in Definition 4.1 and for the Cartan matrix in Eq. (85); please disambiguate, for example by writing Cart_j for the Cartan matrix.","section":"Equation (85)"},{"comment":"The abstract and introduction state that the paper proves the Chapoton--Ladkani--Rognerud conjecture; in light of the dependence on [14] noted above, this claim should be qualified or the external results should be proved.","section":"Abstract and Introduction"}],"recommendation":"major_revision","confidential_remarks":"The paper's new results appear sound and well structured, and the finite checks are a genuine asset. The main theorem, however, is currently a reduction to results in the recent arXiv preprint [14] (version 1, 2025) that are not proved here, together with a sketch-level self-oppositeness lemma. If the journal accepts reductions to unpublished preprints, these dependencies should at minimum be stated very precisely and the conventions matched explicitly; otherwise the author should supply proofs of the cited junction results. The unusual acknowledgments passage should be removed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new material is the relative interval-tilting theorem (Thm 3.9) and its staircase iteration (Thm 4.5). This is a real extension of CLR's mechanism to finite linear categories with arbitrary finite-dimensional Hom spaces, non-semisimple diagonals, and zero composites. The proof via exact right Kan extensions is detailed and I see no gap in the main induction. Corollary 5.8, the Dyck-corner equivalence, is also new, requires no coprimality, and looks solid; the identification of the Dyck poset with the staircase Omega_{a,b} is concrete and the corner multiplication is handled explicitly.\n\nThe soft spot is exactly where the stress-test note lands. Theorem 6.6, the CLR conjecture, is a chain whose load-bearing middle junction is Xing's equivalence B0^{(a+b)} ≃ A^a_{b+1} and the identification of B0 with the staircase corner. These are cited from a 2025 preprint that is not reproved here. Lemma 6.5 (self-oppositeness of A^a_{b+1}) is also only a one-paragraph sketch; the reversal map preserving relations is asserted rather than demonstrated. Section 8's finite checks stop before this junction, so they validate the new staircase and Dyck-corner constructions but do not independently verify the Xing step. These are real limitations, but they are stated honestly and the authors do not hide them.\n\nI should also note the acknowledgments contain a bizarre self-referential paragraph about adjoint companions and derived categories of curiosity. It does not affect the mathematics, but it is so out of place that an editor may want it removed.\n\nOverall: the paper deserves a serious referee. The relative tilting theorem and the staircase/Dyck-corner results are likely correct and are worth publishing even if the final CLR chain is conditional. The referee should check Xing's preprint and ask for a fuller proof of Lemma 6.5. I would not desk-reject this.","headline":"A substantial paper with a genuinely new relative tilting theorem and a Dyck-corner equivalence that looks right; the CLR proof is conditional on Xing's unverified preprint.","tokens_in":34154,"tokens_out":1594,"would_cite":true,"duration_ms":17419,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G20","18G80","05E16","53D37"],"pacs":[],"model":"deepseek-v4-flash","headline":"For coprime positive integers, the incidence algebra of a rational Dyck poset tensored with a line quiver is derived equivalent to the full lattice of lattice paths, proving a conjecture of Chapoton, Ladkani, and Rognerud.","keywords":["derived equivalence","interval tilting","staircase poset","higher Auslander algebra","rational Dyck path","partially wrapped Fukaya category","Chapoton–Ladkani–Rognerud conjecture","Kan extension"],"falsifier":"Compute the corner algebra and Xing's $B_0$ for a coprime pair not covered by explicit examples, say $(a,b)=(5,6)$: if the quiver with relations of $B_0$ differs from the staircase category $C_{m-1}(\\Omega_{a,b})$ in any forced-zero composite, Proposition 5.5 and Theorem 6.6 collapse. Alternatively, find a coprime pair where the Cartan matrices of the two sides fail to satisfy the derived-invariance checks used in Section 8.","tokens_in":33143,"feed_emoji":"🧮","tokens_out":6477,"duration_ms":54921,"temperature":0.7,"pith_summary":"This paper proves the Chapoton–Ladkani–Rognerud conjecture for every pair of coprime positive integers $a,b$: the incidence algebra of the product of a type-$A$ line quiver with the rational Dyck poset $Dyck_{a,b}$ is derived equivalent to the incidence algebra of the full lattice $L_{a,b}$ of lattice paths in the $a\\times b$ rectangle. The proof supplies the missing link in an existing chain: a tilt from the Dyck poset's incidence algebra to Xing's algebra $B_0$, obtained as the canonical corner of a higher Auslander algebra of type $A$. The main tool is a relative interval-tilting theorem that extends the Chapoton–Ladkani–Rognerud mechanism from incidence categories of posets to arbitrary finite $k$-linear categories, tolerating higher-dimensional Hom spaces, non-semisimple diagonal algebras, and zero compositions. A reader should care because the equivalence connects Catalan-indexed poset combinatorics with representation theory and symplectic geometry, and the method produces explicit tilting complexes rather than an existence proof.","feed_headline":"Coprime Dyck posets tilt to full lattice algebras","feed_subtitle":"A relative interval-tilting construction proves the Chapoton–Ladkani–Rognerud conjecture for all coprime pairs.","key_machinery":"The engine is the relative interval-tilting theorem (Theorem 3.9). Given a finite $k$-linear category $X$, a finite poset $Y$, and a monotone family of full sieves $F(y)\\subseteq X$, it builds two categories $\\Gamma(X,Y,F)$ and $\\Gamma^\\sharp(X,Y,F)$, the second differing by a 'target lies in the source fibre' condition that forces certain composites to vanish. The tilting object is the sum over $y\\in Y$, $x\\in F(y)$ of exact right Kan extensions $(\\iota_y)_* \\mathrm{Hom}_{F(y)}(x,-)$, and the theorem computes its opposite endomorphism category as $\\Gamma^\\sharp$; Rickard's derived Morita theorem then gives the equivalence. Iterating this construction coordinate by coordinate inserts one interlacing inequality and its forced-zero composites at each step, and the last category is the corner of a higher Auslander algebra. A forced-prefix deletion identifies rational Dyck paths with such staircases, and the resulting corner is shown to be Xing's $B_0$.","core_discovery":"The central claim is Theorem 6.6: for coprime positive integers $a,b$ and a field $k$, $D^b(k(\\vec A_{a+b}\\times Dyck_{a,b}^{above})) \\simeq D^b(kL_{a,b})$. The discovery that carries the argument is that every finite coordinate staircase admits a derived equivalence to a higher Auslander corner: iterating the relative interval-tilting theorem one coordinate at a time yields $D^b(k\\Omega(H)) \\simeq D^b(e_\\Omega A e_\\Omega)$ for every finite staircase $\\Omega(H)$. For rational Dyck staircases this corner is exactly Xing's $B_0$ after a forced-prefix deletion, and the above/below convention is tracked by taking an opposite algebra. The new Dyck-corner equivalence fills the gap that the previously known results left open, and only the final identification with $L_{a,b}$ requires coprimality.","pith_inferences":["Going beyond the paper, the relative tilting theorem likely applies to sieve-indexed families in categories that are not directed Schur, so it may produce tilting equivalences for many incidence-like algebras with zero relations outside the staircase family.","Going beyond the paper, replicating the Dyck-corner equivalence in non-coprime cases suggests that an orbit-weighted presilting object might extend the CLR equivalence to all $a,b$; the paper itself notes generation remains open there.","Going beyond the paper, the Fukaya realization makes a concrete prediction: if the twisted complexes $L^{(j)}_v$ can be realized by embedded Lagrangians, the half-square zero relations should correspond to holomorphic polygons meeting the stop, a checkable symplectic computation.","Going beyond the paper, the explicit Cartan and Coxeter checks for the $(3,4)$ and $(4,5)$ staircases suggest that computing Coxeter polynomials of $C_j(\\Omega_{a,b})$ at every tilting step for larger pairs would test whether the local corners inherit fractional Calabi–Yau periodicity."],"forward_implications":["The Chapoton–Ladkani–Rognerud conjecture is true for all coprime $a,b$, so the incidence algebra of every rational Dyck poset, after tensoring with a line quiver, is derived equivalent to the full lattice algebra.","Every finite coordinate staircase incidence algebra is derived equivalent to an idempotent corner of a higher Auslander algebra of type $A$, with an explicitly constructed tilting complex.","The Dyck-corner equivalence requires no coprimality and is compatible with replicated algebras: $D^b(k\\vec A_r \\otimes k Dyck_{a,b}) \\simeq D^b((B^{st}_{a,b})^{(r)})$ and its opposite version for the above-diagonal convention.","Staircase derived categories embed as thick subcategories generated by product Lagrangians in partially wrapped Fukaya categories of stopped-disk symmetric products; for coprime Dyck parameters the same category is modeled by Fukaya–Seidel categories of symmetric Brieskorn–Pham singularities.","Along the equivalence the Serre functor of the full path lattice is transported to rotation of the stopped disk, yielding fractional Calabi–Yau dimension $ab/(a+b+1)$."],"supporting_citations":[{"why":"States the conjecture being proved and supplies the interval-tilting mechanism that the paper extends.","marker":"[1]"},{"why":"Provides Xing's equivalence $B_0^{(a+b)}\\simeq A^a_{b+1}$ and the identification of $B_0$ with the Dyck-indexed corner, the cited external step that completes the coprime chain.","marker":"[14]"},{"why":"Ladkani's replicated-algebra equivalence $k\\vec A_r\\otimes\\Lambda\\simeq\\Lambda^{(r)}$, used to pass to replicated algebras.","marker":"[9]"},{"why":"Gottesman's theorem identifying the lattice $L_{a,b}$ with a higher Auslander algebra of type $A$, the right end of the chain.","marker":"[5]"},{"why":"Rickard's tensor-product theorem for tilting complexes, used to tensor the Dyck-corner equivalence with $k\\vec A_{a+b}$.","marker":"[13]"},{"why":"Rickard's derived Morita theorem, which converts the computed endomorphism algebra into the claimed triangle equivalence.","marker":"[12]"},{"why":"Dyckerhoff–Jasso–Lekili's Fukaya-category realization of higher Auslander algebras of type $A$, used for the symplectic interpretations.","marker":"[3]"}],"fun_headline_variants":["Coprime Dyck tilting proves lattice-algebra conjecture","Interval tilting gives Dyck–Auslander corner equivalence","Coprime Dyck posets tilt to full lattice algebras","Dyck staircase tilting fills Chapoton–Ladkani–Rognerud gap","Rational Dyck posets: tilting to higher Auslander corners"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on accepting two cited equivalences from Xing's work—that the corner algebra built from Dyck paths is exactly $B_0$, and that $B_0$'s replicated version is derived equivalent to the relevant higher Auslander algebra—neither of which is reproved here.","fun_headline_variants_meta":{"raw":{"variants":["Coprime Dyck tilting proves lattice-algebra conjecture","Interval tilting gives Dyck–Auslander corner equivalence","Coprime Dyck posets tilt to full lattice algebras","Dyck staircase tilting fills Chapoton–Ladkani–Rognerud gap","Rational Dyck posets: tilting to higher Auslander corners"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00042,"raw_usage":{"total_tokens":2229,"prompt_tokens":1079,"completion_tokens":1150,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":695,"completion_tokens_details":{"reasoning_tokens":1059}},"tokens_in":695,"tokens_out":1150,"duration_ms":11591,"temperature":1.0,"reasoning_tokens":1059,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:15:50.963834+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the corner algebra and Xing's $B_0$ for a coprime pair not covered by explicit examples, say $(a,b)=(5,6)$: if the quiver with relations of $B_0$ differs from the staircase category $C_{m-1}(\\Omega_{a,b})$ in any forced-zero composite, Proposition 5.5 and Theorem 6.6 collapse. Alternatively, find a coprime pair where the Cartan matrices of the two sides fail to satisfy the derived-invariance checks used in Section 8.","supporting_citations":[{"cited_title":"On derived equivalences for categories of generalized intervals of a finite poset","cited_arxiv_id":"1801.05154","evidence_quote":"States the conjecture being proved and supplies the interval-tilting mechanism that the paper extends."},{"cited_title":"Ladkani,On derived equivalences of lines, rectangles and triangles, J","cited_arxiv_id":null,"evidence_quote":"Ladkani's replicated-algebra equivalence $k\\vec A_r\\otimes\\Lambda\\simeq\\Lambda^{(r)}$, used to pass to replicated algebras."},{"cited_title":"Rickard, Derived equivalences as derived functors,J","cited_arxiv_id":null,"evidence_quote":"Rickard's tensor-product theorem for tilting complexes, used to tensor the Dyck-corner equivalence with $k\\vec A_{a+b}$."},{"cited_title":"Rickard,Morita theory for derived categories, J","cited_arxiv_id":null,"evidence_quote":"Rickard's derived Morita theorem, which converts the computed endomorphism algebra into the claimed triangle equivalence."}],"review_version":1}