{"id":"7b4c0fba-ccd7-458f-aab6-ec6df52dc5e8","arxiv_id":"2506.08858","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Edge-labelled polygonal lattices carry preorders on square-equivalence classes of maximal chains that descend to contractions under lattice quotients, yielding new structural results for Cambrian lattices and the Kapranov-Voevodsky map.","lead":"This paper develops a general theory of preorders on equivalence classes of maximal chains in polygonal lattices, using edge labellings, and shows these preorders descend to lattice quotients as contractions. It applies the theory to Coxeter weak orders and Cambrian lattices, proving that the Kapranov-Voevodsky map from two-dimensional higher Bruhat orders to higher Stasheff-Tamari orders has connected fibres.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.11's proof assumes a maximal chain contains the meet of two minimal elements of the next fibre; without connected fibres, the contraction claims do not follow.","rationale":"The reader's weakest-assumption analysis identifies exactly the same load-bearing gap: the connectedness of the fibres of MC(q) in Lemma 2.11 is asserted through a step that requires a maximal chain to contain a meet it need not contain. I verified the local logic of the proof and found the variable shadowing (redefining y as the meet) followed by the claim 'Since C_L also passes through ˇx and y' to be the precise point where the argument breaks. This is not a cosmetic issue: Theorem 3.8(2) uses Lemma 2.11 twice, once to connect elements of MC(q)^{-1}(C) and once to lift the equivalence-class-level connectedness, and Corollary 8.7's 'connected fibres' conclusion is a direct consequence. The rest of the paper is careful and the applications are plausible, but this unproven lemma is load-bearing. A conditional verdict requiring a corrected proof or a formal verification of Lemma 2.11 is appropriate. I do not see a stronger concern that would justify rejection: the framework is coherent, the quotient edge-labelling construction is independent of the gap, and the Coxeter-specific results (Theorems 6.3 and 6.15) are built on more detailed arguments. The proposed computational test would settle whether the lemma is false or merely underproved.","tokens_in":47428,"tokens_out":7455,"duration_ms":83898,"concrete_test":"Implement an exhaustive search over all finite polygonal lattices of size up to, say, 8 (or all distributive lattices on up to 4 generators) and all lattice congruences θ. For each quotient q: L → L/θ, list every maximal chain C of L/θ and compute the fibre MC(q)^{-1}(C); check whether the fibre is connected under polygon moves. A single disconnected fibre disproves Lemma 2.11. In addition, for the specific proof step, search for two chains C_L, C'_L in one fibre such that min_{j+1}(C_L) ∧ min_{j+1}(C'_L) is not an element of C_L; this directly exhibits the unjustified assertion. If the exhaustive check finds no counterexample, the lemma may still be true, but the written proof would need a corrected argument before the contraction theorem can be accepted.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central contraction result (Theorem 3.8(2)) and its applications (Corollary 8.7 for the Kapranov–Voevodsky map) rely on Lemma 2.11, which asserts that every fibre of MC(q) is connected by polygon moves. In the proof of Lemma 2.11, after choosing the first index j where two chains C_L and C'_L in the same fibre differ, the authors set x := max_j(C_L) ∧ max_j(C'_L) and y := min_{j+1}(C_L) ∧ min_{j+1}(C'_L), overwriting the original variables. They then assert that C_L passes through ˇx and y (the new meet) and define bC_L := (C_L \\ [ˇx,y]) ∪ bC. However, C_L contains min_{j+1}(C_L), not necessarily the meet of the two minima; a maximal chain in a lattice interval need not contain the meet of two of its elements. For example, in a Boolean lattice, the chain ∅ < {1} < {1,2} < {1,2,3} does not contain {2}, the meet of {1,2} and {2,3}. Consequently the replacement may not be a chain at all, and the induction step fails. Since this is the only place where connectedness of fibres of MC(q) is established, the contraction property itself is unsupported by the proof as written. The gap is concrete and local, not a matter of disagreement with existing consensus.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general order-theoretic framework for preorders on square-equivalence classes of maximal chains in finite polygonal lattices, using forcing-consistent polygonal edge labellings. It proves that such labellings descend to lattice quotients and that the induced map on classes is an order-preserving surjection; with an additional polygon-completeness condition it is a contraction of preordered sets (Theorem 3.8). The framework is applied to lattices of regions of simplicial hyperplane arrangements and to finite Coxeter groups, where it yields partial orders on commutation classes of reduced expressions of the longest element. In the Cambrian setting the contraction is described in terms of c-stable roots, and the authors connect the construction to maximal green sequences of preprojective and hereditary algebras and to the Kapranov–Voevodsky maps from higher Bruhat to higher Stasheff–Tamari orders. The paper also claims connectedness of the fibres of the latter map and gives a non-simply-laced version of a categorical interpretation of Cambrian congruences.","tokens_in":47678,"tokens_out":16352,"duration_ms":192460,"significance":"If Theorem 3.8 and its applications are correct, the paper provides a substantial unifying framework with several new results: the Kapranov–Voevodsky map from B(n,2) to S(n+2,3) is a contraction (Corollary 8.7), Cambrian contractions admit a stability-theoretic description (Corollary 6.11), and there is a new proof of a categorical interpretation of Cambrian congruences beyond the simply-laced case (Proposition 7.15). The paper is carefully organised and relies on established external results rather than on curve-fitting or self-generated predictions, and its conjectures are clearly separated from theorems. However, the central contraction theorem depends on Lemma 2.11, whose proof currently has a gap; since Theorem 3.8(2), Theorem 6.15, Corollary 8.7, and Proposition 7.15 all invoke that lemma, the main results are not yet established as written.","major_comments":[{"comment":"The proof of connectedness of the fibres of MC(q) contains an unjustified step. After redefining x := x ∧ x' and y := y ∧ y', the text asserts that 'C_L also passes through ˇx and y' and then defines bC_L := (C_L \\ [ˇx,y]) ∪ bC. But C_L contains min_{j+1}(C_L), not necessarily the meet y = min_{j+1}(C_L) ∧ min_{j+1}(C'_L); a maximal chain in a lattice interval need not contain the meet of two of its elements, as illustrated by the Boolean chain ∅ < {1} < {1,2} < {1,2,3}, which does not contain {2}, the meet of {1,2} and {2,3}. The proof gives no property of the fibre that would force this meet to lie on C_L, so bC_L may fail to be a chain and the induction step does not go through. This lemma is the only place where connectedness of fibres of MC(q) is established, and it is explicitly invoked in the proof of Theorem 3.8(2); consequently the contraction conclusions in Theorem 6.15, Corollary 8.7, and Proposition 7.15 are affected. The lemma may be true, but the proof as written needs a repair or a different argument.","section":"Section 2.2, Lemma 2.11"}],"minor_comments":[{"comment":"The definition of contraction congruence would be clearer if it stated explicitly that the relation R in 'the transitive closure −→R of the quotient relation R' is the quotient relation defined in the preceding paragraph, since the notation R is reused.","section":"Section 2.1.2"},{"comment":"Reduced words are set in bold in Section 5.1.2, but in Proposition 7.15 the same symbol w0 is used both for the longest element and for a reduced expression of it; this overloading should be disambiguated.","section":"Section 5.1.2 / Proposition 7.15"},{"comment":"The sentence 'One can use the technique of Lemma 2.6 to extend C and C′' is very terse; spelling out the extension step would improve readability and make the proof easier to verify.","section":"Lemma 2.9"}],"recommendation":"major_revision","confidential_remarks":"The paper is potentially valuable and the external scaffolding is solid, but the gap in Lemma 2.11 sits exactly under the main theorem. I would be comfortable with major revision: I do not recommend rejection because the claim is plausible and the issue is local, but I would not want acceptance until the connectedness lemma has a complete proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know two things about arXiv:2506.08858. First, the paper is a serious piece of work: it sets up a general framework for preorders on square-equivalence classes of maximal chains in polygonal lattices, proves a contraction theorem for lattice quotients, and then uses it to organize a lot of existing material (higher Bruhat orders, Cambrian lattices, maximal green sequences) and to get genuinely new results, like the connectedness of fibres of the Kapranov–Voevodsky map in dimension two and a categorical Cambrian quotient in non-simply-laced types. Second, there is a real gap in one load-bearing lemma, Lemma 2.11, and the paper should not be accepted without fixing it.\n\nThe framework itself is the main selling point. Theorem 3.8 is the engine: under forcing-consistent polygon-complete edge labellings, the induced map on equivalence classes of maximal chains is a contraction. The authors use it to explain why the KV map f: B(n,2) → S(n+2,3) has connected fibres, and the c-stability criterion (Theorem 6.3) gives a clean description of which labels survive Cambrian contraction. The non-simply-laced categorical Cambrian quotient (Prop 7.15) is a nice bonus. The paper is careful about attributing prior results, and the proofs are mostly detailed.\n\nNow the soft spot, and it is not a manufactured one. Lemma 2.11 claims that every fibre of MC(q) is connected by polygon moves. In the proof, the authors reuse the letters x and y for meets: x = max_j(C_L) ∧ max_j(C'_L) and y = min_{j+1}(C_L) ∧ min_{j+1}(C'_L). They then assert that C_L passes through ˇx and y, and replace C_L by (C_L \\ [ˇx,y]) ∪ bC. But C_L contains min_{j+1}(C_L), not necessarily the meet of the two minima; a maximal chain in an interval does not have to contain the meet of two of its elements. The stress-test example in a Boolean lattice is exactly right. So the proof as written does not establish connectedness, and since Theorem 3.8(2) and Corollary 8.7 lean on Lemma 2.11, that part is currently unsupported.\n\nThat said, I think the gap is likely repairable: one can work in [ˇx, min_{j+1}(C_L)] and choose a maximal chain bC that contains the two meets, rather than asserting C_L contains them. The argument then has a chance of going through. But that needs to be written out carefully.\n\nWho this is for: people working on higher Bruhat/Stasheff–Tamari orders, Cambrian lattices, and maximal green sequences will want to read it. It deserves a serious referee, conditional on the repair of Lemma 2.11.","headline":"Substantial contraction framework for maximal-chain preorders, but Lemma 2.11's proof has a real gap in the connected-fibres step that needs fixing before acceptance.","tokens_in":48306,"tokens_out":7117,"would_cite":true,"duration_ms":67442,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["06A07","16G20","17B22","13F60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Lattice quotients of polygonal lattices induce contractions on preordered sets of maximal chains, unifying higher Bruhat orders, Cambrian lattices, and maximal green sequences.","keywords":["preorders","polygonal lattices","edge labellings","lattice quotients","Cambrian lattices","higher Bruhat orders","higher Stasheff-Tamari orders","maximal green sequences"],"falsifier":"A concrete check is to look for a finite polygonal lattice $L$, a lattice congruence $\\theta$, and a maximal chain $C$ of $L/\\theta$ such that the fibre of $\\operatorname{MC}(q)$ above $C$ contains two maximal chains not connected by any sequence of polygon moves, or an interval in which the common lower bound used in Lemma 2.11 lies on no maximal chain. Finding either would disprove Lemma 2.11 and remove the connected-fibres conclusion, and with it the contraction conclusion.","tokens_in":47160,"feed_emoji":"🧩","tokens_out":12247,"duration_ms":117757,"temperature":0.7,"pith_summary":"This paper builds one general construction behind several familiar orders on maximal chains: given a finite polygonal lattice with a forcing-consistent polygonal edge labelling, it defines a preorder on square-equivalence classes of maximal chains. It proves that every lattice quotient induces an order-preserving surjection between these preordered sets, and that when the labelling is polygon-complete the induced map is a contraction of preordered sets, meaning it is surjective, has connected fibres, and lifts covering relations. In Coxeter arrangements the construction recovers the weak Bruhat order and produces a distinct partial order for each Coxeter element $c$; the Cambrian quotient is then a contraction of posets. In simply-laced types a covering relation is contracted exactly when its root label is not $c$-stable, so the quotient chain is labelled by the sequence of $c$-stable roots. The same framework shows that the classical map from the two-dimensional higher Bruhat order to the three-dimensional higher Stasheff--Tamari order is a contraction, and gives new partial orders on maximal green sequences of preprojective algebras.","feed_headline":"Every Cambrian quotient contracts its poset of maximal chains","feed_subtitle":"The same edge-labelling construction orders higher Bruhat orders, higher Stasheff–Tamari orders, and maximal green sequences.","key_machinery":"The load-bearing objects are finite polygonal lattices---lattices whose Hasse diagrams are glued from polygons with exactly two maximal chains---equipped with a forcing-consistent polygonal polygon-complete edge labelling $\\lambda$ valued in a poset; polygon-complete means that increasing polygon moves are covering relations of the preorder. A maximal chain is a sequence of covering relations; two chains are square-equivalent when related by moves across square facets, and $\\operatorname{gMC}_\\lambda(L)$ is the resulting preorder whose comparabilities come from replacing the ascending chain of a non-square polygon by the descending chain. The quotient edge labelling $\\lambda_\\theta$ is defined on the quotient lattice using forcing-equivalence classes of covering relations, and the induced map $\\operatorname{gMC}(q)$ is shown to be order-preserving, surjective, fibre-connected, and surjective on covering relations---exactly the defining properties of a contraction of preordered sets. In the Coxeter case the edge labels are positive roots ordered by the heap poset $\\operatorname{Heap}(w_0(c))$, and $c$-stability of a root $\\beta$---that every rank-two subsystem containing $\\beta$ is ordered along the chain as it is in that heap---governs which labels survive Cambrian contraction.","core_discovery":"On the paper's own terms, the central claim is Theorem 3.8 (Theorem A): if $L$ is a finite polygonal lattice with a forcing-consistent polygon-complete polygonal edge labelling $\\lambda$, then for every lattice quotient $q\\colon L\\to L/\\theta$ the induced map $\\operatorname{gMC}(q)\\colon \\operatorname{gMC}_\\lambda(L)\\to \\operatorname{gMC}_{\\lambda_\\theta}(L/\\theta)$ is a contraction of preordered sets. The quotient edge labelling $\\lambda_\\theta$ is inherited from $\\lambda$ through forcing-equivalence, and the map is order-preserving, surjective, connected in fibres, and surjective on covering relations. In the Coxeter special case this yields Theorem 6.15: the Cambrian quotient $\\operatorname{gMC}(q_c)\\colon \\operatorname{gMC}_{\\lambda_{w_0(c)}}(W)\\to \\operatorname{gMC}_{\\lambda_{\\theta_c}}(W_c)$ is a contraction of posets. In simply-laced types, Theorem 6.3 and Corollary 6.11 identify the non-contracted covering relations exactly as those whose edge label is $c$-stable, so the image of a chain is its sequence of $c$-stable roots. The paper also obtains Theorem 7.14: for each Coxeter element $c$, there is a partial order on equivalence classes of maximal green sequences of the preprojective algebra $\\Pi$ for which the map to the maximal-green-sequence poset of the hereditary algebra $\\Lambda_c$ is a contraction of posets.","pith_inferences":["The dimension-two contraction result makes the paper's higher-dimensional conjecture---that the maps $B(n,d)\\to S(n+2,d+1)$ are contractions for all $d$---directly testable: the first obstruction would be a disconnected fibre in some higher dimension.","The $c$-stability criterion is deliberately analogous to the stable objects of a Rudakov stability condition; one could try to prove that the $c$-stable roots of a chain are exactly the stable objects of an explicit stability condition on the module category, which would explain why the quotient is a contraction and not merely a map.","Because the fibres are connected but not always intervals, the quotient of maximal-chain posets is genuinely weaker than an order congruence; this suggests that connected fibres, rather than interval fibres, is the natural quotient notion for many representation-theoretic and combinatorial maps.","If the flawed step in Lemma 2.11 cannot be repaired, the theorem may still be true in the hyperplane-arrangement and Coxeter settings where extra geometry supplies the needed connectivity; identifying exactly which inputs replace the order-theoretic step would delineate the true scope of Theorem A."],"forward_implications":["The classical map from the two-dimensional higher Bruhat order $B(n,2)$ to the three-dimensional higher Stasheff--Tamari order $S(n+2,3)$ is a contraction of posets, so its fibres are connected even though they are not always intervals.","For every Coxeter element $c$ of a finite Coxeter group, the Cambrian quotient $W\\to W_c$ induces a contraction of posets $\\operatorname{gMC}(q_c)\\colon \\operatorname{gMC}_{\\lambda_{w_0(c)}}(W)\\to \\operatorname{gMC}_{\\lambda_{\\theta_c}}(W_c)$.","In simply-laced types, a covering relation of a chain survives the Cambrian contraction if and only if its label is a $c$-stable root, so the quotient chain is the ordered list of $c$-stable roots of the original chain.","For each Coxeter element $c$, the equivalence classes of maximal green sequences of the preprojective algebra $\\Pi$ carry a partial order for which the canonical quotient map to the hereditary algebra $\\Lambda_c$ is a contraction of posets.","The categorical interpretation of Cambrian congruences through torsion-free classes of preprojective and path algebras holds in non-simply-laced crystallographic types as well."],"supporting_citations":[{"why":"It supplies the definitions of polygonal lattices, polygon moves, and the forcing preorder, along with the facts that maximal chains are connected by polygon moves and that quotients of polygonal lattices are polygonal.","marker":"[Rea16b]"},{"why":"It provides the notion of contraction of preordered sets and the collapse lemma used to turn the Coxeter and Cambrian maps into contractions of posets.","marker":"[CF22]"},{"why":"It introduces Cambrian lattices and Cambrian congruences, the specific quotients to which the main theorem is applied.","marker":"[Rea06]"},{"why":"It establishes c-sortability, c-alignment, and the Euler-form ordering of rank-two subsystems that the c-stability criterion is built from.","marker":"[RS11]"},{"why":"It defines the higher Bruhat and higher Stasheff--Tamari orders and the map between them that the paper shows to be a contraction in dimension two.","marker":"[KV91]"},{"why":"It shows that the higher Bruhat-to-Stasheff--Tamari map is not an order congruence and records the fibre properties that the connectedness result improves.","marker":"[Tho02]"},{"why":"It provides the earlier posets of equivalence classes of maximal green sequences that the paper reinterprets through edge-labelled maximal chains.","marker":"[GW23]"},{"why":"It gives the brick labelling and the torsion-free-class lattice quotient underlying the algebraic realization of Cambrian contractions.","marker":"[Dem+23]"},{"why":"It identifies the maximal-green-sequence poset of the linearly oriented path algebra with the three-dimensional higher Stasheff--Tamari order, completing the comparison.","marker":"[Wil22]"}],"fun_headline_variants":["Cambrian quotients contract maximal chain posets","Edge labellings dictate chain-order contractions","Maximal chains: contractions from Bruhat to Stasheff","Every lattice quotient contracts its chain preorder","Contractions unify Cambrian and green-sequence orders"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"All maximal chains mapping to one chain in a quotient can be linked by polygon moves; the proof of that claim contains a step where a chosen maximal chain is assumed to contain the common lower bound of two elements, but a maximal chain need not contain that common lower bound.","fun_headline_variants_meta":{"raw":{"variants":["Cambrian quotients contract maximal chain posets","Edge labellings dictate chain-order contractions","Maximal chains: contractions from Bruhat to Stasheff","Every lattice quotient contracts its chain preorder","Contractions unify Cambrian and green-sequence orders"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001034,"raw_usage":{"total_tokens":4473,"prompt_tokens":1182,"completion_tokens":3291,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":798,"completion_tokens_details":{"reasoning_tokens":3217}},"tokens_in":798,"tokens_out":3291,"duration_ms":31226,"temperature":1.0,"reasoning_tokens":3217,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:02:20.048255+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check is to look for a finite polygonal lattice $L$, a lattice congruence $\\theta$, and a maximal chain $C$ of $L/\\theta$ such that the fibre of $\\operatorname{MC}(q)$ above $C$ contains two maximal chains not connected by any sequence of polygon moves, or an interval in which the common lower bound used in Lemma 2.11 lies on no maximal chain. Finding either would disprove Lemma 2.11 and remove the connected-fibres conclusion, and with it the contraction conclusion.","supporting_citations":[],"review_version":1}