{"id":"8c2f35e1-f58a-4802-bd25-5c4b38bcc606","arxiv_id":"2507.18230","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Echelonmotion, defined via the Bruhat decomposition of a poset's Cartan matrix, coincides with rowmotion on semidistributive lattices and on trim lattices under vertebral linear extensions, and echelon-independent lattices are exactly semidistributive.","lead":"The paper studies echelonmotion, a rowmotion-like bijection on finite posets built from Bruhat decompositions of Cartan matrices. It proves echelonmotion equals rowmotion on semidistributive lattices and on trim lattices with a suitable 'vertebral' ordering, and it introduces a new class of echelon-independent posets.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the only fragile dependency, Lemma 2.3, is directly supported by the cited DW23 results.","rationale":"The reader's verdict is ACCEPT with high confidence, and their weakest assumption is the imported Lemma 2.3. I agree this is the most load-bearing external dependency for the central theorem, but it does not constitute a real gap: the lemma follows directly from the cited DW23 results by considering the interval [Pop_L(x), x]. I also reviewed the other potentially delicate steps: the existence of the linear extensions in Theorem 1.8, the use of Proposition 3.1 in constructing Ech_σ(j) = m, and the inner induction in the trim-lattice proof of Theorem 1.4. In each case the argument is valid, though the trim-lattice proof's phrase 'proceed by induction on the lattice L' is somewhat terse and is best read as a well-founded induction on the element x. Because the central claim is supported by published, peer-reviewed results and the internal proofs are consistent, no adjustment to the reader's verdict is warranted.","tokens_in":20697,"tokens_out":13985,"duration_ms":146300,"concrete_test":"Brute-force compute μ_L(Pop_L(x), x) for every element x of every semidistributive lattice with at most 8 elements, using a lattice enumeration package (e.g., Sage), and confirm that the value is always ±1. This directly tests the imported Lemma 2.3 in the exact regime used by Theorem 1.3; a vanishing value would invalidate the hypothesis of Proposition 3.3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The forward direction of Theorem 1.3 rests on Proposition 3.3, whose hypothesis μ_L(Pop_L(x), x) ≠ 0 is supplied by Lemma 2.3, imported from Defant--Williams [DW23]. I checked the derivation of Lemma 2.3 from the cited results. For Q = [Pop_L(x), x], Q is semidistrim by [DW23, Theorem 7.8], and the minimum of Q is Pop_L(x). Since Pop_Q(max) = Pop_L(x) = min(Q), the interval does not satisfy the contractibility condition in [DW23, Corollary 8.2], so the order complex of Q \\ {min, max} is homotopy equivalent to a sphere and the Möbius value is ±1. Thus the lemma supplies the required nonzero value. The other input, equation (5) identifying max Υ(x) with Row_L(x), is also an established published result. The remaining arguments in Theorem 1.3, including the converse via Theorem 1.8, are internally consistent; I found no unproved assumption beyond standard published facts.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces echelonmotion, a bijection Ech_σ on a finite poset R defined from the Bruhat decomposition of the Cartan matrix of R with respect to a linear extension σ. It proves that on semidistributive lattices this bijection coincides with rowmotion for every linear extension (Theorem 1.3), and that a lattice is echelon-independent exactly when it is semidistributive. For trim lattices, the authors define vertebral linear extensions and prove that echelonmotion with respect to any such extension equals rowmotion (Theorem 1.4). It further shows that echelonmotion on Eulerian posets is an involution (Theorem 1.5), and gives several structural results for echelon-independent connected posets, including boundedness (Theorem 1.6), absence of fixed points (Theorem 1.7), and semidistributivity of the MacNeille completion (Theorem 1.8). An efficient algorithm for testing echelon-independence is presented and used to report that the Bruhat order on S_n is echelon-independent for n ≤ 5 but not for n = 6.","tokens_in":20885,"tokens_out":9350,"duration_ms":99110,"significance":"The paper substantially generalizes the distributive-lattice result of Klász–Marczinzik–Thomas and connects rowmotion to a concrete linear-algebraic construction via Bruhat decomposition. The characterization of echelon-independent lattices as semidistributive is clean and nontrivial, and the trim-lattice result provides a new instance where echelonmotion matches rowmotion even outside the semidistributive setting. The proofs are systematic and transparent, using the rank criterion (3) and the pop-stack operator. A particular strength is that the main external dependencies, especially Lemma 2.3 imported from Defant–Williams [DW23] and equation (5), are published results that do not assume the target equality; I checked the derivation of Lemma 2.3 from [DW23, Corollary 8.2 and Theorem 7.8] and it is valid. The paper also provides concrete computational evidence, including the S_6 counterexample and tests for modular lattices, making the claims easy to verify.","major_comments":[],"minor_comments":[{"comment":"In the displayed equation after 'Since u ∧ x ≠ Pop_L(x)', the summation set is written as Δ_Qx(u ∧ v), but v is not defined; it should be Δ_Qx(u ∧ x).","section":"Section 3, proof of Proposition 3.3"},{"comment":"After invoking [Sch16, Proposition 8.26], the assertion 'This implies that J_L and M_L are contained in R' would be clearer with a one-sentence justification: a join-irreducible element that is a join of elements of R must be one of those elements, and dually for meet-irreducibles.","section":"Section 4, proof of Theorem 1.8"},{"comment":"The proof of Lemma 2.3 is only sketched through a chain of citations to [DW23]. Since this lemma is load-bearing for Theorems 1.3 and 1.4, a slightly fuller explanation of the interval reduction to [DW23, Corollary 8.2] would improve readability, even though the cited result is published and correct.","section":"Section 2, Lemma 2.3"}],"recommendation":"accept","confidential_remarks":"The manuscript is a strong, self-contained contribution modulo standard published external results. The typo in the proof of Proposition 3.3 and the minor clarity points listed can be handled during copyediting and do not require another round of review."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a genuine generalization of KMT25: echelonmotion, defined via the Bruhat decomposition of Cartan matrices, is shown to coincide with Barnard's rowmotion on every semidistributive lattice, independent of the chosen linear extension. That is the main result, and it is new. The trim-lattice theorem is also new: for every trim lattice there is a specially constructed 'vertebral' linear extension making echelonmotion agree with rowmotion. The Eulerian involution theorem is a short, elegant linear-algebra argument, and the structural results on echelon-independent posets (bounded, no fixed points, semidistributive MacNeille completion) are useful.\n\nThe proofs are carefully constructed. Proposition 3.3 is a clean reduction: if the Möbius values mu(Pop(x), x) are nonzero and x maps to max_sigma Upsilon(x) bijectively, then Ech_sigma equals that map. In the semidistributive case, equation (5) identifies that maximum with Row(x), so the result follows. The trim case is a longer induction, but the logic is visible and the use of [DW23, Proposition 9.5] and Lemma 6.3 is sound. The counterexample to the converse of Theorem 1.8 is explicit and convincing.\n\nThe main dependency is Lemma 2.3, imported from Defant--Williams, asserting mu(Pop(x), x) is +/-1 for semidistrim lattices. If that failed, the central equivalence would collapse. I checked the derivation: for Q = [Pop(x), x], the interval's maximum is Pop(x), the interval's minimum, so the contractibility condition in [DW23, Corollary 8.2] fails, forcing the order complex to be sphere-like and the Möbius value nonzero. The dependency is solid. The other external input, equation (5), is also an established published result.\n\nThe only soft spot worth naming is that the vertebral linear extension construction in Section 6 is somewhat heavy, and the proof there is intricate; an independent check of the induction would be wise. But the paper is honest about what is imported, the computational checks are clearly labeled as checks, and I found no circularity or fitting. The citation pattern is reasonable: prior work by the same group is cited for exactly the needed lemmas, and those lemmas are published.\n\nThis paper deserves a serious referee. It is not a desk reject. I would cite it in future work on rowmotion and lattice dynamics, and I would bring it to reading group.","headline":"Genuine generalization of KMT25: echelonmotion equals rowmotion on semidistributive lattices, with a solid trim-lattice analogue and a clean Eulerian involution result; the main external dependency checks out.","tokens_in":21482,"tokens_out":1930,"would_cite":true,"duration_ms":21650,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["06A07","06B05","15A23"],"pacs":[],"model":"deepseek-v4-flash","headline":"Echelonmotion and rowmotion are the same bijection on every semidistributive lattice, and the converse characterizes echelon-independence.","keywords":["echelonmotion","rowmotion","semidistributive lattices","trim lattices","Eulerian posets","Bruhat decomposition","echelon-independence","linear extensions"],"falsifier":"Find a finite lattice that is echelon-independent but not semidistributive, which would refute the converse half of Theorem 1.3; more directly, compute $\\mu_L(\\mathrm{Pop}_L(x), x)$ on any semidistributive or trim lattice and look for a zero, or compare $\\mathrm{Ech}_\\sigma(x)$ with $\\mathrm{Row}_L(x)$ across all linear extensions and elements of a single lattice.","tokens_in":20487,"feed_emoji":"🔁","tokens_out":14834,"duration_ms":127387,"temperature":0.7,"pith_summary":"Echelonmotion is a matrix-factorization recipe that assigns to a finite poset and a linear extension a bijection of the poset. The paper proves that on semidistributive lattices this bijection is exactly rowmotion, the standard dynamical map that had previously been understood mainly on distributive lattices. It further proves the converse: a lattice is echelon-independent if and only if it is semidistributive. For trim lattices, which need not be semidistributive, the paper identifies a class of vertebral linear extensions for which the equality with rowmotion still holds, and it shows that on Eulerian posets echelonmotion is an involution. The paper thereby gives rowmotion a uniform matrix realization on much broader poset families and an efficient test for when that realization is canonical.","feed_headline":"Echelonmotion equals rowmotion on semidistributive lattices","feed_subtitle":"For these lattices any linear extension gives the same rowmotion map; the property characterizes semidistributivity.","key_machinery":"The central object is echelonmotion: for an $n$-element poset $R$ and a linear extension $\\sigma$, form the $n \\times n$ Cartan matrix $W_{R,\\sigma}$, factor it under the double-coset decomposition $\\mathrm{GL}_n(\\mathbb{C}) = \\bigsqcup_{P \\in S_n} BPB$, and let the unique permutation matrix define the bijection $\\mathrm{Ech}_\\sigma \\colon R \\to R$. The key mechanism is Proposition 3.3: if a lattice $L$ satisfies $\\mu_L(\\mathrm{Pop}_L(x), x) \\neq 0$ for every $x$ and the map $x \\mapsto \\max_\\sigma(\\Upsilon_L(x))$ is a bijection, then $\\mathrm{Ech}_\\sigma(x) = \\max_\\sigma(\\Upsilon_L(x))$, where $\\Upsilon_L(x)$ is the set of elements whose meet with $x$ equals $\\mathrm{Pop}_L(x)$. A known characterization of rowmotion identifies this maximum with $\\mathrm{Row}_L(x)$ on semidistributive lattices, so the proposition converts echelonmotion into rowmotion; the same proposition, applied with vertebral linear extensions, handles trim lattices, and a separate argument shows the permutation matrix is its own inverse on Eulerian posets.","core_discovery":"The paper's central claim is Theorem 1.3: if $L$ is a finite semidistributive lattice, then $\\mathrm{Ech}_\\sigma = \\mathrm{Row}_L$ for every linear extension $\\sigma$ of $L$, where $\\mathrm{Row}_L$ is the rowmotion operator for semidistributive lattices. In particular, every semidistributive lattice is echelon-independent, meaning the chosen linear extension is irrelevant. The converse is also proved: every echelon-independent lattice is semidistributive. For trim lattices, the paper shows that a specially chosen vertebral linear extension still makes $\\mathrm{Ech}_\\sigma$ agree with rowmotion, and for Eulerian posets it proves that $\\mathrm{Ech}_\\sigma$ is an involution for every linear extension.","pith_inferences":["Editorial inference: the paper's rank-based test could be used as a fast certificate that a lattice is not semidistributive: a single pair of linear extensions with different echelonmotion would refute semidistributivity without constructing the full canonical join complex.","Editorial inference: the vertebral-linear-extension construction is a natural candidate to transfer from trim lattices to independence posets, in which case the equality between echelonmotion and rowmotion would give that larger class a canonical matrix realization.","Editorial inference: because echelonmotion is defined for any finite-dimensional algebra with invertible Cartan matrix, the lattice-theoretic results point toward an algebraic analogue of rowmotion whose incidence-algebra special case is the poset echelonmotion studied here."],"forward_implications":["Every semidistributive lattice is echelon-independent, so its rowmotion bijection can be obtained from any linear extension by a single double-coset factorization of its Cartan matrix.","An echelon-independent connected poset is bounded, its echelonmotion is fixed-point-free when the poset has at least two elements, and its MacNeille completion is semidistributive.","Trim lattices that are not semidistributive still have a vertebral linear extension on which echelonmotion and rowmotion coincide.","Eulerian posets, including face lattices of polytopes and intervals in the strong Bruhat order, carry an echelonmotion involution for every linear extension.","Echelon-independence can be tested by computing ranks of at most $16n$ matrices rather than enumerating all linear extensions of an $n$-element poset."],"supporting_citations":[{"why":"Proved the distributive-lattice case that echelonmotion, there studied as the inverse Coxeter permutation, agrees with rowmotion; this is the base result the paper generalizes.","marker":"[KMT25]"},{"why":"Supplies Lemma 2.3 on nonzero Möbius values in semidistrim lattices and the identity max $\\Upsilon_L(x) = \\{\\mathrm{Row}_L(x)\\}$ used to match echelonmotion with rowmotion.","marker":"[DW23]"},{"why":"Defines rowmotion on semidistributive lattices through canonical join label sets, the target map named in Theorem 1.3.","marker":"[Bar19]"},{"why":"Introduces trim lattices and the spine and join-meet bijection that underlie the construction of vertebral linear extensions.","marker":"[Tho06]"},{"why":"Defines rowmotion on trim lattices and supplies the Galois-graph and decomposition facts used in the proof of Theorem 1.4.","marker":"[TW19b]"},{"why":"Gives the meet-semidistributivity criterion via maximal elements of $\\Upsilon_L(j)$, used to show echelon-independent lattices are semidistributive.","marker":"[FJN95]"},{"why":"Provides the theory of MacNeille completions and the structural fact about join- and meet-irreducible elements used in Theorem 1.8.","marker":"[Sch16]"},{"why":"Relates Möbius values to reduced Euler characteristics of order complexes, the bridge needed for Lemma 2.3.","marker":"[Sta97]"}],"fun_headline_variants":["For semidistributive lattices, echelonmotion equals rowmotion","Echelon-independence characterizes semidistributive lattices","Echelonmotion is an involution on Eulerian posets","Trim lattices get echelonmotion-rowmotion agreement"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that echelonmotion equals rowmotion rests on the imported fact that in every semidistrim lattice the Möbius value $\\mu_L(\\mathrm{Pop}_L(x), x)$ is never zero; if some semidistributive or trim lattice violated that fact, the identification would not follow from Proposition 3.3.","fun_headline_variants_meta":{"raw":{"variants":["For semidistributive lattices, echelonmotion equals rowmotion","Echelon-independence characterizes semidistributive lattices","Echelonmotion is an involution on Eulerian posets","Trim lattices get echelonmotion-rowmotion agreement"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000747,"raw_usage":{"total_tokens":3333,"prompt_tokens":955,"completion_tokens":2378,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":571,"completion_tokens_details":{"reasoning_tokens":2297}},"tokens_in":571,"tokens_out":2378,"duration_ms":18368,"temperature":1.0,"reasoning_tokens":2297,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:38:32.758286+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a finite lattice that is echelon-independent but not semidistributive, which would refute the converse half of Theorem 1.3; more directly, compute $\\mu_L(\\mathrm{Pop}_L(x), x)$ on any semidistributive or trim lattice and look for a zero, or compare $\\mathrm{Ech}_\\sigma(x)$ with $\\mathrm{Row}_L(x)$ across all linear extensions and elements of a single lattice.","supporting_citations":[],"review_version":1}