{"id":"bdadcfeb-d48b-4e49-bf51-22a2c8382145","arxiv_id":"2606.13518","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper establishes equivalence between two prior definitions of weak order on ASM(n) and gives three explicit combinatorial descriptions of its covering relations plus a fiber characterization via bumpless pipe dreams.","lead":"This paper proves that two different definitions of weak order on alternating sign matrices coincide and supplies explicit rules for computing covering relations using ASMs, monotone triangles, and bumpless pipe dreams. It also shows that the fibers of the weak order operators are sublattices inside the strong Bruhat order.","discovery_kind":"unification","skeptic_critique":{"model":"grok-4.3","headline":"Compatibility requires explicit bijections to map covering relations identically between the two weak-order definitions on ASM(n).","rationale":"The reader's weakest_assumption correctly isolates the bijection-preservation step as the point where the two independently defined orders could fail to coincide. Because the full manuscript supplies the explicit rules, the natural next check is an exhaustive small-n verification that does not rely on the paper's internal arguments.","tokens_in":1831,"tokens_out":309,"duration_ms":10480,"concrete_test":"For n=3, enumerate all 7 ASMs, compute the covering relations using each of the three explicit rules given in the paper, and check whether the resulting directed graphs are identical; additionally confirm that the poset matches the image of the weak order on S_3 under the standard embedding. Discrepancy on any edge falsifies compatibility.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the authors' a priori weak order on ASMs coincides with the order induced from Hamaker-Reiner via the standard ASM-monotone-triangle bijection (and likewise for bumpless pipe dreams). This holds only if the three explicit covering-relation rules (on ASMs, on triangles, on pipe dreams) are shown to be compatible under the known bijections; any mismatch on even one pair of objects falsifies the compatibility statement. The paper supplies the rules but the load-bearing step is the verification that they are transported correctly by the bijections.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript establishes compatibility between an a priori definition of weak order on ASM(n) and the definition induced from Hamaker-Reiner weak order on monotone triangles via the standard bijection. It supplies three explicit combinatorial rules for computing covering relations (directly on ASMs, on monotone triangles in a manner distinct from Hamaker-Reiner, and on bumpless pipe dreams) and, in the bumpless-pipe-dream model, characterizes the fibers of the weak-order operators as sublattices of the strong Bruhat order on ASM(n).","tokens_in":1940,"tokens_out":539,"duration_ms":18137,"significance":"If the compatibility statement holds, the work unifies previously separate combinatorial models of weak order on alternating sign matrices and supplies practical, explicit covering rules that can be used for computation. The fiber characterization in the bumpless-pipe-dream language is a concrete combinatorial contribution that strengthens the link to the authors' earlier results on ASM varieties and their K-theoretic representatives. The explicitness of the three covering rules is a strength of the manuscript.","major_comments":[{"comment":"The central compatibility claim requires that the three covering rules coincide under the known bijections between ASMs, monotone triangles, and bumpless pipe dreams. The manuscript states the rules and asserts compatibility, but the load-bearing verification that each pair of rules is transported identically by the bijections (for arbitrary n and arbitrary pairs of objects) is not carried out in sufficient detail; a mismatch on even one pair would falsify the claim that the a priori ASM order coincides with the induced order.","section":"§3"},{"comment":"§5 (fiber characterization): the proof that each fiber is a sublattice of the strong Bruhat order relies on the covering rules being compatible with the bijections; without an explicit check that the bumpless-pipe-dream covering rule is the image of the ASM covering rule, the sublattice statement remains conditional on the unverified transport.","section":"§5"}],"minor_comments":[{"comment":"Notation for the three covering relations is introduced separately; a single comparative table or diagram showing the three rules side-by-side would improve readability.","section":null},{"comment":"The manuscript refers to 'the standard bijection' between ASMs and monotone triangles without restating its definition; including a brief recall (or reference to a numbered equation) would make the transport arguments self-contained.","section":"§2"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for highlighting the need for greater explicitness in the compatibility arguments. We respond to the major comments below.","responses":[{"response":"We agree that the verification of compatibility under the bijections would benefit from a more explicit and self-contained presentation. In the revised manuscript we will insert a dedicated subsection (or short appendix) that records the image of each covering rule under the standard bijections, with a general outline for arbitrary n together with illustrative diagrams for small n to confirm that the three rules are transported identically.","revision_made":"yes","referee_comment":"[§3] The central compatibility claim requires that the three covering rules coincide under the known bijections between ASMs, monotone triangles, and bumpless pipe dreams. The manuscript states the rules and asserts compatibility, but the load-bearing verification that each pair of rules is transported identically by the bijections (for arbitrary n and arbitrary pairs of objects) is not carried out in sufficient detail; a mismatch on even one pair would falsify the claim that the a priori ASM order coincides with the induced order."},{"response":"The argument in §5 is indeed conditional on the compatibility of the covering rules. Once the enhanced verification described in our response to the preceding comment is added, the sublattice claim follows directly. We will also add a short clarifying sentence in §5 that explicitly references the updated compatibility argument.","revision_made":"yes","referee_comment":"[§5] §5 (fiber characterization): the proof that each fiber is a sublattice of the strong Bruhat order relies on the covering rules being compatible with the bijections; without an explicit check that the bumpless-pipe-dream covering rule is the image of the ASM covering rule, the sublattice statement remains conditional on the unverified transport."}],"tokens_in":1453,"tokens_out":403,"duration_ms":15828,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that the a priori weak order the authors used in their earlier ASM variety work matches the one induced from monotone triangles, and they supply direct covering rules in three models along with a fiber characterization.\n\nWhat is new is the compatibility proof, the covering rules on ASMs themselves, a different rule for monotone triangles than Hamaker-Reiner gave, the bumpless pipe dream version, and the result that each fiber is a sublattice of the strong Bruhat order. These pieces extend the 2018 work without simply rephrasing it.\n\nThe paper does a clean job of turning the abstract definitions into concrete algorithms that a reader can apply by hand or in code. It treats the standard bijections between ASMs, triangles, and pipe dreams as established input and focuses on transporting the order.\n\nThe soft spot is the verification that the three explicit rules coincide exactly under those bijections. The central claim stands or falls on that step; any mismatch would falsify compatibility. The abstract states the proofs are there, but without the full derivations it is not possible to check for missed cases or off-by-one issues in the bijection arguments.\n\nThis is for specialists already working on alternating sign matrices, pipe dreams, or weak orders in this corner of combinatorics. A reader in that area gets usable tools and a clearer link between models. It deserves a serious referee because the results are a direct, non-routine extension that connects prior definitions and supplies explicit constructions.","headline":"The paper shows the two weak orders on ASMs are the same and gives explicit covering rules on ASMs, triangles, and pipe dreams plus a sublattice description of the fibers.","tokens_in":2457,"tokens_out":382,"would_cite":false,"duration_ms":19737,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Two definitions of weak order on alternating sign matrices coincide and admit explicit covering rules in three models.","keywords":["alternating sign matrices","weak order","monotone triangles","bumpless pipe dreams","Bruhat order","covering relations","combinatorial bijections"],"falsifier":"A concrete pair of n by n alternating sign matrices that cover each other under one definition of weak order but fail to cover under the other.","tokens_in":2714,"feed_emoji":"","tokens_out":634,"duration_ms":17350,"temperature":0.7,"pith_summary":"The paper reconciles an order on alternating sign matrices coming from monotone triangles with an earlier direct definition used for geometric purposes. Establishing that the two orders agree lets combinatorial and algebraic results move freely between the models. The authors supply concrete rules for deciding when one matrix covers another, phrased directly on the matrices, on the triangles (by a method different from earlier work), and on bumpless pipe dreams. In the pipe-dream language they further show that the sets of matrices mapping to the same element under the weak-order operators form sublattices inside the stronger Bruhat order.","feed_headline":"Weak orders on alternating sign matrices coincide","feed_subtitle":"Compatibility gives explicit covering rules on matrices, triangles and pipe dreams; fibers form Bruhat sublattices","key_machinery":"The bijections among alternating sign matrices, monotone triangles, and bumpless pipe dreams that preserve weak-order covering relations.","core_discovery":"The weak order on ASM(n) induced from monotone triangles via the standard bijection is identical to the a priori different weak order previously defined directly on ASMs. Explicit covering relations are given on ASMs themselves, on monotone triangles by a rule distinct from Hamaker-Reiner, and on bumpless pipe dreams. In the bumpless-pipe-dream model the fibers of the weak-order operators are each a sublattice of the strong Bruhat order on ASM(n).","pith_inferences":["The compatibility may let geometric or K-theoretic statements proved in one model be restated combinatorially in another.","The sublattice property on fibers could be used to study chain decompositions or representation-theoretic multiplicities inside the Bruhat order.","Analogous compatibility statements might hold for other partial orders defined on the same three families of objects."],"forward_implications":["Covering relations can be read off directly from the pattern of entries in an alternating sign matrix.","A distinct rule computes covers when the same matrices are viewed as monotone triangles.","Bumpless pipe dreams supply a third explicit combinatorial rule for the covers.","Each fiber of a weak-order operator is a sublattice inside the Bruhat order when realized by bumpless pipe dreams."],"fun_headline_variants":["Weak orders on ASMs coincide from all definitions","ASM weak orders identical under multiple definitions","Weak order fibers form Bruhat sublattices on ASMs","Covers for weak order on ASMs triangles and pipe dreams"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The standard bijections between alternating sign matrices, monotone triangles, and bumpless pipe dreams preserve the covering relations of the two independently defined weak orders.","fun_headline_variants_meta":{"raw":{"variants":["Weak orders on ASMs coincide from all definitions","ASM weak orders identical under multiple definitions","Weak order fibers form Bruhat sublattices on ASMs","Covers for weak order on ASMs triangles and pipe dreams"]},"model":"grok-4.3","cost_usd":0.0053,"raw_usage":{"total_tokens":2581,"prompt_tokens":707,"num_sources_used":0,"completion_tokens":61,"cost_in_usd_ticks":52999500,"prompt_tokens_details":{"text_tokens":707,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1813,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":707,"tokens_out":61,"duration_ms":11102,"temperature":1.0,"reasoning_tokens":1813,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T06:15:08.102681+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete pair of n by n alternating sign matrices that cover each other under one definition of weak order but fail to cover under the other.","supporting_citations":[],"review_version":1}