{"id":"735b40d8-24d4-4bf2-b88f-031d3ecbfa90","arxiv_id":"2512.19034","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Type D Brion atoms are described explicitly and uniformly with the other classical types, giving complete indexing sets for Brion's cohomology formula.","lead":"Brion's formula expresses orbit-closure cohomology classes as sums of Schubert classes; this paper makes the indexing terms explicit for all classical groups, completing the difficult type-D cases. It also defines involution Schubert polynomials for every classical type and states several numerical conjectures.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.15's weak-order edge criterion is the hinge for the type-D main theorems, yet its proof is only a citation to Wyser's thesis; a translation error there would invalidate the new Brion-atom decomposition.","rationale":"The reader's weakest assumption coincides with the main risk I see. The paper's novelty is type D, and its proofs do substantial work in Section 7, but the foundation—the weak-order edge criterion—is imported from Wyser's PhD thesis without a self-contained verification. This is especially delicate because the parametrization of K-orbits and the Richardson–Springer map are also taken from Wyser, so a mismatch between the paper's formulas and Wyser's conventions is easy to introduce. The concrete test of small-rank enumeration would resolve this directly. I also note smaller gaps (the deferred equality in Proposition 2.26 and the cited erratum in [10] for d_z), but they are less central: if Theorem 2.15 is correct, the rest of the type-D argument appears coherent; if not, no amount of internal consistency in Section 7 can save the claim. Therefore the conditional verdict is appropriate: the paper should be accepted only after the weak-order translation is verified.","tokens_in":61332,"tokens_out":9102,"duration_ms":81253,"concrete_test":"Enumerate the weak-order graph for a small rank type-D example (e.g., DI n=4, DII n=3, DIII n=4, DIV n=3) directly from Wyser's thesis [31, §§5.1.2, 5.2.2, 5.3.2], and compare each edge and doubling to the conditions (a)–(c) and doubled-edge rules in Theorem 2.15, focusing on t_{-1} edges and on edges where C(s)⊆M(β). If every edge matches, the translation is validated and the main theorem's input is secure; any mismatch would show the type-D formulas in Section 7 are built on an incorrect premise.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorems 1.3–1.4 reduce the computation of Brion atom sets to the weak-order graph of Section 1.4. Theorem 2.15 is the precise edge criterion used throughout, and in type D it introduces new t_{-1} edges and doubled-edge conditions (e.g., Fix(t0 z) rather than Fix(z) for type DII) that have no analogue in the paper's earlier A/B/C material. The proof of Theorem 2.15 is not a derivation: it says the statement 'follows by carefully comparing Wyser's description of the weak order' [31]. No case check or translation table is provided. Since Lemma 2.21, Proposition 2.26, and every type-D proposition in Section 7 (7.21–7.40) invoke this edge criterion, a single mis-translated condition—especially the Fix(t0 z) doubling rule or the C(s)⊆M(β) case for t_{-1}—would propagate into W^G_K(γ) and make the main decomposition false. The concern is not internal inconsistency; it is that the central input is an unverified new parsing of a dense external source.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper gives a uniform combinatorial description of the sets of Weyl-group elements (Brion atoms) that appear in Brion's cohomology formula for closures of symmetric-subgroup orbits on flag varieties, for all classical types. The authors define extended Brion atoms E^G_K(z), decompose them into shape blocks E^G_K(z,M) indexed by noncrossing symmetric perfect matchings, and identify each block with the principal filter of an explicit generator under an explicit partial order (Theorems 1.3 and 1.4). Types A, B, and C are largely reformulated from prior work; the main new content is type D, developed in Sections 6 and 7. The paper also proves classifications of multiplicity-free orbit closures and introduces involution Schubert polynomials and Stanley symmetric functions with several conjectures.","tokens_in":61600,"tokens_out":6558,"duration_ms":70319,"significance":"If the main theorems are correct, they turn Brion's abstract formula (1.4) into a concrete finite algorithm for all classical symmetric pairs, and they unify and extend earlier work on types A/B/C. The explicit matchings, generators, rank functions, and the detailed type-D propositions are substantial contributions. The paper also provides falsifiable conjectures about Schur P/Q expansions. It is not accompanied by machine-checked proofs or code; the strengths are the explicit combinatorial framework, the detailed structural lemmas in Section 7, and the concrete applications in Section 8. However, several load-bearing steps are delegated to external sources or to 'easy to check' assertions, so the result is not yet established to the standard required for acceptance.","major_comments":[{"comment":"This theorem is the explicit edge criterion for the weak-order graph that underlies Lemma 2.21 and hence all type-D propositions in Section 7. Its proof consists solely of the statement that the criterion 'follows by carefully comparing Wyser's description of the weak order' [31]. In type D the new conditions—the C(s)⊆M(β) case for t_{-1}, the doubled-edge conditions governed by Fix(t0 z) rather than Fix(z), and the sign-restriction conditions in (b)–(c)—have no derivation and no case-by-case translation table. A single mis-translated condition would propagate immediately into W^G_K(γ) and invalidate Theorems 1.3 and 1.4. Please supply a proof, or at least a complete type-by-type comparison table for Section 2.5, before the result can be regarded as verified.","section":"Section 2.6, Theorem 2.15"},{"comment":"The foundational description of E_D(z) rests on Theorem 7.4, whose proof invokes [16, Thm. 3.17] and then says it is a 'straightforward but somewhat tedious exercise' to translate the reduced-word relations (7.4)–(7.5) into the word relation ≪_D. This is not a local remark: Proposition 7.13, Corollary 7.14, and all subsequent shape results for types DI–DIV depend on Theorem 7.4. The missing translation is load-bearing for the main type-D claims and should be written out, or at least organized into a table of cases, rather than left as an exercise.","section":"Section 7.1, Theorem 7.4"},{"comment":"The equality E^G_K(z) = (the set listed in Table 4) is deferred. The proof of Proposition 2.26 says the converse containment 'will also follow from our proof of Theorem 1.3 in Sections 4, 5, and 7,' and that one must observe every matching occurs as an aligned shape for some γ. In type A this existence observation is Lemma 4.6, and for types B/C it is supplied by the cited results in [10]. For type D, however, no analogous lemma is stated or proved: Propositions 7.22, 7.32, and 7.38 show that each matching has a generator in E_D(z), but they do not show that for every matching M there is some γ with ψ(γ)=z and M∈Aligned(γ). Without that lemma, the first displayed disjoint-union equality in Theorem 1.3 is not fully established for type D.","section":"Section 2.7, Proposition 2.26 and Section 7"},{"comment":"The equivalence between (a) and (c) is the bridge from weak-order paths to the matching-alignment condition used in Theorem 1.3. Its proof invokes [10, Lem. 7.4] and compresses the obstruction analysis into the phrase 'we see that P fails to lift to a path ... if and only if ...'. Given how much of Section 7 depends on this lemma, a fuller verification of the obstruction cases—especially the type-BI alternation and the type-CI/DIII/DIV sign conditions—would be appropriate. This is less severe than the previous points but still a significant proof gap.","section":"Section 2.6, Lemma 2.21"}],"minor_comments":[{"comment":"The claimed list of 14 distinct clans for base set {1,2,3} contains the entry (-,-,+) twice and omits (-,-,-). The count should be 14, but the displayed list is incorrect.","section":"Example 2.2"},{"comment":"Typo: 'subroups' should be 'subgroups'. In the abstract, 'as a linear combinations' should be 'as linear combinations.'","section":"Abstract and Section 1.1"},{"comment":"The map from W_n^+ to W_{n+1}^+ is called embed_{n,k}^{DII} in Lemma 7.27, but the proof of Proposition 7.29 refers to it as embed_{n,k}^{DI}. This appears to be a notational slip.","section":"Section 7.4, proof of Proposition 7.29"},{"comment":"Typo: 'multiplicy-free' should be 'multiplicity-free' in the first sentence of Section 8.1.","section":"Section 8.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious and substantial contribution if the indicated gaps are filled. The main risk is the unverified translation of Wyser's weak-order description in Theorem 2.15, which is the hinge for all type-D results. I would encourage the editor to require the author to add a full proof or a detailed verification appendix for that theorem, as well as the missing type-D analogue of the 'every matching is aligned' lemma. I do not see a fundamental obstruction, and the type-D framework is plausible, but in its current form the central claim is not fully independent of external assertions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is a serious completion of a program: it gives a uniform combinatorial description of Brion atoms for all classical symmetric pairs, and the genuinely new part is type D. The type A/B/C material is largely recycled from the author's own previous papers, but the type D treatment (Section 7) is detailed and structurally coherent. The main theorems 1.3 and 1.4 reduce Brion's formula to an explicit finite algorithm, and the definitions of shapes, matchings, and generators are independent of the formula itself; there is no circular fitting. The embedding arguments that deduce DII from DI and DIV from DIII are useful, and the author's acknowledgment of the erratum in a cited source is a good sign.\n\nThe soft spot is real and it is Theorem 2.15. That theorem gives the weak-order edge criterion for all non-A types, and the type-D main propositions depend on it at every step. But its proof is a single sentence: it follows by carefully comparing Wyser's thesis. For types DI and DII, the criterion introduces new features—t_{-1} edges, doubled-edge conditions involving Fix(t_0 z) instead of Fix(z)—that have no analogue in the earlier types. The stress-test note is right: a translation error there would propagate into every type-D result. I did not find an actual error, and the rest of the paper is consistent with the theorem being true, but the paper as written asks the reader to take the most load-bearing input on faith.\n\nAnother smaller issue: Proposition 2.26 defers its equality proof to the later proof of Theorem 1.3, which is logically circular as presented until that later proof is understood. And several case checks in Section 7 are delegated to \"easy to check\" or \"straightforward but tedious\" verification. These are not fatal; they are presentation gaps. I also noticed typos, but nothing that obscures the math.\n\nWho benefits: specialists in Schubert calculus and symmetric-space orbit theory. It is not a branch re-organization, but it is a real contribution that completes a research program and enables concrete classifications (e.g., multiplicity-free orbit closures).\n\nMy recommendation: send it to peer review. The referee should be asked to verify Theorem 2.15 from Wyser's thesis—either present a complete proof in the paper or provide a detailed translation table. If that holds, the paper is solid. As it stands, the type-D claims are credible but not yet fully established.","headline":"A complete and mostly convincing type-D extension of the Brion-atom program, but the key weak-order edge criterion is imported from Wyser's thesis without proof and deserves close scrutiny.","tokens_in":62089,"tokens_out":2798,"would_cite":true,"duration_ms":31001,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14M15","05E05","22E46"],"pacs":[],"model":"deepseek-v4-flash","headline":"Brion's orbit-closure formula becomes an explicit finite algorithm for every classical symmetric pair, with type D finally included. The paper proves that each Brion atom set decomposes into principal filters generated by explicitly defined","keywords":["Brion atoms","symmetric pairs","orbit closures","flag varieties","Schubert classes","type D","weak order graph","involution Schubert polynomials"],"falsifier":"Compute the weak order graph for a small type D example, such as SO(8) with the symmetric pair of type DI, directly from the orbit parametrization and compare the t_{-1} edge and doubling rules in Theorem 2.15 against the explicit list; any mismatch in edge presence or doubling would falsify the claimed description.","tokens_in":61177,"feed_emoji":"🧮","tokens_out":1853,"duration_ms":21663,"temperature":0.7,"pith_summary":"This paper gives a uniform combinatorial description of the sets of Weyl group elements, called Brion atoms, that index the nonzero terms in Brion's cohomology formula for symmetric subgroup orbit closures in flag varieties, covering all classical types including the previously untreated type D. It proves that each Brion atom set is a disjoint union of intervals in an explicit partial order, each interval generated by a single Weyl group element attached to a noncrossing matching. If correct, Brion's abstract formula becomes a finite, concrete algorithm for computing the cohomology class of every K-orbit closure in classical types. The main novelty is a thorough treatment of type D, where new edge rules and doubled edges appear.","feed_headline":"Type D falls in line: Brion atoms get a uniform description","feed_subtitle":"The paper turns Brion's orbit-closure formula into an explicit finite algorithm for all classical symmetric pairs, including the previously","key_machinery":"The engine is the weak order graph on K-orbit indices (clans), whose edges are labeled by simple generators and carry a degree of 1 or 2. The paper translates this graph into explicit combinatorial data: for each twisted involution z, a set of noncrossing perfect matchings on the negated points, a shape operator assigning each atom to a matching, and a transitive relation ≾ on the Weyl group. The generator gen(z,M) is the minimal element of the atom interval of shape M, built from the cycles of z and the matching M via one-line representations and Demazure conjugation formulas.","core_discovery":"For every classical symmetric pair (G,K), each Brion atom set W^G_K(gamma) decomposes as the disjoint union over aligned shapes M of the sets E^G_K(z,M), and each E^G_K(z,M) is exactly the principal filter {w : gen^G_K(z,M) ≾ w} in a graded partial order. The paper constructs the matchings, shape operators, aligned shapes, generators, and partial orders uniformly across types A, B, C, and D, and proves the decomposition theorems (Theorems 1.3 and 1.4). Consequently, Brion's formula becomes an explicit finite sum over pairs (M,w) satisfying the generator inequality, with the coefficient determined by a rank function that depends only on z and w.","pith_inferences":["If the type D edge criterion in Theorem 2.15 passes an independent check, the same decomposition machinery may extend to orbit closures in other spherical varieties whose weak order graphs are known.","The explicit interval description suggests a fast algorithm for computing the Schubert class of any orbit closure by generating the interval bottom-up rather than traversing the full weak order graph.","The conjectured Stanley symmetric function identities for types DI, DII, DIII, and DIV could be tested by computer up to rank 8, providing a direct check of the theorem independent of the proof.","The graded partial orders defined here may have applications to Kazhdan-Lusztig theory or to the study of Hessenberg varieties attached to symmetric orbits."],"forward_implications":["Brion's cohomology formula (1.4) becomes an explicit finite algorithm in all classical types, making it straightforward to test multiplicity-free criteria and single-term expansions.","The paper classifies when orbit closures are multiplicity-free in terms of the parameters of the symmetric pair (Proposition 8.1) and when the Brion atom set equals the full extended atom set (Proposition 8.6).","A notion of involution Schubert polynomials is introduced for all classical types, and several conjectures about their Stanley symmetric function expansions are stated.","The balanced-type cases produce explicit identities such as F_DI = S_{delta} at the longest element, unifying earlier type A results."],"fun_headline_variants":["Uniform Brion atoms for all classical symmetric pairs","Brion atoms: type D falls in line with A, B, C","Explicit finite algorithm for Brion's orbit-closure classes","Involution Schubert polynomials for every classical type","Brion atoms decomposed into principal filters uniformly"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The type D edge rules in Theorem 2.15 are asserted to follow by carefully comparing Wyser's thesis rather than being fully derived, so if that translation contains an error, the type D Brion atom formulas collapse.","fun_headline_variants_meta":{"raw":{"variants":["Uniform Brion atoms for all classical symmetric pairs","Brion atoms: type D falls in line with A, B, C","Explicit finite algorithm for Brion's orbit-closure classes","Involution Schubert polynomials for every classical type","Brion atoms decomposed into principal filters uniformly"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000263,"raw_usage":{"total_tokens":1412,"prompt_tokens":693,"completion_tokens":719,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":437,"completion_tokens_details":{"reasoning_tokens":640}},"tokens_in":437,"tokens_out":719,"duration_ms":6883,"temperature":1.0,"reasoning_tokens":640,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T14:48:36.306370+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the weak order graph for a small type D example, such as SO(8) with the symmetric pair of type DI, directly from the orbit parametrization and compare the t_{-1} edge and doubling rules in Theorem 2.15 against the explicit list; any mismatch in edge presence or doubling would falsify the claimed description.","supporting_citations":[],"review_version":1}