{"id":"ae813996-d996-4608-821d-96e2d72aa1e6","arxiv_id":"2607.02985","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Cellular A1-homology of split semisimple flag varieties equals reindexed Bruhat boundary matrices times η in Milnor-Witt K-theory, for types A, Bn/Cn/Dn (n≤7), F4, E6, E7.","lead":"The paper computes cellular A1-homology of split flag varieties by lifting known signed Bruhat boundary matrices to Milnor-Witt coefficients. This yields explicit chain complexes that recover classical real flag homology under real realization and feed Chow-Witt calculations.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The Reader correctly isolates Def 2.1 as the only structural limitation and correctly treats it as an explicit, checkable range restriction rather than a flaw in the comparison. The local lemmas cover every rank-two subsystem that appears under that hypothesis (A1\times A1, A2, B2=C2), the B2 coefficient-2 bookkeeping (Lemmas 2.2–2.3) shows the unit 2 never multiplies a non-zero motivic boundary factor, and the vertical-determinant argument (Lemma 4.8 / Prop 4.9) justifies the restrict-and-project step for partial flags. Real realization recovers the ordinary signed boundary after reindexing, matching the abstract claim. Because the external matrices are used only as integer input and the paper supplies an independent local A^{1}-degree calculation, an error in [6,7] would simply falsify the numerical output for that type without invalidating the lift itself. No stronger load-bearing concern appears; the Reader’s ACCEPT / low-risk assessment stands.","tokens_in":26111,"tokens_out":728,"duration_ms":6338,"concrete_test":"Independently recompute the single non-zero A^{1}-coefficient for the cover s1s2 ⋗ s2 in SL3/B (Section 6 table) from Lemmas 4.2–4.6 alone: deleted root α1+β, ht(γ^∨)=2 so (ht+1)_ϵ η=η, deletion sign and deg^{A^{1}} from the fixed word s1s2s1; confirm the matrix entry is -η (or isomorphic after basis change). Agreement with the displayed ∂^{A^{1}}_2 matrix verifies the local comparison on a complete low-rank example.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Thm 4.11 / Cor 4.12) is that the oriented cellular A^{1}-differential equals η times the length-reindexed half-boundary matrix E^Θ_i, with coefficients ⟨-1⟩^I deg^{A^{1}}(Φ^{-1} Φ^{\\I}) (ht(γ^∨)+1)_ϵ η. The local comparison is assembled from Lemmas 4.1–4.8 (purity model, rank-two Jacobians for A1\times A1/A2/B2, normal power map t^{ht+1}, deletion sign, vertical determinant ⟨1⟩) and is restricted by Def 2.1 to types whose signed integer matrices are already known. That restriction is stated explicitly and is not a hidden gap: the paper never claims to recompute the combinatorial tables of [6,7] or to treat G2/non-split forms. Within the stated range the argument is self-contained, the real-realization check is consistent, and the SL3/B and F4 applications match the expected 2-torsion pattern. No internal inconsistency or missing local case was found that would undermine the lift.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper lifts known integral Bruhat boundary matrices of split flag varieties G/P_Θ to the Morel–Sawant cellular A^{1}-chain complex over a perfect field k of characteristic ≠ 2. Under the boundary-data hypothesis (Def. 2.1), which takes as input the type-A coefficient formula and the normal-form matrices of [6,7] for B_n,C_n,D_n (n≤7) and F_4,E_6,E_7, the main theorem (Thm 4.11 / Cor 4.12) identifies the oriented cellular A^{1}-differential with ∂^{A^{1}}_i = η E^Θ_i for i≥2 and ∂^{A^{1}}_1 = 0, where E^Θ_i is the length-reversing reindexing of half the ordinary signed Bruhat boundary matrix. The coefficient of a cover is ⟨-1⟩^I deg^{A^{1}}(Φ^{-1} Φ^{\\I}) (ht(γ^∨)+1)_ϵ η. Smith normal form of the integer complex then yields an explicit description of H^{cell}_i in terms of free K^{MW}_i summands and kernels/cokernels of multiplication by qη. Real realization recovers the known 2-torsion pattern of real flag manifolds; detailed checks are given for SL_3/B and the full F_4 flag.","tokens_in":26345,"tokens_out":1097,"duration_ms":8205,"significance":"The result supplies a uniform, chain-level bridge from combinatorial Bruhat boundary data to cellular A^{1}-homology and, via the Morel–Sawant cochain complex, to additive Chow–Witt groups of split flag varieties. Within the stated range the comparison is concrete and immediately usable: every available matrix D^Θ_i lifts by multiplication by η after reindexing, and the Smith formula converts existing Poincaré polynomials and 2-torsion ranks into sheaf decompositions over any perfect k of char ≠ 2. The local analysis of normalized Chevalley coordinates, rank-two Jacobians, and vertical determinants is carefully restricted to A_1\times A_1, A_2 and B_2=C_2, so the argument is self-contained once the external integer matrices are accepted. The SL_3/B cover-by-cover table and the F_4 application illustrate that the lift is computable and recovers the expected topological pattern. This is a solid, usable contribution to explicit A^{1}-homology computations.","major_comments":[],"minor_comments":[{"comment":"In the abstract and introduction the phrase “we compute the cellular A^{1}-homology” could be sharpened to “we identify the cellular A^{1}-chain complex with the reindexed half-boundary matrices of [6,7] and deduce the homology via Smith form,” so that the dependence on external integer input is visible at first reading.","section":null},{"comment":"Section 6 (SL_3/B): the table of covers is clear, but a one-line statement of the chosen ordering of basis elements for the matrix of ∂^{A^{1}}_2 would make the displayed 2\times2 matrix immediately reproducible without re-deriving the signs.","section":null},{"comment":"Section 7 / Table 1: the recurrence T_i = c_i - β_i - T_{i-1} is standard, yet a brief citation or parenthetical reminder that it follows from rank-nullity for a complex with only unit elementary divisors would help readers who consult only the F_4 application.","section":null},{"comment":"Notation: the symbols D^Θ_i, E^Θ_i, δ^Θ_i and c^Θ(w,w') are introduced across Sections 3–4; a short notational summary at the end of Section 3 would reduce the need to flip back when reading the main theorem.","section":null},{"comment":"A few typographical inconsistencies appear (e.g., spacing around A^{1}, occasional missing thin spaces in K^{MW}_i). These are purely cosmetic.","section":null}],"recommendation":"accept","confidential_remarks":"The paper is a clean, correctly scoped lift of existing combinatorial tables into the Morel–Sawant formalism. Its novelty is the local MW comparison rather than new boundary matrices; that is disclosed honestly via Def. 2.1. Fit for a specialized algebraic-geometry / A^{1}-homotopy venue is good. No integrity or citation concerns."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know is that this paper turns the signed integer Bruhat boundary matrices of real flag manifolds into an explicit cellular A1-chain complex over perfect fields of odd characteristic. The differentials are η times the length-reindexed half-boundary matrices E_i, with coefficients written out in terms of deletion signs, A1-local degrees of Chevalley coordinate changes, and the coroot-height factor (ht(γ∨)+1)_ϵ η.\n\nWhat is new is the comparison itself (Thm 4.11 / Cor 4.12), not the matrices. The authors take the type-A formula and the computed tables for Bn/Cn/Dn (n≤7) and F4/E6/E7 as external input under Def 2.1, then prove that the Morel–Sawant attaching maps refine those integers by η. The local work is careful: normalized Chevalley coordinates in A1×A1, A2 and B2=C2, Jacobian determinants of braid moves, the normal power map t^{ht+1}, deletion signs, and the vertical-determinant check for partial flags. The SL3/B cover-by-cover calculation and the F4 Smith table are concrete and match the expected 2-torsion pattern after real realization. The Smith formula then gives the homology sheaves in terms of free KMW summands, KM summands, and η-kernels, which also feed additive Chow–Witt groups.\n\nThe soft spot is exactly the one they flag: dependence on the external tables of [6,7], exclusion of G2 subsystems and non-split forms, and the need for char ≠ 2 in the B2 coordinates. That is a range restriction, not a hidden gap. Within the stated range the argument looks self-contained; real realization is only a consistency check. Citation pattern is appropriate.\n\nThis is for people who compute A1-homology, Chow–Witt groups, or motivic cohomology of homogeneous spaces and want explicit chain-level answers rather than existence statements. It is solid computational work, not a foundational rewrite. I would send it to referees; the central claim is supported and the limitations are explicit. Worth engaging if you work in this area.","headline":"Clean, usable lift of known Bruhat boundary matrices into cellular A1-homology; the comparison is the real contribution, and the range restriction is honest.","tokens_in":26977,"tokens_out":552,"would_cite":true,"duration_ms":5198,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F42","14M15","55N10","20F36"],"pacs":[],"model":"grok-4.5","headline":"Cellular A1-homology of split flag varieties is the Bruhat boundary matrix times η.","keywords":["cellular A1-homology","split flag varieties","Bruhat decomposition","Milnor–Witt K-theory","Smith normal form","real flag manifolds"],"falsifier":"For any cover in the $SL_3/B$ table or the $F_4$ tables, recompute the $\\mathbb{A}^1$-local degree of the Chevalley face map and the height factor; if the product is not exactly half the ordinary real coefficient times $\\eta$, the main comparison fails.","tokens_in":26980,"feed_emoji":"🏳️","tokens_out":753,"duration_ms":4998,"temperature":0.7,"texified_at":"2026-08-05T21:13:39.276762+00:00","pith_summary":"This paper lifts the signed integer boundary matrices of real Bruhat cells to the cellular $\\mathbb{A}^1$-homology of split flag varieties over perfect fields of characteristic not 2. For flag varieties $G/P_\\Theta$ in type A, classical types $B_n, C_n, D_n$ with $n$ at most 7, and exceptional types $F_4, E_6, E_7$, the oriented cellular $\\mathbb{A}^1$-differential is multiplication by $\\eta$ of a reindexed half of the ordinary real Bruhat boundary matrix. Real realization recovers the classical integral homology of real flag manifolds, so topological 2-torsion becomes the sheaf summands $K^M$ and $\\eta K^{MW}$. The result turns existing tables of Poincaré polynomials and torsion ranks into concrete sheaf decompositions of cellular $\\mathbb{A}^1$-homology, with full worked examples for $SL_3/B$ and the full $F_4$ flag.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":6489,"prompt_tokens":665,"completion_tokens":5824,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":665,"completion_tokens_details":{"reasoning_tokens":5155}},"feed_headline":"A1-homology of flags is the Bruhat matrix times η","feed_subtitle":"Real 2-torsion of flag manifolds lifts to Milnor–Witt sheaves over any perfect field","key_machinery":"The Bruhat–Milnor–Witt comparison: local attaching maps of Bruhat covers, written in normalized Chevalley coordinates, produce the Milnor–Witt coefficient $\\langle -1 \\rangle^I \\, \\deg^{\\mathbb{A}^1}(\\Phi^{-1} \\Phi^I) (\\operatorname{ht}(\\gamma^\\vee)+1)_\\epsilon \\eta$, which equals $(c/2) \\eta$ of the ordinary real boundary entry.","core_discovery":"For a split flag variety $X_\\Theta = G/P_\\Theta$ satisfying the boundary-data hypothesis, the Morel–Sawant cellular $\\mathbb{A}^1$-differential is $\\partial^{\\mathbb{A}^1}_i = \\eta E^\\Theta_i$ for $i \\ge 2$ and vanishes in degree 1, where $E^\\Theta_i$ is the length-reversing reindexing of one half of the ordinary signed Bruhat boundary matrix. Equivalently each attaching coefficient is the product of a deletion sign, an $\\mathbb{A}^1$-local degree of the Chevalley coordinate change, and the Milnor–Witt height factor $(\\operatorname{ht}(\\gamma^\\vee)+1)_\\epsilon \\eta$.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["A1-flag differentials equal η times half Bruhat boundary matrix","Cellular A1-homology of split flags from Bruhat matrices times η","Morel–Sawant A1-differentials: η E_i from length-reversed Bruhat data","Split semisimple flag A1-chains: differentials η times Bruhat boundaries","A1-homology of G/P_Θ given by η-scaled Bruhat boundary matrices"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The signed integer Bruhat boundary matrices and normal-form deletion data are taken as external input from existing type-A formulas and tables for $B_n$–$D_n$ ($n \\le 7$) and $F_4, E_6, E_7$; the paper does not recompute those integers and excludes $G_2$ and non-split forms.","fun_headline_variants_meta":{"raw":{"variants":["A1-flag differentials equal η times half Bruhat boundary matrix","Cellular A1-homology of split flags from Bruhat matrices times η","Morel–Sawant A1-differentials: η E_i from length-reversed Bruhat data","Split semisimple flag A1-chains: differentials η times Bruhat boundaries","A1-homology of G/P_Θ given by η-scaled Bruhat boundary matrices"]},"model":"grok-4.5","effort":"low","cost_usd":0.00498,"raw_usage":{"total_tokens":1408,"prompt_tokens":771,"num_sources_used":0,"completion_tokens":98,"cost_in_usd_ticks":49800000,"prompt_tokens_details":{"text_tokens":771,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":539,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":771,"tokens_out":98,"duration_ms":4137,"temperature":1.0,"reasoning_tokens":539,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T05:37:26.227085+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"For any cover in the $SL_3/B$ table or the $F_4$ tables, recompute the $\\mathbb{A}^1$-local degree of the Chevalley face map and the height factor; if the product is not exactly half the ordinary real coefficient times $\\eta$, the main comparison fails.","supporting_citations":[],"review_version":1}