# Verification report — OWR-14213-008 (Karpenko: the special Schubert class e_n and the gamma filtration on K of a maximal orthogonal Grassmannian) Verification date: 2026-10-20. **Verdict.** The note shows that the conjecture is **false**; every statement it makes about it is proved, and the scope is as follows. - **Settled.** For every field F of characteristic 0, every n in {8, 9, 11} and every non-degenerate (2n+1)-dimensional form φ' over F with C_0(φ') a division algebra, the class e_n lies in the term number n+1−[(n+1)/2] of the gamma filtration on K(X'_{F(S)}) (term 5 for n = 8, 9; term 6 for n = 11) (Theorem 1.2). Such forms exist (Lemma 4.1). Since the conjecture is a universal statement ("for n ≥ 8"), it is false. - **Not new mathematics.** The proof is a short consequence of printed results, not found stated in print: Karpenko's own printed argument that his Conjecture 4.7 implies his Conjecture 3.3 (Math. Z. 285 (2017), end of Sect. 4), written out in a general form (Proposition 3.2) with two implicit steps filled in, together with Karpenko's Theorem 4.3 (topological = gamma filtration on K(X'_{F(S)})), Totaro's torsion index of Spin(20) and Spin(24) (u(9) = 4, u(11) = 5; Duke 2005) for n = 9, 11, and Karpenko's theorem on 18-dimensional forms (St. Petersburg Math. J. 36 (2025), Abstract and Cor. 4.4, computer-free) for n = 8. For n = 9 and 11 the disproof was available as soon as the 2017 paper appeared. - **Computer-assisted supplement (tested, not part of the proof).** On the split OGr(n, 2n+1): e_8 lies in gamma^7 but not gamma^8, e_9 in gamma^8 but not gamma^9 (Theorem 5.1), by exact integer and 2-adic computation, reproduced by an independent model and program. For n = 9 a computer-assisted printed result (Publ. Mat. 67 (2023), Sect. 8) also gives the sixth term (Remark 4.3); the computer-free bound is the fifth term. - **Open.** The conjecture's statement for n = 10, n = 12 and every n ≥ 13 is not decided. Totaro's bound does not suffice there: u(n) = [(n+1)/2] for n = 10, 12, 13 and u(n) > [(n+1)/2] for n ≥ 14 (Remark 4.5); an argument along these lines would need smaller indexes of orthogonal Grassmannians of generic forms (for n = 10, 12, 13: index 2^{u(n)−1} of the u(n)-th one, as Karpenko 2025 proves for n = 8), which the note does not use. The note is unrefereed. ## Statement checked - **Primary sources.** N. Karpenko, "Around 16-dimensional quadratic forms in I^3_q", Math. Z. 285 (2017) 433–444, doi:10.1007/s00209-016-1714-x, Conjecture 4.7; and N. Karpenko, "Grothendieck ring of maximal orthogonal Grassmannian", in Oberwolfach Report 5/2016, Oberwolfach Rep. 13 (2016) no. 1, 215–265, Karpenko's abstract pp. 226–227, doi:10.4171/OWR/2016/5, Conjecture 3. - The two statements are word for word the same: for n ≥ 8, the special Schubert class e_n ∈ K(X_{F(S)}) does not belong to the term number n+1−[(n+1)/2] of the gamma filtration. Setting (Math. Z. Sect. 4): φ non-degenerate of dimension 2n+1, X its maximal orthogonal Grassmannian, ind C_0(φ) = 2^n, S the Severi–Brauer variety of C_0(φ), e_n the class of the special Schubert variety of lowest dimension ("corresponding to the class of a rational point on the quadric"). - The 2017 paper was read in the author's final preprint (revised 4 June 2016); one verification run also read the preprint of 29 November 2015 and found the cited parts unchanged. The journal version was not compared. - **Corpus record.** UnsolvedMath (Hugging Face, ulamai/UnsolvedMath, v1.8.0), record OWR-14213-008, status open. The record text agrees with the source; nothing is truncated. - **Companion.** Conjecture 3.3 of the 2017 paper (= Conjecture 4 of the report; corpus record OWR-14213-009): for any even m ≥ 18, the index of the [m/4]-th orthogonal Grassmannian of any generic m-dimensional form in I^3_q is equal to 2^{[m/4]}. The 2017 paper proves Conjecture 4.7 ⇒ Conjecture 3.3 (same m = 2n+2). ## Readings - "Term number c" = the c-th term gamma^c (the numbering of Theorem 4.3, Remarks 4.4 and 4.8 of the 2017 paper). Term numbers: 5 (n = 8, 9), 6 (n = 10, 11), 7 (n = 12). - The conjecture quantifies over all n ≥ 8 and all admissible (F, φ); one n and one admissible φ with e_n in the term refute it. Theorem 1.2 covers every admissible φ' in characteristic 0, so no reading of "generic" changes the answer. By Remark 4.6 of the 2017 paper (final version: K(X_{F(S)}) → K(X_L) preserves the gamma filtration; the topological filtration is not preserved), the conjecture is a statement about the λ-ring K of the split Grassmannian. - e_n ∈ K(X_{F(S)}) is the preimage of [O_{Σ_p}] under the isomorphism K(X_{F(S)}) → K(X̄) (Lemma 4.1 of the 2017 paper with C_0 split over F(S)). ## Results in the paper - **Lemma 3.1.** h^{D−i} = 2l_i − l_{i−1} in K of the split quadric of dimension D = 2n (1 ≤ i ≤ n−1), hence 2^{j−1}l_{j−1} = l_0 + Σ_{i [(n+1)/2] for n ≥ 14. - **Theorem 5.1 (computer-assisted).** λ(8) = 7, λ(9) = 8, where λ(n) = max{k : e_n ∈ gamma^k K(OGr(n,2n+1))}. Model (M1)–(M4) with Lemma 5.2 (gamma^k = Z-span of monomials of weight ≥ k in the gamma-Chern classes of L and T^vee); odd primes never obstruct; 2-adic lattices modulo 2^25 with the certificate 2^{a−1}T^k ⊂ M_k + 2^aK. ## Computations (sanity checks; programs and outputs in reproducibility/) No proof of Sections 3 and 4 depends on a computation. - **Program written with the note** (`check_note.py`, standard library, under 1 s; with `--n7` about 2 min; TOTAL failures: 0). Totaro's u(l) from Theorem 0.1 against his table (l ≤ 18); the n in [8, 400] with u(n) < [(n+1)/2] are 9 and 11; the inequalities of Remark 4.5; term numbers and the j of the proof; the g.c.d. step; Lemma 3.1 via Chern characters (exact power series, n ≤ 12) and the telescoped identity against the printed one; the explicit forms of Lemma 4.1 (printed); and, with an implementation of the model of Section 5 written for the note, χ(O) = χ(e_n) = 1, the K lattice (rank 2^n, all pivots 1), λ(n) = 1, 2, 2, 4, 4, 4, 5 and the g.c.d. of χ on gamma^D = 2^{u(n)} with u = 0, 0, 1, 1, 1, 2, 3 for n = 1..7. - **Programs with which the results were first obtained** (`original/`): Totaro's exponents; gamma levels for n = 1..7 (exact HNF over Z), n = 8 (p-local for all p ≤ 47, level 7, only p = 2 obstructs; about 3 min), n = 9 (p = 2, 3; level 8; about 17 min); the torsion-index check g.c.d. χ(gamma^D) = 2^{u(n)} for n = 1..8. - **Verification run A** (`verification_run_A/`): its own implementation (different generator set, exact integer HNF): levels 1, 2, 2, 4, 4, 4 and torsion exponents 0, 0, 1, 1, 1, 2 for n = 1..6; its n = 7, 8 runs were stopped before the gamma stage; re-run of the original n = 8 program (level 7). - **Verification run B** (`verification_run_B/`): its own model (Borel presentation, Atiyah–Bott localisation, ch(e_n) from the projection formula and HRR, no GRR formula assumed) and lattices modulo 2^64 with a printed exponent certificate: levels 1, 2, 2, 4, 4, 4, 5, 7, 8 and torsion exponents 0, 0, 1, 1, 1, 2, 3, 4, 4 for n = 1..9; its ch(e_n) equals (M3) for n = 1..9; deliberately wrong models (L replaced by L^2, or no line-bundle generator) fail the Totaro check. It also computed n = 10 (e_10 in gamma^8, not gamma^9; 338 s); this is not claimed in the note. - **Third verification run** (`independent_run_2/`, on the final text): `run2_checks.py` (standard library; Totaro's u(l), the lists for n ≤ 400 including u(n) = [(n+1)/2] exactly for n = 8, 10, 12, 13 and u(n) > [(n+1)/2] for 14 ≤ n ≤ 400, term numbers, the row d = 18 of Karpenko 2025, the g.c.d. step, Lemma 3.1 and (2) by Chern characters on the split quadrics of dimension 2n ≤ 24; TOTAL failures 0) and `run2_gamma.py` (python-flint): its own model in integral coordinates Σ d! ch_d, λ^i from Adams operations, all lattices EXACT over Z by Hermite normal forms; levels 1, 2, 2, 4, 4, 4, 5, 7 and g.c.d. χ(gamma^D) = 2^{u(n)} with u = 0, 0, 1, 1, 1, 2, 3, 4 for n = 1..8; [T^k : M_k] a power of 2 for all k; χ(L^k), χ(λ^i T^vee), χ(e_n L^k), χ(e_n λ^i T^vee) agree with Borel–Weil and the Weyl dimension formula on X and on Σ_p (tests (M2), (M3)); deg e_1^D = number of standard shifted tableaux of staircase shape. - **Re-runs.** Verification run B re-ran all original programs (including n = 9) with identical outputs up to time stamps. When the note was written, `check_note.py` was run, and `totaro_vs_conjecture.py`, `gamma_level_v2.py 1 6`, `gamma_level_v3.py 8 8` (168 s), `gamma_level_v5.py 1 7 --index`, `vA_gamma.py 1 5`, `own_totaro.py` and `own_lattice.py 8 8` were run again from a copy of the package; all agree with the recorded outputs (`reproducibility/RERUN_LOG.txt`). The third verification run re-ran, from an extracted copy of `source.zip`, `check_note.py` (also `--n7`), `totaro_vs_conjecture.py`, `gamma_level_v2.py 1 7`, `gamma_level_v3.py 8 8`, `gamma_level_v5.py 1 7` and `8 8 --index`, `gamma_level_v4.py 9 9` (1004 s), `vA_gamma.py 1 5`, `own_totaro.py` and `own_lattice.py 3 7`, `8 8`, `9 9`: all identical to the recorded outputs (own_lattice.py prints one documented extra field). `own_lattice.py` must be started with NMIN ≥ 2 (for n = 1 a sensitivity test added after the recorded run stops with an AssertionError); this is now stated in the README. ## Independent verification runs The results were first obtained with the proofs of Sections 3 and 4 and the programs in `original/`. Two independent verification runs, both AI-assisted, followed, each with its own copies of the sources and its own programs; they examined the first write-up. A third AI-assisted run (run 3) then examined the final text of the note. - **Run A** compared the statement with the report and the 2017 paper; re-derived the printed implication and Proposition 3.2 step by step (found the one implicit g.c.d. step correctly filled); checked the inputs against Totaro (Theorem 0.1, table, Theorem 1.1, the statement for forms in the introduction) and Karpenko 2025 (Abstract, Cor. 4.4, Sect. 1); checked Lemma 4.1 and the Clifford-algebra formulas; re-derived (M1)–(M4), Lemma 5.2, the odd-prime reduction and the 2-adic certificate; ran its own programs and re-ran the original n = 8 program. Verdict: confirmed. Required fixes (all made): use Totaro's statement for forms for n = 9, 11 and say that torsor surjectivity is needed only for the specialisation routes; mark the computer-assisted input for the n = 9 term-6 remark and the n = 9 row of the table; state the Karpenko–Merkurjev 2022 table remark as unverified or drop it (dropped); justify the independence of l_{j−1} and the separability in Proposition 3.2(i); call the Euler-form check redundant. - **Run B** read both preprint versions of the 2017 paper; wrote an independent model and program (above); re-ran all original programs; repeated the literature search (Semantic Scholar citing list, OpenAlex (HTTP 429), zbMATH, Karpenko's publication list with 15 PDFs text-searched, arXiv, one web search); checked the bibliography with Crossref. Verdict: confirmed with fixes. Required fixes (all made): Devyatov–Karpenko–Merkurjev, Forum Math. Sigma 9 (2021) e34, does mention the 2017 paper (Theorem 7.2 confirms Conjecture 3.3 for m ≥ 34), so Conjecture 3.3 is false for m = 18, 20, 24 by printed results and true for m ≥ 34, and this route cannot refute Conjecture 4.7 for large n; Baek–Devyatov is Ann. Inst. Fourier 76 (2026) no. 3, 907–943, on a different conjecture; cite the final version of the 2017 paper and use its wording on the change of field; novelty wording; say that n = 10, 12 are not decided and keep its n = 10 computation unclaimed. - Neither run found a gap in the proof. - **Run 3** (third verification run, AI-assisted, on the final text of the note, as if nothing had been checked) read again the cited parts of the report, of the final 2017 preprint, of Totaro, Karpenko 2025, Karpenko 2023, Karpenko–Merkurjev 2022 and Devyatov–Karpenko–Merkurjev 2021 and compared every quoted statement; re-derived Lemma 3.1, Proposition 3.2, the four steps of the proof of Theorem 1.2, Lemma 4.1, (M1)–(M4), Lemma 5.2 and the 2-adic certificate line by line; recomputed the finite claims and the computational supplement with programs of its own (above); re-ran the package from the archive; checked all 17 DOIs with Crossref; repeated the literature search. Verdict: confirmed, no gap. Required fixes (all made): the abstract's "the method gives nothing for n ≥ 13" replaced by "nor is any n ≥ 13 decided"; Remark 4.5 and Sect. 1.3 now include n = 13 (u(13) = 7 = [(13+1)/2]) and u(n) > [(n+1)/2] for n ≥ 14, and say what an argument along these lines would need; the quotation of Conjecture 3.3 ("any generic", "is equal to"); "neither of these two papers mentions Conjecture 3.3 or 4.7"; Remark 4.2 lists all uses of characteristic 0; the run is recorded in the note and here; the package documentation (own_lattice.py NMIN ≥ 2, the new folder) was completed. ## Relation to the literature, novelty and scope - **Searches (20 October 2026).** Semantic Scholar list of the nine works citing the 2017 paper (OpenAlex citing list refused with HTTP 429, not retried); Karpenko's publication list and later papers (text search); arXiv listings of Karpenko, Totaro and Yagita; zbMATH "gamma filtration" from 2016 on; two web searches; Crossref for all DOIs. Repeated by run 3 on the same day: arXiv API queries (gamma and topological filtrations, special Schubert classes in K-theory, Karpenko, Spin(18)/Spin(20)); Crossref (17 DOIs, all consistent); Semantic Scholar (the same nine citing works); OpenAlex (K17 record found; citing list again HTTP 429); zbMATH; Karpenko's publication list of 14 Sep 2026 (his 2024 NYJM and 2026 PAMS papers on critical and subcritical exponents read in the relevant parts; they do not mention Conjecture 3.3 or 4.7); one further web search (only papers on the CH = gr K conjecture). - No printed answer to the question was found, and no printed statement that Conjecture 4.7 or Conjecture 3.3 is false. - Closest items: Devyatov–Karpenko–Merkurjev 2021 (Conjecture 3.3 holds for m ≥ 34; nothing on Conjecture 4.7 or the gamma filtration); Karpenko, Publ. Mat. 2023 and St. Petersburg Math. J. 2025 (their index results contradict Conjecture 3.3 at m = 18 without mentioning it); Totaro's table (u(9) = 4) already contradicts Conjecture 3.3 at m = 20, a result the 2017 paper cites. Baek–Devyatov (Ann. Inst. Fourier 2026), Yagita, Baek and others treat a different conjecture of Karpenko (Chow ring vs graded K of generic flag varieties). - The failure of Conjecture 3.3 at m = 18 was reported earlier in the follow-up report FU119 to the UnsolvedMath collection, on record OWR-14213-009. - **What was read.** Karpenko's abstract in the Oberwolfach report (publisher's file); the 2017 paper (author's final preprint); Totaro (author's file: introduction, Theorem 0.1 and table, Theorem 1.1); Karpenko 2025 (preprint of 28 Aug 2023: abstract, Sects. 1, 2, 4); Karpenko 2023 (preprint revised 23 Nov 2021: Sects. 1, 5–8); Karpenko–Merkurjev 2022 (Sect. 1); Devyatov–Karpenko–Merkurjev 2021 (introduction, remark after Theorem 7.2); Baek–Devyatov (abstract). Not read: Karpenko 1990, Swan, Pittie, Steinberg, Panin, Buch–Ravikumar, Karpenko–Merkurjev 2006, and the cited parts of Elman–Karpenko–Merkurjev and Lam; what is used from them is taken from the 2017 and 2025 papers or is standard. The proofs of the cited theorems (Theorem 4.3, Proposition 4.2, Lemma 4.1 of the 2017 paper; Totaro's Theorem 0.1; Karpenko 2025 Cor. 4.4) were not checked. - **Caveats.** A search that finds nothing is not a proof of novelty. The journal versions of the 2017 and 2025 papers were not compared with the preprints. - **Scope.** The conjecture is refuted at n = 8, 9, 11 (every admissible form, characteristic 0). The claim of the note is only to state this consequence, to fill the two implicit steps, to give Proposition 3.2, and the computer-assisted supplement. No priority is claimed beyond that. ## Public release and license This paper, its source files, and this verification report are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/