{"id":"8946e09e-0668-45b7-b841-15549532230a","arxiv_id":"2607.18188","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Among the recursive Weyl invariants in the Benson–Wood generating sets, only q_3 (Spin(10)) and f_2 (Γ⁺(7)) lie in the Chow characteristic image; all others fail explicit Steenrod tests.","lead":"For the spin groups Spin(n) and special Clifford groups Γ⁺(n), this paper determines exactly which of the standard algebraic-cycle classes (the recursive Weyl invariants q_i and f_i) actually come from algebraic cycles: only q₃ for Spin(10) and f₂ for Γ⁺(7). The classification completes Karpenko's Steenrod-operation program and, over the complex numbers, yields a sharp statement of which cohomology classes of these classifying spaces are algebraic modulo torsion.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Exceptional inclusions in Theorems 1.1 and 1.4 depend entirely on the unverified claim that t(Spin(10)) = t(Γ+(7)) = 2; if the cited torsion-index arithmetic fails, the iff collapses.","rationale":"I agree with the reader that the weakest assumption is the torsion-index arithmetic. My independent pass through the Steenrod machinery found no countervailing defect: Lemma 3.2's exponents are integral for the parities used because 2^i ≡ 2 (mod 6) for odd i and ≡ 4 (mod 6) for even i; Lemma 3.1's argument is sound; the degree conventions in Corollaries 1.2–1.3 are a notational bug (2i+1 should be 2^{i+1}) that does not affect Theorem 1.1. The paper's proof structure is otherwise coherent: nonexceptional exclusions follow from (2) and the odd-degree lemma plus the recurrence, and the positive inclusions are constructive conditional on [18]. Thus the single decisive test is whether the cited torsion-index arithmetic is accurate. If it is, the classification likely proves; if not, the iff becomes only-if. This does not change the reader's CONDITIONAL verdict, so I mark UNCHANGED.","tokens_in":14829,"tokens_out":23346,"duration_ms":205139,"concrete_test":"Verify in Totaro [18, Theorem 0.1] the exact torsion index of Spin(10) (and Spin(7)), and confirm Theorem 1.3(1) gives 't(G)·CH(BT)^W ⊂ Im Φ_G' with t(G) the torsion index. If t(Spin(10))>2 or the annihilation statement is weaker, Lemma 4.1 fails and Theorem 1.1's 'if' direction is unsupported. If the cited values are correct, the central claim stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 4.1's proof of q3∈ImΦ10 constructs d=q3−Φ10(a)=2y with y∈CH(BT)^W and then asserts 'The torsion index of Spin(10) is 2 and annihilates the cokernel of Φ10 [18, Theorems 0.1 and 1.3(1)]. Therefore 2 CH(BT)^W ⊂ Im Φ10.' Lemma 5.3 makes the parallel assertion for Γ+(7): same torsion index 2, hence 2 CH(BeT)^W ⊂ Im eΦ7. These two torsion-index claims are the only mechanism establishing the positive inclusions in Theorems 1.1 and 1.4; every nonexceptional direction is a Steenrod obstruction that does not need them. If the actual torsion index of Spin(10) (or of Γ+(7)) is a higher power of 2, or if Theorem 1.3(1) only annihilates the cokernel by a multiple of the index, then the inclusions d∈ImΦ10 and f2∈Im eΦ7 do not follow. The paper does not re-derive [18] or offer an independent check, so the iff is exactly as secure as this external arithmetic. This is the load-bearing external input.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Chow characteristic homomorphism for split spin groups Spin(n) and special Clifford groups Γ+(n), n≥7. Theorem 1.1 classifies the recursive Benson–Wood invariants q_i in the Chow characteristic image, proving that q_i∈ImΦ_n iff (n,i)=(10,3). Theorem 1.4 gives the analogous classification for the recursive invariants f_i, proving f_i∈ImΦ̃_n iff (n,i)=(7,2). Corollary 1.2 translates these statements into a classification of algebraic classes modulo torsion in the integral cohomology of the classifying space over C, and Corollary 1.3 realizes the corresponding classes on a single smooth projective variety for each n, with a Bockstein–Steenrod obstruction preventing algebraicity even after adding torsion. The non-inclusion directions are proved by a Steenrod-stability criterion together with Lemmas 3.1 and 3.2 controlling odd-degree classes in the mod-2 image; the exceptional inclusions are constructive, using the Euler class c_5 for Spin(10), the fourth Chern class of an extended spin representation for Γ+(7), and Totaro's torsion-index results.","tokens_in":14847,"tokens_out":29084,"duration_ms":262856,"significance":"If the results are correct, the paper completes the description of the recursive generators in the Chow characteristic image for all spin groups n≥7 and for the special Clifford groups, substantially extending Karpenko's work. The proof architecture is coherent and modular: the nonexceptional exclusions are reduced to explicit Steenrod coefficient computations, and the exceptional inclusions are genuinely constructive. I checked several of the displayed Steenrod identities, including (13), the St^7 computation for n=11, and the St^1 computation for φ_4 in Lemma 5.2, and found them consistent. The paper also gives a clean simultaneous smooth-projective realization of the Hodge-theoretic consequences. The main weakness is that the two positive inclusions rest on an external torsion-index input, which is cited but not unpacked; see the major comment.","major_comments":[{"comment":"The positive inclusions q_3∈ImΦ_{10} and f_2∈ImΦ̃_7 are the only places where the '⇐' direction of Theorems 1.1 and 1.4 is established. In both lemmas the key line is: 'The torsion index ... is 2 and annihilates the cokernel of Φ [18, Theorems 0.1 and 1.3(1)]. Therefore 2 CH(BT)^W ⊂ Im Φ.' This is load-bearing: if the torsion index of Spin(10) or Γ+(7) is a higher power of 2, or if the cited theorem only annihilates the cokernel by a proper multiple of the index, then the inclusions do not follow. I ask the author to state precisely which statement in [18] gives the annihilation of the cokernel of this specific map, and to write out the few lines connecting the torsion index to the inclusion 2 CH(BT)^W ⊂ ImΦ_{10} (and the corresponding statement for Γ+(7)). For Γ+(7), the identification of the torsion index via Γ+(7)/B̃ ≃ Spin(7)/B should also be made explicit. This is not a request to r","section":"§4, Lemma 4.1; §5, Lemma 5.3"}],"minor_comments":[{"comment":"The notation H^{2i+1} in Corollaries 1.2 and 1.3, and 'degree 2i' in Lemma 3.2, should be H^{2^{i+1}} and degree 2^i. As typeset, the degrees are ambiguous and the exponent formulas in Lemma 3.2 are hard to parse.","section":"Corollaries 1.2, 1.3; Lemma 3.2"},{"comment":"The exponents c_6^{2a} and c_6^{2a+1} are difficult to read in the current formatting. Please ensure superscripts are clearly typeset, and consider adding a one-line explanation of why T/c_4 and T/(c_2c_6) are the only possible square quotients.","section":"Lemma 3.2 proof"},{"comment":"Reference [6] is cited as an arXiv preprint (2009). If a published version of Ekedahl's projective approximation theorem exists, the citation should be updated.","section":"References"},{"comment":"The derivation of ρ(φ_3)=c_8+c_2c_6+c_3c_5+c_4^2+c_2c_3^2 would be easier to verify if the expansion of ϵ(E_3) using (8) and δ_1,δ_3,δ_4 were displayed explicitly.","section":"Lemma 5.2(i)"}],"recommendation":"major_revision","confidential_remarks":"The paper is carefully argued, and the Steenrod obstruction machinery is convincing. The make-or-break point is the two torsion-index citations used for the exceptional inclusions. If the editor or a second referee can confirm that Totaro's theorem indeed gives exactly 2 CH(BT)^W ⊂ ImΦ_{10} and 2 CH(BT̃)^W ⊂ ImΦ̃_7, I would be happy to accept after the requested clarifications. My recommendation of major revision is solely to make that load-bearing step explicit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper finishes the job Karpenko left open: for Spin(n), n≥7, exactly q_3 for Spin(10) lies in the Chow characteristic image among the recursive Benson–Wood invariants q_i, and for the special Clifford groups exactly f_2 for Γ^+(7) does. The non-inclusion direction is proven by the Steenrod-stability criterion (2) plus Lemma 3.1's control of odd-degree classes, with i≥4 handled by the coefficient-recursion Lemma 3.2. I spot-checked (13), the St_7 computation for n=11, the 2f_2=f_1^2−p_2 identity, and the c_4(Δ) identity in (20); they check out. The Hodge-theoretic corollaries also give a clean single-variety simultaneous realization, and the Bockstein–Steenrod obstruction showing nonalgebraicity persists after adding torsion is a real improvement over earlier counterexamples.\n\nNow the soft spots, in proportion. First, the manuscript as received has a cluster of notational errors that make several statements literally unreadable: the degree of q_i and F_i is 2^i, not 2i; Corollaries 1.2–1.3 and Lemma 4.2 place u_i in H^{2i+1}(BG;Z) when the correct degree should be H^{2^{i+1}}; and Lemma 3.2's exponents (2i−2)/6 and (2i−4)/6 are non-integral for most i, indicating missing superscripts. These are all fixable, but a referee will need a clean version to verify the degree bookkeeping.\n\nSecond, both affirmative exceptions (Spin(10) q_3 and Γ^+(7) f_2) are built on Totaro's torsion-index theorem: t=2 and 2·CH(BT)^W ⊂ Im Φ. The authors cite [18, Theorems 0.1 and 1.3(1)] rather than re-derive it. If that citation is misapplied, the iff collapses to only-if. I do not regard this as a circular or internal flaw — it is a standard external input — but the paper should state explicitly why Totaro's result gives the cokernel annihilation in the needed degrees, and a referee should double-check it. That is the one genuinely load-bearing external step.\n\nOverall, the mathematics is coherent and the classification is new. The paper deserves a serious referee. My advice: send it to peer review with a request to fix the grading typos and tighten the torsion-index citation.","headline":"Solid new classification of recursive Weyl invariants in Chow characteristic images; the main results hold up, but the text has fixable degree typos and the two exceptional inclusions rest on Totaro's torsion-index theorem.","tokens_in":15686,"tokens_out":4788,"would_cite":true,"duration_ms":59778,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C25","20G15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For all n ≥ 7, the only recursive Weyl invariant q_i in the Chow characteristic image of Spin(n) is q_3 for Spin(10); for special Clifford groups, the only one is f_2 for Γ⁺(7).","keywords":["Chow ring of a classifying space","spin group","special Clifford group","Weyl invariants","Steenrod operations","integral Hodge conjecture","algebraic cycles modulo torsion","torsion index"],"falsifier":"Compute the cokernel of Φ_10 in degree 8 (and of the characteristic map of Γ⁺(7) in degree 4) and test whether multiplication by 2 kills it. If not, q_3 or f_2 is not in the image, contradicting the classification.","tokens_in":14394,"feed_emoji":"🎯","tokens_out":11562,"duration_ms":103970,"temperature":0.7,"pith_summary":"The paper asks which of the recursively defined integral Weyl invariants q_i — the extra generators in a standard presentation of the invariant ring of a maximal torus — are restrictions of algebraic cycles on the classifying space of Spin(n). That map, from the Chow ring of the classifying space to the torus invariants, is the Chow characteristic homomorphism. The paper proves that, for every n ≥ 7 and over any field, q_i lies in its image exactly for (n,i)=(10,3), and that the analogous invariants f_i for the special Clifford group Γ⁺(n) lie in the image exactly for (n,i)=(7,2). Over the complex numbers, this becomes the statement that the corresponding integral cohomology classes are algebraic modulo torsion precisely in those two exceptional cases, and that every non-exceptional class is detected by a Bockstein–Steenrod operation even after adding torsion. This completes the classification of which recursive generators survive as algebraic classes, a question previously settled only in a numerical range.","feed_headline":"Spin groups: just one invariant class is algebraic","feed_subtitle":"Among the q_i for n ≥ 7, only q_3 in Spin(10) survives; for Clifford groups, only f_2 in Γ⁺(7).","key_machinery":"The argument runs on two tools. The first is the Steenrod-stability criterion: if x∈CH(BT)^W lies in the image of Φ_n, then the reduction of any Steenrod operation on x must lie in the image of the mod-2 restriction map ρ from the classifying space. Lemma 3.1 tells what that image can contain: for odd n or n divisible by 4 it has no nonzero odd-degree elements at all, and for n=2m with m odd its only odd-degree class is a multiple of the Euler generator c_m. The second tool is a coefficient recurrence (Lemma 3.2) which, from the inductive definition of the q_i, produces in every non-exceptional case a specific monomial, c_3 c_6^{(...)} or c_5 c_6^{(...)}, whose Steenrod square is nonzero, so","core_discovery":"For a split spin group G=Spin(n) with n≥7, let Φ_n carry algebraic cycles on the classifying space BG to the W-invariant part of the Chow ring of a maximal torus. The q_i are the recursive generators in a standard presentation of that invariant ring, and the main theorem is a complete classification: q_i lies in Im Φ_n if and only if (n,i)=(10,3). The same statement for the special Clifford group Γ⁺(n) says f_i lies in the image of its characteristic map exactly when (n,i)=(7,2). Over C, naturality of cycle-class maps identifies membership in the Chow image with algebraicity modulo torsion of the corresponding class u_i in H^{2^{i+1}}(BG;Z)/torsion, so the classification transfers verbatim.","pith_inferences":["This sharpens the contrast with topological cohomology: in the topological image all q_i in the finite range occur in most congruence classes, while in the Chow image only one does; the paper's result shows the failure is integral and torsion-sensitive, not rational.","The Bockstein–Steenrod obstruction that resists adding torsion suggests a general recipe for producing non-torsion integral Hodge classes that fail the integral Hodge conjecture modulo torsion on smooth projective approximations of reductive groups; applying the recipe to other families, such as orthogonal groups or the even special Clifford groups, would be a natural test.","The exceptional ranks 10 and 7 coincide with small spins and low-degree Chern classes of spin representations entering the invariant ring; one might conjecture that no further exceptions occur for larger n, but proving that would require checking the same Steenrod recurrences, which the paper does in the full range.","A direct computation of the cokernel of Φ_10 and ̃Φ_7 in the relevant degrees would turn the external torsion-index input into a self-contained verification; the paper's classification makes that computation a well-posed question."],"forward_implications":["The complete list of recursive q_i in Im Φ_n is reduced to the single pair (n,i)=(10,3); all other q_i are excluded by explicit Steenrod operations.","The same complete list for the special Clifford group is the single pair (n,i)=(7,2).","Over C, algebraicity modulo torsion of the cohomology classes u_i is equivalent to membership in the Chow image, so u_i is algebraic modulo torsion exactly in these two exceptional cases.","For every n, one smooth projective variety X_n realizes all classes α_{n,i} simultaneously; in non-exceptional cases each α_{n,i} is non-algebraic modulo torsion and stays non-algebraic after adding any torsion class.","The Steenrod obstruction now covers the full generating range, not just the earlier numerical bound 2^i+1 < n/2."],"fun_headline_variants":["Only q_3 for Spin(10) survives as algebraic invariant","Spin(10)'s q_3 and Γ⁺(7)'s f_2: the sole algebraic invariants","Classification: exactly two algebraic invariants in spin and Clifford groups","Spin and Clifford: only one algebraic invariant each"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The classification rests on the external arithmetic fact that the torsion index of Spin(10) (and of Γ⁺(7)) is 2 and annihilates the cokernel of the Chow characteristic map in the degree of the exceptional invariant; if that fact is wrong in those degrees, the exceptional inclusions fail and only the non-inclusion half of the theorem remains.","fun_headline_variants_meta":{"raw":{"variants":["Only q_3 for Spin(10) survives as algebraic invariant","Spin(10)'s q_3 and Γ⁺(7)'s f_2: the sole algebraic invariants","Classification: exactly two algebraic invariants in spin and Clifford groups","Spin and Clifford: only one algebraic invariant each"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000234,"raw_usage":{"total_tokens":1345,"prompt_tokens":770,"completion_tokens":575,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":514,"completion_tokens_details":{"reasoning_tokens":493}},"tokens_in":514,"tokens_out":575,"duration_ms":6425,"temperature":1.0,"reasoning_tokens":493,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T15:48:51.679980+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the cokernel of Φ_10 in degree 8 (and of the characteristic map of Γ⁺(7) in degree 4) and test whether multiplication by 2 kills it. If not, q_3 or f_2 is not in the image, contradicting the classification.","supporting_citations":[],"review_version":1}