{"id":"aa981553-f9cc-4c3d-a7b9-e3ddfc7237d3","arxiv_id":"2504.15014","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The Spin-Sp(4), Spin-SU(8), and Spin-Spin(16) bordism groups in dimensions up to 7 are stated with explicit generators, but the key Adams spectral sequence computation is omitted.","lead":"This paper computes the Spin-G bordism groups in dimensions up to 7 for G = Sp(4), SU(8), and Spin(16), with explicit generators such as HP1, CP2, and SU(3)/SO(3). It is a technical algebraic topology computation relevant to global anomaly cancellation in string and supergravity theories, but the central spectral sequence calculation is largely asserted rather than shown.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 3.7 is not checkable: Tables 5–7, the Adams E2-terms, are blank, and the A1-modules in Figures 1–3 are specified only up to degree 8, which cannot determine Ext on the t−s≤7 diagonal. The claimed bordism groups are therefore unsupported.","rationale":"The reader rejected the paper because the Adams spectral sequence computation is not present: the E2-term tables are blank, the differentials are asserted to vanish, and the A1-module structures are stated without derivation. My stress-test agrees and sharpens this. The reason the blank tables matter is not merely expositional: the displayed A1-module data in Figures 1–3 stop at degree 8, while the E2-term on the t−s≤7 diagonal requires module information in higher degrees due to arbitrarily high filtration s. Thus the theorem is genuinely unproven as written, even if the final groups happen to be correct. The concrete test above asks the author or a referee to compute the missing higher-degree input and the resulting E2 entries; this would settle whether the claimed computation is correct. I do not see a reason to move away from the reader's REJECT verdict: the main claim is unsupported, though there is independent evidence in the paper (explicit generators, partial SS arguments) that the result may be true. A revised version supplying the filled E2 pages and a complete differential analysis could change the verdict.","tokens_in":16396,"tokens_out":14704,"duration_ms":132827,"concrete_test":"Fill in the E2-term for at least one case, say Spin-Sp(4), on the diagonal t−s=7 by computing Ext_{A1}^{s,t}(H^*_ko(Mkof0,x2),Z2) for (s,t)=(2,9),(3,10),(4,11), using the actual A1-module in degrees 9–11 obtained from Theorem 3.3 and [6, Theorem 2.28(3)]. If any of these groups is nonzero, the claimed E2-term and the conclusion Ω_7=0 cannot hold as stated. If all are zero, the author still must provide the corresponding entries for Spin-SU(8) and Spin-Spin(16) and verify the h0-linearity differential argument against a filled E2 chart; without that, the proof of Theorem 3.7 remains incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem, Theorem 3.7, is proved by asserting an Adams spectral sequence computation. In §3.4, the author replaces Mkof0,x2 by an A1-module that is required only to match the true module in dimensions ≤7 as a Z2-vector space and to have matching A1-action in dimensions ≤8. From this truncated module, the E2-term is declared to be the blank Tables 5–7. This is not a valid inference. For a connective module, the entry Ext^{s,t}_{A1}(M,Z2) with t−s≤7 can require module data in degrees up to t. Since s is unbounded for fixed t−s (there are infinite h0-towers), entries such as (s,t)=(2,9), (3,10), (4,11) need module degrees 9, 10, and 11, which Figures 1–3 do not specify and which no argument in the paper determines. The differential collapse is also asserted by h0-linearity alone: from d_r(h0 x)=h0 d_r(x) one cannot conclude d_r=0 without knowing that the h0-towers in the relevant E2-range are finite or otherwise controlled, and no E2 chart is shown to check this. The invocation of [6, Theorem 2.28(3)] for non-vector-bundle Thom spectra is also not verified against its hypotheses for Sp(4)/Z2, SU(8)/Z2, and Ss(16), but the missing E2 computation is the immediate load-bearing gap. Secondary typos (e.g., π0(Sp(4))≅Z in §3.1) do not change this.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to compute the Spin-G bordism groups Omega_k^{Spin-Sp(4)}, Omega_k^{Spin-SU(8)}, and Omega_k^{Spin-Spin(16)} for k=0,...,7, together with explicit manifold generators. The method combines the Spin-G bordism isomorphism with Madsen-Tillmann spectra, the Adams spectral sequence for connective ko-homology, and the Debray-Yu theorem for non-vector-bundle Thom spectra. The main results are stated in Theorem 1.1 and proved in Theorem 3.7, with additional geometric arguments in Section 4 identifying generators such as HP^1, CP^2, SU(3)/SO(3), and products of CP^1 and CP^2 factors.","tokens_in":16854,"tokens_out":4975,"duration_ms":42033,"significance":"If the computations were correct and fully justified, the paper would be a useful addition to the program of computing physics-motivated Spin-G bordism groups, and the explicit generator descriptions are valuable. The paper also demonstrates awareness of the modern non-vector-bundle Thom spectrum technology. However, the central Adams spectral sequence computation is not actually presented: the E2-term tables are blank, the truncation argument for the A1-modules is not sufficient, and the differential collapse is asserted rather than proved. As written, the main theorem is unsupported, so the significance cannot be assessed.","major_comments":[{"comment":"The Adams E2-terms are not displayed: Tables 5, 6, and 7 are blank. The proof merely says that 'using the concrete example from Section 4 of [3]' the desired E2-term is obtained. This is the central computation of the paper, and without the actual Ext groups, the reader cannot verify the claimed bordism groups.","section":"Section 3.4, Theorem 3.7 and Tables 5-7"},{"comment":"The proof replaces Mko f0,x2 by an A1-module that is required only to agree with the true module in degrees at most 7 as a Z2-vector space and to have matching A1-action in degrees at most 8. This is insufficient to determine Ext_{A1}^{s,t}(M,Z2) for t-s at most 7: for a fixed small t-s, t can be as large as s+7, and entries with s>0 can depend on module data in degrees above 8. The paper does not explain why the truncated module determines the relevant Ext groups.","section":"Section 3.4, A1-module replacement"},{"comment":"The proof asserts that all differentials vanish because they commute with the Ext_{A1}(Z2,Z2)-action and multiplication by h0. From d_r(h0 x)=h0 d_r(x) one cannot conclude d_r=0 without further information about the h0-towers in the relevant E2-range. The paper provides no E2 chart and no analysis of h0-towers, so the collapse is not justified.","section":"Section 3.4, differential collapse"},{"comment":"The proof contains a concrete false statement: it claims pi_0(Sp(4)) is isomorphic to Z (and similarly for SU(8) and Spin(16)), but pi_0 of a connected Lie group is trivial. In addition, the proof that f and g are 4-equivalences lists pi_4(SU(8)) and pi_4(Spin(16)) but omits pi_4(Sp(4)) and the induced map on pi_4; the cohomological argument about H^4 is therefore not a complete proof of the claimed 4-equivalence.","section":"Section 3.1, Proposition 3.1"},{"comment":"The paper relies on [6, Theorem 2.28(3)] for non-vector-bundle Thom spectra, but does not verify that the hypotheses of that theorem hold for the three cases (G,H) = (Sp(4),Sp(4)/Z2), (SU(8),SU(8)/Z2), and (Spin(16),Ss(16)). This verification is load-bearing because the entire computational framework depends on it.","section":"Sections 2.3 and 3.4, use of Debray-Yu Theorem"}],"minor_comments":[{"comment":"The word 'Hurewitz' should be 'Hurewicz'.","section":"Section 3.1"},{"comment":"The statement that Omega_*^{Spin-Spin(16)} has no odd torsion because H^*(BSpin(16);Z) has none does not immediately follow; H^*(BSs(16);Z) is a quotient of a free ring and could in principle have torsion. The argument needs a justification for the quotient.","section":"Section 3.2, Proposition 3.2"},{"comment":"The text refers to a 'blue Z2' and a 'purple Z2' in Table 5, but Table 5 is blank, so these color-coded references are unverifiable.","section":"Section 4.2"},{"comment":"The notation U in the figures is never defined; it appears to denote the free A1-module on one generator, but this should be stated explicitly.","section":"Figures 1-3"},{"comment":"Reference [10] is a URL to nLab/MathOverflow; this should be replaced by a proper citation to the literature.","section":"Reference [10]"},{"comment":"The statement that the next relation lies in dimension 13 and is not addressed is too vague; either provide the computation or remove the remark.","section":"Remark 3.6"}],"recommendation":"reject","confidential_remarks":"The paper appears to be an early-stage manuscript: the main computational tables are blank, and the proof of the central theorem is an assertion rather than a derivation. The false pi_0 statement and the incomplete pi_4 discussion in Proposition 3.1 reinforce the impression that the technical core is not yet in place. The topic is suitable for the journal, but the current version does not meet the standard of a publishable computation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: the Spin-Sp(4) part is new and the generator analysis is concrete, but the proof of the main theorem is not checkable. The Adams E2-term tables are blank, the differential collapse is asserted in one sentence, and the module truncation argument is invalid on its face: matching the A1-module up to degree 8 does not determine Ext^{s,t} on the t−s≤7 diagonal because the h0-towers have unbounded s.\n\nWhat's good: The paper addresses bordism groups that are relevant to maximal supergravity, and the Spin-Sp(4) computation appears to be genuinely new. The SU(8) and Spin(16) cases are mostly recycled from Debray–Yu [7] and from Tachikawa/Kneißl, but the author does give a unified presentation and explicit generators. Section 4 does real work: it identifies HP1, CP2, the Wu manifold, and products as generators using characteristic number arguments. The bibliography is appropriate.\n\nWhere it falls short: The central theorem, Theorem 3.7, rests on an unshown Adams calculation. The author never displays the E2 chart, so there is no way to verify the stated groups. The claim that all differentials vanish by h0-linearity is not a proof; without knowing the h0-towers are finite or controlled, one cannot draw that conclusion. The stress-test's point that low-degree module data cannot give Ext on the diagonal is correct: for fixed t−s, s is unbounded, and the required module degrees can exceed 8. So the asserted replacement module does not determine the E2-term. There are also smaller problems: Proposition 3.1 says π0 of Sp(4), SU(8), Spin(16) is Z, which is wrong for connected Lie groups, and the π4 discussion is incomplete. The paper also fails to separate new from old results, which makes it hard to identify the original contribution.\n\nThe result may well be true, especially the Spin-Sp(4) part, and the author clearly knows the tools. But as written, I would not rely on it or cite it. The fair path is to send it to a referee who can ask for the full computation; if the author can supply the E2 and differential details, the paper could become a useful contribution to a specialist audience.","headline":"New Spin-Sp(4) computation, but the proof's central Adams calculation is absent — blank E2 tables and an invalid truncation argument leave the theorem uncheckable.","tokens_in":17323,"tokens_out":9947,"would_cite":false,"duration_ms":93467,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57R90"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper computes three Spin-G bordism groups through dimension seven.","keywords":["Spin-G bordism","bordism groups","Adams spectral sequence","Sp(4)","SU(8)","Spin(16)","Thom spectra","manifold generators"],"falsifier":"Recompute the $E_2$-term $\\mathrm{Ext}_{A_1}(-, \\mathbb{Z}_2)$ for the three $A_1$-modules in Figures 1-3 in total degrees up to 7 and check each possible differential $d_r$ against the $h_0$-action; since the paper's $E_2$-tables are blank, this independent calculation would settle Theorem 1.1. A second check would be to compute the three homomorphisms in (18) in degree 6 and verify that $CP^1\\times CP^1\\times CP^1$ indeed evaluates to $-2$ on the integral class $z_6$.","tokens_in":16158,"feed_emoji":"🔄","tokens_out":9696,"duration_ms":78027,"temperature":0.7,"pith_summary":"This paper determines the Spin-G bordism groups in dimensions up to 7 for three symmetry groups that arise as maximal compact subgroups of hidden symmetries in supergravity: $Sp(4)$, $SU(8)$, and $Spin(16)$. The main result is a complete list of groups, with explicit manifolds ($HP^1$, $CP^2$, $SU(3)/SO(3)$, and products with circles) generating each nonzero summand. These groups classify the possible global anomalies carried by spin manifolds with those twistings, so the computation is a step toward deciding which string-theory backgrounds are anomaly-free in low dimensions.","feed_headline":"Spin-G bordism groups computed through dimension 7","feed_subtitle":"Explicit generators are identified for the Sp(4), SU(8), and Spin(16) twistings.","key_machinery":"The load-bearing device is the Adams spectral sequence for connective $ko$-homology of the Madsen-Tillmann spectra $MT(\\mathrm{Spin}\\text{-}G)$, arrived at through the Anderson-Brown-Peterson splitting. Because the relevant Thom spectra are not vector-bundle Thom spectra, the paper invokes a cited theorem that lets the standard $M\\mathrm{ko}_{f_0,x_2}$ model still be used. The computation then rests on $A_1$-module presentations ($A_1$ is the subalgebra of the Steenrod algebra generated by $Sq^1$ and $Sq^2$) for $H^*_{ko}(M\\mathrm{ko}_{f_0,x_2})$ up to degree 8, obtained from the $\\mathbb{Z}_2$-cohomology rings of the classifying spaces; on the resulting $\\mathrm{Ext}_{A_1}(-, \\mathbb{Z}_2)$ $E_2$-terms; and on the claim that all Adams differentials vanish because they commute with the $h_0$-action. The Leray-Serre spectral sequence supplies the cohomology rings, and a theorem on Eilenberg-Mac Lane spectra is used to split off extensions at the end.","core_discovery":"The central claim is Theorem 1.1: for $k=0,\\dots,7$, the three groups are $\\Omega_k^{\\mathrm{Spin}\\text{-}\\mathrm{Sp}(4)} = (\\mathbb{Z},0,0,0,\\mathbb{Z}\\oplus\\mathbb{Z},\\mathbb{Z}_2\\oplus\\mathbb{Z}_2,\\mathbb{Z}_2\\oplus\\mathbb{Z}_2,0)$, $\\Omega_k^{\\mathrm{Spin}\\text{-}\\mathrm{SU}(8)} = (\\mathbb{Z},0,0,0,\\mathbb{Z}\\oplus\\mathbb{Z},\\mathbb{Z}_2,\\mathbb{Z}\\oplus\\mathbb{Z}_2,0)$, and $\\Omega_k^{\\mathrm{Spin}\\text{-}\\mathrm{Spin}(16)} = (\\mathbb{Z},0,0,0,\\mathbb{Z}\\oplus\\mathbb{Z},\\mathbb{Z}_2,\\mathbb{Z}_2,0)$. The free part in degree 4 is generated by $HP^1$ and $CP^2$ for all three; in degree 5, $SU(3)/SO(3)$ generates a $\\mathbb{Z}_2$ in all three cases and $HP^1\\times S^1$ gives a second $\\mathbb{Z}_2$ for $\\mathrm{Spin}\\text{-}\\mathrm{Sp}(4)$; in degree 6, $HP^1\\times S^1\\times S^1$ and $CP^2\\times CP^1$ generate $\\mathbb{Z}_2\\oplus\\mathbb{Z}_2$ for $\\mathrm{Spin}\\text{-}\\mathrm{Sp}(4)$, while $\\mathrm{Spin}\\text{-}\\mathrm{SU}(8)$ has a free summand generated by $CP^1\\times CP^1\\times CP^1$ plus a $\\mathbb{Z}_2$ generated by $CP^2\\times CP^1$, and $\\mathrm{Spin}\\text{-}\\mathrm{Spin}(16)$ has only the $\\mathbb{Z}_2$ generated by $CP^2\\times CP^1$. The paper also proves that the three theories are isomorphic in degrees up to 4 and have no odd-prime torsion.","pith_inferences":["Pushing the same computation beyond degree 7 would require including the 13-dimensional relation noted in Remark 3.6, so the higher groups are not determined by this paper.","The same fiberwise arguments may give a uniform computation for $\\mathrm{Spin}\\text{-}\\mathrm{Sp}(2n)$, $\\mathrm{Spin}\\text{-}\\mathrm{SU}(2n)$, and $\\mathrm{Spin}\\text{-}\\mathrm{Spin}(4n)$ for larger $n$, where the cohomology rings are known but the $A_1$-module complexity grows.","If these bordism classes are realized in string theory, the nonzero $\\mathbb{Z}_2$ classes in degrees 5 and 6 would imply that certain five- and six-dimensional backgrounds cannot be made anomaly-free by adding local counterterms alone."],"forward_implications":["In dimensions 0-7 the three Spin-G bordism groups are completely known, so any Spin-Sp(4), Spin-SU(8), or Spin-Spin(16) manifold in these dimensions is either bordant to one of the listed generators or to a boundary.","Because the three theories are isomorphic through degree 4, the first genuinely different torsion appears in degree 5.","The explicit generators give concrete representatives: $HP^1$ and $CP^2$ generate the degree-4 free part, $SU(3)/SO(3)$ generates a $\\mathbb{Z}_2$ in degree 5 in every case, and circle products generate the extra torsion for $\\mathrm{Spin}\\text{-}\\mathrm{Sp}(4)$.","The vanishing in degrees 1, 2, 3, and 7 means no nontrivial bordism obstructions exist for these twistings in those dimensions."],"supporting_citations":[{"why":"Supplies the theorem that Adams spectral sequence computations remain valid for non-vector-bundle Thom spectra, letting the M ko_{f0,x2} model stand in for the Spin-G Madsen-Tillmann spectra.","marker":"[6]"},{"why":"Provides the prior Spin-SU(8) computation and identifies SU(3)/SO(3) as the degree-5 generator, which is reused here.","marker":"[7]"},{"why":"Establishes the Spin-G bordism framework as spin bordism of a Thom spectrum and the h0-commutativity argument that kills the Adams differentials.","marker":"[5]"},{"why":"Supplies the worked guide and concrete example used to identify the E2-term Ext_{A1}(-, Z2) for the three modules.","marker":"[3]"},{"why":"The cited theorem on Eilenberg-Mac Lane spectra is invoked to split off the circled nodes in the E2-term with no nontrivial extensions.","marker":"[13]"},{"why":"Provides the integral cohomology of the Lie groups and the inclusion maps used to compute the Z2-cohomology rings of the classifying spaces.","marker":"[15]"},{"why":"Used to rule out odd-prime torsion in the Spin-Spin(16) case from the integral cohomology of BSpin(16).","marker":"[12]"}],"fun_headline_variants":["Spin-Sp(4), Spin-SU(8), Spin-Spin(16) bordism to dim 7","Explicit generators for three Spin-twisted bordisms up to 7D","Spin-G bordism: three twistings computed explicitly to 7","Spin-Sp(4), Spin-SU(8), Spin-Spin(16) bordism groups to 7D","Three Spin-twisted bordism groups solved explicitly through dimension 7"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The computation rests on the unshown algebraic claim that the $A_1$-module structures in Figures 1-3 are exactly correct up to degree 8 and that every Adams differential vanishes; if either part fails, the listed groups change.","fun_headline_variants_meta":{"raw":{"variants":["Spin-Sp(4), Spin-SU(8), Spin-Spin(16) bordism to dim 7","Explicit generators for three Spin-twisted bordisms up to 7D","Spin-G bordism: three twistings computed explicitly to 7","Spin-Sp(4), Spin-SU(8), Spin-Spin(16) bordism groups to 7D","Three Spin-twisted bordism groups solved explicitly through dimension 7"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002249,"raw_usage":{"total_tokens":8743,"prompt_tokens":1042,"completion_tokens":7701,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":658,"completion_tokens_details":{"reasoning_tokens":7585}},"tokens_in":658,"tokens_out":7701,"duration_ms":50956,"temperature":1.0,"reasoning_tokens":7585,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:35:18.720057+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the $E_2$-term $\\mathrm{Ext}_{A_1}(-, \\mathbb{Z}_2)$ for the three $A_1$-modules in Figures 1-3 in total degrees up to 7 and check each possible differential $d_r$ against the $h_0$-action; since the paper's $E_2$-tables are blank, this independent calculation would settle Theorem 1.1. A second check would be to compute the three homomorphisms in (18) in degree 6 and verify that $CP^1\\times CP^1\\times CP^1$ indeed evaluates to $-2$ on the integral class $z_6$.","supporting_citations":[{"cited_title":"What bordism-theoretic anomaly cancellation can do for U","cited_arxiv_id":null,"evidence_quote":"Provides the prior Spin-SU(8) computation and identifies SU(3)/SO(3) as the degree-5 generator, which is reused here."},{"cited_title":"Campbell","cited_arxiv_id":null,"evidence_quote":"Supplies the worked guide and concrete example used to identify the E2-term Ext_{A1}(-, Z2) for the three modules."},{"cited_title":"Eilenberg-Mac Lane Spectra","cited_arxiv_id":null,"evidence_quote":"The cited theorem on Eilenberg-Mac Lane spectra is invoked to split off the circled nodes in the E2-term with no nontrivial extensions."},{"cited_title":"Topology of Lie groups, I and II","cited_arxiv_id":null,"evidence_quote":"Provides the integral cohomology of the Lie groups and the inclusion maps used to compute the Z2-cohomology rings of the classifying spaces."},{"cited_title":"On the integral cohomology of BSpin(n)","cited_arxiv_id":null,"evidence_quote":"Used to rule out odd-prime torsion in the Spin-Spin(16) case from the integral cohomology of BSpin(16)."}],"review_version":1}