{"id":"90e7b1d8-efbe-4637-9300-6ba32b45b479","arxiv_id":"2412.10876","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors provide a machine-generated dataset and proof tables for Adams spectral sequence computations that support the resolution of the Last Kervaire Invariant Problem in dimension 126.","lead":"This paper describes a large computational dataset of Adams differentials and extensions for 49 CW spectra, with machine-generated proof tables. The data was used, together with further arguments in a companion paper, to resolve the Last Kervaire Invariant Problem in dimension 126.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof log's soundness hinges on unverified applications of two companion-paper theorems; a bug or missing hypothesis in any Syn/SynCs row could invalidate the stem-126 differentials.","rationale":"The reader's weakest_assumption identifies essentially the same locus: correctness depends on theorems from the companion paper [7] and on unverified software. My concern is a refinement rather than a different objection. The key additional point is that the proof log's most powerful inference types (Syn, SynCs, SynIn) are precisely the ones whose logical content and hypothesis checks are not auditable from the present document alone. This is a genuine load-bearing concern, but it does not by itself refute the central claim. Relying on a companion paper for theorems is standard mathematical practice, and the authors are transparent that the theorems come from [7]. The right response is to require an independent audit of the proof log before the machine-proof claim is accepted as settled. Since the reader's verdict is already CONDITIONAL, my analysis does not move the verdict; it sharpens the condition that should be imposed: independent verification of the Syn-family rows used in the stem-126 argument. I do not see an internal inconsistency that would justify REJECT or UNVERDICTED on the basis of this document alone.","tokens_in":22060,"tokens_out":5385,"duration_ms":56491,"concrete_test":"Write an independent checker for the proof log: parse proofs.db, build the full dependency graph of the 21 million rows, and (1) reject any cycle; (2) for every row with reason 'Syn', 'SynCs', or 'SynIn', check the info-column conditions against the formal statements and hypotheses of the Generalized Leibniz Rule and Generalized Mahowald Trick exactly as stated in [7]; (3) recompute the Section 5 differentials in stems 122-127 of S0 and stem 126 of S0/nu using only axiom rows (reason d2, N, G, XX, XY, CsCm) plus the verified Syn rows. If the checker passes on every differential used in [7], the machine-proof subclaim is sound; if it rejects any one of those differentials, the central claim is unsupported as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central artifact of the paper is the Table of Proofs: 21 million rows that are supposed to constitute machine-generated proofs of Adams differentials and extensions. The highest-risk rows are those with reason 'Syn', 'SynCs', and 'SynIn' (Section 4), which invoke the Generalized Leibniz Rule and the Generalized Mahowald Trick from the companion paper [7]. The present document does not state these theorems or their hypotheses, and the only evidence that the hypotheses are satisfied is the program's own assertion that it 'checked' them, recorded in the info column. No independent verifier is provided for these checks. Since Section 5 presents differentials in stems 122-127 of S0 and stem 126 of S0/nu as ingredients for the Last Kervaire Invariant Problem in [7], a single invalid Syn row in that range could break the advertised Kervaire conclusion. There is also an implicit circularity risk: [7] is the same paper that uses these machine-generated results, and if its proofs of the Generalized Leibniz Rule or Generalized Mahowald Trick depend on any of the very differentials the program derives using them, then the logical chain would be circular. The paper gives no dependency graph or machine-readable audit trail that would rule this out.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper describes a computational dataset of Adams E2-pages, d2 differentials, Adams differentials, and extension data for 49 CW spectra, 180 maps, and 61 cofiber sequences, together with a 21-million-row Table of Proofs generated by the programs ./Adams and ./ss. The text specifies the data formats, the naming conventions for spectra and maps, the encoding of spectral sequence differentials, and the structure of the machine-generated proofs. Section 5 displays machine-generated Adams differentials for the sphere spectrum in stems 122–127 and for S0/nu in stem 126. The abstract states that these results, with additional human ad hoc arguments, resolve the Last Kervaire Invariant Problem in dimension 126; that solution is the subject of the companion paper [7]. The present document is therefore primarily a data paper and proof-log guide, but it also advertises a major mathematical outcome.","tokens_in":22341,"tokens_out":8226,"duration_ms":78722,"significance":"If the dataset and proof logs are sound, this is a valuable and unusually transparent computational resource for the Adams spectral sequence, with per-differential proof records and public data on Zenodo and GitHub. The paper also represents a concrete step toward a machine-assisted resolution of the Kervaire invariant problem in dimension 126, which would be a landmark result. The strengths include the reproducibility of the data artifacts and the explicit documentation of proof structures. However, the significance is conditional on two unresolved points: the correctness of the unverified programs that produced the proofs, and the independence of the companion-paper theorems used as inference rules. As written, the machine proofs are not independently auditable in a formal sense, and the possible circularity with [7] is not addressed. These concerns are load-bearing because the advertised Kervaire conclusion depends on the reliability of specific differential rows.","major_comments":[{"comment":"The machine proofs invoke the Generalized Leibniz Rule and the Generalized Mahowald Trick from the companion paper [7], but this document neither states these theorems nor their hypotheses, and it does not explain why the conditions recorded in the info column are sufficient for their application. Since [7] is the same paper that uses the machine-generated differentials of this document to solve the Kervaire problem, the absence of a dependency analysis leaves open a circularity risk: if the proof of either theorem uses any differential that the program derives with it, the logical chain is circular. Please state the precise theorems, their hypotheses, and an explicit argument that they are independent of the machine results that the program proves using them.","section":"§1, §4 (reason='Syn', 'SynCs', 'SynIn')"},{"comment":"The central artifact is a table of 21 million proof rows produced by the unverified program ./ss. There is no independent proof checker, no formal specification of the inference rules, and no soundness theorem for the program; a 'proof' row consists of assumption rows (reason='T') and deduction rows (reason='D') with natural-language justifications in the info column. The program's own assertion that a condition was 'checked' is the only evidence. To deserve the term 'machine proofs', the paper should provide an auditable mechanism: for example, a formal checker for the proof logs, an independent reimplementation, or a precise operational semantics of ./ss's inference steps together with a theorem that every generated row is valid in the Adams spectral sequence.","section":"§4, Table of Proofs"},{"comment":"The data encoding assumes that all visible nontrivial differentials have length shorter than 1000. This assumption is stated but not justified or verified. If a differential of length at least 1000 occurs in the computed range, the level encoding would misclassify its target as a permanent cycle (level=9000), making the dataset interpretation unsound. Please provide a programmatic check that no such differential exists up to the stated internal degrees, or restrict the completeness claims of the dataset.","section":"§3, Notation 3.5"},{"comment":"Several basis elements have d2=[NULL] (Section 2.1, Notation 2.3) and several displayed differentials are marked with '?' (for example, Table 25 rows 'x126,8,4+x126,8 d6 ?' and 'h6(C'+X2) d17 ?'; Table 24 row 'x126,18+e0x109,14,2 d7 ?'). The paper should clarify which entries are asserted to be fully resolved with machine proofs and which remain undetermined, and it should state explicitly whether the Kervaire-critical rows used in [7] belong to the resolved class. As it stands, the reader cannot determine whether an unresolved '?' row is part of the advertised chain.","section":"§2.1, §2.3, §5"},{"comment":"The naming convention for several CW spectra explicitly allows multiple homotopy types when the stated cofiber sequences exist, and the paper says it does not assume which one is picked. However, the Adams E2-page and subsequent differentials generally depend on the actual homotopy type, not only on the existence of cofiber sequences. The paper should either prove that the computed E2 data and proof logs are independent of these choices, or specify the precise models for the spectra in the dataset. Without this, the dataset's rows may not be well defined for every named spectrum.","section":"§2, Remark 2.6"}],"minor_comments":[{"comment":"The text contains numerous typos and missing spaces (e.g., 'Table1provides', 'Thecolumnname', 'Theprogram', and the word 'and' appearing as 'A' in the sentence beginning '...and we can get...'). A careful proofreading pass is needed.","section":"Throughout"},{"comment":"The convention that x_i denotes ring generators and v_i denotes module generators is not consistently reflected in the tables of Section 5, where many symbols (e.g., x123,9, h6(C'+X2), [B4], M, g) are used without a glossary. Please add a reference to the definitions in the dataset or include a notation appendix.","section":"§2.1, Notation 2.1"},{"comment":"The rules for determining the unique suspension k and the unique inclusion or quotient for maps named X__Y are stated informally; please give the precise criterion used by the program to select among candidates.","section":"§2.2, Notation 2.8"},{"comment":"The examples of level encodings (e.g., level=9998, 9000, 2, 10000) are not annotated; a step-by-step decoding of one row from each table would greatly improve readability.","section":"§3, Tables 15–16"},{"comment":"The descriptions of reasons 'ToCs', 'OutCsI', and 'CsCm' are terse and would be considerably easier to trust with one concrete worked example for each reason, parallel to the examples already given for 'T' and 'D'.","section":"§4, row reasons"},{"comment":"Only 22 of the 49 spectra listed in Table 1 appear in Table 11 for computed d2 differentials; please state whether d2 is trivial, unknown, or simply omitted for the remaining spectra, and explain how ./ss handles missing d2 values in the proof system.","section":"Table 11"},{"comment":"The tables list differentials without cross-references to the corresponding proof row ids in the Table of Proofs; adding an id column would make the individual machine proofs directly auditable.","section":"§5, Tables 18–28"},{"comment":"Since the Generalized Leibniz Rule and the Generalized Mahowald Trick are load-bearing for the proofs, the citation [7] should include specific theorem numbers, and the paper should note whether [7] is a preprint or accepted for publication.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"This manuscript is essentially a data companion to the authors' solution of the Last Kervaire Invariant Problem in [7], and its standalone value rests on the reliability and auditability of the proof logs. The two main risks are the unverified proof generator and the circularity/dependency on [7]. Both are addressable in principle, but the current text does not provide the necessary statements, hypotheses, or audit trail. I recommend major revision with a strong request for an independent verification mechanism and an explicit independence argument. Editors may also wish to consider whether the scope of the journal supports a paper whose main theorem is delegated to a companion preprint."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: this is a companion resource paper, and it is genuinely useful. The new thing is the scale: 49 CW spectra, 180 maps, 61 cofiber sequences, and a 21-million-row Table of Proofs recording machine-generated Adams differentials and extensions. That is not a rehash of earlier computations; no public dataset of this size for Adams spectral sequences exists. The paper is also unusually candid. It marks [NULL] entries, labels some differentials as undetermined with '?', and explicitly says which rows come from human-inserted facts, like the image-of-J differentials and the Bruner–Rognes tmf differential. That honesty matters, and it makes the resource usable.\n\nThe proof format is real work, not a black box: each row states the spectrum, bidegree, differential, and the reason (d2, naturality, Leibniz, Try/Deduction). The d2 rows are tied to secondary Steenrod operations with published references. If you want to check a particular differential, you can query proofs.db and trace the reason chain. That is a serious contribution.\n\nThe soft spots are real but not disqualifying. The proofs rely on the Generalized Leibniz Rule and Generalized Mahowald Trick from the authors' companion paper [7], and this document does not state those theorems or their hypotheses. So the Table of Proofs is not self-contained. A skeptical reader cannot verify the highest-risk rows (reason Syn, SynCs, SynIn) from this PDF alone. The stress-test worry about circularity does not actually land based on the text: nothing here shows that [7] proves its theorems using the very differentials that ./ss derives from them. But the paper should explicitly rule that out, ideally with a dependency diagram or a sentence in [7] confirming independence. Also, the assumption that all visible nontrivial differentials have length below 1000 is a computational truncation, not a theorem; it is stated, not justified. For the Kervaire range that may be harmless, but as written it is an open loop. And the programs ./Adams and ./ss are not formally verified; these are machine-generated proofs, not Lean/Coq-checked proofs. That is a limitation, not a fatal one.\n\nWho is this for? Stable homotopy theorists working with Adams spectral sequences, and anyone who needs to rely on the Kervaire resolution in [7]. It should be peer-reviewed, but the referee should have access to both papers and should check whether the companion theorems are independent of the dataset. Conditional acceptance with a request for an explicit independence statement would be a fair outcome.\n\nMy own view: treat the dataset as a large computational resource with honest caveats, not as a closed formal proof. Send it to a serious referee.","headline":"A big, honest computational dataset for Adams spectral sequences; it deserves a serious referee, but acceptance should be tied to clarifying its dependency on the companion paper.","tokens_in":22789,"tokens_out":1828,"would_cite":true,"duration_ms":19674,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55T15","55Q45","55P42"],"pacs":[],"model":"deepseek-v4-flash","headline":"Machine-generated, checkable proofs of Adams differentials and extensions, with a few human arguments, resolve the Last Kervaire Invariant Problem in dimension 126.","keywords":["Adams spectral sequence","Adams differentials","extension problems","CW spectra","Kervaire invariant problem","machine-generated proof","stable homotopy groups of spheres","spectral sequence data"],"falsifier":"An independent recomputation of the Adams spectral sequence of $S^0$ in stems 122-127 and of $S^0/\\nu$ in stem 126 would falsify the central claim if any listed $d_r$ value has a different target, or if a 'Try' row that is supposed to end in contradiction instead propagates consistently; equivalently, checking the machine proofs of the three hand-added differentials would reveal whether the Table of Proofs uses them only where the paper says.","tokens_in":1798,"feed_emoji":"🧮","tokens_out":4706,"duration_ms":120906,"temperature":0.7,"pith_summary":"This paper is the computational companion to a solution of a long-standing stable homotopy problem: it presents the dataset and machine-generated proofs behind the resolution of the Last Kervaire Invariant Problem in dimension 126. The dataset covers 49 connective CW spectra, 180 maps between them, and 61 cofiber sequences, with Adams $E_2$ pages, $d_2$ differentials, higher differentials, and extensions computed by two programs. The central claim is that every computed Adams differential and extension in the dataset carries a machine-generated proof, organized in a searchable Table of Proofs, and that these results, together with a few hand-added differentials, close the dimension-126 case. A reader should care because the claims are not left as black-box computer output: each differential is documented as a nested case analysis that a reader can inspect row by row.","feed_headline":"Machine proofs resolve the last Kervaire case at 126","feed_subtitle":"A dataset of 49 spectra and 180 maps carries the checkable differential proofs behind the dimension-126 result.","key_machinery":"The engine is the program ./ss, which stores an Adams spectral sequence as nested subgroups $0 = B_1^{s,t} \\subseteq B_2^{s,t} \\subseteq \\cdots \\subseteq Z_2^{s,t} \\subseteq E_2^{s,t}$ and treats each differential as an isomorphism between quotient groups, storing both $d_r$ and its inverse $d_r^{-1}$. The same data structure is applied to a cofiber sequence $X \\to Y \\to Z$ to encode extensions as maps between filtration subgroups, so extensions become differential-like objects with their own filtration jump and source and target degrees. From the $d_2$ differentials computed by the program ./Adams, the program derives new differentials and extensions using the Leibniz rule, naturality, and two theorems from the companion paper - the generalized Leibniz rule and the generalized Mahowald trick - which give conditions under which auxiliary differentials or a cofiber sequence force the value of a target differential. Every step is recorded as a row in the Table of Proofs, with reason labels 'T' for a trial assumption and 'D' for the deduction that rules out candidates.","core_discovery":"The paper's central claim is that the Adams spectral sequence data for a carefully chosen collection of CW spectra can be produced by machine together with machine-readable proofs of every computed differential and extension. The programs compute the $d_2$ differentials from secondary Steenrod operations, then derive all higher differentials and extensions by propagating through the Leibniz rule, naturality, and the generalized inference rules introduced in the companion paper. The proof of each nontrivial differential is a nested case analysis recorded in the Table of Proofs: candidate values are assumed one by one, each assumption is propagated until it contradicts an earlier computation or degree bound, and the candidate that survives is the value the machine records. Three differentials are put in by hand - two from the image of $J$ and one from power operations on $\\mathrm{tmf}$ - and these are the only non-machine inputs. On top of this computed data, a few ad hoc human arguments complete the argument that resolves the Last Kervaire Invariant Problem in dimension 126.","pith_inferences":["Run the same machinery on a spectrum outside the current list, such as a longer stunted real projective space, and compare the computed differentials with known stable homotopy groups; agreement would test the two generalized rules.","The Table of Proofs is a structured collection of case analyses, so an independent proof checker could turn each machine proof into a formally verified certificate.","Extending the computation to higher stems would test the length-shorter-than-1000 assumption directly; a longer visible differential would require revising the display convention.","The same nested proof format could be carried over to other spectral sequences once analogues of the generalized Leibniz rule and generalized Mahowald trick are established there."],"forward_implications":["If the companion paper's human arguments hold, the Last Kervaire Invariant Problem in dimension 126 is closed.","Every differential and extension in the dataset can be audited row by row, so the computational portion of the solution is not a black box.","The same pipeline - $d_2$ via secondary Steenrod operations followed by the generalized inference rules - can be applied to other connective CW spectra to produce $E_2$ pages, differentials, extensions, and proofs over a large stem range.","Encoding differentials as isomorphisms between nested quotient groups gives a uniform, memory-efficient way to organize and share spectral sequence computations."],"supporting_citations":[{"why":"Supplies the generalized Leibniz rule and generalized Mahowald trick, the inference rules the program uses to derive new differentials and extensions.","marker":"[7]"},{"why":"Supplies the computer programs that produce the $E_2$ data, differentials, extensions, and proof tables.","marker":"[4]"},{"why":"Is the accompanying dataset containing the spectra, maps, cofiber sequences, and the Table of Proofs described here.","marker":"[6]"},{"why":"Provides the construction of the $E_3$ page used in the algorithm that computes $d_2$ differentials via secondary Steenrod operations.","marker":"[3]"},{"why":"Provides the secondary Steenrod algebra operations on which the $d_2$ computation rests.","marker":"[8]"},{"why":"Gives the tmf power-operation differential used as one of the hand-added inputs.","marker":"[2]"},{"why":"Frames the Kervaire invariant problem whose dimension-126 case is resolved using the dataset.","marker":"[1]"},{"why":"Establishes the strong Kervaire invariant problem in dimension 62, which is used for the existence of several spectra in the dataset.","marker":"[9]"}],"fun_headline_variants":["Machine proofs settle Kervaire at dimension 126","Last Kervaire case resolved by machine-checkable proofs","Machine proofs end 126-dimensional Kervaire hunt","Computational proofs crack final Kervaire invariant","49 spectra, 180 maps: machine proofs for Kervaire 126"],"cache_read_input_tokens":25088,"weakest_assumption_plain":"The whole edifice rests on the correctness of the two theorems imported from the companion paper, on the correctness of the implementations of the two programs, and on the assumption that every visible nontrivial differential in the data has length shorter than 1000; this document supplies no independent proof of any of those.","fun_headline_variants_meta":{"raw":{"variants":["Machine proofs settle Kervaire at dimension 126","Last Kervaire case resolved by machine-checkable proofs","Machine proofs end 126-dimensional Kervaire hunt","Computational proofs crack final Kervaire invariant","49 spectra, 180 maps: machine proofs for Kervaire 126"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00047,"raw_usage":{"total_tokens":2276,"prompt_tokens":822,"completion_tokens":1454,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":438,"completion_tokens_details":{"reasoning_tokens":1387}},"tokens_in":438,"tokens_out":1454,"duration_ms":8986,"temperature":1.0,"reasoning_tokens":1387,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:30:48.689735+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"An independent recomputation of the Adams spectral sequence of $S^0$ in stems 122-127 and of $S^0/\\nu$ in stem 126 would falsify the central claim if any listed $d_r$ value has a different target, or if a 'Try' row that is supposed to end in contradiction instead propagates consistently; equivalently, checking the machine proofs of the three hand-added differentials would reveal whether the Table of Proofs uses them only where the paper says.","supporting_citations":[{"cited_title":"Github repo.https://github.com/WayneLin92/SSeqCpp Github Webpage 2024","cited_arxiv_id":null,"evidence_quote":"Supplies the computer programs that produce the $E_2$ data, differentials, extensions, and proof tables."},{"cited_title":"Machine proofs for Adams differentials and extension problems among CW spectra","cited_arxiv_id":null,"evidence_quote":"Is the accompanying dataset containing the spectra, maps, cofiber sequences, and the Table of Proofs described here."},{"cited_title":"On the secondary Steenrod algebra.New York J","cited_arxiv_id":null,"evidence_quote":"Provides the secondary Steenrod algebra operations on which the $d_2$ computation rests."},{"cited_title":"Bruner and John Rognes","cited_arxiv_id":null,"evidence_quote":"Gives the tmf power-operation differential used as one of the hand-added inputs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Frames the Kervaire invariant problem whose dimension-126 case is resolved using the dataset."},{"cited_title":"The strong Kervaire invariant problem in dimension 62.Geom","cited_arxiv_id":null,"evidence_quote":"Establishes the strong Kervaire invariant problem in dimension 62, which is used for the existence of several spectra in the dataset."}],"review_version":1}