{"id":"453b67d8-7876-44b5-b027-74d419328593","arxiv_id":"2605.26199","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"All partial groups of order ≤10 are enumerated, and two theorems are proved: high-dimension indecomposables are group skeleta, and degree-≤2 partial groups are 2-coskeletal.","lead":"The paper enumerates all partial groups (in Chermak’s sense) of order at most 10 and proves two structural theorems suggested by that census. A smart generalist might care because the data-to-theorem loop yields clean classification results about when such objects are just groups or are coskeletal.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"Supplied full text is the wrong paper; the census that underpins both theorems remains uninspectable.","rationale":"The reader already diagnosed the manuscript mismatch and correctly set the verdict to UNVERDICTED with low confidence. The same mismatch is still present, so the load-bearing computational foundation of the central claim remains uncheckable. No deeper mathematical objection can be raised until the correct text is supplied; the reader’s weakest-assumption diagnosis is therefore unchanged and still decisive.","tokens_in":8952,"tokens_out":419,"duration_ms":13252,"concrete_test":"Obtain the genuine source of arXiv:2605.26199; re-implement or re-run the enumeration for order ≤5 and check that the list of indecomposables matches the complete list claimed to appear in the paper. Any mismatch falsifies the census and undermines the theorems drawn from it.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The strongest claim (indecomposables of dimension |G|-2 are precisely group skeleta; degree ≤2 partial groups are 2-coskeletal) is presented as having been first observed in a complete computer census of Chermak partial groups of order ≤10 (123 650 of order ≤9, 178 937 003 of order 10) and then proved. That claim therefore rests on the enumerator correctly realising Chermak’s definition together with the derived notions of dimension, indecomposability, higher Segal degree and coskeletality. The CACHEABLE manuscript body, however, is an unrelated GR paper on black strings in PFDM (arXiv:2605.26198). Neither the algorithm, the invariants used for isomorphism filtering, the verification method, nor the subsequent proofs can be examined. Any systematic bug in the enumerator would simultaneously invalidate the published counts and the empirical basis of the two theorems.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript claims a complete computer enumeration of all partial groups (in Chermak’s sense) of order at most 10, reporting 123,650 such objects of order ≤9 and 178,937,003 of order 10, together with an explicit list of the indecomposable ones of order ≤5. Inspection of that census is said to have suggested, and then led to proofs of, two structural theorems: (i) the indecomposable partial groups whose dimension is two less than their order are precisely the skeleta of ordinary groups of that order, and (ii) every partial group of higher Segal degree at most 2 is 2-coskeletal. The supplied body text, however, is an unrelated general-relativity paper on black strings in perfect-fluid dark matter (arXiv:2605.26198), so none of the enumeration algorithm, invariants, verification method, or proofs can be examined.","tokens_in":9102,"tokens_out":628,"duration_ms":6234,"significance":"If the census is correct and the two theorems are proved as claimed, the work would supply a concrete computational foundation for the still-young theory of Chermak partial groups and would give two clean structural characterisations (group skeleta at codimension 2, and 2-coskeletality for low Segal degree). Those results would be of genuine interest to researchers working on fusion systems, higher-categorical groupoids and related combinatorial algebra. At present, however, the significance remains purely prospective: the only available text is a different paper, so neither the computational claims nor the proofs can be assessed.","major_comments":[{"comment":"The body supplied under the arXiv identifier 2605.26199 is in fact the unrelated manuscript “Black string immersed in perfect fluid dark matter” (arXiv:2605.26198). Consequently every load-bearing claim of the abstract—the enumeration algorithm, the isomorphism invariants, the notions of dimension/indecomposability/higher Segal degree/coskeletality, the dataset counts, and the two subsequent proofs—is completely absent. Without that material the paper cannot be refereed.","section":null},{"comment":"Even granting the abstract’s numerical claims, the correctness of the two theorems rests on the uninspectable assertion that the enumerator realises Chermak’s definition faithfully. A systematic bug would simultaneously invalidate the published counts (123 650 / 178 937 003) and the empirical basis from which the conjectures were drawn. The manuscript as submitted therefore contains no verifiable mathematical content supporting its central claims.","section":null}],"minor_comments":[],"recommendation":"reject","confidential_remarks":"The supplied full-text body is a completely different paper (gr-qc black-string work). This appears to be a packaging or arXiv-identifier error rather than a mathematical submission that can be evaluated. I recommend the editor contact the authors to obtain the correct source before any further consideration; until then the only possible recommendation is reject for lack of content."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing you need to know first: the manuscript body we were handed is not Hackney’s partial-groups paper. It is an unrelated GR note on black strings in PFDM (2605.26198). Everything below is therefore an abstract-only read of arXiv:2605.26199.\n\nWhat looks new is a complete computer enumeration of Chermak partial groups through order 10, with explicit cardinalities (123 650 of order ≤9, 178 937 003 of order 10) and a shipped dataset, plus the full list of indecomposables through order 5 printed in the paper itself. Inspection of that data produced two clean theorems the author then proved: indecomposables of dimension |G|−2 are precisely the skeleta of ordinary groups of that order, and partial groups of higher Segal degree ≤2 are already 2-coskeletal. That is honest specialized progress—usable census plus two structural lemmas—not a routine parameter scan.\n\nThe soft spot is load-bearing and exactly the one the stress-test flags. Both theorems were first observed in the census, so they rest on the enumerator correctly realizing Chermak’s definition together with dimension, indecomposability, higher Segal degree and coskeletality. The abstract asserts the counts and the subsequent proofs but exhibits none of the algorithm, isomorphism invariants, or verification method. Until the correct PDF and dataset appear we cannot check soundness or reproducibility; a systematic bug would sink both the numbers and the empirical origin of the lemmas.\n\nIf the real paper delivers what the abstract promises, it is for people working on partial groups, fusion systems or higher Segal conditions who need small-order examples or structural shortcuts. On the abstract alone it is formally grounded enough and evidentially sharp enough to deserve referee time rather than a desk reject. I would not cite it myself in the next year (wrong corner for my work) and I would not bring it to reading group until we have the actual text. Recommendation: request the correct manuscript; if it matches the abstract, send it to peer review.","headline":"Abstract promises a usable small-order census of Chermak partial groups plus two clean structural lemmas, but the supplied full text is the wrong paper, so the enumerator and proofs stay uncheckable.","tokens_in":9728,"tokens_out":528,"would_cite":false,"duration_ms":16538,"reading_group":"no","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20N99"],"pacs":[],"model":"grok-4.5","headline":"Indecomposable partial groups of dimension two less than their order are exactly the skeleta of ordinary groups, and partial groups of higher Segal degree at most 2 are 2-coskeletal.","keywords":["partial groups","Chermak","enumeration","coskeletal","Segal degree","skeleta","indecomposable"],"falsifier":"An explicit partial group of order at most 10 that is indecomposable of dimension |X|-2 yet is not the skeleton of a group, or one of Segal degree ≤2 that fails to be 2-coskeletal; or a reproducible discrepancy showing the published counts are incomplete.","tokens_in":9773,"feed_emoji":"🔢","tokens_out":852,"duration_ms":23780,"temperature":0.7,"pith_summary":"The paper carries out a complete computer enumeration of all partial groups (in Chermak’s sense) of order at most 10, producing a public dataset of 123650 examples up to order 9 and more than 178 million of order 10, together with an explicit list of the indecomposable ones of order at most 5. Inspection of that census suggested two structural patterns that the authors then proved: an indecomposable partial group whose dimension equals its order minus two must be the skeleton of a group of the same order, and every partial group whose higher Segal degree is at most 2 is automatically 2-coskeletal. The result therefore supplies both a definitive small-order classification and two theorems that isolate precisely when a partial group collapses to ordinary group data or satisfies a strong coskeletal condition. A reader interested in fusion systems or higher-categorical algebra cares because these theorems mark a sharp boundary between classical groups and genuinely partial objects.","feed_headline":"Partial groups of dim n-2 are exactly group skeletons","feed_subtitle":"A full census to order 10 turns observed patterns into two proved structure theorems.","key_machinery":"The exhaustive computer enumeration of partial groups of order ≤10 (together with the resulting dataset of isomorphism types), which both suggested the two theorems and supplied the small-order evidence used to formulate them.","core_discovery":"After enumerating every partial group of order at most 10, the authors prove that the indecomposable partial groups of dimension two less than their order are precisely the skeleta of groups of that order, and that every partial group of higher Segal degree at most 2 is 2-coskeletal.","pith_inferences":["The jump to more than 178 million partial groups already at order 10 indicates that most examples are far from groups, so theorems that isolate the group-like ones become especially useful for navigation of the census.","The same enumeration pipeline could be used to test whether the coskeletal property for fixed Segal degree d extends beyond d=2.","The two theorems together suggest that only a small number of numerical invariants (dimension and Segal degree) control the gap between partial groups and ordinary groups."],"forward_implications":["Any indecomposable partial group of order n and dimension n-2 can be identified with the skeleton of a group of order n.","A partial group of higher Segal degree at most 2 is completely determined by its 2-coskeleton, so higher simplicial data need not be checked separately.","The complete lists up to order 10 become a definitive reference against which further structural conjectures can be tested by machine.","The explicit classification of indecomposables of order ≤5 can be used by hand in low-order arguments."],"fun_headline_variants":["Indecomposable partial groups of dim n-2 equal group skeleta","Order-10 census proves dim n-2 partial groups are group skeleta","Partial groups of Segal degree ≤2 are always 2-coskeletal","Enumeration to order 10 yields two proved partial-group theorems","Dim-equals-order-minus-2 partial groups are precisely group skeleta"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The computer enumeration correctly implements Chermak’s definition of partial groups and finds every isomorphism type up to order 10.","fun_headline_variants_meta":{"raw":{"variants":["Indecomposable partial groups of dim n-2 equal group skeleta","Order-10 census proves dim n-2 partial groups are group skeleta","Partial groups of Segal degree ≤2 are always 2-coskeletal","Enumeration to order 10 yields two proved partial-group theorems","Dim-equals-order-minus-2 partial groups are precisely group skeleta"]},"model":"grok-4.5","effort":"low","cost_usd":0.004578,"raw_usage":{"total_tokens":1251,"prompt_tokens":636,"num_sources_used":0,"completion_tokens":81,"cost_in_usd_ticks":45780000,"prompt_tokens_details":{"text_tokens":636,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":534,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":636,"tokens_out":81,"duration_ms":5024,"temperature":1.0,"reasoning_tokens":534,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T15:57:24.882055+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"An explicit partial group of order at most 10 that is indecomposable of dimension |X|-2 yet is not the skeleton of a group, or one of Segal degree ≤2 that fails to be 2-coskeletal; or a reproducible discrepancy showing the published counts are incomplete.","supporting_citations":[],"review_version":2}