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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.

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2026-07-12 15:57 UTC pith:D46IVHZ5

load-bearing objection 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. the 2 major comments →

arxiv 2605.26199 v2 pith:D46IVHZ5 submitted 2026-05-25 math.GR math.ATmath.CO

On partial groups of small order

Philip Hackney This is my paper
classification math.GR math.ATmath.CO MSC 20N99
keywords partial groupsChermakenumerationcoskeletalSegal degreeskeletaindecomposable
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

Core claim

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.

What carries the argument

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.

Load-bearing premise

The computer enumeration correctly implements Chermak’s definition of partial groups and finds every isomorphism type up to order 10.

What would settle it

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.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 0 minor

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.

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 (2)
  1. 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.
  2. 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.

Circularity Check

0 steps flagged

No circularity: abstract describes a standard census-then-conjecture-then-prove workflow; supplied body is the wrong paper and cannot exhibit any reduction.

full rationale

The only inspectable claim text for arXiv:2605.26199 is the abstract. It states that a computer enumeration of Chermak partial groups of order ≤10 produced a dataset, that inspection of that dataset suggested two statements, and that those statements were then proved. That is the ordinary mathematical pattern (empirical observation → conjecture → independent proof), not a self-definitional or fitted-input loop: the theorems are not claimed to be true because the enumerator said so, nor is any fitted parameter renamed as a prediction. No uniqueness theorem is imported from the authors’ prior work to force the result; no ansatz is smuggled in via self-citation; no known empirical pattern is merely renamed. The CACHEABLE full manuscript is an unrelated GR paper (black string in PFDM, arXiv:2605.26198) and therefore supplies no equations, definitions, or self-citations belonging to the partial-groups derivation that could be reduced to their inputs. Uninspectability of the enumerator is a verification/correctness concern, not circularity under the stated criteria. With no quotable reduction of a claimed prediction or first-principles result to its own inputs, the circularity score is 0 and the steps list is empty.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Abstract-only review of a pure-math enumeration paper. Load-bearing background is Chermak’s definition of partial groups and the standard notions of dimension, indecomposability, higher Segal degree, and coskeletality. No free numerical parameters are fitted. No new physical entities are invented. The main unproved-from-here inputs are the correctness of the computer search and the ambient definitions from the literature.

axioms (3)
  • domain assumption Chermak’s definition of partial groups (and the associated notions of product, units, and morphisms) is the correct ambient category being enumerated.
    The entire census and both theorems are stated ‘in the sense of Chermak’; if that formalization differs from another common one, the counts and theorems shift.
  • ad hoc to paper The computer enumeration of all partial groups of order ≤10 is complete and correctly classifies isomorphism type, indecomposability, dimension, and higher Segal degree.
    The dataset sizes and the conjectures-turned-theorems rest on this computational claim; the abstract does not supply a proof of completeness.
  • standard math Standard facts about groups, skeleta, and coskeletal simplicial objects used to state the two theorems.
    Background algebraic topology / category theory assumed known to the reader.

pith-pipeline@v1.1.0-grok45 · 12997 in / 2519 out tokens · 28337 ms · 2026-07-12T15:57:24.882055+00:00 · methodology

0 comments
read the original abstract

We explain a computer enumeration of all partial groups (in the sense of Chermak) of order at most 10. An accompanying dataset contains a full list, consisting of 123,650 partial groups of order at most 9 and 178,937,003 partial groups of order 10; the paper itself contains a complete list of indecomposable partial groups of order at most 5. Inspection of the data led us to conjecture and then prove two results: that indecomposable partial groups of dimension two less than their order are precisely skeleta of groups of that order, and partial groups of (higher Segal) degree at most 2 are 2-coskeletal.

Figures

Figures reproduced from arXiv: 2605.26199 by Philip Hackney.

Figure 1
Figure 1. Figure 1: Compatibility graph of atomic BPGs for (2, 0) (Here Aut([−a, a]) and Aut(X1) are groups of pointed automorphisms – automor￾phisms preserving the basepoint 0). Altogether Aut(I) has a! · 2 a · b! elements. 4.2. Binary partial groups with a given underlying involutive set. In this section we describe an algorithm for finding all binary partial groups whose nonidentity elements come from a fixed involutive se… view at source ↗
Figure 2
Figure 2. Figure 2: Compatibility graph of atomic BPGs for (1, 3) 1 · 1 = −2 1 · 1 = −1 1 · 1 = 2 2 · 2 = 1 2 · 2 = −2 2 · 2 = −1 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Representatives for isomorphism classes of cliques We now have excellent combinatorial control over the collection of BPGs on I. The standard Bron–Kerbosch algorithm for finding cliques in a graph may be used to find the full collection of BPGs [BK73]. This means we’ve completed Step 2. However, we are only interested in enumerating partial groups up to isomorphism. As I is fixed, this means we should quot… view at source ↗

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Coskeletality and the higher Segal conditions

    math.AT 2026-07 conditional novelty 6.0

    A simplicial set is upper/lower d-Segal iff it is (d+1)-coskeletal and satisfies the d-Segal conditions in dimensions d+1 and d+2 (with the proved 'if' direction using (d+2)-coskeletality).

Reference graph

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