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arxiv: 2604.13809 · v1 · submitted 2026-04-15 · 🧮 math.RA · math.GR

Recognition: unknown

Symbolic computation in cubic Jordan matrix algebras and in related structures

Torben Wiedemann

Pith reviewed 2026-05-10 11:53 UTC · model grok-4.3

classification 🧮 math.RA math.GR
keywords cubic Jordan matrix algebrasGAP packagesymbolic computationF4-graded groupscommutator relationsLie-theoretic structuresJordan algebras
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The pith

A GAP package enables symbolic computation in cubic Jordan matrix algebras to derive relations in F4-graded groups

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

The paper presents CubicJordanMatrixAlg, a GAP package designed for symbolic computations involving cubic Jordan matrix algebras and related Lie-theoretic structures. It applies this package to calculate commutator relations in F4-graded groups constructed from the algebras. A sympathetic reader would care as it provides a computational aid for handling intricate algebraic operations that support theoretical work in algebra and group theory.

Core claim

The central claim is that the CubicJordanMatrixAlg package facilitates symbolic computation in cubic Jordan matrix algebras and related structures, and its application yields computed commutator relations in F4-graded groups built from these algebras by De Medts and the author.

What carries the argument

The CubicJordanMatrixAlg GAP package, which carries out the symbolic operations on the algebras to enable the relation computations.

If this is right

  • The package makes it possible to compute specific commutator relations in the F4-graded groups.
  • Symbolic computation becomes practical for Lie-theoretic structures linked to Jordan algebras.
  • This tool supports further investigations into the properties of these graded groups.

Where Pith is reading between the lines

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

  • The approach could be adapted to compute relations in other types of graded groups derived from similar algebraic constructions.
  • It may encourage the development of computational tools for additional classes of nonassociative algebras.
  • Verification through this package could lead to new insights or corrections in the understanding of F4-graded groups.

Load-bearing premise

The package correctly implements all algebraic operations and relations of the cubic Jordan matrix algebras without errors.

What would settle it

A discrepancy between the relations computed by the package and those obtained from direct theoretical analysis or another independent computational method would challenge the package's accuracy.

Figures

Figures reproduced from arXiv: 2604.13809 by Torben Wiedemann.

Figure 1
Figure 1. Figure 1: Each rectangle represents one of the elements of G0 2 ⊆ Z 2 . Big rectangles cor￾respond to short roots and small rectangles to long roots. Each four-digit number ϵ1ϵ2ϵ3ϵ4 represents the element P4 i=1 ϵiei of F 0 4 where ¯2 and ¯1 stand for −2 and −1, respectively. For any α ∈ F 0 4 , π(α) corresponds to the rectangle in which α lies. We now turn to (computations in) the Lie algebra L from [DMW26] that ca… view at source ↗
read the original abstract

We present CubicJordanMatrixAlg, a GAP package for symbolic computation in cubic Jordan matrix algebras and in related Lie-theoretic structures. As an application, we use it to compute certain (commutator) relations in $F_4$-graded groups that were constructed by De Medts and the author from cubic Jordan matrix algebras.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper introduces CubicJordanMatrixAlg, a GAP package for symbolic computation in cubic Jordan matrix algebras and in related Lie-theoretic structures. As an application, it uses the package to compute certain commutator relations in F4-graded groups that were constructed from cubic Jordan matrix algebras by De Medts and the author.

Significance. Should the package be correctly implemented and the computed relations verified, this work would provide a practical computational tool for exploring algebraic identities in Jordan algebras and exceptional Lie groups. It could enable researchers to handle complex symbolic computations that are otherwise tedious, building upon existing constructions in the literature and potentially leading to new insights in the field.

major comments (2)
  1. §3 (application section): The specific commutator relations computed by the package are not listed, tabulated, or compared to any theoretical predictions from the De Medts construction, preventing assessment of whether the output is new or correct.
  2. §2 (package description): No implementation details are given for how the cubic Jordan product, the cubic form, or the Jordan identity are encoded symbolically in GAP, and no test cases or verification against known low-dimensional identities are supplied. This directly undermines the claim that the package yields reliable relations in the F4-graded groups.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. We address each major comment below and outline revisions that will strengthen the paper's clarity and verifiability.

read point-by-point responses

  1. Referee: §3 (application section): The specific commutator relations computed by the package are not listed, tabulated, or compared to any theoretical predictions from the De Medts construction, preventing assessment of whether the output is new or correct.

    Authors: We agree that explicit listing and comparison would improve the manuscript. In the revised version, we will expand §3 to include a table or enumerated list of the key commutator relations computed by the package. Each relation will be accompanied by a direct reference to the corresponding theoretical prediction in the De Medts construction, with notes on any discrepancies or new computational insights. This will allow readers to verify correctness and assess novelty. revision: yes

  2. Referee: §2 (package description): No implementation details are given for how the cubic Jordan product, the cubic form, or the Jordan identity are encoded symbolically in GAP, and no test cases or verification against known low-dimensional identities are supplied. This directly undermines the claim that the package yields reliable relations in the F4-graded groups.

    Authors: We acknowledge the need for greater transparency here. The revised manuscript will augment §2 with a new subsection detailing the GAP implementation of the cubic Jordan product, the cubic form, and the Jordan identity (including relevant code structures or pseudocode). We will also add a verification subsection containing test cases against known low-dimensional identities, such as those satisfied by 3×3 Jordan algebras over split octonions or smaller associative cases. These tests will precede the F4 application and support the reliability of the computed relations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; software artifact and computations are self-contained

full rationale

The manuscript presents a new GAP package implementing algebraic operations on cubic Jordan matrix algebras and uses it to derive specific commutator relations in F4-graded groups whose construction is cited from prior independent work. No load-bearing mathematical derivation, prediction, or first-principles result is claimed that reduces by construction to fitted inputs, self-definitions, or unverified self-citations. The package itself constitutes an external, executable artifact whose correctness is independently testable against the algebraic axioms, and the cited group construction serves only as application context rather than a premise that the present results are forced to reproduce.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The contribution is a software implementation rather than a derivation from new axioms or parameters; the abstract invokes standard properties of Jordan algebras and the GAP computer algebra system but introduces no free parameters, additional axioms, or invented entities.

pith-pipeline@v0.9.0 · 5330 in / 1035 out tokens · 37298 ms · 2026-05-10T11:53:01.697764+00:00 · methodology

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Reference graph

Works this paper leans on

5 extracted references · 2 canonical work pages

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    [JMO92] Pokrass Jacobs, Sekhar V

    Association for Computing Machinery. [JMO92] Pokrass Jacobs, Sekhar V. Muddana, and A. Jefferson Offutt. A computer algebra system for nonassociative identities. InHadronic mechanics and nonpotential interactions, Part 1 (Cedar Falls, IA, 1990), pages 185–195. Nova Sci. Publ., Commack, NY,

    1990

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    [Wid98] Alfred Widiger

    Cor- rected reprint of the 1966 original. [Wid98] Alfred Widiger. Deciding degree-four-identities for alternative rings by rewriting. InSymbolic rewriting techniques (Ascona, 1995), volume 15 ofProgr. Comput. Sci. Appl. Logic, pages 277–288. Birkhäuser, Basel,

    1966

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    [Wie26] Torben Wiedemann

    PhD Thesis,https://doi.org/ 10.22029/jlupub-18373. [Wie26] Torben Wiedemann. CubicJordanMatrixAlg,

    work page doi:10.22029/jlupub-18373