Discrete octonionic analysis: a unified approach to the split-octonionic and classical settings

Anastasiia Legatiuk; Dmitrii Legatiuk; Rolf S\"oren Krau{\ss}har

REVIEW 3 major objections 4 minor 1 cited by

Discrete octonionic analysis: a unified approach to the split-octonionic and classical settings

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper claims that a single abstract identity covers discrete octonionic and split-octonionic analysis, the split case differing only by one family of index sets.

desk verdict The unified packaging of discrete octonionic analysis is clean and useful, but the new split-octonionic results rely on an index set D7 that contradicts the paper's own multiplication table, so the split-case theorems are not established as printed. read the letter →

arxiv 2502.02227 v1 pith:HK6JZBGB submitted 2025-02-04 math.CV math.OA

Rolf Sören Krau{ss}har , Anastasiia Legatiuk , Dmitrii Legatiuk This is my paper · ORCID

classification math.CVmath.OA MSC 30G3539A1217A75

keywords discreteoctonionicanalysissplit-octonionsCauchy-RiemannoperatorsassociatorStokesformulaBorel-PompeiuCauchyWeylcalculusdiscretization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's central claim is that the discrete function theory of the octonions and of the split-octonions is one theory. It introduces two abstract discrete associators $\mathcal{A}_1$ and $\mathcal{A}_2$ together with an abstract Stokes operator $\mathcal{S}$, and states that the discrete Stokes, Borel-Pompeiu and Cauchy formulas all take the single form $\mathcal{S}(f,g,D_1,D_2,N)=\mathcal{A}_i(f,g,N,K)$ for both algebras and for both the forward/backward and Weyl-calculus discretisations. The split-octonionic case is included by replacing the octonionic index families $C=\{C_1,\dots,C_7\}$ with the split-octonionic families $D=\{D_1,\dots,D_7\}$, so the whole difference between the two settings is a relabelling of the index sets in the associator. A reader should care because split-octonions are the algebraic structure behind recent formulations of fluid Maxwell and plasma equations, and a discrete operator calculus is the step that makes such eight-dimensional formulations available for computation.

What carries the argument

The load-bearing object is the abstract discrete associator $\mathcal{A}_i$, a sum over triples $(j,\ell,k)$ of indices drawn from the seven index sets $C_j$ (octonions) or $D_j$ (split-octonions) that encode the multiplication tables of $\mathbb{O}$ and $\mathbb{O}'$. The associator is the term that records where $(f e_\ell)g$ differs from $f(e_\ell g)$ in the discrete product rule, and it supplies the correction that must be added to the naive discrete Stokes identity. It is paired with the abstract Stokes operator $\mathcal{S}(f,g,D_1,D_2,N)=\sum_{x\in N}[(g(x)D_1)f(x)+g(x)(D_2 f(x))]h^8$ and with boundary operators $\mathcal{B}_1,\mathcal{B}_2$ describing the two boundary layers of the half-spaces; choosing $i=1$ or $i=2$ selects the forward/backward or Weyl discretisation, and choosing $K=C$ or $K=D$ selects the algebra.

What would settle it

Take a small finite box of lattice points, choose generic finite-support functions $f$ and $g$ with values in the split-octonions, and evaluate both sides of $\mathcal{S}(f,g,D_+,D_-,N)=\mathcal{A}_1(f,g,N,D)$ numerically; a mismatch for any index triple outside the listed sets $D_1,\dots,D_7$ would disprove the unified formula. A second check is to verify directly that $D_+ E_h^+(m h)=\delta_h(m h)$ holds at every lattice point in the split-octonions, since any nonzero value away from $m=0$ would break the derivation of the Borel-Pompeiu formula.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the non-associativity of the two eight-dimensional algebras can be packaged, once and for all, in two abstract associators $\mathcal{A}_1(f,g,N,K)$ and $\mathcal{A}_2(f,g,N,K)$ whose only algebra-dependent input is a set $K=\{C,D\}$ of index families. With these objects, the discrete Stokes formula for the whole lattice and for the half-spaces has the uniform statement $\mathcal{S}(f,g,D_1,D_2,N)=\mathcal{A}_i(f,g,N,K)$, up to the boundary terms $\mathcal{B}_i$ that live on the three boundary layers of the half-space. Substituting the discrete fundamental solution $E_h^+(\cdot-mh)$ turns this into discrete Borel-Pompeiu formulas, and restricting to backward monogenic $f$ turns it into discrete Cauchy formulas. The explicit formulas in Section 4 show that the octonionic and split-octonionic versions differ only in the seven index sets $C_j$ versus $D_j$; everything else, including the boundary operators, is identical.

Load-bearing premise

The load-bearing premise is that the seven index lists assigned to split-octonions capture every correction term that non-associative multiplication produces, and that the Fourier-transform fundamental solution still works in an algebra with zero divisors.

Editorial extensions

If this is right

Every formula proved for classical discrete octonions in the authors' earlier work has a split-octonionic counterpart obtained by replacing the index sets $C_j$ with $D_j$.
The explicit half-space Borel-Pompeiu and Cauchy formulas in Section 4 are claimed to hold for split-octonions with the displayed $D$ index sets and the same three-layer boundary structure.
The Weyl-calculus discretisation is covered by the same abstract Stokes identity, so algorithms written against $\mathcal{A}_2$ apply to octonions and split-octonions without structural change.
Discrete Hardy spaces for split-octonions can be defined in the same way as in the classical case, giving the same function-space setting for boundary value problems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the $D_j$ index sets are correct, a direct numerical check of the identity $\mathcal{S}(f,g,D_+,D_-,N)=\mathcal{A}_1(f,g,N,D)$ on a small box would independently confirm them before any analytic proof is completed.
The same construction suggests a recipe for other eight-dimensional non-associative algebras: recompute the seven index sets from the multiplication table and the entire Stokes-Borel-Pompeiu-Cauchy formalism carries over unchanged.
The paper leaves the Weyl-calculus version of the split-octonionic boundary operators implicit; spelling out $\mathcal{B}_2$ for $K=D$ is a direct extension that would make the umbrella fully explicit.
Because split-octonions contain zero divisors, the discrete Fourier transform construction of $E_h^\pm$ is the part most likely to fail; if it does, the Borel-Pompeiu and Cauchy formulas may need a different kernel even though the Stokes identity itself would survive.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The genuinely new thing here is the split-octonionic discrete setting, wrapped in a unified notation. That part does not survive a direct check. The paper does something useful: it proposes abstract associators A1, A2 and boundary operators B1, B2 that express Stokes, Borel-Pompeiu, and Cauchy formulas uniformly for forward/backward and Weyl discretizations. For classical octonions, this is a compact reformulation of the authors' earlier results [14,16,17], and the unified notation genuinely helps keep track of cases.

The soft spot is the split-octonionic case. It is asserted, not proved. Theorem 2's proof says the split case follows 'analogously' after 'working out carefully these index sets.' But the index sets printed in Section 3 are not correct. The paper gives D7 = {4,5,6}, which is the classical Fano line. In the split multiplication table (Table 1, right), e4e5 = e1, e5e6 = e3, and e6e4 = -e2. None of these products lies in {4,5,6}. The correct split-octonionic line containing 4 and 5 is {3,5,6}, since e3e5 = e6, e5e6 = e3, and e6e3 = -e5. With the printed sets, the associator A1 does not encode the non-associative remainder correctly, so Theorem 1 for split-octonions, and Theorems 3-5 that reuse these sets, are false as stated. This is not a question of zero divisors or convergence; it is a concrete algebraic mismatch in the core construction.

Minor issue: the boundary operators in eq. (6) and the explicit formulas in Theorem 4 contain repeated summation blocks—the same boundary term appears three times. That is probably a typography error, but it adds to the impression that the explicit formulas were not carefully checked.

Overall, the unified framework is plausible and likely repairable by correcting D7 and rewriting the split-case proof at least once. As it stands, the central new claim is not established. I would send it to a referee, because the approach is of interest to the discrete Clifford and octonionic analysis community and the flaw is concrete and fixable. But I would not rely on any split-case formula from this version.

Referee Report

3 major / 4 minor

Summary. The paper proposes a unified framework for discrete octonionic analysis intended to cover both classical octonions and split-octonions under the same abstract formalism. The framework is built from two abstract associators A1 and A2, which depend on index sets C (classical) and D (split), and two abstract boundary operators B1 and B2. On this basis the paper states discrete Stokes, Borel-Pompeiu, and Cauchy formulas for the whole lattice and for half-spaces. The classical-octonionic results are quoted from the authors' previous works [14,16,17], while the split-octonionic versions are asserted to follow analogously after changing the index sets from C to D.

Significance. If the split-octonionic statements were established, the paper would offer a convenient notational umbrella for several discrete octonionic settings and would be a useful reference for applications using split-octonions. The abstract associator/boundary-operator formalism is a reasonable organizational idea, and the explicit formulas in Section 4 make the intended computations concrete. However, the split-octonionic case is the main new contribution, and its verification is not supplied: the proof of Theorem 2 only says that the split case follows 'analogously,' and the one concrete piece of information used for the split case, the index set D7, is inconsistent with the paper's own split-octonionic multiplication table. In addition, the discrete fundamental solution is asserted to carry over to split-octonions without addressing the changed squares of basis elements. The paper should receive credit for being explicit about its reliance on [17], but as printed the central split-case claims are not supported.

major comments (3)

[Section 3, definition of D_i; Theorem 1] The index set D7 is printed as {4,5,6}, but Table 1 (right) shows that in the split-octonions e4e5=e1, e5e6=e3, and e6e4=-e2, none of which belongs to {4,5,6}. Thus D7 is not a line of the split-octonionic multiplication table; the set {3,5,6} is the evident split-octonionic line containing 5 and 6. Since the associator A1(f,g,Z8,D) is defined by summing over triples with i,j in D_q and k outside D_q, replacing the classical C7={4,5,6} by the same set for the split case does not encode the non-associative contributions correctly. Consequently the split-case identity in Theorem 1, S(f,g,D^+,D^-,Z8)=A1(f,g,Z8,D), is not correct as stated, and Theorems 3-5, which reuse the same D sets, inherit the problem.
[Section 2, Eq. (3) and Definition 2] The discrete fundamental solution E_h^± is recalled in Eq. (3) via the discrete Fourier transform with the Euclidean star-Laplacian symbol sigma=4/h^2 sum sin^2(ξ_j h/2). The text states that the split-octonionic case is 'completely analogous.' This is not automatic: in the split-octonionic table, e4^2=e5^2=e6^2=+1 rather than -1, so the Fourier symbol of D^+D^- is not the Euclidean symbol used in Eq. (3). A separate argument is needed to show that the same formula satisfies Definition 2 in the split case, and zero divisors may invalidate the standard construction. As written, Eq. (3) does not obviously define a fundamental solution for O', and this gap affects the split-case Borel-Pompeiu and Cauchy formulas.
[Theorem 2, proof] The proof of the split-octonionic Stokes formula for half-spaces consists solely of the statement that the split case follows 'analogously' after 'working out carefully these index sets.' For a paper whose novelty is precisely the split-octonionic case, an unproved analogy is insufficient, especially because the only concrete assertion about the split index sets is incorrect (D7 as discussed above). The reference to [17] cannot fill the gap, since [17] treats only the classical octonions and the split multiplication table differs exactly where D7 is wrong.

minor comments (4)

[Eq. (6), definition of B1] In the formula for B1, the three displayed summation blocks are printed identically, each with i ranging over {1,6}. The analogous formula for B2 suggests that the intended index ranges are {1,6}, {2,5}, and {3,4}; as printed, B1 is ambiguous or incorrect.
[Theorem 4, explicit Borel-Pompeiu formulas] The boundary sums in the explicit octonionic and split-octonionic formulas of Theorem 4 also display three identical summation blocks. If this is a typesetting artifact, the formulas need correction; if not, the explicit formulas do not match the boundary operator B1 defined in Eq. (6).
[Theorem 3] In Theorem 3 the right-hand side still contains the function g in A1(f,g,Z8_+,C) and A1(f,g,Z8_+,D), although the substitution g = E_h^+(yh-xh) is made in the Stokes operator on the left. The associator arguments should be updated consistently.
[Theorem 4, final line] The sentence 'The discrete Borel-Pompeiu formulae for the lower half-space Z8_-+ are defined analogously' contains a typo: the half-space should be Z8_- (or 'the lower half-space case is defined analogously').

Circularity Check

2 steps flagged · score 6.0 of 10

The split-octonionic 'prediction' is written into the definition of the abstract associator and then asserted by analogy with the authors' prior work, so the new result partially reduces to its own input.

self definitional [Section 3, after Eq. (5); restated in Theorem 1]
"By the help of the abstract discrete associators (4) and the abstract Stokes' operator (5), we can now formally write the general discrete Stokes' formula: S(f,g,D1,D2,N)=A_i(f,g,N,K), with i=1,2. This formula now covers simultaneously all the cases we want to consider, namely the Weyl calculus-based approach and the approach based on forward and backward Cauchy-Riemann operators both for octonions O and split-octonions O'."

The split-octonionic Stokes identity is introduced here as a formal writing-out before any proof: the associators A1, A2 are already parameterized by K={C,D}, with D standing for split-octonions. Theorem 1 later states the same identity for O' as S=A1(...,D). Thus the split-case 'prediction' is not derived from the multiplication table; it is the content that was placed into the definition of the umbrella. The proof of Theorem 2 confirms this by saying only that the O' case follows 'analogously' after 'working out carefully these index sets'.
ansatz smuggled in via citation [Section 3, Remark 1 and the definition of D_i following Eq. (4)]
"Analogously, the split-octonionic case then leads to the index sets D_i, i=1,...,7."

The D index sets are the only place where split-octonionic multiplication enters the unified theory, yet they are not derived in this paper: they are imported by analogy from [17], which treats only classical octonions. Since Theorems 3-5 reuse these same D sets, the entire split-octonionic chain rests on this self-cited ansatz. The paper's own Table 1 (right) gives e4e5=e1, e5e6=e3, e6e4=-e2, so the printed D7={4,5,6} contradicts the input multiplication table; the correct split-octonionic line is {3,5,6}. The ansatz is therefore both load-bearing and unverified as printed.

full rationale

The classical-octonionic part is legitimately supported by the authors' earlier proof in [17]; that citation is real evidence and is not by itself circular. The circularity enters with the claimed split-octonionic extension. Equation (4) defines the abstract associators using the index sets K={C,D}, and the paper then 'formally write[s]' the general Stokes formula S=A_i before proving it; Theorem 1 for O' merely instantiates this formal identity with K=D. No independent calculation connects the split multiplication table to A1(...,D): Theorem 2's proof says only that the split case is 'analogous' and that the D_i must be 'worked out carefully'. A direct check against Table 1 shows D7 as printed is not a split-octonionic triple (e4e5=e1, e5e6=e3, e6e4=-e2), so the advertised index-set replacement is not merely an unproved analogy but is inconsistent with the paper's own input. This is a partial circularity--the split-case prediction is built into the definition of the umbrella--rather than a fully independent derivation; the D7 error is a correctness risk that reinforces, but is not the basis for, the circularity finding.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No empirically fitted parameters appear; this is a pure operator calculus paper. The load-bearing assumptions are imported from the authors' prior octonionic papers, especially the octonionic Stokes formula [17], plus two ad hoc elements for the new split case: the index sets D_i and the validity of the DFT fundamental solution in an algebra with zero divisors. No new physical entities are introduced.

assumptions (5)

domain assumption Correctness of the discrete octonionic Stokes formula proved in the authors' previous work [17].
Theorem 1 for O is quoted from [17], and all split-octonionic theorems are stated to follow from it analogously. The paper does not reproduce the proof.
ad hoc to paper The index sets D_i in Section 3 correctly encode all non-associative contributions for split-octonions.
These sets are introduced by table without a derivation; Theorem 2's proof says they were obtained by carefully working out the multiplication rules.
ad hoc to paper The discrete fundamental solution (3), built via discrete Fourier transform, is a valid fundamental solution for split-octonions despite zero divisors.
Section 2 says the split case is 'completely analogous' to the octonionic case, but the paper itself notes zero divisors require additional attention in the continuous setting.
domain assumption Convergence of all series appearing in the Stokes, Borel-Pompeiu and Cauchy formulas.
Theorems 1-5 all end with 'under the condition that the series converge'; no growth or decay conditions are specified.
domain assumption The three-layer discrete boundary representation used by the boundary operators B1 and B2.
The boundary structure is imported from discrete Clifford analysis references [5,16] without further justification in this paper.

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Cite this review

Pith. "Pith review of Discrete octonionic analysis: a unified approach to the split-octonionic and classical settings." pith.science (2026). https://pith.science/paper/HK6JZBGB

@misc{pith2026250202227,
  author       = {Pith},
  title        = {Pith review of: Discrete octonionic analysis: a unified approach to the split-octonionic and classical settings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HK6JZBGB}},
  note         = {Machine review of arXiv:2502.02227}
}

read the original abstract

Various problems of mathematical physics consider octonions and split-octonions as a mathematical structure, which underpins the eight-dimensional nature of these problems. Therefore, it is not surprising that octonionic analysis has become an area of active research in recent years. One of the main goals of octonionic analysis is to develop tools of an octonionic operator calculus for solving boundary value problems of mathematical physics that benefit from the use of the octonionic structure. However, when we want to apply the operator calculus in practice, it becomes evident that adequate discrete counterparts of continuous constructions need to be defined. In previous works, we have proposed several approaches to discretise the classical continuous octonionic analysis. However, the split-octonionic case, which is particularly important for practical applications concretely investigated in the last years, has not been considered until now. Therefore, one of the goals of this paper is to explain how to particularly address the discrete split-octonionic setting. Additionally, we propose a general umbrella to cover all different discrete octonionic settings in one unified approach that also encompasses the different eight-dimensional algebraic structures.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

The Dirac equation in (split-)octonions: origins, variants, and modern context
math-ph 2026-06 conditional novelty 5.0 of 10

Four approaches to the Dirac equation in (split-)octonions are classified; a 2024 split-octonionic equation is shown identical to the 2006 2-factor form by structure-preserving rotation and relabeling.

Reference graph

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

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R.S. Kraußhar, A. Legatiuk, D. Legatiuk, The discrete octonionic Stokes’ formula revisited , Research Perspectives: Ghent Analysis and PDE Center, Spri nger, 2024

work page 2024

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work page Pith review arXiv 2016

[3] [2]

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Brackx, F., De Schepper, H., Sommen, F., Van de Voorde, L.: Discrete Cliﬀord analysis: a germ of function theory. Hypercomplex Analysis, Birkh ¨auser Basel, 37-53, 2009

work page 2009

[4] [3]

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Cerejeiras, P ., K ¨ahler, U., Ku, M., Sommen, F.: Discrete Hardy spaces. Journa l of Fourier Analysis and Applications, 20(4), 715-750, 2014

work page 2014

[5] [4]

Cerejeiras, P ., K ¨ahler, U., Legatiuk, A., Legatiuk, D.: Boundary values of di screte monogenic functions over bounded domains in R3. in D. Alpay, M. Vajiac (eds) Linear Systems, Signal Processing and Hypercomplex Analysis. Operator Theory: Ad vances and Applications, vol

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work page 2016