{"id":"815b0997-ddcd-4f69-bec7-4d4097fa15bb","arxiv_id":"2502.02227","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proposes a unified discrete operator calculus for octonions and split-octonions, extending earlier octonionic results to the split case by swapping a set of index sets.","lead":"Octonions are eight-dimensional numbers with unusual multiplication, and split-octonions are a variant used in some physics models. This paper puts the discrete, grid-based versions of both into one shared framework, with the main new case being split-octonions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The split-octonionic index set D7={4,5,6} contradicts Table 1 (right): e4e5=e1, which is not in D7; the corresponding split Fano line is {3,5,6}. Since D_q carries the non-associative remainder, the split-case Stokes formula in Theorem 1 is false as printed.","rationale":"The reader's verdict is CONDITIONAL and identifies two unproved premises for the split-octonionic theorems: the correctness of the D index sets and the validity of the DFT fundamental solution in the presence of zero divisors. I agree with the first premise and regard it as the more decisive, because it is directly falsifiable from the paper's own multiplication table. The printed D7={4,5,6} is not the split-octonionic analogue of the classical C7={4,5,6}; the split table makes {3,5,6} the corresponding Fano line. Since the associator A1 is defined by summing over D_q, a wrong D_q changes the non-associative remainder and invalidates the claimed equality in Theorem 1, and therefore also the split Borel-Pompeiu and Cauchy formulae that reuse the same index sets. This is a stronger objection than the zero-divisor concern in one respect: it does not depend on subtle questions of Fourier inversion or distribution theory, only on the multiplication table printed in the paper. That said, the error is localized and likely repairable, so the appropriate disposition remains conditional: the manuscript needs a corrected D7 (and a check of the other D_q at the same level of detail), after which the split-case claims can be re-examined. The zero-divisor/fundamental-solution concern is real and should also be addressed, but the index-set mismatch is the single most load-bearing issue because it breaks the central identity as written. I therefore do not change the reader's conditional verdict, but I sharpen the reason: not merely 'unproved index sets' but a specific incorrect set that a direct table check reveals.","tokens_in":14260,"tokens_out":26038,"duration_ms":249573,"concrete_test":"Using Table 1 (right), verify the closure property that each D_q must satisfy in the proof of [17]: for every pair of distinct indices i,j in D_q, the product e_i e_j must lie, up to sign, in D_q. D7={4,5,6} fails because e4e5=e1, so it is not a split-octonionic Fano line; the line {3,5,6} passes. Then recompute the split Stokes identity in Theorem 1 for a generic pair f,g supported on two adjacent sites, comparing S(f,g,D^+,D^-,{0,e7}) with A1(f,g,{0,e7},D): with D7={4,5,6} the two sides differ, while replacing D7 by {3,5,6} restores the identity. This settles whether the printed index sets or only the surrounding notation need to be corrected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central new claim is that the split-octonionic case is obtained from the classical one by replacing the index sets C with D in the associator A1. This is the entire content of Theorem 1 for O', and it is explicitly unproved: Theorem 2's proof says only that the split case follows after 'working out carefully these index sets.' A direct check against the paper's own Table 1 shows the sets were not worked out correctly. For classical octonions, C7={4,5,6} is a Fano line: e4e5=-e6, e5e6=-e4, e6e4=-e5. For split-octonions, the printed D7 is the same set {4,5,6}, but Table 1 (right) gives e4e5=e1, e5e6=e3, e6e4=-e2: none of the products lies in {4,5,6}. The correct split-octonionic line is {3,5,6}, since e3e5=e6, e5e6=e3, e6e3=e5. Thus the associator A1(f,g,Z8,D) with the printed D7 does not encode the split-octonionic non-associative contributions correctly, so the equality S(f,g,D^+,D^-,Z8)=A1(f,g,Z8,D) in Theorem 1 is not established and is false in general. Since Theorems 3-5 reuse these D sets in the Borel-Pompeiu and Cauchy formulae, the split-case results in the paper are unsupported as written. This is not a question of zero divisors or convergence; it is a concrete algebraic mismatch in the core construction. The reference to [17] cannot fill the gap because [17] treats only the classical octonions, and the split multiplication table is different exactly where D7 is wrong.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14737,"tokens_out":14051,"duration_ms":133202,"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":[{"comment":"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":"Section 3, definition of D_i; Theorem 1"},{"comment":"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.","section":"Section 2, Eq. (3) and Definition 2"},{"comment":"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.","section":"Theorem 2, proof"}],"minor_comments":[{"comment":"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.","section":"Eq. (6), definition of B1"},{"comment":"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).","section":"Theorem 4, explicit Borel-Pompeiu formulas"},{"comment":"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.","section":"Theorem 3"},{"comment":"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').","section":"Theorem 4, final line"}],"recommendation":"major_revision","confidential_remarks":"The central new claim of the paper depends on the split-octonionic index sets and on the validity of the fundamental solution in the split case. The D7 error is a concrete algebraic mismatch that can in principle be repaired, and a corrected treatment of the fundamental solution may also be possible, so I do not recommend rejection. However, the authors must provide a genuine derivation of the split-case formulas rather than an assertion of analogy, and they must verify all seven D_i against Table 1. The editor may also wish to check whether the boundary operator B1 in Eq. (6) was correctly typeset, since the repeated blocks suggest a production error."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nMinor 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.\n\nOverall, 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.","headline":"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.","tokens_in":15211,"tokens_out":5434,"would_cite":false,"duration_ms":44337,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["30G35","39A12","17A75"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["discrete octonionic analysis","split-octonions","discrete Cauchy-Riemann operators","associator","discrete Stokes formula","Borel-Pompeiu formula","Cauchy formula","Weyl calculus discretization"],"falsifier":"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.","tokens_in":14077,"feed_emoji":"🧮","tokens_out":13054,"duration_ms":112110,"temperature":0.7,"pith_summary":"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.","feed_headline":"Split-octonions enter discrete octonionic calculus under one umbrella","feed_subtitle":"The same associator identity gives Stokes, Borel-Pompeiu and Cauchy formulas for both algebras and both discretizations.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It proves the classical octonionic discrete Stokes formula whose index sets $C_j$ the split-octonionic case is claimed to follow from by substitution.","marker":"[17]"},{"why":"It introduces the discrete forward and backward Cauchy-Riemann operators and their fundamental solution, the constructions the unified scheme abstracts.","marker":"[14]"},{"why":"It sets up the Weyl-calculus discretisation of discrete octonionic analysis that the associator $\\mathcal{A}_2$ is designed to cover.","marker":"[16]"},{"why":"It supplies the three-layer boundary description and the discrete Hardy-space framework used for the half-space formulas.","marker":"[5]"},{"why":"It provides the discrete Fourier transform scheme used to construct the fundamental solutions $E_h^\\pm$ for both algebras.","marker":"[18]"},{"why":"It gives the earlier forward/backward discretisation of octonionic operators that the present paper extends to split-octonions.","marker":"[13]"},{"why":"It motivates split-octonions as the algebra behind fluid Maxwell equations, the practical target of the discrete calculus.","marker":"[7]"}],"fun_headline_variants":["Associators unify discrete Stokes, Borel-Pompeiu, Cauchy formulas","Split-octonionic discrete calculus joins the classical framework","Uniform discrete formulas for octonions and split-octonions","Non-associativity distilled: one associator for discrete settings","Discrete split-octonionic analysis covered by associators"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Associators unify discrete Stokes, Borel-Pompeiu, Cauchy formulas","Split-octonionic discrete calculus joins the classical framework","Uniform discrete formulas for octonions and split-octonions","Non-associativity distilled: one associator for discrete settings","Discrete split-octonionic analysis covered by associators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000737,"raw_usage":{"total_tokens":3315,"prompt_tokens":988,"completion_tokens":2327,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":604,"completion_tokens_details":{"reasoning_tokens":2236}},"tokens_in":604,"tokens_out":2327,"duration_ms":16243,"temperature":1.0,"reasoning_tokens":2236,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T12:54:57.019962+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Kraußhar, A","cited_arxiv_id":null,"evidence_quote":"It proves the classical octonionic discrete Stokes formula whose index sets $C_j$ the split-octonionic case is claimed to follow from by substitution."},{"cited_title":"Kraußhar, D","cited_arxiv_id":null,"evidence_quote":"It introduces the discrete forward and backward Cauchy-Riemann operators and their fundamental solution, the constructions the unified scheme abstracts."},{"cited_title":"Kraußhar, A","cited_arxiv_id":null,"evidence_quote":"It sets up the Weyl-calculus discretisation of discrete octonionic analysis that the associator $\\mathcal{A}_2$ is designed to cover."},{"cited_title":"Complex Analysis and Operator Theory, 15:4, 2021","cited_arxiv_id":null,"evidence_quote":"It supplies the three-layer boundary description and the discrete Hardy-space framework used for the half-space formulas."},{"cited_title":"Stummel, Elliptische Differenzenoperatoren unter Dirichletrandbe dingungen, Mathematis- che Zeitschrift, 97, 169-211, 1967","cited_arxiv_id":null,"evidence_quote":"It provides the discrete Fourier transform scheme used to construct the fundamental solutions $E_h^\\pm$ for both algebras."},{"cited_title":"Kraußhar, A","cited_arxiv_id":null,"evidence_quote":"It gives the earlier forward/backward discretisation of octonionic operators that the present paper extends to split-octonions."},{"cited_title":"Demir and M","cited_arxiv_id":null,"evidence_quote":"It motivates split-octonions as the algebra behind fluid Maxwell equations, the practical target of the discrete calculus."}],"review_version":1}