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  pdftitle={KASVD Spectral Geometry and Seven-Dimensional Anticommutative Algebras},
  pdfauthor={Ivan B. Kurpishev},
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\title{KASVD Spectral Geometry and Seven-Dimensional Anticommutative
Algebras}
\usepackage{etoolbox}
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\makeatother
\subtitle{A Matrix Bridge for KLT-RBD Reper Artifacts}
\author{Ivan B. Kurpishev}
\date{Kaliningrad, 2026}

\begin{document}
\maketitle

\section{Abstract}\label{abstract}

We construct a rigorous mathematical layer connecting a finite typed
graph of a Reper artifact with a matrix \(M\in M_3(\mathbb R)\), its
singular spectrum, and a stratified KASVD state space. In parallel, we
study the seven-dimensional anticommutative algebra

\[
A_M=\mathbb R^3\oplus\mathbb R^3\oplus\mathbb Rh,
\]

in which the same matrix \(M\) defines a mixed central bilinear block.
We prove invariance of the normalized SVD passport, a rank-gap
statement, a rank-birth criterion for rank-one updates, the exact
rank-one edit distance, a formula for the dimension of the derivation
algebra, and rigidity with respect to the Binary-Lie, Malcev, and Sagle
varieties:

\[
A_M\in\mathrm{BinaryLie}\iff A_M\in\mathrm{Malcev}\iff A_M\in\mathrm{Sagle}\iff M=0.
\]

We also introduce Kurpishev's authorial variational functional on a
spectrally safe region. The mathematically proved spectral-algebraic
layer is explicitly separated from the prospective/blind infrastructure
of KLT-RBD. No empirical predictive validation is claimed in this paper.

\textbf{Keywords:} KASVD; KLT-RBD; Reper; singular value decomposition;
anticommutative algebra; Binary-Lie; Malcev; Sagle; rank-one
perturbation; variational principle.

\textbf{MSC 2020:} 17A30, 15A18, 15A60, 65F35.

\section{1. Introduction}\label{introduction}

In the project LOGIC OF KURPISHEV 2, a Reper object is written as
\(\operatorname{Rep}=(R,I,U;D)\), while KLT-RBD is used as a
machine-auditable memory of sources, sufficient grounds, transitions,
blockers, and statuses. The present paper isolates a self-contained
mathematical problem from that larger corpus: to construct a small
matrix carrier on which rank, spectral symmetry, perturbation stability,
and elementary structural transitions can be treated rigorously.

The central object is a matrix \(M\in M_3(\mathbb R)\). It appears in
two independent ways. First, it is an operator assembled from typed
edges of a finite Artifact graph. Second, it is the mixed bilinear block
of a seven-dimensional anticommutative algebra \(A_M\). These two levels
are not semantically identified. Their mathematical relation is a common
SVD invariant.

The classical background consists of singular value decomposition, the
Eckart-Young-Mirsky theorem, standard singular-value perturbation
bounds, and the established definitions of Binary-Lie, Malcev, and Sagle
algebras {[}1-6{]}. The authorial layer consists of the
Artifact-to-KASVD construction, the KASVD state stratification and
transition interpretation, the use of one matrix \(M\) as a common
module for the Artifact operator and \(A_M\), and the variational
functional \(\mathcal J_K\).

\section{2. Artifact operator and
KASVD}\label{artifact-operator-and-kasvd}

\subsection{Definition 2.1. Artifact
matrix}\label{definition-2.1.-artifact-matrix}

Let \(G_A=(V_A,E_A,\tau,w)\) be a finite typed weighted graph. Fix
orthonormal three-dimensional channel spaces

\[
X=\operatorname{span}\{R,I,U\},\qquad
Y=\operatorname{span}\{D,P,G\}.
\]

Here \(P\) and \(G\) are analytical proof/evidence and gap/blocker
channels; they are not declared to be additional components of the Reper
itself. To each edge \(e\) assign \(x_e\in X\), \(y_e\in Y\), and a
weight \(w(e)\). Define

\[
T_A=\sum_{e\in E_A}w(e)x_e\otimes y_e,
\qquad
M_A=[T_A]=\sum_{e\in E_A}w(e)x_e y_e^T.
\]

\subsection{Definition 2.2. KASVD
passport}\label{definition-2.2.-kasvd-passport}

Let \(M\neq0\) and let

\[
\sigma_1(M)\ge\sigma_2(M)\ge\sigma_3(M)\ge0
\]

be its singular values. Set

\[
\rho(M)=\|M\|_F,
\qquad
s_2=\frac{\sigma_2}{\sigma_1},
\qquad
s_3=\frac{\sigma_3}{\sigma_1}.
\]

We call

\[
\operatorname{SpecRep}(M)=(\rho;s_2,s_3)
\]

the spectral passport, and \((s_2,s_3)\) the KASVD point. It lies in the
closed triangle

\[
0\le s_3\le s_2\le1.
\]

\subsection{Proposition 2.3. Shape
invariance}\label{proposition-2.3.-shape-invariance}

Under

\[
M\mapsto cAMB^T,
\qquad A,B\in O(3),\quad c\neq0,
\]

the KASVD point \((s_2,s_3)\) is preserved, while \(\rho\) is multiplied
by \(|c|\).

\textbf{Proof.} Orthogonal factors preserve singular values and
multiplication by \(c\) scales all singular values by \(|c|\). Their
ratios to \(\sigma_1\) are therefore unchanged. \(\square\)

\subsection{Theorem 2.4. Rank-gap
theorem}\label{theorem-2.4.-rank-gap-theorem}

If the active edges of the Artifact graph use at most two independent
input directions in \(X\), or at most two independent output directions
in \(Y\), then

\[
\operatorname{rank}M_A\le2,
\qquad
\sigma_3(M_A)=0.
\]

\textbf{Proof.} In the first case the effective domain of \(T_A\) has
dimension at most two; in the second case the image of \(T_A\) is
contained in a subspace of dimension at most two. Hence the rank is at
most two. \(\square\)

The converse is false: \(\sigma_3=0\) may also arise from linear
dependence or cancellation of weights. Semantic, structural, and
spectral gaps must therefore be stored separately.

\section{3. Rank-one dynamics}\label{rank-one-dynamics}

Let

\[
M'=M+E,
\qquad
E=\alpha xy^T,
\]

so that \(\operatorname{rank}E\le1\).

\subsection{Theorem 3.1. Rank-birth
criterion}\label{theorem-3.1.-rank-birth-criterion}

Assume \(\operatorname{rank}M=2\), and let unit vectors \(\ell\) and
\(r\) span \(\ker M^T\) and \(\ker M\), respectively. Then

\[
\operatorname{rank}(M+\alpha xy^T)=3
\]

if and only if

\[
\alpha\neq0,
\qquad
\ell^Tx\neq0,
\qquad
y^Tr\neq0.
\]

\textbf{Proof.} A rank-one update raises the rank of a rank-two matrix
exactly when the new left direction lies outside \(\operatorname{Col}M\)
and the new right direction lies outside \(\operatorname{Row}M\). In
codimension one these conditions are equivalent to the two stated
nonzero scalar products. \(\square\)

\subsection{Theorem 3.2. Rank-change
bound}\label{theorem-3.2.-rank-change-bound}

For a rank-one update,

\[
\left|\operatorname{rank}(M+E)-\operatorname{rank}M\right|\le1.
\]

Thus one elementary edge update cannot carry a rank-one state directly
to a rank-three state.

\subsection{Theorem 3.3. Exact rank-one edit
distance}\label{theorem-3.3.-exact-rank-one-edit-distance}

The minimum number of rank-one summands required to pass from a fixed
matrix \(M\) to a fixed matrix \(N\) is

\[
\boxed{d_1(M,N)=\operatorname{rank}(N-M).}
\]

\textbf{Proof.} A sum of \(k\) rank-one matrices has rank at most \(k\),
giving the lower bound. Conversely, the SVD of \(N-M\) writes it as a
sum of exactly \(r=\operatorname{rank}(N-M)\) rank-one terms.
\(\square\)

\section{4. KASVD stratification and
adjacency}\label{kasvd-stratification-and-adjacency}

For a nonzero \(3\times3\) matrix we use seven strata

\[
S_1,
S_{2g},S_{2e},
S_{3g},S_{3,12},S_{3,23},S_{3i}.
\]

The index gives the rank; \(g\) denotes pairwise distinct positive
singular values; \(e\) denotes equality of the two positive singular
values in rank two; and \(12\), \(23\), and \(i\) denote the
corresponding multiplicities in rank three.

\subsection{Theorem 4.1. Existential rank-one
adjacency}\label{theorem-4.1.-existential-rank-one-adjacency}

For two KASVD strata there exists at least one pair of representatives
\(M\in S_\alpha\), \(N\in S_\beta\) such that

\[
\operatorname{rank}(N-M)\le1
\]

if and only if

\[
\left|\operatorname{rank}S_\alpha-\operatorname{rank}S_\beta\right|\le1.
\]

\textbf{Proof.} Necessity follows from the standard rank inequality. For
ranks one and two, diagonal representatives
\(\operatorname{diag}(a,0,0)\) and \(\operatorname{diag}(a,b,0)\)
suffice, including the case \(a=b\). For transitions from rank two to
rank three, diagonal representatives cover generic and double strata.
The transition \(S_{2g}\to S_{3i}\) is realized, for example, by
\(I-uv^T\to I\) with \(u=e_1\), \(v=e_1+e_2\); the source has singular
values \((\sqrt2,1,0)\). Within the rank-three block, most cases are
obtained by changing one diagonal coordinate; the transition
\(S_{3i}\to S_{3g}\) is realized by \(I+e_1(e_1+e_2)^T\), which has
three distinct singular values. Reverse arrows follow by replacing \(E\)
with \(-E\). \(\square\)

The theorem is existential. For specific matrices \(M\) and \(N\), the
exact criterion remains \(d_1(M,N)=\operatorname{rank}(N-M)\).

\section{5. Spectral stability}\label{spectral-stability}

For \(M'=M+E\), the standard singular-value perturbation bound gives
{[}3,4{]}

\[
|\sigma_k(M')-\sigma_k(M)|\le\|E\|_2.
\]

In the generic full-rank stratum define

\[
m_R=\sigma_3,
\qquad
m_{12}=\frac{\sigma_1-\sigma_2}{2},
\qquad
m_{23}=\frac{\sigma_2-\sigma_3}{2},
\]

and

\[
\boxed{r_{\mathrm{safe}}(M)=\min\{m_R,m_{12},m_{23}\}.}
\]

\subsection{Corollary 5.1. Safe-radius
guarantee}\label{corollary-5.1.-safe-radius-guarantee}

If

\[
\|E\|_2<r_{\mathrm{safe}}(M),
\]

then \(M+E\) remains full rank and does not cross either surface
\(\sigma_1=\sigma_2\) or \(\sigma_2=\sigma_3\).

For a sequence of updates \(E_1,\ldots,E_n\), absence of intermediate
crossings is guaranteed, for example, by the stronger condition

\[
\sum_{j=1}^n\|E_j\|_2<r_{\mathrm{safe}}(M_0),
\]

because it controls every prefix.

\section{6. The seven-dimensional anticommutative
algebra}\label{the-seven-dimensional-anticommutative-algebra}

Let

\[
A_M=V\oplus W\oplus L,
\qquad
V\cong W\cong\mathbb R^3,
\qquad
L=\mathbb Rh,
\]

and define the anticommutative product

\[
[(v,w,t),(v',w',t')]
=
(v\times v',\;w\times w',\;v^TMw'-v'^TMw).
\]

The vector \(h\) spans a central annihilator. For \(M=0\) one obtains
the Lie algebra

\[
A_0\cong\mathfrak{so}(3)\oplus\mathfrak{so}(3)\oplus\mathbb Rh.
\]

For \(M\neq0\), the mixed block generally violates the Jacobi identity.

\subsection{Proposition 6.1. Orientation-preserving SVD
equivalence}\label{proposition-6.1.-orientation-preserving-svd-equivalence}

If \(A,B\in SO(3)\) and the central coordinate is rescaled by a nonzero
factor, the mixed block transforms as

\[
M\mapsto cAMB^T,
\qquad c\neq0.
\]

Hence \((s_2,s_3)\) is an invariant of this equivalence.

The Artifact operator admits the larger \(O(3)\times O(3)\) action,
whereas the cross product is naturally equivariant under \(SO(3)\). Thus
the common object of the two levels is the SVD invariant, not an
identity of the full transformation groups.

\section{7. Derivations}\label{derivations}

\subsection{Theorem 7.1. Stabilizer of the mixed
block}\label{theorem-7.1.-stabilizer-of-the-mixed-block}

Let \(M\neq0\). Every derivation of \(A_M\) induces on \(V\oplus W\) a
block \(P\oplus Q\) with

\[
P,Q\in\mathfrak{so}(3),
\]

and

\[
PM=MQ.
\]

If the positive singular values of \(M\) have multiplicities
\(m_\alpha\), and

\[
k=\dim\ker M,
\]

then

\[
\boxed{
\dim\operatorname{Der}(A_M)
=
\sum_\alpha {m_\alpha\choose2}+k(k-1).
}
\]

\textbf{Proof.} The annihilator \(L\) is characteristic. On the quotient
\(A_M/L\cong\mathfrak{so}(3)\oplus\mathfrak{so}(3)\) the induced
derivation is inner, hence is represented by \(P,Q\in\mathfrak{so}(3)\).
Compatibility with the central bilinear form has the form \(cM=-PM+MQ\).
Taking the Frobenius inner product with \(M\) gives \(c\|M\|_F^2=0\),
because \(\operatorname{tr}(M^TPM)\) and \(\operatorname{tr}(M^TMQ)\)
vanish for skew-symmetric \(P,Q\). Hence \(c=0\). After SVD, the
equation \(PM=MQ\) splits into blocks of equal positive singular values
and the kernel. A positive block of multiplicity \(m_\alpha\)
contributes \(\binom{m_\alpha}{2}\) parameters, while two independent
skew blocks on a \(k\)-dimensional kernel contribute
\(2\binom{k}{2}=k(k-1)\). \(\square\)

\section{8. Binary-Lie, Malcev, and Sagle
rigidity}\label{binary-lie-malcev-and-sagle-rigidity}

Let

\[
J(x,y,z)=[[x,y],z]+[[y,z],x]+[[z,x],y]
\]

be the Jacobian.

\subsection{Theorem 8.1. Binary-Lie
rigidity}\label{theorem-8.1.-binary-lie-rigidity}

For the family \(A_M\),

\[
\boxed{A_M\in\mathrm{BinaryLie}\iff M=0.}
\]

\textbf{Proof.} In anticommutative notation the Binary-Lie identity is
equivalent to {[}1,2{]}

\[
J(x,y,[x,y])=0.
\]

For every \(i,j\in\{1,2,3\}\) choose any \(k\neq j\) and set

\[
x=f_k,
\qquad
y=e_i-f_j.
\]

A direct calculation gives

\[
\boxed{J(x,y,[x,y])=M_{ij}h.}
\]

Thus all nine entries \(M_{ij}\) vanish. Conversely, \(M=0\) yields the
Lie algebra \(A_0\), and every Lie algebra is Binary-Lie. \(\square\)

\subsection{Corollary 8.2. Malcev
rigidity}\label{corollary-8.2.-malcev-rigidity}

\[
\boxed{A_M\in\mathrm{Malcev}\iff M=0.}
\]

\textbf{Proof.} Every Malcev algebra is Binary-Lie {[}1,2{]}. Apply
Theorem 8.1; the converse follows from the Lie case. \(\square\)

\subsection{Theorem 8.3. Sagle
rigidity}\label{theorem-8.3.-sagle-rigidity}

Write the Sagle identity in the form {[}1{]}

\[
[J(x,y,z),w]
=
J(w,z,[x,y])+J(w,y,[z,x])+J(w,x,[y,z]).
\]

Then

\[
\boxed{A_M\in\mathrm{Sagle}\iff M=0.}
\]

\textbf{Proof.} Define the Sagle defect

\[
\mathcal S(x,y,z,w)=
[J(x,y,z),w]
-J(w,z,[x,y])-J(w,y,[z,x])-J(w,x,[y,z]).
\]

For the standard bases \(e_1,e_2,e_3\) of \(V\) and \(f_1,f_2,f_3\) of
\(W\), direct substitution gives:

\begin{longtable}[]{@{}lll@{}}
\toprule\noalign{}
coefficient & \((x,y,z,w)\) & \(\mathcal S(x,y,z,w)\) \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
\(M_{11}\) & \((e_1,e_2,f_1,e_2)\) & \(M_{11}h\) \\
\(M_{12}\) & \((e_1,e_2,f_2,e_2)\) & \(M_{12}h\) \\
\(M_{13}\) & \((e_1,e_2,f_3,e_2)\) & \(M_{13}h\) \\
\(M_{21}\) & \((e_1,e_2,f_1,e_1)\) & \(-M_{21}h\) \\
\(M_{22}\) & \((e_1,e_2,f_2,e_1)\) & \(-M_{22}h\) \\
\(M_{23}\) & \((e_1,e_2,f_3,e_1)\) & \(-M_{23}h\) \\
\(M_{31}\) & \((e_1,e_2,f_2,f_3)\) & \(-M_{31}h\) \\
\(M_{32}\) & \((e_1,e_2,f_1,f_3)\) & \(M_{32}h\) \\
\(M_{33}\) & \((e_1,e_2,f_1,f_2)\) & \(-M_{33}h\) \\
\end{longtable}

If \(A_M\) is Sagle, every defect is zero, so \(M_{ij}=0\) for all
\(i,j\). Conversely, \(M=0\) yields a Lie algebra, and Lie algebras
satisfy the Sagle identity. \(\square\)

Thus, inside the family \(A_M\), three broader nonassociative classes
collapse to the unique Lie layer:

\[
\boxed{
A_M\in\mathrm{BinaryLie}
\iff
A_M\in\mathrm{Malcev}
\iff
A_M\in\mathrm{Sagle}
\iff M=0.
}
\]

\section{9. The common SVD-moduli
bridge}\label{the-common-svd-moduli-bridge}

\subsection{\texorpdfstring{Theorem 9.1. Artifact/KASVD - \(A_M\)
bridge}{Theorem 9.1. Artifact/KASVD - A\_M bridge}}\label{theorem-9.1.-artifactkasvd---a_m-bridge}

Let the same nonzero matrix \(M\) be used

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  as the matrix of an Artifact operator \(T_A:X\to Y\);
\item
  as the mixed central block of the anticommutative algebra \(A_M\).
\end{enumerate}

Then the normalized spectrum

\[
(s_2,s_3)
=
\left(
\frac{\sigma_2}{\sigma_1},
\frac{\sigma_3}{\sigma_1}
\right)
\]

is a common invariant of the orthogonal-scale geometry of the Artifact
operator and of the orientation-preserving orthogonal-scale equivalence
of the algebraic mixed block.

\textbf{Proof.} At the Artifact level the singular ratios are invariant
under \(O(3)\times O(3)\) and a common nonzero scale. At the algebraic
level the continuous basis action contains \(SO(3)\times SO(3)\)
together with the same nonzero rescaling of the central coordinate. In
both cases all singular values are changed only by a common factor;
hence their ratios agree. \(\square\)

The theorem does not assert that a semantic Artifact is an algebra. It
asserts only that the two constructions possess a rigorous common matrix
invariant.

\section{10. Weighted KASVD and the variational
layer}\label{weighted-kasvd-and-the-variational-layer}

For a fixed matrix \(M\) and a target stratum \(T\), define the local
cost

\[
c_M(T)=
\inf\{\|E\|_2:\operatorname{rank}E\le1,\ M+E\in T\}.
\]

For a full-rank matrix the minimum cost of rank loss is

\[
\boxed{c_M(R_2)=\sigma_3,}
\]

attained by \(E=-\sigma_3u_3v_3^T\). In the reverse direction, from rank
two to rank three, the infimal cost is zero: a new independent mode can
be created by an arbitrarily small nonzero perturbation whose left and
right directions both leave the column and row spaces.

On the generic region

\[
\Omega_{gen}=\{\sigma_1>\sigma_2>\sigma_3>0\}
\]

define

\[
V_K(M)=
-\alpha_R\log m_R
-\alpha_{12}\log m_{12}
-\alpha_{23}\log m_{23},
\qquad \alpha_i>0,
\]

and Kurpishev's authorial mathematical variational functional

\[
\boxed{
\mathcal J_K[M]
=
\int_{t_0}^{t_1}
\left(
\frac12\|\dot M\|_F^2+V_K(M)
\right)dt.
}
\]

For simple singular values, \(\nabla\sigma_i=u_iv_i^T\), so inside a
regular stratum the formal Euler-Lagrange equation is

\[
\ddot M=\nabla_FV_K(M).
\]

\subsection{Proposition 10.1. A well-posed existence
setting}\label{proposition-10.1.-a-well-posed-existence-setting}

The logarithmic barrier alone is not coercive on the unbounded matrix
space. On the compact safe region

\[
\Omega_\delta
=
\{M:\|M\|_F=1,\ m_R,m_{12},m_{23}\ge\delta\},
\]

a finite-dimensional discrete fixed-endpoint functional attains a
minimum whenever the admissible trajectory set is nonempty.

\textbf{Proof.} \(\Omega_\delta\) is compact, a finite product of copies
of it is compact, the fixed-endpoint admissible set is closed, and the
discrete functional is continuous. The Weierstrass theorem applies.
\(\square\)

\section{11. Prospective certificate and the boundary of empirical
claims}\label{prospective-certificate-and-the-boundary-of-empirical-claims}

For KLT-RBD a computational contour has been constructed:

\[
\text{Source}
\to\text{Intake}
\to\text{Evidence-D}
\to G_t
\to M_t
\to\operatorname{SVD}
\to\text{Transition}
\to\text{Certificate}\vee\text{ABSTAIN}.
\]

The append-only ledger, hash-bound materialization, blind split, and
scoring protocol make the computation reproducible, but they are not
evidence of predictive power. Synthetic tests are used to test
executability of the model, not as a substitute for a real holdout. At
the time of this paper, real prospective validation has not been
performed; empirical predictive claims therefore remain outside the
proved layer.

\section{12. Novelty and limitations}\label{novelty-and-limitations}

The following are classical and are not claimed as authorial
discoveries: SVD, rank inequalities, the Eckart-Young-Mirsky theorem,
standard perturbation bounds, the established
Lie/Malcev/Binary-Lie/Sagle theory, and compactness arguments.

The authorial mathematical construction of I. B. Kurpishev in this paper
consists of:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  the map from a finite Reper Artifact graph to a three-channel operator
  \(M_A\) and its KASVD passport;
\item
  KASVD stratification as a KLT-RBD state layer together with its
  operational rank-one interpretation;
\item
  the use of one matrix \(M\) as a common module for the Artifact
  operator and the family \(A_M\);
\item
  explicit rigidity theorems for \(A_M\), including nine Binary-Lie and
  nine Sagle witnesses;
\item
  weighted KASVD and the functional \(\mathcal J_K\);
\item
  the strict separation between a computed certificate and truth-status.
\end{enumerate}

Open problems include the full isomorphism group of \(A_M\) including
discrete components, the global geometry of KASVD moduli near repeated
singular values, and independent real validation of the prospective
prediction layer.

\section{13. Conclusion}\label{conclusion}

A single rigorous matrix layer has been obtained in which the same
\(3\times3\) matrix carries two different structures: a spectral state
of a Reper Artifact operator and the mixed central block of a
seven-dimensional anticommutative algebra. The SVD ratios \((s_2,s_3)\)
form a common invariant of these constructions. For the algebraic
family, a rigidity result is proved: every nonzero \(M\) moves \(A_M\)
simultaneously outside the Binary-Lie, Malcev, and Sagle classes. For
the Artifact layer, exact rank-one distances and safe spectral
thresholds are obtained. The variational layer provides a mathematical
trajectory model, while its empirical interpretation remains a separate
prospective-verification problem.

\section{Appendix A. Nine Binary-Lie
witnesses}\label{appendix-a.-nine-binary-lie-witnesses}

For \(i,j\in\{1,2,3\}\) choose any \(k\neq j\) and set

\[
x=f_k,\qquad y=e_i-f_j.
\]

Then

\[
J(x,y,[x,y])=M_{ij}h.
\]

For instance one may fix \(k(1)=2\), \(k(2)=3\), \(k(3)=2\), yielding
nine deterministic tests of all entries of \(M\).

\section{Appendix B. Nine Sagle
witnesses}\label{appendix-b.-nine-sagle-witnesses}

Let \(\mathcal S(x,y,z,w)\) denote the Sagle defect with all terms moved
to one side. The nine quadruples listed in Theorem 8.3 yield
respectively

\[
M_{11},M_{12},M_{13},-M_{21},-M_{22},-M_{23},-M_{31},M_{32},-M_{33}
\]

in the central \(h\) coordinate. Thus the Sagle identity annihilates
each coordinate of \(M\) separately.

\section{References}\label{references}

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\item
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