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\signature{Lior Horesh \\ (on behalf of all authors)}
\address{IBM Research \\ Yorktown Heights, NY, USA}

\begin{document}

\begin{letter}{}
%{The Editors \\ \textit{Nature}}
\opening{Dear Editors,}

We submit for your consideration our manuscript entitled
\textbf{``Group-Algebraic Tensors: Provably-optimal Equivariant Learning
and Physical Symmetry Discovery.''}

The dominant approach to incorporating symmetry in machine learning relies
on equivariant neural networks (ENNs), which embed each symmetry into the
network architecture.  While powerful, this paradigm requires bespoke
engineering for every new symmetry, offers no optimality guarantees, and
cannot compose multiple symmetries without redesigning the entire
architecture.  We propose a fundamentally different approach: rather than
constraining the architecture, we change the algebra.

The paper introduces the $\star_G$ tensor algebra, in which any finite
group $G$ defines the multiplication rule.  Our main contributions are:

\begin{enumerate}[leftmargin=1.8em,itemsep=4pt,topsep=4pt,parsep=2pt]
\item \textbf{Provably optimal decomposition.}  We prove that the
$\star_G$-SVD achieves Eckart--Young optimality, the first such result
for symmetry-preserving tensor approximation.

\item \textbf{Compositional symmetry.}  Product groups compose via
Kronecker factorization of the Fourier transform with no architectural
changes.  We demonstrate that $\mathbb{Z}_6 \times \mathbb{Z}_4$ achieves
$R^2 = 1.000$ while each factor alone recovers at most 23\%.

\item \textbf{Physical symmetry discovery from data alone.}  Without any
quantum-mechanical theory as input, we decompose QM9 molecular geometry
over the octahedral group and recover the Wigner--Eckart selection rules
governing angular momentum coupling: the T$_1$/A$_1$ predictive-power
ratio is $5\times$ larger for vector than scalar observables, and the
isotropic polarizability is uniquely insensitive to the $l\!=\!1$
channel, exactly as the representation-theoretic decomposition of
symmetric rank-2 tensors demands.  These selection rules, cornerstones
of atomic spectroscopy since 1931, emerge here as empirical consequences
of an algebraic decomposition applied to molecular geometry data.

\item \textbf{Algebraic disentanglement of tensor components.} On the
QM7-X tensorial polarizability benchmark, the same algebra delivers a
result no equivariant neural network we tested can reproduce. Trained on
a single octahedral irrep at a time, $\star_G$-SVD~+~Ridge (two
parameters) achieves cross-selectivity between the $E_g$ and $T_{2g}$
components above $96\%$, while MACE, SchNet, and an e3nn-based SE(3)-
equivariant network ($10^5$--$10^6$ parameters each) all sit below
$1.1\%$ cross-selectivity despite achieving comparable per-component
$R^2$. The algebra exposes a representation-theoretic structure that
the neural architectures, by their end-to-end design, cannot.

\item \textbf{Parameter efficiency in the data-scarce regime.}  On full
QM9 (130{,}831 molecules), $\star_G$-SVD with ridge regression
delivers $R^2 = 0.998$ on ZPVE and $R^2 = 0.909$ on isotropic
polarizability with 144 parameters, comparable to MLP baselines at
$20$--$40\times$ more parameters. While equivariant neural networks
achieve higher pooled $R^2$ at $10^5$--$10^6$ parameters, our
within-isomer audit shows the gap is largely a size-prediction effect
that vanishes once chemistry is controlled for; $\star_G$ retains its
parameter-efficiency advantage in the data-scarce regime where neural
training is infeasible.

\item \textbf{Machine-verified proofs.}  All core algebraic results are
formalized in Lean~4 (600 lines, zero unresolved goals, five standard
axioms), to our knowledge the first machine-verified Eckart--Young-type
optimality theorem for symmetry-preserving tensor approximation.
\end{enumerate}

Beyond the specific results reported, the framework opens capabilities that
were not previously available.  Because the irreducible representation
decomposition reveals \emph{which} angular momentum channels carry
information about \emph{which} observables, the $\star_G$ algebra
functions as a symmetry spectroscope for empirical data.  This makes it
possible to (i)~identify the physical symmetry content of observables
directly from measurements, without solving the Schr\"odinger equation or
invoking any quantum-mechanical theory; (ii)~perform meaningful molecular
property prediction from as few as 100 molecules, a regime relevant to
rare materials, radioactive compounds, and exotic states of matter where
large training sets are unavailable; and (iii)~test candidate symmetry
groups against data to determine which group best describes a system, a
capability that could accelerate the study of materials whose symmetries
are unknown, approximate, or under debate (e.g., quasicrystals,
frustrated magnets, and biological macromolecules with pseudo-symmetry).

The work bridges two classical results that share a common author in Carl
Eckart: the Eckart--Young theorem (optimal low-rank matrix approximation,
1936) and the Wigner--Eckart theorem (angular momentum selection rules,
1931), unifying them through a single algebraic construction.  That the
same mathematics which delivers provably optimal compression also recovers,
without any physics input, the selection rules that govern atomic
spectroscopy suggests that the $\star_G$ algebra captures something
fundamental about how symmetry organizes physical information.  We believe
this connection, together with the practical demonstrations on molecular
data and the machine-verified proofs, makes the paper suitable for
\textit{Nature Communications'} broad readership across mathematics, physics, chemistry,
and machine learning.

The manuscript has not been submitted elsewhere.  All authors have reviewed
the final version and approved its submission.  We declare no competing
interests.

We suggest the following potential referees:

\begin{itemize}[leftmargin=1.6em,itemsep=3pt,topsep=4pt,parsep=1pt]
 \item \textbf{Michael W.\ Mahoney} (UC Berkeley),
  randomized linear algebra, matrix approximation theory,
  and scientific machine learning.
\item \textbf{Petros Drineas} (Purdue University),
  randomized matrix and tensor methods, low-rank approximation
  guarantees, and spectral algorithms.
\item \textbf{Lek-Heng Lim} (University of Chicago),
  tensor rank, tensor decomposition complexity, and algebraic
  aspects of multilinear approximation.
\item \textbf{Laurent Demanet} (MIT),
  computational harmonic analysis, Fourier methods, and spectral
  decomposition in scientific computing.
\item \textbf{Soledad Villar} (Johns Hopkins University),
  mathematical foundations of equivariant machine learning,
  invariant theory, and group symmetry in learning.
\item \textbf{Stefanie Jegelka} (MIT / TU Munich),
  geometric and combinatorial structure in machine learning,
  invariances, and generalization theory.
\item \textbf{Shaul Mukamel} (UC Irvine),
  theoretical chemical physics, nonlinear spectroscopy, and
  angular momentum selection rules.
\end{itemize}

\closing{Sincerely,}

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