Exact Symmetry as Algebra: A Machine-Verified Tensor Calculus that Enforces Physical Selection Rules

תקציר מקורי באנגליתarXiv:2605.20440v2 Announce Type: replace Abstract: Symmetry is central to the physical sciences, yet machine learning usually captures it only approximately, leaving a residual per-step equivariance error $\varepsilon$ that compounds with depth $M$ as $M\varepsilon$, whereas exact equivariance holds at unbounded depth; we demonstrate this divergence at fourteen orders of magnitude. We show that a symmetry can be made exact by construction, as the multiplication rule of a tensor algebra. In the resulting $\starG$ algebra, defined by any finite group $G$, the group-Fourier transform block-diagonalizes every tensor into irreducible-representation blocks, making equivariance intrinsic; requiring equivariance conversely \emph{forces} this suitably normalized transform, so the algebra is determ