The algebra and geometry ofSU(3) matrices

Fan Hong-Yi

R. Aldrovandi

Janez Povh

In this paper, we explain the relations between combinatorial optimization and real algebraic geometry with a special focus to the quadratic assignment problem. We demonstrate how to write a quadratic optimization problem over discrete feasible set as a linear optimization problem over the cone of completely positive matrices. The latter formulation enables a hierarchy of approximations which rely on results from polynomial optimization, a sub-eld of real algebraic geometry.

Goce Chadzitaskos, J. Tolar

David Mulnix

This paper extends the multiscale spectral algebra—originally developed for Hadamard matrices and the discrete {±1}-geometry—into the unitary group U(n). The original algebra provided a closed‑form, dimension‑independent description of collapse phenomena in binary matrices, revealing how spectral invariants, dyadic block structure, and multiscale operators govern descent toward extremal Hadamard structure. The present work tests whether those same mechanisms retain meaning inside the curved, complex geometry of unitary operators. The extension is operational rather than symbolic. A multiscale block‑sum invariant, originally defined for {±1}-matrices, is transplanted directly into U(n). A collapse flow is constructed using coordinate perturbations and QR re‑unitarization, and its behavior is evaluated using strict quantum‑validity criteria: unit spectral radius, unimodular eigenvalues, determinant magnitude 1, and machine‑precision unitarity. Across dimensions n=4,8,12,16, the multiscale flow remains stable, monotone, and fully compatible with unitary geometry. The results show that the dyadic multiscale algebra possesses a genuine structural universality. Its invariants, operators, and collapse mechanisms—originally discovered in the discrete Hadamard landscape—extend coherently to continuous operator space. The same multiscale energy that organizes collapse in {±1}-matrices defines a new invariant for unitary operators; the same projector‑aligned descent mechanisms produce structured flows inside U(n); and the same spectral‑dynamical principles provide new analytic tools for quantum operator geometry. The paper formalizes the unitary edition of the multiscale spectral algebra, defines the block‑sum energy on U(n), constructs the collapse flow, and analyzes its stability and geometric consequences. The results demonstrate that the multiscale spectral algebra is not domain‑specific but broadly general: a closed‑form, dimension‑independent spectral‑dynamical system capable of organizing collapse phenomena across both discrete and continuous settings.