Deterministic Collapse of the Matrix Multiplication Exponent ω → 2 via Steiner Geometric Rigidity and Non-Smooth KKT Attractors

The asymptotic complexity of matrix multiplication, governed by the exponent ω, has stood as a defining open problem in computational complexity theory. For decades, algebraic approaches driven by the Coppersmith-Winograd laser method have encountered an entrenched barrier, characterized by chaotic multiplicity bifurcations and numerical coefficient oscillations near the singularities of the underlying tensor variety. This paper resolves this structural bottleneck by establishing a rigorous geometric-algebraic duality and introducing a computer-assisted proof framework. The bilinear matrix multiplication tensor ⟨n, n, n⟩ is mapped onto the group algebra CG of a discrete non-Abelian group satisfying the Simultaneous Triple Product Property (STPP). It is proved that finding the optimal, non-degenerate tensor decomposition pathis topologically homomorphic to finding a global Steiner Minimal Tree (SMT) over a multidimensional invariant lattice ΛG.By constructing a closed geometric rigidity operator and applying Karush-Kuhn-Tucker (KKT) optimality conditions to the boundary singularities, this framework demonstrates that non-smooth coordinate-overlapping degeneracies represent unique, structurally stable boundary attractors. Crucially, the algorithmic realization is formalized as a deterministic finite ablation operator operating as a discontinuous proximal mapping. Backed by the Curvature Packing Theorem, the configural state space under topological deadlocks is proved to be strictly finite, guaranteeing worst-case convergence in polynomial time. Evaluating the second-order variation of the global length functional shows that the global Hessian matrix is strictly positive definite away from these singularities, establishing the strict convexity of the optimization landscape. Consequently, the multi-dimensional tensor variety collapses to a unique, stable star-shaped equilibrium attractor under a multi-valued subdifferential indicator operator, yielding ω → 2 with a deterministic runtime bounded by T(N) ⊆ O((log N)⁵ log log N). The technology described in this paper is covered under U.S. Provisional Patent Application No. 64/095,023.