Algebraic Crystallization of Toeplitz's Square Peg Problem: 4-Fold Orthogonal Eigenstates on Rough Boundaries via ROA

Toeplitz’s square peg problem conjectures that every continuous simple closed curve contains four points that form the vertices of a square. Classical geometric and topological approaches fundamentally break down when the curve becomes infinitely rough or fractal. In this paper, we bypass Euclidean geometry by redefining the arbitrary Jordan curve as a non-commutative rough lattice boundary ∂Ω_R within the Rough Operator Algebra (ROA) framework. By introducing the localized 4-orthogonal star-commutator P_4[Z,R_π/2]_∗RP_4, we prove that geometric asymmetries are strictly absorbed by the anti-symmetric Seonggil Torsion TensorT_ijk. Driven by the conservation of the non-commutative Chern-Simons invariant and the Z_4 symmetry naturally emerging from the 3 + 1 topological dimension, the cyclic trace mandates the algebraic crystallization of four perfectly symmetric orthogonal eigenstates, definitively proving the universal existence of inscribed squares.