A Mathematical Curiosity — XII Universal Computation via T-Gate Synthesis in S₄ Subgroups

Universal Computation via T-Gate Synthesis in S4 Subgroups We investigate the computational complexity class of the finite monoid M_ext. We demonstrate that the algebraic structure supports a framework for universal quantum computation. 1. Clifford Group Mapping: The base structure (S4, Order 24) is shown to be isomorphic to the Clifford group operations acting on qubits. 2. Universal Resource: The symmetry-breaking boundary condition provides the necessary non-Clifford resource (analogous to Magic State injection or T-Gate) required for universality. 3. Turing Completeness: The braiding of algebraic generators allows for the approximation of unitary target gates, defining a complete computational model.