1.5 Algebraic Action S(σ)

This paper defines the Algebraic Action , and proves that the action coefficient is determined by the fixed point of Spin(8) triality. The action is purely algebraic — it contains no transcendental functions, and its minimum provides algebraic constraints on physical constants. This paper further provides a complete differential-geometric proof of (based on the heat kernel expansion on the constraint product manifold and the fixed point argument via extrinsic curvature), the variational principle of the action, numerical verification of , and the landscape structure of the action on the Birkhoff Polytope.