Deterministic Symmetry Breaking and Mass Gap Generation in Cℓ4 Yang-Mills Gauge Theory via the Isomorphic Model

1) The quantum Yang-Mills existence and mass gap problem represents one of the central open challenges in mathematical physics. In this paper, we construct a deterministic, algebraic-geometric framework based on the four-dimensional Clifford algebra Cℓ4 to model non-Abelian gauge fields without relying on non-deterministic symmetry-breaking mechanisms. 2) We represent the pure gauge state as a graded multivector |Ψ4D⟩ ∈ Cℓ4 under local Spin(4) rotor transformations R(θ) = exp(−θB/2). We introduce a contractive projection operator P3D(A) = 1/2(A − ew A ew) that projects the hyperspherical state onto the local physical interface Cℓ3. 3) We prove that this deterministic projection breaks the 4D orthogonal symmetry, inducing a lower energy bound ∆m·c² > 0 governed by the gradient of the von Neumann entropy flux dS/dθ. This establishes an exact algebraic origin for the mass gap ∆m > 0 in Cℓ4 quantum gauge theory.