Riemann's Zeta Function on the Critical Line from Symmetric Variation Geometry: Heat-Kernel Asymptotics and Tauberian Forcing

This symmetric-variation geometric framework analyzes the Riemann zeta function via short-time heat-trace asymptotics Θ(t) ∼ t⁻¹ + oscillatory terms indexed by nontrivial zeros, deriving an explicit Chebyshev formula ψ(x) - x = x1/2 ∑ A_γ cos(γ log x + ϕ_γ) + O(x1/2 log x) and using Tauberian forcing to establish that all nontrivial zeros lie on Re(s)=1/2, with optimal error ψ(x) = x + O_ε(x1/2 + ε).