The Energy Separation Theorem established that the spectral energy of the TAC operator on the prime Hilbert space decomposes into an S-symmetric component (vacuum) and an S-antisymmetric component (matter excitations). States with definite S-parity correspond to zeros of the Riemann zeta function on the critical line \operatorname{Re}(s) = 1/2. This paper proves that the S-symmetric subspace is the unique attractor of the Steering dynamics. The S-variance functional \mathbb{E}[\psi] = \|\mathcal{S}\psi - \psi\|^2 drives the asymmetric component to zero exponentially in meta-time: \|\psi_-(\tau)\|^2 = \|\psi_-(0)\|^2 e^{-16\kappa\tau}, where \kappa is the Steering rate. The symmetric subspace \mathcal{H}_+ is the global minimum of \mathbb{E} and is exponentially stable under the gradient flow. What this paper proves: · Exponential convergence to S-invariance. Under the Steering dynamics d\psi/d\tau = -\kappa \delta\mathbb{E}/\delta\psi, the asymmetric component decays as \psi_-(\tau) = \psi_-(0) e^{-8\kappa\tau}. The S-variance decays as e^{-16\kappa\tau}.· The symmetric subspace as global attractor. \mathcal{H}_+ is the unique global attractor of the Steering dynamics. Any perturbation in \mathcal{H}_- decays exponentially. The S-variance is a strict Lyapunov function: d\mathbb{E}/d\tau = -16\kappa\mathbb{E} \leq 0, with equality only on \mathcal{H}_+.· Spectral gap determines convergence rate. The convergence rate is proportional to the spectral gap \Delta E between the lowest eigenvalue in \mathcal{H}_- and the vacuum energy in \mathcal{H}_+. From the TAC operator structure, \Delta E \approx \log 2, the smallest prime frequency.· Exact parallel with the cosmological attractor. The Steering dynamics for the cosmological constant are d\Lambda/d\tau = -8\kappa(\Lambda - \Lambda_{\min}). The mathematical structure is identical. Both are exponential relaxations to an S-invariant attractor. A complete correspondence table shows that the asymmetric energy E_- in the prime system is the analog of the matter density \Omega_m in cosmology.· Connection to the Riemann Hypothesis. If the meta-time available to the observable universe is sufficient for the Steering dynamics to reach the attractor (\tau \gg 1/\kappa), then the ground state of \hat{H}_{\text{TAC}} lies in \mathcal{H}_+, and all zeros of \xi(s) corresponding to the observable universe's spectral resolution lie on the critical line.· What this does not prove. The theorem establishes a mechanism—it explains why zeros would be on the critical line, not that they are. Whether the attractor is actually reached depends on initial conditions, the total meta-time available, and the Steering rate. The Riemann Hypothesis remains an empirical question about the observable universe, not a logical consequence of axioms. Why this matters: The S-invariant attractor is a unifying principle across the canvas model. The same mathematical mechanism that explains why the universe is asymptotically de Sitter also explains why the Riemann zeros approach the critical line. This unification is testable: if future surveys detect a deviation of the dark energy equation of state from w = -1, the same framework predicts a corresponding deviation in the spectral statistics of the Riemann zeros—specifically, a deviation from GUE pair correlations at very high heights. Keywords: S-invariant attractor, Steering dynamics, Riemann zeros, critical line, prime Hilbert space, TAC operator, spectral gap, meta-time, Lyapunov function, exponential convergence, cosmological attractor, de Sitter space, Hilbert-Pólya conjecture