Containment as Catalyst: Topological Bounds on Representation Geometry as a Mechanism for Forced Parameter Efficiency

We prove that hard topological boundaries on neural network representation spaces — enforced via Morse-barrier projected gradient descent — increase Fisher Information Density (Tr(F)/d_intrinsic) without architectural modification. The mechanism is geometric: containment eliminates the radial degree of freedom, forcing the optimizer to compress discriminative information into angular coordinates within a bounded domain. We establish two theoretical results: (1) CFL-bounded random walks in Morse-barrier domains converge to reflected Brownian motion with uniform stationary distribution, and (2) the CFL process provides strict containment with equilibrium safety margin O(σ²/C). Empirically, containment increases Fisher density by 104–578% on MNIST/MLP (p = 2 × 10⁻⁶), scales through 256 dimensions on CIFAR-10/CNN without decay, and transfers to GPT-2 fine-tuning with ellipsoidal Mahalanobis barriers at 768D. A wake-sleep rumination protocol demonstrates that bounded representations support autonomous competence recovery after perturbation. The framework is architecture-agnostic, requiring only a differentiable barrier function and a projection operator.