Equilibrium Diffusion: LIM-RPS x Discrete Token Diffusion

> A unified framework for motion-conditioned music generation via coupled equilibrium systems. A DEQ replaces L explicit layers with a single implicit layer defined by its fixed point: **Banach Fixed-Point Theorem**: If f is a contraction mapping (Lipschitz constant L < 1), then: 1. A unique fixed point z* exists 2. The iteration z_{k+1} = f(z_k, x) converges to z* for any initial z_0 3. Convergence is exponential: ||z_k - z*|| <= L^k ||z_0 - z*|| | | LIM-RPS Fixed-Point | Diffusion Denoising | |---|---|---| | Iteration | z_{k+1} = z_k - gamma * B(z_k) + prox_pull | x_{t-1} = x_t + epsilon * s(x_t, t) + noise | | Vector field | B: CrossModalOperator (96 -> 96) | s_theta: score network | | Contraction | Spectral norm(B) <= 1 (1-Lipschitz) | Bounded Jacobian of s_theta | | Attractor | Unique z* (deterministic) | Data manifold (statistical) | | Convergence proof | Banach theorem (spectral norm < 1) | Score matching loss -> true score | | Without noise | Deterministic equilibrium | Probability flow ODE | The operator B in LIM-RPS plays the same mathematical role as -s_theta in diffusion. Both are vector fields pushing the state toward an attractor. The spectral normalization of B is equivalent to requiring the score function to have bounded Jacobian, which is exactly the condition needed for diffusion convergence.