Imaginative Generative Modeling via Spectral Diversity Regularization Beyond the Entropy Wall

Generative models are primarily designed to imitate the data distribution, an objective that neither effectively corrects diversity lost by a learned generator nor explicitly defines how generation should extend beyond the data distribution itself. We introduce the framework called Imaginative Generative AI (IGA), which makes diversity part of the target-distribution design problem: among distributions close to a reference, IGA selects one whose spectral entropy, the von Neumann entropy of its kernel covariance, reaches a specified level. The spectral entropy of the data distribution defines an Entropy Wall. Below the wall, IGA performs diversity repair, and the target is provably no farther from the data than the base model; beyond it, IGA attains greater representation-relative spectral diversity. These regimes form a single regularization path from imitation to spectral extrapolation and define an i.i.d. target distribution at every diversity level along this path. We develop the theory of this entropy-constrained projection and show that, under a KL anchor to a pretrained generator, the optimum satisfies a self-consistent exponential-tilt relation. This characterization leads to IGA Guidance, a retraining-free inference-time method for score-based and diffusion models. Our numerical results on synthetic and vision benchmarks indicate diversity repair below the Entropy Wall and controlled spectral extrapolation beyond it.