Bridging Algorithmic Information Theory and Machine Learning, Part IV: Solomonoff Gaussian Hilbert Spaces, Solomonoff Gaussian Processes, and Solomonoff Gaussian Fields

This paper asks how Solomonoff-style program-length weighting can be represented by positive semidefinite kernels and Gaussian-process covariance operators. The answer developed here is the Solomonoff Feature Mixture (SFM), a kernel templatek(x,y)=∫Ωuω(x)uω(y)dπ(ω),π(ω)∝2−ℓ(ω),where uω may be uncentered, prior-centered, or data-centered as defined explicitly below. This includes ideal Solomonoff kernels, D2KE-based KC-kernel surrogates, Occam mixtures, and centered covariance kernels as special cases.The SFM construction induces a Solomonoff Kernel Covariance Operator (SKCO) and, when used as a covariance, classical Gaussian-process objects: Solomonoff Gaussian Processes, Solomonoff Gaussian Hilbert Spaces, and Solomonoff Gaussian Fields. The main result is structural rather than empirical or rate-optimal: under explicit non-degeneracy assumptions, the SKCO spectrum is controlled by program-length statistics, while truncation, landmark, and compressor-based approximations give computable surrogates. The paper also makes explicit the limits of this bridge: Gaussian processes do not reproduce Solomonoff induction at the level of discrete hypothesis probabilities or universal dominance; they preserve a second-order, operator-level form of algorithmic bias.