Explorable Multiverses in Reproducing Kernel Hilbert Spaces: A Universal Framework for Semantic Navigation and Content Variation

Abstract

We present RKHS Multiverses, a mathematical framework for creating infinite explorable knowledge spaces by grounding semantic embeddings in Reproducing Kernel Hilbert Space theory. Traditional embedding systems treat vectors as static representations for retrieval; we reconceptualize them as functions in a complete Hilbert space (H, ⟨·,·⟩) where content items live on the unit sphere S^(d-1) ⊂ R^d, enabling: (1) kernel-based navigation via cosine similarity K(x,y) = ⟨f(x), f(y)⟩ without LLM inference, (2) continuous variation operators preserving normalization, (3) branching multiverses where users fork items to create variations with quantified divergence, and (4) LLM-powered multiverse generation coherently mixing locally-factual (entities in our reality node) and non-locally-factual (entities from alternate nodes) content. Our universal .rkhs.json format encodes nodes (768D embeddings + 3D PCA positions), edges (kernel similarities), and graph structure in a domain-agnostic specification working for literary works, life timelines, creative writing states, and LLM-generated alternate completions. Zero-cost exploration operators navigate semantic space via pre-computed kernel matrices with no API calls: k-nearest neighbors (similarity), greedy max-min selection (diversity), stochastic random walks (serendipity). Fork operators create variations through continuous transformations T: S^(d-1) → S^(d-1)—style-shift operators interpolate toward style centroids T_style(f, s, α) = normalize((1-α)f + α·c_s), complexity operators apply directional vectors, all preserving unit norm, continuity, and composability. Since factual answers are part of our multiverse node and "hallucinations" are non-locally-factual entities from alternate nodes, LLMs systematically generate and navigate infinite alternate realities while measuring semantic distance. The framework maintains dual-representation architecture with 768D semantic fidelity and 3D human-interpretable visualization, handling 28K+ nodes with batch matrix operations. Demonstrations include CodexSpace v1 (28,602 pre-1919 books navigable by semantic similarity), personal timelines forking life decisions, creative workflows tracking variations, and LLM multiverse generation. Fully open-source with mathematical formalization, example universes, and conversion pipeline enabling domain-specific multiverses.

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