Stable Self-Improving AI under Value-Anchored Natural-Law Gradient Flows

This preprint develops a mathematically grounded framework for stable self-improving AI by embedding the update dynamics of an autonomous learner into a value-anchored natural-law gradient flow. The design space is a Wasserstein space P2 (Θ) P₂ () P2 (Θ) over a metric hypothesis space (Θ, dΘ) (, d_) (Θ, dΘ), equipped with a “value shortfall” functional Val∗ (μ) =∫V (θ) μ (dθ) Val^* () = V () \, (d) Val∗ (μ) =∫V (θ) μ (dθ) that aggregates deficits in alignment, performance, or physical feasibility. Under standard assumptions from Ambrosio–Gigli–Savaré theory—λλ-convexity and coercivity of Val∗Val^*Val∗ along W2W₂W2-geodesics—the associated EVIλ_λ gradient flow defines a unique value-anchored equilibrium μ∗^*μ∗ that is globally attracting. Self-improvement is modeled as a Markov kernel–induced update TTT that approximates a time-hhh step ShSₕSh of this law-level gradient flow. The main stability theorem shows that if each update satisfies WΘ (Tμ, Shμ) ≤εW_ (T, Sₕ) Θ (Tμ, Shμ) ≤ε, then the entire discrete self-improvement loop stays within a controlled Wasserstein neighbourhood of μ∗^*μ∗, with explicit bounds obtained via the contraction of EVIλ_λ flows and a discrete Grönwall argument. The paper does not claim new curvature bounds or entropy inequalities; instead, it packages existing gradient-flow theory into a reusable specification principle that can host concrete value functionals derived from persistence-first holographic systems (PFHS), holographic observation quotients (HOQ), and gradient-flow–based compute–performance trade-offs. This yields a law-level template for designing AI systems whose self-modifications remain stably anchored to a mathematically explicit notion of value.