Stable Self-Improving AI under Value-Anchored Natural-Law Gradient Flows
Takahashi, K · Zenodo (CERN European Organization for Nuclear Research) · 2025
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)P_2(\Theta)P2(Θ) over a metric hypothesis space (Θ,dΘ)(\Theta,d_\Theta)(Θ,dΘ), equipped with a “value shortfall” functional Val∗(μ)=∫V(θ) μ(dθ)\mathrm{Val}^*(\mu)=\int \mathcal{V}(\theta)\,\mu(d\theta)Val∗(μ)=∫V(θ)μ(dθ) that aggregates deficits in alignment, performance, or physical feasibility. Under standard assumptions from Ambrosio–Gigli–Savaré theory—λ\lambdaλ-convexity and coercivity of Val∗\mathrm{Val}^*Val∗ along W2W_2W2-geodesics—the associated EVIλ_\lambdaλ gradient flow defines a unique value-anchored equilibrium μ∗\mu^*μ∗ that is globally attracting. Self-improvement is modeled as a Markov kernel–induced update TTT that approximates a time-hhh step ShS_hSh of this law-level gradient flow. The main stability theorem shows that if each update satisfies WΘ(Tμ,Shμ)≤εW_\Theta(T\mu,S_h\mu)\le\varepsilonWΘ(Tμ,Shμ)≤ε, then the entire discrete self-improvement loop stays within a controlled Wasserstein neighbourhood of μ∗\mu^*μ∗, with explicit bounds obtained via the contraction of EVIλ_\lambdaλ 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.