Intrinsic Bayesian Self-Improvement on Entropic Law Spaces

This preprint develops a measure-theoretic framework for intrinsic Bayesian self-improvement on entropic law spaces. Building on a previous entropic temporal-interface complexity (ETIC) perspective, a “law” is treated as a full implementation-level stochastic mechanism: internal state dynamics, an interface to the environment, and an internal Bayesian module that runs only on its own observable history. The paper formalizes the intrinsic filtration generated by internal states, interface outputs, and law-time indices, and imposes a strict no-meta constraint: once the initial law, prior, and move set are fixed, all subsequent belief updates and law-space moves must be measurable with respect to this intrinsic information alone. On this foundation, the work defines an entropic efficiency functional J(L) as an expectation-valued functional of the interface path under a frozen law, and models candidate laws as a parametric (or nonparametric) family {L_theta} with prior Pi_0. Under explicit assumptions—stationarity and ergodicity under frozen laws, posterior consistency, compactness of a relevant parameter region, and bounded continuity of J and local move-effect functionals—internal Bayesian estimators of J(L) and of move effects Delta J(L, m) are shown to converge almost surely in sufficiently long frozen-law phases. A no-meta proposition formalizes that, given the same prior and the same intrinsic filtration, the internal posterior mean is already Bayes-optimal under squared loss; an external supervisor with no extra information cannot systematically improve the self-evaluation. The paper then introduces intrinsic Bayesian move policies on law space and studies the resulting law-time self-improvement dynamics. Under additional structural assumptions (compactness and continuity on a restricted law region, a qualitative exploration condition, and a monotone “no harmful moves” regime with a null move), the main theorem establishes a local criticality property: any accumulation point of the law-time process is almost surely such that no allowed move has strictly positive true gain in J. A stronger local optimality pattern—where accumulation points coincide with local maxima of J relative to the move set—is formulated as a conjecture rather than a theorem, to keep the mathematical claims aligned with what can currently be proved. To illustrate the assumptions and their practical reading, the framework is instantiated on simple finite settings, including bandit-like agents and representation-choice laws, and a complexity discussion provides explicit cost estimates for posterior updates and move evaluation in finite-state cases. The analysis highlights that the scheme is best interpreted as a conceptual, implementation-level blueprint for intrinsic self-evaluation and self-modification under entropic objectives, not yet as a scalable algorithm. Robustness under approximate inference, finite frozen phases, and large law spaces, as well as full scheme-level optimisation of joint entropic efficiency and law-time complexity, are posed as explicit open problems for future work.