Information--Dimensional Law Selection under No--Meta Constraints

This preprint introduces a mathematically explicit framework for law selection based purely on internal information–theoretic scalars, without any external or meta-level reward (“no-meta” setting). We work on symbolic configuration spaces X=AZdX = AZ^dX=AZd and use Kolmogorov–Sinai entropy to define an information dimension dim⁡I=hKS/∣log⁡r∣_I = hKS/|log r|dimI=hKS/∣logr∣ for both the underlying law and a finite family of observers. For each law TTT we extract structural scalars a state information dimension dstate(T)dₛₜₐₜₑ(T)dstate(T), observer–specific visible dimensions dobsi(T)dobs^i(T)dobsi(T), nonnegative visibility gaps Gapi(T)=dstate(T)−dobsi(T)Gap_i(T) = dₛₜₐₜₑ(T) - dobs^i(T)Gapi(T)=dstate(T)−dobsi(T),together with an abstract law complexity Comp(T)Comp(T)Comp(T). We then formulate a class of no-meta local objectives that depend only on these internal scalars and satisfy axioms of monotonicity and observer separability. Within this class we design a canonical family of linear local surrogates and show how they can be optimized over probability distributions on law space via the multiplicative weights update (MWU) algorithm, with a clean variance-based proof of monotone improvement. To prevent probability mass from escaping to laws with arbitrarily large hidden structure, we introduce superlinear gap penalties and prove that any maximizing sequence has uniformly bounded expected gap costs under mild linear growth assumptions. Finally, we specialize to i.i.d. universes, where information dimensions reduce to single–site entropies, and show that—under a natural state–gap balance condition—our canonical surrogate becomes a weighted “mutual information minus complexity” objective over the observer channels. The framework thus provides a fully internal, information–dimensional formulation of law selection that links symbolic dynamics, KS entropy, mutual information, MWU dynamics, and complexity regularization in a single no-meta design.