An MDL-Style Cost Functional KC, Distribution-Preserving Reductions ($A2^d$), and an $AC^0$+log Lower Bound for 3SAT via Balanced 3XOR

Abstract

We introduce a model-agnostic MDL-style cost functional $K_C$ for resource-bounded classifiers and prove a Total-Variation stable reduction lemma ($A2^d$) for distribution-preserving many-to-one reductions. On a balanced distribution of random 3XOR instances (with co-rank $t'=\Theta(n)$) we obtain a size-aware lower bound against P-uniform AC^0+log models: $\Pr[M=\chi] \le \frac{1}{2} + s(N)\exp(-\alpha_d m^{c/d})$ with an absolute $c \in (0,1)$ (e.g., $c=1/3$ gives $\beta_d=1/(3d)$). A deterministic, injective 3XOR->3SAT translation (four 3-clauses per XOR, no auxiliaries) is $\delta=0$ measure-preserving on its image window; by $A2^d$ the bound transfers to 3SAT. This yields, to our knowledge, the first explicit $K_C$-reading of such size-aware bounds under a $\delta=0$ measure-preserving reduction in small-depth circuit lower bounds. We provide artifacts (generator -> DIMACS -> verification) with match-rate 1.0.