Bridging Shannon and Turing: A Formal Entropy–Complexity Equivalence over Canonical Grammars

We establish a formal correspondence between Shannon entropy and Kolmogorov complexity
in symbolic domains governed by constrained rewrite systems. By defining a class of canonical
representations for symbolic structures, we construct a bridge between statistical compressibility
and algorithmic compressibility. We prove that for any computable distribution over irreducible
forms, the expected Kolmogorov complexity deviates from Shannon entropy by at most a con-
stant plus a structural degeneracy term. This yields tight two-sided bounds unifying probabilistic
and algorithmic theories of information. Applications include lossy compression lower bounds,
grammar-constrained learning, and complexity-preserving communication protocols.