Radioactive Information: How Uncomputability Ensures O(1) Precision for Non-Shannon Inequalities

Shannon entropy and Kolmogorov complexity describe complementary facets of information. We revisit Q2 from 27 Open Problems in Kolmogorov Complexity: whether all linear information inequalities including non‑Shannon‑type ones admit $\mathcal{O}(1)$-precision analogues for prefix‑free Kolmogorov complexity. We answer in the affirmative via two independent arguments. First, a contradiction proof leverages the uncomputability of $K$ to show that genuine algorithmic dependencies underlying non‑Shannon‑type constraints cannot incur length‑dependent overheads. Second, a coding‑theoretic construction treats the copy lemma as a bounded‑overhead coding mechanism and couples prefix‑free coding (Kraft's inequality) with typicality (Shannon-McMillan-Breiman) to establish $\mathcal{O}(1)$ precision; we illustrate the method on the Zhang-Yeung (ZY98) inequality and extend to all known non‑Shannon‑type inequalities derived through a finite number of copy operations. These results clarify the structural bridge between Shannon‑type linear inequalities and their Kolmogorov counterparts, and formalize artificial independence as the algorithmic analogue of copying in entropy proofs. Collectively, they indicate that the apparent discrepancy between statistical and algorithmic information manifests only as constant‑order effects under prefix complexity, thereby resolving a fundamental question about the relationship between statistical and algorithmic information structure.