The Algorithmic Copy Lemma Fails for Prefix Kolmogorov Complexity

The copy lemma is the central tool for proving non-Shannon-type linear information inequalities. Its algorithmic analogue -- often coined *artificial independence* -- would, if it held with constant overhead, allow one to translate those inequalities to prefix Kolmogorov complexity K with O (1) precision, thereby settling Problem Q2 from *27 Open Problems in Kolmogorov Complexity*. We prove that even the simplest instance of the algorithmic copy lemma (the empty-background case) is false: for any fixed constant c, there exist strings x for which no string u can simultaneously satisfy the required complexity equalities with error bounded by c. The argument uses only the existence of Martin-Löf random strings and the symmetry of information for K. Consequently, all known non-Shannon-type inequalities, whose proofs rely on the copy lemma, cannot be lifted to their K counter-part with O (1) precision accuracy via the classical technique. Additionally, we connect this impossibility to Levin's broader *Forbidden Information* framework and to the known non-reducibility of conditional prefix complexity, showing that the empty-background copy lemma fails precisely because it would require an impossible reduction of conditional to unconditional algorithmic information. Finally, we suggest a refined question of whether *any* non-Shannon-type inequality holds with O (1) precision for K.