Information-theoretic Limits on Programmatic Specification of Biological Systems

Abstract

The central problem of biology is the origin of biological organization. We show, using information theory, that an organism does not contain enough organism-specific information to specify its own fully functioning microscopic organization. The organized machinery of life is therefore not the execution of a fully prewritten organism-specific program under favorable conditions. Rather, it is the compilation of a coarse organism-specific specification by a shared physical background that is constitutive of biological organization. We formalize this as a coarse-graining information threshold on biological specification, with two complementary entropy faces -- a Shannon face controlling stochastic generation and a Hartley face controlling zero-error deterministic addressability. Above the threshold, organism-controlled information is sufficient to specify structural and functional ensembles; below it, programmed microstate determinism is impossible: deterministic addressability fails by pigeonhole, and any algorithm producing sub-threshold outputs must consume runtime randomness proportional to the information deficit. The threshold follows from two information-theoretic constraints -- finite specification capacity and causal locality -- supplemented by a mixing lemma showing that initial-condition information decays exponentially under thermal dynamics. We establish the threshold as a family of maximal capacity-compatible coarse-grainings, distinguish the proven impossibility below the threshold from the empirically realized coarse mappings above it, and locate the threshold empirically through worked cases of protein folding, E. coli, Drosophila early development, and C. elegans, together with computational verification using AlphaFold-2, the JCVI-syn3A 4D whole-cell simulation, and canonical stochastic gene network models. We further show that no known naturally realized environmental channel can close the gap. The result rules out programmed microstate determinism while leaving physical determinism untouched, reframes the genome as a generator specification rather than a trajectory program, and unifies gene-centric, developmental, and field-theoretic (bioelectric, morphogenetic, and related continuum) views of biological specification under a single coarse-graining framework.