The origin and evolution of biological systems are deeply and inseparably linked to the storage, transmission, and processing of information. The final and most complex step in the flow of genetic information is protein synthesis during the translation process. Microbial phylogenetic studies have revealed that translation was highly developed at the root of the universal phylogenetic tree, suggesting its key role in the abiotic-biotic transition. The simplest abstract description of translation, the genetic code, is characterized by hierarchical organization and various symmetries. These symmetries are not mere mathematical peculiarities but are deeply intertwined with the molecular mechanisms governing gene expression and the faithful decoding of genetic information, indicating that certain symmetries can be evolutionarily very distant. Remarkably, the hierarchical organization of the genetic code and its symmetries are describable through arithmetic regularities in nucleon numbers of its constituents. Here we will show that the invariance within these arithmetic regularities provides a mathematical foundation for scale-free computation. Its emergence from symmetries across evolutionary epochs of the genetic code at the progenotic stage, coupled with its grounding in nucleon (mass) regularities, suggests its persistent relevance and a nontrivial origin for living organisms at the mesoscopic scale. Furthermore, this connection between scale-free computing and nucleon organization in living matter may support theories of emergent gravity.