Geometric Structure of Genomes Across the Tree of Life: Toward a Geometric Theory of Sequence Structure

This work develops a geometric and statistical framework for analyzing the structure of biological sequences and explores its implications for understanding the emergence and evolution of life. Motivated by questions concerning the transition from prebiotic chemistry to living systems, the quantification of negentropy in organic matter, and the distinction between random and biologically viable sequences, we introduce mathematical descriptors that measure deviation from linearity and related geometric irregularities of self-replicating macromolecules. These descriptors reveal a pronounced geometric separation between biological DNA and random sequences, underscoring the non-random structural organization characteristic of living systems. Using these descriptors, we compare a broad range of species across the Tree of Life and examine how geometric complexity varies between primitive and more advanced organisms. We further investigate whether these measures provide a natural way to compare organismal complexity, characterize the structure of viable sequence space, and identify potential constraints on evolutionary trajectories. The framework also offers an initial perspective on how natural selection and stochastic mutations may jointly influence genomic organization. Finally, we outline speculative connections between increasing geometric irregularity and the emergence of biological complexity, suggesting that such geometric transitions may offer insight into the origins of life and the theoretical limits of evolutionary development.