Information and the living tree of life: A theory of measurement grounded in biology

Abstract: We extend a formal framework that previously derived time from the multifractal structure of biological lineages (Hudnall & D'Souza, 2025). That work showed that time itself is multifractal -- not a universal background dimension, but an observer-dependent geometry. Here we develop the corresponding theory of measurement: showing that a multifractal conception of time not only permits measurement, but grounds it more rigorously in the structure of biology. The tree of life is modeled as the outcome of stochastic, convex branching, and we show how information-theoretic and fractal measures render its multifractal geometry into measurable, observer-relative time intervals. At the core is a dilation equation that expresses relative time elapse between entities as dimensionless ratios. Operational standards such as the SI second remain valid, but our framework makes explicit their lineage-dependence. This framework unifies measurement theory with biological form, preserves full compatibility with established science, and provides a biologically grounded theory of observation. It enables comparative analyses of duration and kinematics across lineages, with predictions that are directly open to experimental validation.