Cross-Scale Geodesics in Biological Systems: Signalling, Quantum-Classical Boundary, and Morphogenesis

The “tyranny of scales” problem in biology — the inability of any single mathematical framework to account for behaviour across all spatial scales simultaneously — is resolved within the four-dimensional scale space (x,y,z,s) framework, in which physical scale is a genuine spatial coordinate with metric dσ2 = e2s/L(dx2 + dy2 + dz2) + α2ds2. We develop three applications to biological systems. First, the quantum-classical boundary at s∗≈−24 nats — derived in a companion paper from the metric alone, without a decoherence postulate — places the transition zone at the protein and DNA length scale, coinciding precisely with the known domain of quantum biology. Second, all categories of biological signalling (gene cascades, paracrine, endocrine, neural) are unified as geodesics in (x,y,z,s), distinguished only by their ratio of spatial to scale displacement. The maximum spatial reach of any cross-scale signal is ∆xmax = Lpi/(2m's); we show this single formula correctly orders the spatial ranges of all four signalling categories and predicts a specific inverse relationship between signalling range and scale displacement that is testable by experiment. Third, the scale quantum number ns is identified with protein conformational breathing modes: the folding funnel is a scale potential well, the native state is ns = 0, conformational transitions are ns excitations, and misfolding corresponds to localisation in the wrong scale minimum. Morphogenesis is reframed as scale extent expansion from ∆s≈8 nats (fertilised egg) to ∆s≈18 nats (adult), providing a quantitative measure of organismal complexity. All predictions are derived from the geometry of (x,y,z,s) and are stated in forms testable by existing experimental methods.