Paper 5: Four-Dimensional Extended Objects: Scale Pressure, Boundary Conditions, and Fractal Biological Architecture

We develop the four-dimensional extended object framework within the scale space (x, y, z, s) formalism, in which physical scale is a genuine spatial coordinate. Physical objects have genuine, irreducible extent in the scale coordinate s, just as they have extent in (x, y, z): a living cell is not a point in scale space but a 4D object spanning ∆s ≈6 nats. The stress-energy tensor of such objects gains new components: scale pressure Ps (resisting changes in scale extent) and scale shear Tⁱs (coupling spatial and scale deformation). We show that the gravitational compactness parameter Leff = Rc²/GM of the companion Paper 1 is the equilibrium scale length at which internal scale pressure balances the gravitational scale potential gradient — giving M/R compactness a direct physical interpretation in terms of the 4D stress-energy tensor, and providing a geometric constraint on the constitutive relations connecting Ps to the scale strain field. We introduce three scale boundary conditions — free (Ps = 0), fixed (us = 0), and periodic (χ (smin) = χ (smax) ) — and derive their physical and biological consequences. The periodic boundary condition produces discrete standing wave modes in the scale direction and is shown to be the geometric origin of biological fractality: organisms with fractal architecture (lung, vascular tree, neural dendrites) satisfy periodic scale boundary conditions, and the number of fractal branching orders equals the number of standing wave modes k^ (n) s = 2πn/∆sbio. Life is characterised as the maintenance of non-equilibrium scale pressure against thermodynamic relaxation. Two disease predictions follow directly from boundary condition failure: cancer is loss of the upper scale boundary (smax), corresponding to uncontrolled scale expansion; and neurodegeneration is failure of the lower scale boundary (smin), corresponding to collapse of molecular-scale organisation. Both predictions are stated in forms measurable by neutron scattering and are falsifiable by existing experimental techniques.