The Sustainability Condition: A Conservation Law for Trajectory Commitment in Self-Organising Systems

Three axioms—state-dependence, additivity, and scale invariance—uniquely yield a sustainability condition for dissipative structures: V(1−α) ≥ ln(N/α), where V is the logarithmic driving, α the dissipation fraction, and N the organisational complexity. The equation has zero free parameters. At saturation, it yields γ + β = 1, constraining the transport exponent γ and complexity exponent β to be complementary. The maximum entropy principle—derived from the same axioms—then determines the partition: γ = 1/d_eff, where d_eff is the number of independent dynamical degrees of freedom. The theory is confirmed across six systems spanning twenty-seven orders of magnitude in scale: Darcy–Bénard convection (d_eff = 1), Rayleigh–Bénard convection (d_eff = 3), elastic turbulence (d_eff = 4), biological scaling under Rubner's law (d_eff = 3), biological scaling under Kleiber's law (d_eff = 4), and the cosmological energy partition (d_eff = 3). All predictions match measurement within uncertainty. None involve adjustable parameters. The framework predicts a stable cosmological attractor at Ω_Λ/Ω_m = 2, constituting a falsifiable departure from ΛCDM, which predicts Ω_Λ/Ω_m → ∞. Recent DESI observations of weakening dark energy are consistent with this prediction. The Rubner-to-Kleiber transition—from d_eff = 3 to d_eff = 4 as organisms evolve internal transport networks—is a d_eff transition predicted by the sustainability condition, analogous to the classical-to-ultimate regime transition in Taylor–Couette flow.