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/dₑff, where dₑff 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 (dₑff = 1), Rayleigh–Bénard convection (dₑff = 3), elastic turbulence (dₑff = 4), biological scaling under Rubner's law (dₑff = 3), biological scaling under Kleiber's law (dₑff = 4), and the cosmological energy partition (dₑff = 3). All predictions match measurement within uncertainty. None involve adjustable parameters. The framework predicts a stable cosmological attractor at Ω_Λ/Ωₘ = 2, constituting a falsifiable departure from ΛCDM, which predicts Ω_Λ/Ωₘ → ∞. Recent DESI observations of weakening dark energy are consistent with this prediction. The Rubner-to-Kleiber transition—from dₑff = 3 to dₑff = 4 as organisms evolve internal transport networks—is a dₑff transition predicted by the sustainability condition, analogous to the classical-to-ultimate regime transition in Taylor–Couette flow.