∴ Disequilibrium is Eternal

Eternal Disequilibrium in Infinite Space

Equilibrium requires S = S_max.
For infinite space with non-zero entropy density s(x):

S = ∫_ℝ³ s(x) d³x = ∞
S_max = ∫_ℝ³ s_max(x) d³x = ∞

When S_max = ∞, the condition S = S_max becomes meaningless. Equilibrium is undefined, not unreachable.

Consequence: ∞ ⇒ ∇

Infinity implies gradients. These drive dissipative processes, which generate organized structures (Prigogine), which create new gradients. Without a finite maximum, entropy has no attractor state. Gradients persist because there’s no state for them to relax into.

∇ → Dissipation → Organization → ∇

This cycle is permanent. Disequilibrium is not a transient state approaching equilibrium—it’s the only possible state for infinite systems. In infinite space this cycles perpetually because equilibrium (∇ = 0 everywhere) is impossible.

Welcome to the Eternal Recurrence.

S_max is undefined ∞
∴ ∇φ ≠ 0 (always)