Exact Cosmological Solution of the Information–Einstein Equations — A Unified Critical Mechanism for Inflation and Dark Energy

Abstract

This paper presents an exact solution of the information–Einstein equations of the Order Parameter Spacetime Theory in an FLRW background. The core discovery is that when the feedback coefficient c₁ is exactly equal to 1, the Hubble parameter H completely disappears from the equation — 0 · H² = 0, and the Universe can expand at an arbitrary rate. This critical condition is naturally satisfied by the spontaneous evolution of the order-parameter field: as it approaches a Type-B singularity, c₁ increases and passes through 1, triggering inflation; as it moves away from the singularity, c₁ decays and falls back below 1, terminating inflation. Inflation is a transient phase through the single point c₁ = 1 — its duration is determined by the relaxation time of the order-parameter field at the Type-B singularity. The inflation condition c₁ = 1 is equivalent to the product of the quantum dissipation length and the soliton scale reaching a critical value, tracing the onset of inflation back to a microscopic scale matching. When c₁ is close to but not equal to 1, the same equation produces dynamic dark energy; the deviation of w(z) is completely determined by the redshift evolution of c₁ — this closed-form expression possesses exclusive testability. Inflation and dark energy are therefore not two independent problems, but manifestations of the critical feedback of the order-parameter field at different epochs of the Universe. The solution presented here is an algebraic critical condition, not a complicated differential equation solution — it reveals that the physical essence of inflation is a critical phase transition in the structure of the equation.