This paper establishes the following results:
First, cosmic expansion is shown to be the geometric interpretation of increasing informational entropy,
with the underlying entropy trajectory given by the exact, closed-form Ehrenfest relaxation curve
p(τ) = 
(1 −e−2τ). Beginning from the zero-entropy state identified in Paper II [meskanen2002] as the
informational singularity, this process drives the systematic expansion of the spatial manifold
without requiring a cosmological constant, ad-hoc inflaton fields, or hard-coded equations of
motion.
Second, the abundance of any emergent micro-structure follows one of exactly three combinatorially determined shapes — monotone increasing, monotone-flattening, or a genuine interior peak — fully predictable in advance from the composition (a,b) of the pattern being counted. Moreover, the hump-shaped regime (a < b) is not merely one of three equally likely outcomes but the typical one: weighting compositions by their multiplicities shows it dominates the composition-weighted probability mass across essentially the entire relaxation trajectory, ceding majority status to the non-hump regimes only in the immediate approach to final equilibrium, a window that narrows as pattern length grows. Physically-motivated matter-detection filters should have their composition checked against this classification to confirm they fall in the peak-producing regime, which the present result shows is the generic, expected case rather than a special one.
Third, the reciprocal transform X(τ) = (1 − 2p(τ))−1 = e2τ reproduces the de Sitter scale factor exactly under a linear identification of bit-flip time with cosmological time, resolving the apparent shape incompatibility between this model’s (bounded) entropy curve and cosmological expansion. This correspondence is exact in form but not yet independently motivated from the underlying bitstring-to-geometry decoding, and does not by itself resolve the quantitative calibration question — addressed, and left open, in Paper IV [meskanen2026iv].