The identification of cosmic expansion with entropy increase is not merely analogical. The geometric and informational descriptions of a physical configuration are the same object viewed through different lenses. The statement that the universe is expanding is, in this framework, the geometric reading of the statement that the entropy of the underlying configuration is increasing. These are not two correlated phenomena but one phenomenon with two descriptions.
This has a sharp implication. The arrow of time, the thermodynamic second law, and the expansion of the universe are not three independent features of the observed world. They are one feature.
Penrose’s Weyl curvature hypothesis [penrose1979] proposes that the thermodynamic arrow of time originates in the asymmetry between initial and final singularities: the initial singularity had vanishing Weyl curvature (smooth, ordered), while final singularities (black holes) have diverging Weyl curvature (chaotic, disordered). This is a geometric statement about the initial state, motivated by the need to explain the arrow of time.
The present results suggest that the initial state is not merely smooth within quantum limits but perfectly smooth: zero entropy, uniquely determined (up to the 0L∕1L labelling convention noted in Section 1), admitting no fluctuations. Yet qualitative features of matter organisation and spatial expansion can emerge. No residual quantum irregularities are required.
The exact solution of Section 2 gives 1 − 2p(τ) = e−2τ. Its reciprocal,
is an exponentially growing quantity, unbounded as τ →∞ (as p → 1∕2). Identifying τ = 
t
— a strictly linear relationship between rescaled bit-flip time and cosmological time t —
gives
an exact match to the de Sitter scale factor, valid at every τ, not merely asymptotically or in some restricted regime. This resolves what would otherwise be a structural mismatch between this model and de Sitter expansion: entropy itself, S(τ) = H2(p(τ)), is bounded and cannot correspond to an unbounded scale factor under any reparametrisation of time, but X(τ) is a different, unbounded transform of the same underlying relaxation variable p(τ), and it is X(τ), not S(τ), that plays the role of scale factor here.
The correspondence has a clean physical reading: as τ →∞ (equilibrium, i.e. heat death), X →∞, so the model predicts unbounded expansion that never actually completes in finite τ — the same asymptotic structure as the usual heat-death-as-limit picture, recovered here without being assumed.
Two points bound the scope of this result and are stated here rather than left implicit. First, the specific choice of X(τ) = (1 − 2p)−1, rather than some other transform of p, is not yet independently derived from the bitstring-to-geometry decoding maps of Section 2; establishing why this transform in particular should represent physical spatial extent, rather than simply being the transform that reproduces the target curve, is left for future work. Second, this is a statement about the shape of the correspondence (that an exact match exists under a linear time identification), not about its quantitative calibration. Substituting τ = traw∕L into the identification above shows that matching a given physical Hubble rate H requires H = 2∕L exactly, if one raw bit-flip tick is identified with one Planck time.