Entropy Debt as the Geometric Origin of Temporal Irreversibility: A Cosmological Framework Unifying the Thermodynamic and Cosmological Arrows of Time

We propose that temporal irreversibility — the arrow of time — is a necessary geometric consequence of a persistent structural deficit between the maximum entropy capacity of the observable universe and its actual entropy content, a quantity we term entropy debt (ΔS). Because the cosmological horizon expands at a rate far exceeding the speed at which causal processes can fill the available phase space, entropy debt grows without bound under accelerated expansion. This framework resolves the Penrose initial low-entropy problem without invoking special boundary conditions: when spatial volume approaches zero, the holographic entropy bound (Sₘax ∝ a (t) ²) likewise approaches zero, making low initial entropy a geometric inevitability rather than a fine-tuning mystery. Numerical estimates using standard cosmological parameters indicate that the universe has filled only approximately 5 × 10⁻¹⁹ of its entropy capacity, with the ceiling expanding roughly three trillion times faster than actual entropy production. We further argue that the constancy of the speed of light c is naturally interpreted as the causal upper bound on entropy propagation within this geometric structure, and that perceived time is the subjective manifestation of local entropy increase against this background. The framework yields a falsifiable prediction: a sustained deceleration of cosmic expansion should correlate with a weakening of thermodynamic irreversibility.