Geometric Overflow: The Emergence of Irreversibility

In the preceding toy formulation of Selection Geometry, local reduced mixednesswas introduced as a first quantitative proxy for geometric overflow: the structuralexcess that remains inaccessible under observer-dependent realization. In this paper, we argue that such overflow provides a natural route to the emergence of irreversibility. While the global state evolves unitarily and remains fully coherent,the realized local sector is obtained only through observer-dependent reduction,and this reduction is generically non-invertible. As a consequence, realized evolution is not most naturally described by a reversible group structure, but by asemigroup-like ordering of accessible states. We propose that this structural asymmetryunderlies the appearance of local mixedness, loss of reversibility, entropy-likegrowth, and thermality-like behavior without requiring any fundamental breakdownof global quantum evolution or the ad hoc introduction of stochastic collapse mechanisms. On this view, irreversibility is neither imposed by external coarse-grainingnor merely an epistemic artifact of human ignorance, but emerges from geometricoverflow itself: what cannot be locally re-accessed reappears operationally as directionaltemporal order. Geometric overflow is thereby advanced as a structuralbridge between reversible quantum possibility and irreversible realized history.