Precision's arrow of time

The arrow of time is usually attributed to two mechanisms: decoherence through environmental entanglement, and chaos through nonlinear dynamics. Here we demonstrate a third route, Precision-Induced Irreversibility (PIR), requiring neither. No entanglement. No nonlinearity. Just three ingredients: amplification, non-normality, and finite dynamic range, whose interplay yields an operational arrow of time; remove any one and reversibility can be restored. Non-Hermitian evolution remains mathematically invertible, yet beyond a sharp temporal predictability horizon scaling linearly with available precision, distinct states collapse onto identical representations. Echo-fidelity tests confirm this transition across arbitrary-precision calculations and hardware, revealing where formal invertibility and physical reversibility diverge.