The emergence of a stable, classical world from the underlying quantum reality is a central puzzle of modern physics. While decoherence explains the suppression of quantum interference, it does not, by itself, explain the selection of specific, robust classical histories. Here, we present a complete, end-to-end theory for the emergence of classicality, grounded in the thermodynamics of irreversible record-formation and testable on current-generation quantum computers. We first derive a consistent, directed temporal order from thermodynamic principles, resolving known issues with relational time concepts. We then argue that the coherence of histories within this emergent time is unstable to noise unless actively stabilized by a local, error-correcting-like mechanism. We formalize this mechanism as a projection onto a constrained subspace of the history Hilbert space and argue for its stability for a general class of noise models. Finally, we map this mechanism onto the dynamics of a 2D lattice of superconducting qubits, where the constraints are identified with the stabilizers of the surface code. We derive an analytic expression for the phase boundary between stable and unstable regimes, validate it with illustrative calculations based on the analytic theory, and propose a specific experiment on a trapped-ion quantum computer to test a unique, falsifiable power-spectrum signature. This work bridges the gap between foundational theory and experimental reality, providing a concrete, testable mechanism for the emergence of classicality.