Geometry from Irreversibility: Emergent Metric Structure and Proper Time from Quantum Dissipation

Can spatial distance and proper time be defined operationally by how a system leaks information to its environment rather than postulated as primitive geometric quantities? We answer affirmatively for both, within Markovian open quantum dynamics. This paper derives both the spatial triangle inequality and the Lorentzian reversed triangle inequality (the twin paradox) from a single physical ingredient: the irreversible flow of information from a quantum system to its environment. Key findings Emergent Spatial Metric: Spatial distance is defined as the excess irreversibility generated by superposing two states ($d(A,B)^{2} = \alpha(\psi_{AB}) - \frac{1}{2}[\alpha(A) + \alpha(B)]$). This distance forms a genuine metric of negative type that embeds isometrically into a Hilbert shadow space and recovers the Euclidean metric of $\mathbb{R}^{3}$ for independent position channels. Crucially, metric structure breaks down before decoherence even for orthogonal states, demonstrating that geometry and classicality emerge together via einselection. Proper Time and the Twin Paradox: Proper time is defined as background duration minus the minimal accumulated deficit of an optimal control protocol ($\tau = T - \Lambda^{*}/\kappa$). Because optimal control costs are Bellman-subadditive, subtracting them from an additive duration automatically makes proper time superadditive, yielding the reversed triangle inequality of relativity without geometric postulates. Light Cone as an Information Bound: The exact Minkowski interval corresponds precisely to a dissipation rate that saturates at high transport velocities. The light cone exhibits itself operationally as a fundamental bound on irreversibility production Thermodynamic Anchoring: In an exactly solvable damped-oscillator model, Spohn relative-entropy production coincides identically with Clausius entropy flow. The optimal transport problem is solved in closed form, making the reversed triangle inequality exact at the quantum level and yielding an emergent signal speed fixed by environmental transport properties ($c^{2} = \kappa D$, with $D$ the Einstein diffusion coefficient of the environment) Numerical Verification: All analytical results are thoroughly verified using lattice decoherence models, exact linear-quadratic optimal control, and Fock-space computations