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? Within Markovian open quantum dynamics, we show that: (i) the spatial triangle inequality and (ii) the Lorentzian reversed triangle inequality (twin paradox) both emerge from irreversible information flow. However, while the reversed inequality holds generally, the exact Minkowski interval remains conditional on a saturating dissipation profile that our solvable model does not produce. Key findings Emergent Spatial Metric: Distance is defined as excess irreversibility from superposing two states (d² = α(ψ_AB) − ½[α(A)+α(B)]). This forms a genuine metric of negative type, isometrically embeddable in Hilbert space, recovering the Euclidean metric of ℝ³ for independent position channels. Crucially, geometry breaks down before decoherence—even for orthogonal states—showing metric structure and einselection emerge together (Theorem 3.23). Proper Time Inequality: Proper time τ = T − Λ*/κ (background duration minus optimal dissipation cost) is superadditive by Bellman subadditivity of optimal control. This yields the reversed triangle inequality generally, without geometric postulates (Theorem 4.6, Corollary 5.9). Conditional Light Cone: The exact Minkowski interval requires a dissipation profile g(v) = 1 − √(1−v²/c²) that saturates at velocity c. Our solvable damped-oscillator model produces only Newtonian-order kinetics (g ≈ v²/2c²), so the light cone remains conjectural (Corollary 4.9, Remark 5.11). Thermodynamic Anchoring: In the oscillator model, Spohn relative-entropy production coincides exactly with Clausius entropy flow (Lemma 5.1). The minimal transport cost is solved in closed form (Λ* = ηΔx²/φ(T)), with emergent speed c² = κD where D is the Einstein diffusion coefficient. Universality of c requires the Compton-clock hypothesis κ ∝ m (Conjecture 8.1). Numerical Verification: All analytical results are verified using lattice decoherence models (241-site), exact linear-quadratic optimal control, and Fock-space computations (Tables 4–6). Scope: This is an analog-gravity construction in a preferred frame; boost symmetry is not derived (Section 8.3).