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? This paper answers affirmatively for both, within Markovian open quantum dynamics, building on the survival-expansion framework of the companion paper "Local Curvature of Quantum Information" (doi:10.5281/zenodo.18792424). Spatial distance is defined as the excess irreversibility generated by superposing two states: when a system is superposed over two locations, the environment learns which at a rate growing with their separation. This distance is proven to be a genuine metric — it satisfies the triangle inequality via an isometric embedding into environmental "shadow space", recovers the Euclidean metric of R³ for independent position channels, and becomes undefined before decoherence even for orthogonal states, so that geometry and classicality emerge together. Proper time is defined as elapsed background time minus the minimal dissipation needed to traverse a worldline. The reversed triangle inequality of relativity (the twin paradox) then follows automatically from the subadditivity of optimal costs, and the exact Minkowski interval corresponds to a dissipation rate that saturates at high transport velocity: the light cone as a bound on irreversibility production. Both Lorentzian signatures are traced to a single convex-analytic root: spatial distance is directly a subadditive object (a norm), while proper time is an additive duration minus a subadditive optimal cost, hence superadditive. In an exactly solvable driven damped-oscillator model, the information deficit coincides identically with Clausius entropy production, the minimal transport deficit obeys \( \Lambda^* = \frac{\gamma m}{2 k_B T_b} \frac{\Delta x^2}{T} \), and the emergent light speed is \( c^2 = \frac{\kappa k_B T_b}{\gamma m} = \kappa D \), with $D$ the Einstein diffusion coefficient. The universality of the emergent light cone is shown to be equivalent to a Compton-clock scaling of the dissipation–time conversion rate (\( \kappa = \frac{2 m c^2}{\hbar} \)), with the quantum–classical crossover condition \( k_B T_b = \frac{\hbar \gamma}{2} \) arising as an internal consistency requirement rather than a postulate. A junction-cost theorem establishes that velocity discontinuities incur strictly positive excess dissipation, so the turning twin ages strictly less even at equal speeds. All results are verified numerically on lattice decoherence models, exact linear–quadratic optimal control, and Fock-space computations; the verification scripts accompany the manuscript. Locality of the environmental couplings is assumed, not derived, and the construction has analog-gravity status: a Lorentzian interval in a preferred frame defined by the bath.
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? This paper answers affirmatively for both, within Markovian open quantum dynamics, building on the survival-expansion framework of the companion paper "Local Curvature of Quantum Information" (doi:10.5281/zenodo.18792424). Spatial distance is defined as the excess irreversibility generated by superposing two states: when a system is superposed over two locations, the environment learns which at a rate growing with their separation. This distance is proven to be a genuine metric — it satisfies the triangle inequality via an isometric embedding into environmental "shadow space", recovers the Euclidean metric of R³ for independent position channels, and becomes undefined before decoherence even for orthogonal states, so that geometry and classicality emerge together. Proper time is defined as elapsed background time minus the minimal dissipation needed to traverse a worldline. The reversed triangle inequality of relativity (the twin paradox) then follows automatically from the subadditivity of optimal costs, and the exact Minkowski interval corresponds to a dissipation rate that saturates at high transport velocity: the light cone as a bound on irreversibility production. Both Lorentzian signatures are traced to a single convex-analytic root: spatial distance is directly a subadditive object (a norm), while proper time is an additive duration minus a subadditive optimal cost, hence superadditive. In an exactly solvable driven damped-oscillator model, the information deficit coincides identically with Clausius entropy production, the minimal transport deficit obeys ( \Lambda^* = \frac{\gamma m}{2 k_B T_b} \frac{\Delta x^2}{T} ), and the emergent light speed is ( c^2 = \frac{\kappa k_B T_b}{\gamma m} = \kappa D ), with D the Einstein diffusion coefficient. The universality of the emergent light cone is shown to be equivalent to a Compton-clock scaling of the dissipation–time conversion rate (( \kappa = \frac{2 m c^2}{\hbar} )), with the quantum–classical crossover condition ( k_B T_b = \frac{\hbar \gamma}{2} ) arising as an internal consistency requirement rather than a postulate. A junction-cost theorem establishes that velocity discontinuities incur strictly positive excess dissipation, so the turning twin ages strictly less even at equal speeds. All results are verified numerically on lattice decoherence models, exact linear–quadratic optimal control, and Fock-space computations; the verification scripts accompany the manuscript. Locality of the environmental couplings is assumed, not derived, and the construction has analog-gravity status: a Lorentzian interval in a preferred frame defined by the bath.
The author proposes two postulates for the foundation of time and spacetime in open quantum systems. Postulate P1 (Time): the proper time of an open quantum system is generated by the integral of its boundary structural-persistence functional Φ on the variationally selected autonomous boundary. Postulate P2 (Spacetime): spacetime is the synchronisation pattern of a network of open-system boundary Φ-clocks coupled through shared baths. From P1 the author derives (i) a thermodynamic clock τ_∂ = ∫ Φ ds, whose sign and magnitude classify structure-building, structure-maintaining, and structure-dissolving regimes; and (ii) two channels of local arrow-inversion windows (monitored feedback Φ_fb > 1; non-Markovian backflow Φ_NM > 1), bounded in duration by the local entropy budget and consistent with the global second law (Sagawa-Ueda for feedback; memory-induced negative σ_irr for backflow). From P2 the author derives (iii) a synchronisation functional Σ_ij between two Φ-clocks coupled through a shared bath, with the Connes-Rovelli thermal time hypothesis recovered in a triple limit (joint KMS state + Σ → 1 + Spohn-modular reduction); and (iv) an induced graph distance d_ij = -log|<X_i X_j>|/N that satisfies the triangle inequality on a metric-compatibility regime operationally characterised by joint CP-divisibility and absence of BLP non-Markovianity witnesses on the relevant triangles. The two witness conditions are jointly necessary for the triangle inequality on this regime; their sufficiency is conjectured under qualifications stated in the manuscript and listed as an open problem. The author provides a complete epistemic-status audit of every quantitative claim and an explicit comparison with the principal existing programmes on time and emergent spacetime: Connes-Rovelli thermal time hypothesis, Page-Wootters conditional time, Van Raamsdonk / AdS-CFT entanglement-based emergence, causal set theory (Sorkin), Verlinde / Padmanabhan thermal gravity, and Wolfram hypergraph rewriting. The audit of falsifiable predictions, the experimental discriminators (3-qubit feedforward, directional asymmetry, triangle inequality), and the worked-case reanalysis of Micadei et al. (2019) are the subject of the companion Preprint II.
This note extends the quantum reference-frame framework from charge, time, and position to geometry itself. It studies the consequences of treating spacetime geometry as a quantum variable in exactly solvable models. An observer without access to a superposed geometry register is shown to possess a geometric information deficit: the inaccessible geometry information is quantified by the entropy of the geometry sectors. Conditioning on the geometry register recovers the individual geometric branches exactly, establishing geometry as a quantum reference frame with the same structure as charge, time, and position reference frames. The work studies clocks evolving in superposed gravitational redshifts and derives exact proper-time interference. Different geometric branches generate which-geometry records in the clock, producing decoherence and periodic recoherence according to the same information-transfer structure found throughout the reference-frame framework. The note further investigates the emergence of classical spacetime. Multiple quantum systems coupled to the same geometry register collectively decohere the geometric superposition faster than a single system, while non-universal coupling does not produce the same amplification. This provides an exactly solvable realization of the idea that spacetime classicality can emerge through continuous measurement by matter. Finally, the paper derives the effective energy-product interaction used in gravitationally induced entanglement models. By introducing a dynamical geometry mode sourced by matter energy and eliminating the mediator, the familiar energy-product coupling emerges directly from the quantum geometry model rather than being assumed. The results connect quantum reference frames, proper-time superposition, gravitationally induced decoherence, gravitational entanglement, and the emergence of classical geometry within controlled finite models. The declared limits are finite geometry registers, simplified sourced geometry modes, and no claim of a complete theory of dynamical quantum gravity. Verification and reproducibility materials are included.
We derive geometrical bounds on the irreversibility in both quantum and classical Markovian open systems that satisfy the detailed balance condition. Using information geometry, we prove that irreversible entropy production is bounded from below by a modified Wasserstein distance between the initial and final states, thus strengthening the Clausius inequality in the reversible-Markov case. The modified metric can be regarded as a discrete-state generalization of the Wasserstein metric, which has been used to bound dissipation in continuous-state Langevin systems. Notably, the derived bounds can be interpreted as the quantum and classical speed limits, implying that the associated entropy production constrains the minimum time of transforming a system state. We illustrate the results on several systems and show that a tighter bound than the Carnot bound for the efficiency of quantum heat engines can be obtained.
Overview: This paper develops the state-to-geometry interface of the OT–GKSL architecture for open quantum sources. Its starting point is a familiar low-energy tension: classical general relativity couples spacetime geometry to matter through the Einstein equations, whereas quantum matter may be open, coherent, correlated, preparation dependent, or subject to measurement and environmental records. Semiclassical gravity and stochastic-gravity approaches have long made clear that the relation between quantum sources and classical geometry is not conceptually trivial (Page & Geilker, 1981; Kuo & Ford, 1993; Hu & Verdaguer, 2008). The OT–GKSL architecture addresses this problem by not assuming classical spacetime as part of the native quantum state description. Instead, it separates a persistent quantum/open-state domain from later classical descendants. At the native level, states belong to a faithful finite-dimensional working stratum, ρ ∈ D°(H_eff), and evolve under a declared GKSL dynamics, dρξ/dξ = L*(ρξ). The completely positive Markovian structure follows the standard theory of Gorini, Kossakowski & Sudarshan (1976) and Lindblad (1976). On a faithful reversible detailed-balance component, the dissipative generator admits the noncommutative optimal-transport gradient representation developed in the quantum Markov setting by Carlen & Maas (2017): L*_db(ρ) = - grad_gOT D(ρ || π). Here g_OT is a positive state-space metric and D(ρ || π) = Tr[ρ(log ρ - log π)] is the relative entropy to a faithful stationary state π. The detailed-balance entropy production is σ_db(ρ) = ||grad_gOT D(ρ || π)||²_gOT ≥ 0. This identity concerns the detailed-balance dissipative component, not automatically the complete GKSL trajectory. If L* = L_coh + L_db + L*_rem, then the full relative-entropy balance retains the coherent and remainder contributions separately. The native evolution parameter ξ, the accumulated entropic ordering variable t_ent, and physical readout time t_ro are therefore distinct: ξ ≠ t_ent ≠ t_ro. A physical clock requires a stabilized and calibrated clock record. Its identification with the temporal sector of the Lorentzian spacetime readout requires the additional clock–coframe and proper-time compatibility conditions of the certified spacetime branch. From records to geometric readout: Classical geometry is not identified directly with the native OT geometry. The architecture first constructs physical records. Protocol-complete certified records determine event identity, while a separate geometric coarse-graining determines which event distinctions are used for localization. A 2 rank-D family then supplies local geometric coordinates. On the observed spacetime branch, D = 4. native states → certified physical records → event classes → geometric coarse-graining → rank-D coordinate records → horizontal readout sector → section and soldering →Lorentzian coframe → classical metricThe important distinction is complete event-record content ≠ rank-D coordinate content.Additional records may encode matter properties, preparation data, measurement outcomes, orientation, gauge information, clock information, or internal structure without becoming additional spacetime dimensions. On a regular geometric patch, the coordinate map defines a vertical sector and its OT-orthogonalhorizontal complement, V = ker DΠ_coord, H = V^(⊥ gOT). A local section selects native representatives and a calibrated soldering map transfers the readablehorizontal data to the physical frame bundle. The physical coframe e^a then defines g_ro = η_ab e^a ⊗ e^b, with η_ab = diag(-1,+1,+1,+1)on the observed Lorentzian branch. The positive metric g_OT and the Lorentzian metric g_ro are therefore different mathematical objects with different domains and physical roles. Lorentzian signature is not obtained by changing the signature of g_OT; it enters through the calibrated readout branch.This distinction is compatible with the broader operational idea that physical reference frames and records may themselves be quantum systems rather than external classical structures (Bartlett, Rudolph & Spekkens, 2007; Giacomini, Castro-Ruiz & Brukner, 2019), while the specific record-to-geometry ordering is a commitment of the OT-GKSL architecture Horizontal connection and holonomy bridge: A central result of the paper is the typed separation between native state-space geometry and readout geometry. Let ∇^OT denote the Levi-Civita connection of the applicable OT metric. The connection relevant for the geometric bridge is the projected connection on the horizontal bundle, ∇^H_X Y = P_H ∇^OT_X Y.Its curvature R^H is not generally the restriction of the ambient OT curvature R^OT. In particular, R^OT ≠ 0 does not by itself imply R^H ≠ 0.The readout bridge therefore transports the horizontal connection, rather than silently identifying ambient OT curvature with spacetime curvature. With a local section s and soldering map Ω, the transferred connection has the standard inhomogeneous transformation structure, Γ_B,s = Ω (s*Γ_H) Ω^(-1) - (dΩ) Ω^(-1).It may then be compared with the independently constructed readout connection. The corresponding bridge defect is controlled on the declared certified domain. For a fixed loop C, a connection-level defect gives a length-type transport estimate of Duhamel form. An area-type small-loop estimate requires the stronger input of curvature control, a suitable small contractible loop, and compatible gauge/trivialization data. The paper therefore distinguishes four statements: ambient OT curvature; horizontal curvature and holonomy; transferred/readout holonomy; detector-visible holonomic response. None is identified with the next without an explicit bridge or visibility condition.3. This separation is related conceptually to earlier geometric treatments of mixed-state parallel transport and quantum holonomy (Uhlmann, 1986; Andersson & Heydari, 2014), although the OT-GKSL horizontal/readout bridge is an architecture-specific construction. Finite holonomy and preparation sensitivity. The consolidated formulation distinguishes a nontrivial finite loop signal from its sensitivity to preparation. A finite holonomy may satisfy H(ρ*) ≠ I while nevertheless Dρ H(ρ*) = 0. Thus anontrivial background holonomy does not prove that the holonomy varies under the preparation tested by an experiment.Two operational observables are accordingly kept distinct.Y_H^fin = L_H^fin(H - I) is a finite-loop detector. A body-fixed or Maurer-Cartan response may instead use Ξ_H = L̂_H(H^(-1) ∇_prep H).These are distinct calibrated observables. They coincide only under an explicitly declared translation/calibration relation. Preparation differentiation must also be gauge controlled, by freezing a physical record-defined frame, differentiating a gauge-invariant functional, or using a covariant derivative.Source-side constitutive branch The source/matter descendant is constructed independently of the geometric descendant. Within the Dirac/HS-NJL working class considered in the paper, an internal operator Y defines the positive quadratic tracer κ_Y = Tr(Y†Y). A frozen calibration maps this quantity to an operational coordinate p_κ. The scalar source-side constitutive law is written μ = μ(p_κ), with local susceptibility β_μ(p_κ) = - ∂_(pκ) ln μ(p_κ). A certification gate may determine whether the working point remains admissible, but its derivative is not part of the physical constitutive susceptibility in the certified interior. The scalar constitutive channel and the holonomic channel are mathematically distinct and possess different experimental symmetry handles. Einstein lock: The observed gravitational branch is constrained by an Einstein lock. The two-derivative gravitational kinetic block is fixed to the Einstein-Hilbert form,S_EH[g_ro; G0] = (c³ / 16πG0) ∫ d⁴x sqrt(-g_ro) R[g_ro], with G0 = constant.This is a branch-selection and recovery condition of the OT-GKSL corpus. It is not claimed to be the unique consequence of GKSL dynamics alone. The architecture forbids a readable preparation-dependent factor multiplying the Einstein-Hilbert curvature term on the locked branch. State dependence is instead carried by independently specified source, response, gauge, interface, readout, or higher-order sectors. The gravitational comparison is performed only after geometry and source data have beenindependently fixed: 4 C_E = G_μν[g_ro] - κ0 T^eff_μν - Δ^if_μν. A gravitational claim requires the residual C_E to satisfy its declared norm and tolerance. TheBianchi identity acts as a compatibility condition on these already constructed sectors; it does not define the source after the fact. The branch-resolved Trilemma: The central result is a local and conditional separation theorem for the visible constitutive and holonomic branches. On a finite-dimensional admissible test sector, define the kinematic analysis using the unweighted source coordinate and the holonomic preparation response, h → (dp_κ[h], ζ_H[h]). The constitutive coefficient β_μ is applied only after this kinematic conditioning. Consequently, testing β_μ,* = 0 does not make the underlying analysis map singular by definition. For a normed visible output space, the declared first-order closure has the form DO_vis[h] = N_E(β_μ,* dp_κ[h]) + N_H(ζ_H[h]) + B_oth[h] + R_aud[h]. Here B_oth contains independently constructed visible branches not removed by the frozen protocol, and R_aud is the declared audit remainder. The factorization of the scalar constitutive term through β_μ is an explicit visible/source closure assumption; it is not derived solely from OT variational calculus. At a working point where β_μ,* = 0, if, for a tested variation h, ||N_H(ζ_H[h])|| > ||B_oth[h] + R_aud[h]||, then DO_vis[h] ≠ 0. The final implication follows from the reverse triangle inequality. The nontrivial content lies upstream: construction of the regular record/geometric quotien
This note studies the emergence of time in a timeless quantum universe using a Wheeler–DeWitt-style constrained model with quantum geometry, clocks, and matter records. The model contains no external time parameter. The total state satisfies a timeless constraint while the geometry register, clock, and matter systems remain fully quantum and entangled. Despite the absence of fundamental time, conditioning on the clock and geometry recovers branch-dependent Schrödinger evolution with the correct redshifted proper-time rates. The work shows that quantum geometry can act as a reference frame for time. An observer without access to the geometry register carries a geometric information deficit, while conditioning on the geometry restores the individual geometric branches exactly. Superposed geometries generate proper-time interference, with matter records determining the transition between quantum time-history interference and classical branch evolution. The central result is an exact emergence-of-time identity: the observable coherence between different geometry branches is equal to the product of geometry coherence and matter-record overlap. When geometry information is fully recorded by matter, the interference between different time histories disappears, and each branch exhibits its own classical Schrödinger evolution. The note also derives a continuous crossover law: the emergence of classical time depends quantitatively on how strongly matter measures the quantum geometry. This connects the relational-time mechanism of Page–Wootters with the semiclassical time program of Kiefer and Zeh in an exactly solvable setting. The results are obtained in finite Wheeler–DeWitt-style toy models with declared assumptions: quantum geometry registers, simplified redshift structure, and no claim of a complete theory of dynamical quantum gravity. Verification and reproducibility materials are included.
The second law does more than set the direction of thermodynamic evolution: it endows nonequilibrium transformations with an underlying geometry. In this work, we provide a unified geometric description of entropy production in classical and quantum thermodynamics based on Wasserstein-2 structures arising from gradient flows of free energy. We review how relaxation to equilibrium, in overdamped diffusions, discrete detailed-balanced Markov chains, and dissipative Lindblad dynamics, can be formulated as a gradient flow on the space of states. The associated Wasserstein-2 distance bounds entropy production, yielding a finite-time refinement of the second law. We extend this framework beyond purely dissipative dynamics by introducing generalized Wasserstein-2 metrics that incorporate conservative (Hamiltonian) dynamics in both classical inertial systems and open quantum systems, yielding intrinsic distances that exactly characterize minimal entropy production under fixed dissipative mobilities. We establish equivalence bounds between purely dissipative and Hamiltonian-dissipative geometries, explicitly quantifying how inertial or coherent dynamics can reduce dissipation. Finally, when restricted to equilibrium distributions, we recover the thermodynamic length of linear response-including the quantum thermodynamic length-thereby linking optimal transport, thermodynamic length, and counterdiabatic protocols within a single geometric framework. All in all, our results extend the Riemannian program of thermodynamics further from equilibrium and provide a geometric foundation for optimal protocols beyond the overdamped setting.