Optimal Transport Gravity: SDW Equations and Axiomatics: Operational OTq Density-Matrix Dynamics with Einstein-Locked Geometry
Gwenolé Bocquet
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
•Feb 22, 2026•0 citations