Optimal-Transport Gravity Trilemma: Holonomy, GKSL Dynamics, and Source-Side Coherence

This manuscript develops a constrained and low-energy testable theory of the state–geometry interface in which classical spacetime geometry is not fundamental, but appears only as a certified readout on a controlled infrared window WaccW_{\rm acc}Wacc. The native layer is formulated on the manifold of full-rank quantum states D∘\mathcal D^\circD∘, with GKSL/open-system dynamics

dρξ/dξ=L(ρξ),

and, in the detailed-balance subclass, an entropic clock defined by

dEntπ(ρξ)/dξ=−σ(ρξ)≤0.

At readout level, the two-derivative gravitational sector is kept strictly Einstein,

Gμν=8πG_0/c4 Tμν_tot, with no term of the form μ(ρ) R,

The manuscript derives a local bridge between state-space holonomy and readout holonomy, together with a conditional trilemma excluding βeff≡0 on the certified window when OT non-flatness, bridge fidelity, and Einstein lock are simultaneously maintained. It also formulates reduced operational equations in which the low-energy response separates into a constitutive branch governed by

βeff(pκ):=−∂_pκ (ln⁡Λ(pκ)),

and an independent holonomic branch controlled by projected curvature. The result is a certified and falsifiable low-energy framework for testing whether preparation-dependent quantum-state structure can induce readable gravitational signatures.