Optimal Transport, Emergent Spacetime, and Source-Only Gravity

/// The version below is outdated. Please read: Foundations //// Optimal Transport, Emergent Spacetime, and Source-Only Gravity - Information-Geometric Foundations and the Seeley–DeWitt Bridge This work proposes a unified theoretical framework in which spacetime geometry and gravitational dynamics emerge from the information-geometric structure of quantum states, rather than being postulated at the fundamental level. The central mathematical tool is optimal transport, extended to quantum states (OTq) and combined with dissipative quantum dynamics of GKSL type. In this setting, dissipation is formulated as a gradient flow of relative entropy on the manifold of density operators, providing an intrinsic arrow of time prior to spacetime emergence. A key result is a structural separation between propagation and sourcing. Using spectral methods and the Seeley–DeWitt expansion, the Einstein–Hilbert term is shown to be universally fixed by the a2 coefficient, while all state-dependent effects enter only through higher-order terms. This leads to a source-only modification of gravity, in which state-dependent factors reweight gravitational sources without renormalizing the gravitational coupling or altering causal propagation. The framework is non-perturbative, background-independent at the pre-geometric level, and naturally incorporates gauge connections, dissipative flows, and entropic curvature bounds (EVI/mLSI). It provides a coherent bridge between quantum information geometry, spectral geometry, and emergent gravitational dynamics, and opens new directions for studying spacetime emergence, relaxation phenomena, and testable deviations in weak-field gravity. Advisory. This manuscript is part of a testable certified-domain OT/GKSL architecture organized in distinct layers: native dynamics, certified readout, Einstein-locked nonlinear closure, and controlled recoveries. The Einstein kinetic sector remains locked, Bianchi-compatible closure is enforced, and readable state dependence is confined to the source/response sector. It should be read as one structured component of a closed operational framework, not as a standalone modified-gravity model. 1. Foundations of the Architecture Foundations | Establishes the core Einstein-locked OT/GKSL architecture for certified geometric readout and coherence-dependent gravitational sourcing. 2. Emergence and Recovery of Classical Physics Exact Reduced OT/GKSL Equations | Demonstrates the controlled recovery of classical Newtonian and gravitational sectors as exact non-linear reductions of the native OT/GKSL state dynamics. Certified Einstein Non-Linear Readout | Develops the full non-linear Einstein-locked readout closure for the metric sector. Non-Linear Dynamics and Readout | Explores the exact reduced non-linear evolution on collective state manifolds. The Seeley–DeWitt Bridge | Formalizes the operational connection between native state dynamics and the effective classical readout. The SDW Bridge: Composite Brout–Englert–Higgs Dynamics, Spectral Separation, and the Emergent Graviton | Formalizes the emergence of the Brout-Englert-Higgs composite scalar and the spin-2 graviton via the Seeley-DeWitt expansion, strictly preserving the Einstein-Lock. Bridge between QCD and OT/GKSL Readout | Connects the Optimal Transport / GKSL framework to Quantum Chromodynamics, exploring the constitutive bridge and effective low-energy dynamics. 3. The Certified Boundary and Structural Limits Certified Spacetime Readout on Finite Support: A Unified Temporal and Geometric Boundary | Unifies the temporal and geometric branches of classical readout into a single certified spacetime problem. Introduces the unified spacetime readout burden and derives the central unified certified-budget inequality, proving that temporal precision, geometric coframe nondegeneracy, and bridge compatibility draw from the same finite entropic and informational resources and cannot be made simultaneously ideal. Certified Causality, Locality, Nonlocality, and Relativity in the Einstein-Locked OT/GKSL Framework | Determines the exact status of causality, locality, nonlocality, and the principle of relativity within the Einstein-locked OT/GKSL architecture. Shows that causal-local spacetime semantics is a certified readout property rather than a primitive native axiom; proves a patchwise gluing theorem for certified local causal structure; and derives a unified finite-budget inequality showing that temporal precision, geometric certification, bridge admissibility, and overlap compatibility all compete for a single residual causal-local headroom on finite effective support. Entropic Tick Cost and Certified Temporal Readout in the Einstein-Locked OT/GKSL Framework | Demonstrates that classical ticks are finite-resource readout objects extracted from native entropic ordering, rather than primitive background parameters. Decomposes the entropic tick cost into native, extraction, and certification branches, and derives a theorem-level certified temporal budget inequality connecting temporal resolution, finite effective support, and certification margins. Entropic Tick Cost & Spectral Budget | Establishes a theorem-strength certified boundary for classical spacetime by proving a fundamental trade-off between entropic tick resolution, coframe stability, and finite informational budget. Optimal-Transport Gravity Trilemma | Identifies the certified operational boundary of geometric readout by proving the fundamental trade-off between temporal resolution, coframe stability, and bridge fidelity. Toy Certified Pipeline from Optimal Transport QCD | Provides a protocol-level implementation and scaling model for certified bridge margins. 4. Cosmological Dynamics & Global Readout Constraints Vacuum-like Residual Energy from Constitutive-Holonomic Balance in a Minimal Reduced OT-C3 Sector | Demonstrates analytically that the macroscopic cosmological constant emerges as a non-zero vacuum-like residual energy resulting from the exact balance between scalar constitutive dissipation (source sector) and the non-commutative holonomic barrier of the Optimal Transport geometry. Homogeneous Closed Readout Dynamics under Finite Spacetime Budget | Constructs a homogeneous and isotropic model (G-FLRW) demonstrating how the spacetime budget acts as a branch-selection mechanism, effectively identifying the vacuum-like sector (Λ) as the maintenance cost of certified spacetime solvability. 5. Experimental Protocols and Testability Testing Source-Side State Dependence in Gravity with Lock-In Atom Interferometry | Proposes a concrete experimental protocol to falsify source-only emergent gravity at low energy. A Lock-in Atom-Interferometric Test (Clock) | Detailed operational implementation of the low-energy readout test for the Einstein-locked framework.