From Optimal Transport (Figalli) to Heat-Kernel EFT: An OT–SDW Bridge with Einstein-Locked Gravity and Gauge Connections from a4

This document is an architecture/specification note that makes explicit, in an operational and unambiguous way, how an Optimal-Transport (OT) layer (in the spirit of dynamic OT and its modern mathematical formulation associated with the Monge–Ampère viewpoint and the works of Figalli) interfaces with a local Effective Field Theory (EFT) organized by the heat-kernel / Seeley–DeWitt (SDW) expansion. On the OT side, a prepared family of source states is indexed by a protocol-defined proxy PopP, mapped monotonically to a normalized state parameter pk=h(Pop)∈[0,1]. OT also provides regime control: GKSL-compatible dissipation is bounded and a precise adiabaticity condition ensures that pk(x,t) can be treated as locally frozen during measurements. On the SDW side, the EFT is organized by coefficients (a0,a2,a4,… ) of a Laplace-type operator H. The central rule is the Einstein lock in d=4: the Einstein–Hilbert prefactor is fixed by the a2 block and is state-independent (G0 remains universal and constant). State-dependence is therefore confined to (i) a source-only reweighting μeff(pk)=1/Φeff and/or (ii) slow pk-dependent Wilson coefficients multiplying dimension-4 operators carried by a4. A key outcome is a clean gauge story across layers: “gauges” in OT are dual multipliers enforcing admissible currents, while gauge fields in EFT are genuine connections in ∇μ; their kinetic term Tr(FμνFμν) appears necessarily at order a4 via Ωμν, not at order a2. The document closes with a minimal testing pipeline translating differential state modulations δpk into bounds on ∂pkln⁡μeff and on pk derivatives of a4-level couplings. A companion glossary documenting terminology and structural conventions is available at DOI: 10.5281/zenodo.18526966. and fundations:10.5281/zenodo.18644600 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. 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.