The Seeley–DeWitt Bridge: From Nambu--Jona-Lasinio Dynamics to the Einstein--Hilbert Action - V1.0

/// The version below is outdated. Please read: Foundations //// Starting from a microscopic NJL-type fermionic theory, we perform a Hubbard–Stratonovich transformation to introduce auxiliary bosonic fields and rewrite the four-fermion interactions. This yields an effective Dirac operator in a background that encodes the relevant condensates and mean fields. The square of this Dirac operator admits a Lichnerowicz–type decomposition, which puts it in Laplace form and makes the underlying geometric structure explicit. Using the Schwinger proper-time / heat-kernel representation of the resulting Laplace-type operator, we obtain a systematic Seeley–DeWitt expansion, whose coefficients {a0,a2,a4,a6,… } encode the effective action and, in particular, the emergent Einstein–Hilbert and higher-curvature terms. The SDW projection can be developed in arbitrary dimension n. In practice, the micro–macro dictionary and metrological anchoring constrain n; empirical consistency of the induced Einstein–Hilbert sector and its operator hierarchy selects n=4 at the classicalized (projection) layer. The scope is deliberately bounded: we address emergent gravitational geometry from open quantum matter dynamics, not the ab initio origin of the full Standard Model spectrum. 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.