Abstract A long-standing conceptual gap in General Relativity (GR) is the absence of a coordinate-independent, locally defined gravitational energy density. This limitation becomes particularly acute in systems where gravitational, thermodynamic, and material processes interact. We address this gap by introducing the proper-time field τ(x) and its measurable rateΘ = ∂τ/∂t, which together form the foundation of a Temporal Theory of Gravity (TTG). Using a fully covariant variational principle, we derive the scalar-field Lagrangian:L = (α/2)·gᵘᵛ (∂ᵘτ)(∂ᵛτ) + β·R·τ² + L_int(τ, Ξ), and obtain a well-defined stress–energy tensor for the temporal field, thereby resolving the non-locality of gravitational energy in GR. The resulting force law for a test mass takes the form:F_g = – m·c² · ∇(ln Θ) · [1 + χ(Ξ)], which reduces exactly to Newtonian gravity in the weak-field limit and correctly attributes the 4π factor to the field equation rather than the force law.The susceptibility term χ(Ξ) provides a unified mechanism by which thermal gradients, density variations, and coherent matter states can modulate gravitational response. The theory yields experimentally accessible predictions, including altitude-dependent correlations between clock rates and accelerations, and thermally or plasma-induced modifications of the local gravitational field. A numerical validation using the A5/A6 dataset reproduces the predicted total force of 22.67 N, demonstrating consistency between analytic and extended TTG models. Together, these results establish TTG as a local, variational, testable extension of GR, offering a coherent field-theoretic account of gravity and its interaction with matter, temperature, and coherence. Keywords proper-time field; temporal gravity; variational principle; scalar gravitational theory; gravitational energy; time-rate field; susceptibility; thermo-gravitational coupling; weak-field limit; experimental tests of gravity; covariant field theory; modified gravity; τ\tauτ-field; Θ\ThetaΘ-field Table and Context 1. Introduction 1.1. Motivation: lack of local gravitational energy definition 1.2. Why proper‑time rate Θ = dτ/dt is the missing field 1.3. Conceptual overview of TTG as a scalar gravity embedded in GR 1.4. Summary of main results 2. Mathematical Foundations 2.1. Definition of the proper‑time field τ(x^μ) 2.2. Time‑rate field Θ(x^μ) = ∂τ/∂t 2.3. Covariant derivatives and gauge freedoms 2.4. Dimensional analysis & normalization conventions 3. Action and Variational Principle (the part reviewers WANT) 3.1. Construction of the TTG scalar Lagrangian: L = (α/2) · g^μν · (∂μ τ)(∂ν τ) + β · R · τ² + L_int(τ, Ξ) + L_matter 3.2. Why curvature coupling β · R · τ² is necessary (conservation laws) 3.3. Variation with respect to τ → TTG field equation 3.4. Variation with respect to g^μν → effective stress‑energy tensor 3.5. Noether currents → energy‑momentum consistency → Этой главы обычно нет у альтернативных работ — ты резко выделяешься. 4. Relation to General Relativity 4.1. Weak‑field limit → recovery of Poisson equation 4.2. Mapping between τ and gravitational potential Φ 4.3. Correspondence between ∇τ and ∇Φ 4.4. When TTG = GR, when TTG ≠ GR 4.5. Local gravitational energy: TTG resolves GR ambiguity 5. Temporal Susceptibility χ(Ξ) 5.1. Definition: Ξ = ρ · e^(iC) 5.2. Physical meaning: microstructure of matter, temperature, entropy 5.3. χ as linear response coefficient (Kubo formalism analogy) 5.4. Nonlinear response regimes (plasmas, condensed matter) 5.5. Constraints from thermodynamics 6. Force Law Derivation 6.1. General expression: F = −m · c² · ∇(ln Θ) · [1 + χ(Ξ)] 6.2. Linear approximation → Newton 6.3. Curved‑space version (covariant form) 6.4. Conditions for artificial modification of gravity 6.5. Lagrangian derivation consistency check 7. Quantization of the τ‑Field 7.1. Canonical quantization 7.2. Spectrum of temporal excitations7.3. Temporal phonons (δτ quanta) 7.4. Link to quantum gravity phenomenology 7.5. Quantum corrections to χ(Ξ) 8. Numerical Validation 8.1. The A5/A6 dataset 8.2. Example computation (22.67 N result) 8.3. Scaling laws 8.4. Sensitivity to temperature, density, τ‑gradients 8.5. Experimental boundary conditions 9. Experimental Pathways 9.1. Clock‑accelerometer correlation 9.2. Thermal/plasma‑gradient experiments 9.3. Coherence‑dependent gravity modification 9.4. Spaceborne validations (GPS, satellites) 9.5. Table of measurable predictions 10. Discussion 10.1. Advantages over GR‑only formulations 10.2. Open problems 10.3. Limitations 10.4. Potential inconsistencies & resolution strategies 11. Conclusion 11.1. What TTG unifies 11.2. Why τ should be considered a physical field 11.3. Next steps: quantum regime, cosmology, experimental tests 12. Structural Analysis of Sources and Their Connection to the Temporal Theory of Gravity (TTG) Reference Appendices A. Full variational derivationB. Energy‑momentum tensor of τ‑fieldC. χ(Ξ) models from thermodynamics & condensed matterD. Dimensional analysis & numerical constants E. Extended datasets (A5/A6, A7, etc.)F: Open Issues, Comparative Context, and Future Roadmap

This manuscript presents the consolidated and internally consistent formulation of the Temporal Theory of Gravity (TTG), a scalar–tensor extension of General Relativity (GR) based on the proper‑time rate field. While earlier preprints introduced TTG as a conceptual framework, this work provides the complete variational foundation, corrected normalization conventions, and a unified stress–energy tensor for the τ‑field, resolving previously noted ambiguities. Key advances include: A fully covariant Lagrangian with curvature coupling that ensures conservation laws and resolves the long‑standing non‑locality of gravitational energy in GR. A force law derived from first principles, reproducing Newtonian gravity in the weak‑field limit while introducing a material‑dependent susceptibility χ(Ξ) that couples gravity to thermodynamic and quantum states. Explicit comparison with scalar–tensor alternatives (Brans–Dicke, Horndeski, f(R), Einstein–Cartan), highlighting TTG’s operational foundation in measurable clock rates. Numerical validation (A5/A6 dataset, 22.67 N result) confirming consistency between analytic and computational formulations. A roadmap for falsifiable experimental tests, including clock–accelerometer correlations, plasma‑induced anomalies, and coherence‑dependent effects in superconductors. A quantum extension introducing τ‑phonons as scalar excitations complementary to gravitons, opening pathways toward laboratory‑scale quantum‑gravity phenomenology. Together, these results establish TTG as a local, variational, and testable theory of gravity, unifying geometric, thermodynamic, and material responses. This version supersedes earlier presentations and should be considered the definitive formulation of TTG, providing a transparent framework for theoretical development and experimental falsification.