Published November 6, 2026
| Version v18
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Time Field Theory (TFT): Axiomatic Foundation, Dynamics and Classical Gravitational Tests — Version 2.0 (Perfect Circle Postulate and Periodic Self-Consistency Ontology)
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Description
### Version 2.1 Update Notes (November 2026)
This is a minor revised version of the TFT foundational monograph. All quantitative results of classical gravitational tests (gravitational time dilation, light deflection, Mercury perihelion precession, GPS clock correction) remain fully unchanged. The revision focuses on improving the rigor of derivation, purifying ontological expression, and strengthening the internal logical unity of the theoretical framework.
#### Changelog
1. **Reconstructed the gravitational light deflection derivation (Section 8)**
Removed the equivalent refractive index formulation entirely. The deflection angle is now derived strictly from first principles via path integration of the transverse gradient of the γ-field metric benchmark. It is explicitly demonstrated that the 1/γ² weight factor arises naturally from the simultaneous double metric scaling of both the local speed of light and the spatial length benchmark, which explains why early precursor calculations obtained only half of the correct deflection value.
2. **Revised the strong-field falsifiability statement (Section 8.5)**
Clarified the mathematical isomorphism of static spherically symmetric null geodesics between TFT and the Schwarzschild solution. The core falsifiable predictions in strong-field regimes are relocated to compact object phenomenology (gravitational wave ringdown echoes, black hole shadow edge structure, multi-body gravitational superposition behavior) instead of the simple light deflection angle.
3. **Strengthened the logical unity of the TFT system**
Supplemented explicit correspondences: the double γ-scaling in light deflection, the γ² denominator in the velocity composition formula, and the global flux-conservation law all share the same monistic metric ontology. All kinematic and gravitational effects are uniformly traced back to the self-consistent distribution of the γ-field.
This paper presents Version 2.0 of Time Field Theory (TFT), which reconstructs the entire axiomatic foundation from the kinematic level to the ontological level of periodic self-consistency and the perfect circle postulate. Starting from three core axioms of stable periodic oscillation (constant modulus, periodic closure, intrinsic dynamics), we rigorously derive the scalar time field definition, the spacetime periodicity law (flux conservation), the kinematic velocity decomposition theorem, the exact static spherically symmetric solution, and the particle dynamical equations, without additional ad hoc assumptions.
All classical gravitational test results, including gravitational time dilation, gravitational light deflection, Mercury's perihelion precession and GPS clock corrections, remain fully consistent with observational data and are identical to Version 1.0. This upgrade deepens the first principle of the theory while preserving all verified physical conclusions, and forms a complete logical closed loop with the TFT metretical program and the endogenous derivation of Riemannian geometry. The theory provides a natural ontological bridge from classical gravitation to quantum mechanics via the Fourier synthesis principle of periodic oscillations.
This work is No.1 in the TFT Series.
### Version 1.1 Update Notes (November 2026)
This is a critical revised version of the redshift monograph. The core ontological framework (unified redshift mechanism, conformal energy conservation, flat spacetime interpretation) remains fully unchanged. The revision fixes multiple derivation errors and logical gaps that would affect the physical conclusion, and aligns all definitions and terminology strictly with the TFT foundational monograph metric system.
#### Changelog
1. **Corrected the frequency scaling relation and replaced the core invariant theorem**
The original text confused coordinate frequency and local proper frequency, and incorrectly concluded that local frequency is proportional to $\gamma$. It is now strictly derived from the fundamental definition of proper time $d\tau=\gamma dt$: the locally measured proper frequency is inversely proportional to the local time-field strength $\gamma$.
Correspondingly, the original "wavelength invariance theorem" is replaced by the **conformal energy conservation theorem** as the core invariant of photon propagation, which completely eliminates the logical contradiction and is fully self-consistent with the metric system of the TFT main paper.
2. **Corrected the cosmic evolution direction of $\gamma(t)$ and the sign of the Hubble parameter**
The original text incorrectly inferred that $\gamma(t)$ decreases monotonically with cosmic time. It is now revised to: the global time field $\gamma(t)$ increases monotonically and accelerates with cosmic coordinate time (clocks ran slower in the early universe and run faster at present), which naturally produces the observed cosmological redshift.
The sign error in the Hubble parameter formula is fixed: the negative sign is removed, and $H(t)=\dot{\gamma}/\gamma$ is adopted to ensure a positive value, consistent with astronomical observation conventions.
3. **Fixed the logical gap between the static self-calibration axiom and the dynamic sourced field equation**
Clarified the applicable levels of Axiom 4 and Axiom 5: Axiom 4 is the self-referential calibration constraint of the pure metric field in the absence of external matter; Axiom 5 is the dynamic equation driven by macroscopic matter sources. The two are not in a degenerate relationship.
It is explicitly stated that in the cosmological derivation of this paper, $\rho(t)$ is treated as the large-scale averaged independent macroscopic matter source, eliminating the mathematical inconsistency.
4. **Improved terminology and expression rigor**
- Corrected the calculus terminology error: revised "concave function" to "convex function" to match the definition of positive second derivative.
- Clearly distinguished between "coordinate frequency (globally conserved)" and "local proper frequency (locally measured)" to eliminate conceptual ambiguity.
- Supplemented the physical definition of the Hubble parameter to clarify its correspondence with the apparent cosmic expansion rate.
All core physical conclusions remain unchanged: both gravitational and cosmological redshift are pure measurement effects caused by differences in local time benchmarks; no spacetime curvature, no spatial expansion, and no dark energy entity are required. After revision, the entire derivation chain is strictly self-consistent and fully matches the TFT theoretical system.
### Citation Note
When citing this work, please specify the version number:
Huang, H. (2026). Time-Field Theory: Unified Derivation of Gravitational Redshift and Cosmological Redshift from Conformal Energy Conservation (Version 1.1).