Unified Axioms — Improved

I. Fundamental Axioms ›

Axiom 1: Complex Spacetime Metric

\(ds^2 = (c\, dt)^2 + (i\, v\, dt)^2\)

(Fundamental Metric)

Where:

Real axis: Time domain (c·dt) — wave-like behavior
Imaginary axis: Space domain (i·v·dt) — particle-like behavior
i² = -1: Complex structure
v: spatial velocity (decoherence parameter)

Axiom 2: Mass as ℏ-Coupling

\(m = \dfrac{\hbar\, \omega}{c^2} = \dfrac{\hbar}{\lambda_C\, c}\)

(Mass Definition)

Mass is not fundamental but emerges from:

Coupling strength between time and space domains
Quantized by Planck constant ℏ
Creates curvature (well) in time-space interface

Axiom 3: Interface Geometry

Particles exist at the interface between domains:

\(\Delta z(r) = \dfrac{\hbar\,\omega}{c^2}\; f\!\left(\dfrac{r}{\lambda_C}\right)\)

(Well Depth Function)

Where \(\lambda_C = \dfrac{\hbar}{m c}\) is the Compton wavelength

Typical form:

\(f(\rho) = \dfrac{e^{-\rho}}{r} \quad\text{with }\rho= r/\lambda_C\)

(Yukawa-type Potential)

II. Unified Forces ›

General Force Law

\(F = \alpha\, \dfrac{(\hbar\omega_1)(\hbar\omega_2)}{c^4\, r^2}\)

(Universal Force)

All forces share this structure with different coupling constants α:

Gravity

\(F_G = G\, \dfrac{(\hbar\omega_1/c^2)(\hbar\omega_2/c^2)}{r^2} = G\,\dfrac{m_1 m_2}{r^2}\)

(Gravitational Force)

\(\alpha_G = G/c^4 \approx 8.26 \times 10^{-45}\;\mathrm{m^2/J}\)

(Coupling Constant)

Electromagnetism

\(F_{EM} = k\dfrac{q_1 q_2}{r^2} \Big[1 + i\,\beta(T)\dfrac{\mathbf{v}_1\cdot\mathbf{v}_2}{c^2}\Big]\)

(Complex EM Force)

Where:

Real part: Electric (Coulomb) force
Imaginary part: Magnetic (drag) force
\(\beta(T)\): Temperature-dependent wake parameter

\(\beta(T) \sim \sqrt{\dfrac{\mu_0(T)}{\epsilon_0(T)}}\)

(Vacuum Properties)

At low T: \(\epsilon_0\downarrow,\; \mu_0\uparrow\) → narrow wakes, laminar flow (superconductivity)

Nuclear Forces

Not separate — just ultra-deep gravitational wells at \(r\sim\lambda_C\):

\(F_{nuclear} \sim G\dfrac{m_p^2}{r^2}\quad\text{at } r\approx 10^{-15}\,\mathrm{m}\)

(Nuclear Binding)

Appears strong due to extreme curvature at nuclear scales

III. Time Dilation (Unified) ›

Optical Refraction Model

Time dilation emerges from refraction of "time ticks" through curved interface:

\( \frac{d\tau}{dt} = \sqrt{1 - \frac{v^2}{c^2} - \frac{2\hbar\omega}{c^3 r}} \)

(Combined Time Dilation)

This unifies:

Kinetic term: \( \frac{v^2}{c^2} \) (Special Relativity)
Gravitational term: \( \frac{2\hbar\omega}{c^3 r} = \frac{2Gm}{r c^2} \) (General Relativity)

Mass-Dependent Prediction

Unlike standard GR, proper time depends on particle's own mass:

\( \frac{d\tau_e}{dt} = \sqrt{1 - \frac{v^2}{c^2} - \frac{2Gm_e}{r c^2}} \)

(Electron)

\( \frac{d\tau_p}{dt} = \sqrt{1 - \frac{v^2}{c^2} - \frac{2Gm_p}{r c^2}} \)

(Proton)

Testable prediction: Proton experiences ×1836 stronger self-gravitational dilation!

Relativistic Mass Enhancement

\( m_{\text{eff}}(v) = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} = m_0 \, \gamma(v) \)

(Moving Particle Well Depth)

Well deepens with velocity → additional gravitational dilation

IV. Quantum Mechanics ›

Wave-Particle Duality

Not mysterious — location in complex plane:

Time domain (\( ds^2 = (c\,dt)^2 \)): Pure wave, \( \omega \) oscillations
Space domain (\( ds^2 \) includes \( (i\,v\,dt)^2 \)): Particle, localized
Interface: Superposition of both

De Broglie Relations

\( \lambda = \frac{h}{m v} = \frac{2\pi\hbar}{m v} \)

(Wavelength)

\( \omega = \frac{m c^2}{\hbar} \)

(Frequency)

Atomic Structure

Electrons in orbitals sit at different depths in nuclear well:

\( m_{\text{eff}}(n) = m_e + \Delta m_{\text{grav}}(r_n) \)

(Effective Mass in Orbital)

\( \Delta m_{\text{grav}}(r_n) \approx -m_e \frac{G m_{\text{nucleus}}}{r_n c^2} \)

(Gravitational Correction)

Spectral lines encode ℏ-coupling changes, not just energy!

Quantum Tunneling

Deep in wells, particles become wave-like in time domain:

Barrier opaque in space → transparent in time
Wave propagates through time domain
Recoheres on other side (resonance cavity effect)

\( P \sim e^{-2 \int \sqrt{\frac{2m(V-E)}{\hbar^2}} \, dr} \rightarrow \text{Phase integral in time domain} \)

(Tunneling Probability)

V. Particle Statistics ›

Bosons vs Fermions

Bosons (Time-Domain Natives)

\( ds^2 = (c\,dt)^2, \; v = 0, \; m \approx 0 \)

No ℏ-coupling wells (or very weak)
Integer spin: \( s = 0, 1, 2, \ldots \)
No Pauli exclusion (share time-domain states)
Examples: photons, gluons, W/Z

Fermions (Space-Domain Trapped)

\( ds^2 = (c\,dt)^2 + (i\,v\,dt)^2, \; m > 0 \)

Have ℏ-coupling wells
Half-integer spin: \( s = \frac{1}{2}, \frac{3}{2}, \ldots \)
Pauli exclusion (can't share spatial states)
Examples: electrons, quarks, neutrinos

Bose-Einstein Condensation

At \( T \to 0 \), fermions recohere into time domain:

\( k_B T_c \sim \hbar \omega_0 \sim m c^2 \)

(Critical Temperature)

\( v \to 0, \; ds^2 \to (c\,dt)^2 \)

(Below \( T_c \))

Spin as Interface Geometry

\( \hat{S} = \hbar \frac{\partial}{\partial \phi} \)

(Spin Operator)

Integer spin: Full symmetry, \( 2\pi \to \) same state
Half-integer: Twisted coupling, \( 4\pi \to \) same state

VI. Cosmology ›

Hubble Expansion

\( H_0 \sim (\text{decoherence rate}) \cdot (\text{interface curvature}) \)

(Hubble Parameter)

Dark Energy

\( \rho_\Lambda = \rho_{\text{time}} \, P(\text{decohere}) \sim 10^{-120} \, \rho_{\text{Planck}} \)

(Vacuum Energy Density)

Hawking Radiation

\( k_B T_H = \frac{\hbar c^3}{8\pi G M} \)

(Temperature)

VII. Experimental Predictions ›

1. Mass-Dependent Time Dilation

\( \Delta \tau_p - \Delta \tau_e \sim \frac{(m_p - m_e) G v^2 t}{r c^2} \)

2. Atomic Spectral Corrections

\( h\nu = \Delta E + c^2 \Delta m_{\text{coupling}} \)

3. Temperature-Dependent EM Constants

\( \varepsilon_0(T), \, \mu_0(T), \, c(T) = \frac{1}{\sqrt{\varepsilon_0(T)\mu_0(T)}} \)

4. BEC Phase Transition

\( k_B T_c = \frac{\hbar^2}{2m} \left( \frac{n}{\zeta(3/2)} \right)^{2/3} \)

IX. Compatibility with Established Theories ›

A. Schrödinger Equation

\( i\hbar \frac{\partial \psi}{\partial t} = \hat{H}\psi = -\frac{\hbar^2}{2m}\nabla^2\psi + V\psi \)

\( \psi = \psi_{\text{time}} + i\,\psi_{\text{space}} \)

B. Dirac Equation

\( (i\gamma^\mu \partial_\mu - m)\psi = 0 \)

\( \eta^{\mu\nu} = \text{diag}(-1, +1, +1, +1) \)

\( ds^2 = (c\,dt)^2 + (i\,dx)^2 + (i\,dy)^2 + (i\,dz)^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2 \)

C. Klein-Gordon Equation

\( \left( \frac{\partial^2}{\partial t^2} - c^2\nabla^2 + \frac{m^2 c^4}{\hbar^2} \right)\psi = 0 \)

D. Quantum Field Theory (QFT)

\( \hat{\phi}(x,t) = \int [a(k)e^{i(kx-\omega t)} + a^\dagger(k)e^{-i(kx-\omega t)}]\,d^3k \)

\( E_{\text{vacuum}} = \sum \frac{\hbar\omega_k}{2} \)

(Vacuum Energy)