Lorentz Symmetry and Special Relativity from Binary Time-Flow

Proof 1: Emergence of Lorentz Symmetry from Discrete TFP Dynamics

1. Discrete Substrate and Lattice

• Sites i with binary states \(F_i \in \{+1, -1\}\)
• Adjacency graph with coordination number \(K = 12\) (icosahedral symmetry)
• Discrete proto-time tick \(\tau_0\)
• Handshake capacity \(H = 132\) per tick per site

Adjacency spacing: \(a_s\)
Maximum speed: \(c = a_s / \tau_0\)

2. Binary Time-Flow Variables

Each site i at tick n has a discrete time-flow orientation: \[ u_i[n] = (u_i[n], v_i[n], w_i[n]) \in \{\pm 1\}^3 \]

Proper time increment per tick: \(W\) (constant)

3. Continuum Limit and Coarse-Graining

Consider long wavelengths \(\lambda \gg a_s\) and long times \(T \gg \tau_0\).

Coarse-grain over sites and ticks to define local averages: \[ \phi(t, x) = \langle F_i \rangle, \quad u(t, x) = \langle u_i[n] \rangle \]

Expand discrete action \(S_\text{disc}\) in small gradients: \(\Delta x / a_s, \Delta t / \tau_0 \ll 1\)

4. Icosahedral Symmetry → Rotational Invariance

Icosahedral group is a finite subgroup of SO(3). At long wavelengths, lattice appears isotropic: correlation functions depend only on \(|x - y|\). Effective action contains only rotationally invariant terms, e.g., \((\nabla \phi)^2\).

5. Emergence of Lorentz Invariance

Maximum speed \(c = a_s / \tau_0\), locality and causality from adjacency and updates. Quadratic effective action: \[ S_\text{eff} = \int d^4x \left[ \frac{1}{2} (\partial_t \phi)^2 - \frac{c^2}{2} (\nabla \phi)^2 + \dots \right] \]

Invariant interval: \[ ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2 \]

Symmetry group: SO(1,3) (Lorentz group)

6. Time Dilation and Length Contraction

Worldline of a site: total proper time \(\tau = N W\). Average 4-velocity: \[ U_\mu = (\gamma, \gamma v / c), \quad \gamma = \frac{1}{\sqrt{1 - v^2 / c^2}} \]

Coordinate time: \(t = \gamma \tau \Rightarrow \Delta t = \gamma \Delta \tau\)

Spatial intervals: \(L = L_0 / \gamma\)

7. Continuous Velocity from Binary Flows

Instantaneous directions: \(u_i[n] \in \{\pm1\}^3\). Continuous velocity emerges from statistical averaging: \[ \langle u \rangle = \sum_{u \in \{\pm1\}^3} u P(u), \quad v = c \cdot \frac{\langle u \rangle}{\sqrt{3}} \] Magnitude satisfies \(|v| < c\)

8. Verification

Transformation between observers: \[ x'_\mu = \Lambda^\mu_\nu x_\nu \] \(\Lambda\) satisfies \(\Lambda^T \eta \Lambda = \eta\), \(\eta = \text{diag}(-1,1,1,1)\)

Propagator: \(G(k) = -k_0^2 + c^2 |k|^2\), Lorentz invariant

9. Conclusion

In the continuum limit (\(a_s \to 0, \tau_0 \to 0, c\) fixed), TFP discrete dynamics yields SO(1,3) invariance. Time dilation, length contraction, and invariant interval emerge naturally.

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Proof 2: Emergence of Special Relativity from Binary Time-Flow (XuYvZw)

1. Proper Time and 4-Velocity

Invariant proper time \(W\). 4-velocity: \[ U = \left( \frac{dt}{dW}, \frac{dx}{dW}, \frac{dy}{dW}, \frac{dz}{dW} \right) = (\gamma, \gamma v_x, \gamma v_y, \gamma v_z) \]

Express spatial components in terms of binary flows u, v, w: \[ \gamma v_x = \gamma |v_x| u, \quad \gamma v_y = \gamma |v_y| v, \quad \gamma v_z = \gamma |v_z| w \]

Here \(u,v,w \in \{\pm1\}\)

2. Invariant Magnitude

Minkowski norm: \[ U \cdot U = (dt/dW)^2 - (dx/dW)^2 - (dy/dW)^2 - (dz/dW)^2 = \gamma^2 - \gamma^2 (v_x^2 + v_y^2 + v_z^2) = \gamma^2 (1 - v^2) = 1 \]

3. Lorentz Transformations

Transformations leaving ds^2 invariant: \[ ds^2 = dt^2 - dx^2 - dy^2 - dz^2 \]

Time dilation: \(\Delta t' = \Delta t / \gamma\) Length contraction: \(L = L_0 / \gamma\)

Continuous velocities emerge via averaging over many discrete steps: \(\langle u \rangle, \langle v \rangle, \langle w \rangle \to v\) continuous

4. Emergent Relativity

Discrete XuYvZw flows produce SR-like kinematics. Proper time W is invariant across observers. 4-velocity magnitude is fixed → γ factor and Lorentz formulas emerge naturally

5. Continuum Limit

Many ticks, coarse-graining: discrete binary flows average to continuous velocities. Lorentz invariance arises statistically, not as a fundamental assumption.

Conclusion

Special relativity (time dilation, length contraction, invariant interval) emerges from the discrete binary time-flow framework XuYvZw.