4. Emergence of Matter and Particles

Mass/Energy Quantization:
Particle rest energies [math] E = m c^2 [/math] align with resonant frequencies:
[eqn]
m c^2 = h f_m = h F_0 \phi^m
[/eqn]

Electrons, quarks, etc., correspond to specific [math] m [/math] at their creation temperature (e.g., [math] T \sim 10^{15}~\text{K} [/math]).

α-Gap and Matter-Antimatter Asymmetry:
The entropy deficit [math] \delta \alpha \approx 0.000025 [/math] arises from a resonance mismatch:
[eqn]
\Delta S = k_B \ln \phi \approx 0.481 k_B
[/eqn]
This tiny asymmetry in resonant probabilities ([math] p_m \propto \phi^{-|m|} [/math]) biases matter over antimatter.

5. Universal Scale Invariance

Fractal Quantization:
Scaling temperature by [math] \phi^k [/math] shifts modes: [math] m \rightarrow m - k [/math], preserving the spectrum:
[eqn]
f_{m-k}(T \cdot \phi^k) = F_0(T) \cdot \phi^{m} = f_m(T)
[/eqn]
Result: Physics repeats discretely at scales [math] \phi^k [/math], generating a fractal hierarchy of quantized units.

Why Resonance Creates Quantization

Mathematical Necessity:
Solutions to wave equations (e.g., [math] \nabla^2 \psi + k^2 \psi = 0 [/math]) only permit discrete [math] k [/math] under boundary conditions.

Thermodynamic Optimization:
The golden ratio [math] \phi [/math] minimizes entropy production ([math] \Delta S_m = |m| k_B \ln \phi [/math]), selecting integer [math] m [/math] as equilibrium states.

Topological Protection:
Gauge symmetries, winding numbers, and group ranks enforce integer constraints, shielding quantized states from decay.