# Structural Uncertainty & Interference

TNFR demonstrates that "Quantum Weirdness" (Uncertainty, Duality) is simply the behavior of coherent wave-packets in the Nodal Dynamics framework.

## 1. The Structural Uncertainty Principle

In Quantum Mechanics, $\Delta x \cdot \Delta p \ge \hbar/2$.
In TNFR, this is a property of the **EPI (Form) vs. Frequency ($\nu_f$)** relationship.

- **EPI** is a structural configuration (Form).
- **$\nu_f$** is the rate of reorganization (Frequency).

To define a precise "Frequency", a pattern must repeat for a long time (High $\Delta t$).
To define a precise "Time/Position", a pattern must be localized (Low $\Delta t$).

**Derivation**:
Since EPI evolution is driven by $\nu_f$, the spectral analysis of the EPI trajectory reveals that:
$$ \Delta t_{EPI} \cdot \Delta \nu_f \ge K $$
This is not a physical law imposed from outside, but a mathematical necessity of signal processing (Gabor Limit) applied to structural information.

## 2. Structural Interference (The Double Slit)

When a TNFR node emits a structural pattern (Operator `AL` + `RA`), it propagates as a phase wave through the network.

- **Wave Nature**: The phase coupling ($\phi_i \to \phi_j$) creates wavefronts.
- **Interference**: When wavefronts from two paths (e.g., two "slit" nodes) meet, they add vectorially (Complex Structural Field $\Psi$).
    - **Constructive**: $\Delta \phi \approx 0$. Coherence increases.
    - **Destructive**: $\Delta \phi \approx \pi$. Coherence cancels out (Dissonance).

This produces interference fringes in the "Sense Index" ($Si$) or Coherence ($C(t)$) of the receiving nodes, reproducing the double-slit experiment without assuming particles are waves. They *are* structural waves.

## Visual Proofs

The script `examples/14_uncertainty_and_interference.py` demonstrates:

- `results/quantum_demo_2/01_uncertainty_principle.png`: Shows that as the Structural Packet width ($\sigma_t$) increases, the Frequency Spectrum width ($\sigma_f$) decreases, maintaining a constant product $\sigma_t \cdot \sigma_f \approx 0.16$.
- `results/quantum_demo_2/02_interference_pattern.png`: Displays the 2D wave field and the accumulated intensity, revealing clear interference fringes behind the double slit.
- `results/quantum_demo_2/03_interference_profile.png`: The intensity profile at the screen shows the characteristic peaks and troughs of wave interference.

These results confirm that **Uncertainty and Interference are intrinsic properties of Nodal Dynamics**, emerging from the wave-like nature of structural propagation and the Fourier limits of information processing.

