This user guide is designed to help navigate the Wand 2 interface, bridging the gap between artistic exploration and the rigorous mathematics of the GCD Oscillator Model.

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1. The Core Philosophy

In this system, Geometry = Stability. We use a 2D Spirograph projection to show how three independent “spins” (frequencies) either conflict or cooperate.

• The Goal: Find the “Natural Order” where the path closes perfectly.
• The Result: A state of Topological Protection where the system is shielded from noise.

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2. The Controls: Setting Your Spins

The Spins Grid

Enter three integers ($s_1, s_2, s_3$). These represent the “gears” of your system.

• Small Integers (e.g., 1, 2, 3): Create simple, elegant loops. These are “Identity” states—the most fundamental but also the most “fragile” (thinnest) protection.
• Tesla Ratios (e.g., 3, 6, 9): These create high-density, symmetric patterns. Because their GCD is 3, they are 3x more structurally sound than the 1-2-3 baseline.

The Permutation Panel ($S_3$ Group)

There are 6 ways to assign three spins to the three axes of a torus.

• Even with the same numbers, order matters.
• Look for the Chernoff Point (usually the natural ascending order like $1 \to 2 \to 3$). This is the “self-bracing arch” where the magnetic moments of the frequencies nestle together with zero friction.

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3. The Dashboard: Reading the “Magic”

Maslov Index ($\mathcal{M}$)

This is the “Greatest Common Divisor” of the differences between your spins.

• If $\mathcal{M} \ge 2$: The system is “Highly Protected.” The geometry is reinforced.
• If $\mathcal{M} = 1$: The system is “Baseline Protected.” It is stable but thin.

Closure Cycles

This tells you how many rotations the “pen” makes before the pattern perfectly overlaps.

• Low Cycles: High efficiency. The system “locks” quickly.
• High Cycles: Low efficiency. The system is “searching” for resonance, requiring more energy (coupling) to stay stable.

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4. The Visual Feedback

The “Closure” Effect

As you adjust your spins, watch the canvas.

• A Protected State: The lines meet up exactly. This is Zero-Framing ($w=0$). In meditation, this represents deep “Flow.” In quantum memory, this represents “Error-Free” storage.
• A Frustrated State: The lines don’t meet, or they create a “messy” overlap. This requires a higher “Coupling” (effort) to fix.

Line Thickness and Glow

The app visually rewards High-GCD states. When you hit a “Tesla-class” resonance (like $3-6-9$), the pattern will appear “bolder” and more symmetric. This represents a thickening of the topological walls protecting the system.

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5. Designing with Harmonics: A Practical Walkthrough

1. Start with the Breath: Input a simple ratio like 1, 2, 3. Observe the simple loop.
2. Find the Resonance: Move to 3, 6, 9. Notice how the pattern gains “mass” and symmetry. This is a “Strong Protection” state.
3. Test the Permutations: Click through the $S_3$ buttons. Notice how some “wobble” the pattern while others (the identity) make it snap into focus.
4. The “Tesla” Challenge: Try to find other high-GCD combinations (like 2, 4, 6 or 5, 10, 15). These are your “Hardware-Level” stable states.

Pro Tip: In the 2D view, look for Symmetry. If the pattern looks like a balanced flower or a star, you have found a Zero-Framing state where the energy cost of maintaining the system is at its absolute minimum.

They do not need your equations. The geometry is the proof.