magicmaslov.com · Wand 2 · Geometric Harmony

The Art of Protection

"The eye recognizes a protected state before the mind can calculate it — and both are right for the same reason."

Three oscillating spins are assigned to the three circles of a torus. Their combined motion traces a curve — whether it closes into a clean mandala or wanders restlessly is determined by a single number: the Maslov 𝓜, the Greatest Common Divisor of the pairwise spin differences. High 𝓜 means topological protection — the geometry snaps shut. Low 𝓜 means topological frustration — the curve searches without finding rest.

The permutation panel shows all orderings of your spins. Arithmetically, all permutations of a protected set share the same 𝓜 — the GCD of differences is invariant under reordering. But on the physical three-torus with unequal radii (r₁=r, r₂=1.2r, r₃=0.8r), the assignment of spins to geometric axes shifts the stability window. Configurations like (1,2,3) are protected over almost the entire parameter range, while permutations like (2,1,3) show narrower windows. The Chernoff Point Conjecture (Paper 2) predicts (1,2,3) is maximally stable — it sits at the centroid of the spin simplex, the most symmetric point under S₃.

⚠ Some rapidly cycling combinations may be uncomfortable for light-sensitive eyes.

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Maslov 𝓜

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Spin differences |s₂−s₁| · |s₃−s₁| · |s₃−s₂| — · — · —

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Closure Cycles

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Adjust the three spins and watch harmony emerge — or fail to.

Three Spins — Torus Assignment

Each spin is assigned to one torus. On the asymmetric torus (r₁=r, r₂=1.2r, r₃=0.8r), which spin occupies which torus changes the stability window even when 𝓜 is the same.

Torus 1

Torus 2

Torus 3

All Permutations

All permutations share the same 𝓜 arithmetically. But on the geometric torus, each ordering is a different physical assignment. Teal marks the natural ascending order — the Chernoff Point prediction.

Preset Gallery

Includes the Chernoff Point (1,2,3) and deliberate permutation pairs to show how geometric assignment changes visual character at equal 𝓜.

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The Chernoff Point Conjecture (Paper 2):
In the GCD oscillator model, three spins sit on three tori with unequal radii r₁=r, r₂=1.2r, r₃=0.8r. Swapping which spin occupies which torus changes the coupling geometry — even when 𝓜 is unchanged. Numerical sweeps show (1,2,3) is protected over almost the entire parameter range, while permutations like (2,1,3) show narrower stability windows or fragility.

(1,2,3) is the Chernoff point on the spin simplex — the centroid of all permutations under S₃, maximally equidistant from every frustrated boundary. The natural ascending assignment is not arbitrary: it is the most symmetric, most protected geometric configuration. This nuance is noted in Paper 1 and fully proven in Paper 2.