Golden Jones Matching

Abstract

The golden square is the first nonintegral low Jones value and the Fibonacci dimension.

Theorem 1.1 (The golden square is the first nonintegral low Jones value).

$l e t J (n) = 4 \cdot cos (\frac{π}{n})^{2}; ((((((((((J (3) = 1 \land J (4) = 2) \land J (5) = ϕ^{2}) \land J (6) = 3) \land ϕ^{2} = \frac{3 + sqrt ( 5 )}{2}) \land 2 < ϕ^{2}) \land ϕ^{2} < 3) \land \frac{13}{5} < ϕ^{2}) \land ϕ^{2} < \frac{131}{50}) \land (\forall n \in Natural (), (3 \leq n \land n < 5) \Rightarrow (\exists m \in Integer (), J (n) = m))) \land (\neg (\exists m \in Integer (), J (5) = m))) \land (\forall d \in Real (), (0 < d \land d^{2} = 1 + d) \Rightarrow (d = ϕ \land d^{2} = ϕ^{2}))$

Proof. Machine-checked in Lean as D5/S3/Observer/GoldenCoding/GoldenJonesMatching.golden_jones_matching (✓ std3). ∎

Source. Repository-derived.

Commentary.

For J(n)=4 cos^2(pi/n), the four explicit values at n=3,4,5,6 are 1, 2, phi^2, and 3. The central identity uses the repository’s pentagon cosine theorem rather than reproving the special value.

The radical identity phi^2=(3+sqrt(5))/2 yields both 2<phi^2<3 and the sharper enclosure 2.6<phi^2<2.62. Irrationality of phi then shows that J(5) is not an integer, while the only earlier indices n>=3 are n=3 and n=4 and have explicit integer witnesses.

The source also describes the self-dual Fibonacci fusion rule. Because the source supplies no category, tensor product, unit, or dimension map, the formal statement records its exact decategorified numerical consequence: every positive d satisfying d^2=1+d equals phi and has squared dimension phi^2.

References

Truth anchor: D5/S3/Observer/GoldenCoding/GoldenJonesMatching.golden_jones_matching
Dependency: D5/S1/FixedPoints/Algebraic/GoldenFixedPoint
Dependency: D5/S3/Constants/PentagonCosines

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Golden Jones Matching

Abstract

References