Fibonacci Substitution Spectrum

Abstract

The Fibonacci substitution has two golden eigenpairs and an exact contracting error.

Theorem 1.1 (Golden eigenpairs and contracting error).

$$\forall n\in \mathbb{N},\mathrm{expandingEigenvector}\ne 0\wedge \mathrm{fibonacciSubstitution}\mathrm{expandingEigenvector}=\phi \mathrm{expandingEigenvector}\wedge \mathrm{contractingEigenvector}\ne 0\wedge \mathrm{fibonacciSubstitution}\mathrm{contractingEigenvector}=\mathrm{contractingEigenvalue}\mathrm{contractingEigenvector}\wedge (F_n\phi -F_{n+1})=-{\mathrm{contractingEigenvalue}}^n$$

Proof. Machine-checked in Lean as D5/S1/Scale/FibonacciEigen.fibonacci_substitution_spec (✓ std3). ∎

Citation. Thomas Koshy (2001). Fibonacci and Lucas Numbers with Applications. DOI: 10.1002/9781118033067.

Commentary.

The explicit substitution matrix has nonzero expanding and contracting eigenvectors, and the same theorem gives the exact signed Fibonacci error for every natural index.

References

• Truth anchor: D5/S1/Scale/FibonacciEigen.fibonacci_substitution_spec
• Dependency: D5/S0/Carrier/GoldenRatio