Fibonacci Convergent Error Ratio

Abstract

Fibonacci convergents have an exact golden residual and a limiting error ratio.

Theorem 1.1 (Exact signed golden residual).

$$\forall n\in \mathbb{N},F_n\phi -F_{n+1}=-{\left(-\frac{1}{\phi }\right)}^n.$$

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

Source. Repository-derived.

Commentary.

For every natural index, multiplying the Fibonacci denominator by the golden ratio and subtracting the next Fibonacci number gives exactly the negative n-th power of the contracting factor -1/phi.

Theorem 1.2 (Adjacent absolute-error ratio).

$$e_n=\phi -\frac{F_{n+2}}{F_{n+1}},\frac{|e_{n+1}|}{|e_n|}=\frac{F_{n+1}}{F_{n+2}}\frac{1}{\phi }.$$

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

Source. Repository-derived.

Commentary.

Let e_n be the signed error of the shifted Fibonacci convergent F_(n+2)/F_(n+1). Its adjacent absolute-error ratio is exactly the shifted ratio F_(n+1)/F_(n+2), divided by the golden ratio.

Theorem 1.3 (Limit of adjacent absolute-error ratios).

$$\underset{n\to \infty }{\lim }\frac{|e_{n+1}|}{|e_n|}=\frac{1}{{\phi }^2}.$$

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

Source. Repository-derived.

Commentary.

The adjacent absolute-error ratios of the shifted Fibonacci convergents tend to the reciprocal square of the golden ratio.

References

• Truth anchor: D5/S1/Scale/FibonacciErrorRatio.fibonacci_convergent_error_ratio
• Truth anchor: D5/S1/Scale/FibonacciErrorRatio.fibonacci_convergent_error_ratio_tendsto
• Truth anchor: D5/S1/Scale/FibonacciErrorRatio.fibonacci_golden_residual
• Dependency: D5/S1/Scale/FibonacciEigen