Golden Fibonacci Approximation Constant

Abstract

Fibonacci approximants attain the reciprocal square-root-five scaled-error limit.

Theorem 1.1 (Scaled Fibonacci approximation errors tend to one over square root five).

$$\underset{n\to \infty }{\lim }{F_n}^2|\phi -\frac{F_{n+1}}{F_n}|=\frac{1}{\sqrt{5}}.$$

Proof. Machine-checked in Lean as D5/S3/AnalyticClosure/GoldenApproximationConstant.golden_fibonacci_approximation_constant_tendsto (✓ std3). ∎

Source. Repository-derived.

Commentary.

For the consecutive Fibonacci approximant F_(n+1)/F_n, multiply the absolute golden-ratio error by the square of its denominator. Once F_n is positive, clearing that denominator identifies this expression with the existing scaled Fibonacci residual score. Its established limit therefore gives exactly 1/sqrt(5).

This closes only the asymptotic constant along the Fibonacci convergents. Global optimality, the first two levels of the approximation spectrum, and the semantic uniqueness claim remain unresolved.

References

• Truth anchor: D5/S3/AnalyticClosure/GoldenApproximationConstant.golden_fibonacci_approximation_constant_tendsto
• Dependency: D5/S3/ObserverMemory/GoldenRevivalScore