Golden Fibonacci Revival Score

Abstract

Fibonacci golden return scores converge to the sharp quadratic-irrational constant.

Theorem 1.1 (Fibonacci revival scores tend to one over square root five).

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

Proof. Machine-checked in Lean as D5/S3/ObserverMemory/GoldenRevivalScore.golden_fibonacci_revival_score_tendsto (✓ std3). ∎

Source. Repository-derived.

Commentary.

At the Fibonacci return time F_n, the scaled golden return error is F_n times the absolute difference between F_n times the golden ratio and F_(n+1). Binet’s formula and the exact contracting residual reduce this score to a geometric correction of 1/sqrt(5), whose correction vanishes. This closes only the Fibonacci extremal subsequence; the full spectrum classification and global optimality remain unresolved.

References

• Truth anchor: D5/S3/ObserverMemory/GoldenRevivalScore.golden_fibonacci_revival_score_tendsto
• Dependency: D5/S1/Scale/FibonacciErrorRatio