The Evolution of the Golden Ratio: A Root-Free Form for Fibonacci Numbers and a Convergent Sequence for √5

Binet's integer form for the Fibonacci numbers requires the irrational constant square root of 5 and, when constants are not precomputed, a square-root operation per evaluation. This paper presents a closed form that evaluates Fn without a square root: correct for every nn tested up to 10,000, returning the same rounded integer as Binet's round encoding up to the stored-precision boundary, and 16.8% faster when the square root would otherwise be recomputed. The same form, iterated from the first Fibonacci ratio φ1=1φ1=1, evolves the golden ratio to its exact value at 1.39 bits per step and generates a sequence converging to the square root of 5 from which the Fibonacci numbers are recovered without a square root at any point.