The golden ratio is the irrational number (1 + √5) / 2. Ratios of consecutive Fibonacci numbers approach it as the sequence grows. section 5.4
A quantum outcome-selection example uses Fibonacci structure to make binary outcomes containing consecutive 1s impossible, giving them probability 0. section 5.4
The Fibonacci connection follows from counting valid binary strings:
- For one bit, both outcomes are valid, so |G1| = 2. section 5.4
- For two bits, 00, 01, and 10 are valid, so |G2| = 3. section 5.4
- Splitting by the first bit gives the recurrence |Gn| = |Gn−1| + |Gn−2|, matching Fibonacci numbers. section 5.4
Computing the approaching ratio
The recursive sequence begins [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]. Dividing neighboring Fibonacci numbers produces ratios that move toward the golden ratio.
