Exercise 3: Fibonacci Pair vs. Non-Fibonacci Pair Step Counts — Possible Solution
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GIVEN
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Consecutive Fibonacci pair: gcd(55, 34)
Non-Fibonacci pair, similar size: gcd(55, 21)

TRACE: GCD(55, 34)
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55 = 34*1 + 21
34 = 21*1 + 13
21 = 13*1 + 8
13 = 8*1 + 5
8  = 5*1 + 3
5  = 3*1 + 2
3  = 2*1 + 1
2  = 1*2 + 0

GCD = 1. Total steps: 8

TRACE: GCD(55, 21)
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55 = 21*2 + 13
21 = 13*1 + 8
13 = 8*1 + 5
8  = 5*1 + 3
5  = 3*1 + 2
3  = 2*1 + 1
2  = 1*2 + 0

GCD = 1. Total steps: 7

RESULT
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gcd(55, 34) takes 8 steps; gcd(55, 21) takes 7 steps. The
consecutive-Fibonacci pair takes ONE MORE step than the similarly-
sized non-Fibonacci pair.

DOES THIS MATCH THIS CHAPTER'S OWN CLAIM?
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Yes. This chapter's own worked example showed the same pattern at
a different scale - gcd(144, 89) (a consecutive Fibonacci pair)
took 10 steps, one more than gcd(144, 55) (a similarly-sized non-
Fibonacci pair) at 9 steps. This exercise reproduces the identical
one-extra-step pattern at a smaller scale, reinforcing that
consecutive Fibonacci numbers genuinely do sit at (or very near)
the worst case for a given input size - every single quotient
along the way turns out to be exactly 1, which is precisely what
forces the maximum possible number of steps for numbers of that
size.

WHY THIS WORKS AS AN ANSWER
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Both traces are carried out in full, step by step, with an honest
step count for each rather than an estimate, and the comparison
explicitly checks the result against this chapter's own stated
claim using its own prior worked example as the point of
comparison, rather than treating the claim as self-evidently true.