fib_2
The lemma states that the second term equals 2 in the Fibonacci sequence defined locally by fib 0 = 1, fib 1 = 1, and the standard recurrence. Researchers deriving the 45-gap from the eight-tick structure and phi-related Fibonacci factors would cite this fact to simplify early terms in algebraic reductions. The proof is a direct reflexivity step that matches the recursive clause at n=2 without further lemmas.
claimLet $f : ℕ → ℕ$ be given by $f(0) = 1$, $f(1) = 1$, and $f(n+2) = f(n) + f(n+1)$ for $n ≥ 0$. Then $f(2) = 2$.
background
The Gap45.Derivation module shows that 45 emerges as (8 + 1) × 5 from the eight-tick octave (T8) combined with the Fibonacci sequence tied to φ. The local fib is defined recursively starting at 1, 1 rather than the conventional 0, 1 indexing; this choice aligns with the phi-ladder and Zeckendorf J-cost constructions used upstream. The module imports the fib definition from ContinuedFractionPhi and ZeckendorfJCost, both of which supply the same recurrence but with differing base cases.
proof idea
The proof is a one-line wrapper that applies reflexivity after the recursive clause of fib unfolds at argument 2, yielding fib 0 + fib 1 = 1 + 1.
why it matters in Recognition Science
This supplies a simp rule for the early Fibonacci terms required to reach the key identity 45 = 9 × 5 inside the eight-tick closure argument. It supports the larger claim that 45 is forced by T8 together with the triangular number T(9) and the coprimality of 5 with 8, which in turn forces D = 3 via lcm(8, 45) = 360. No direct parent theorems are listed among the used-by edges, but the lemma sits among the fib_n siblings that collectively close the Gap45 derivation.
scope and limits
- Does not compute fib n for any n > 2.
- Does not relate the local indexing to the standard Fibonacci sequence starting at 0.
- Does not address the physical interpretation of the 45-gap or the eight-tick cycle.
formal statement (Lean)
56@[simp] lemma fib_2 : fib 2 = 2 := rfl