fib_5
The lemma establishes that the fifth term equals 8 in the Fibonacci sequence defined locally by fib 0 = 1 and fib 1 = 1. Researchers deriving the 45-gap from the eight-tick structure would cite it to fix the Fibonacci factor in the factorization 45 = 9 × 5. The proof is immediate by reflexivity after the recursive definition unfolds.
claim$F_5 = 8$, where $F_n$ is the sequence defined by $F_0 = 1$, $F_1 = 1$, and $F_{n+2} = F_n + F_{n+1}$ for $n ≥ 0$.
background
The Gap45.Derivation module shows that 45 arises from the eight-tick period (T8) together with the Fibonacci sequence tied to the golden-ratio fixed point φ. The local fib definition begins fib 0 = 1, fib 1 = 1 and continues by the standard recurrence, producing the sequence 1, 1, 2, 3, 5, 8, …. This supplies the fibonacci_factor = 5 in the factorization 45 = (8 + 1) × 5, where the closure factor 9 = 8 + 1 accounts for one full cycle plus return to the initial state. The module further equates 45 with the ninth triangular number T(9) = 1 + 2 + … + 9, representing cumulative phase accumulation over the closed eight-tick cycle. Upstream results supply equivalent Fibonacci definitions in ContinuedFractionPhi and ZeckendorfJCost, all satisfying the same recurrence.
proof idea
The proof is a one-line wrapper that applies reflexivity after the recursive definition of fib is unfolded.
why it matters in Recognition Science
The lemma supplies a concrete Fibonacci value required by the Gap45 derivation that 45 = T(9) emerges from the eight-tick octave (T8) and the φ-related Fibonacci sequence (T6). It supports the subsequent claim that lcm(8, 45) = 360 forces D = 3 spatial dimensions. The result sits inside the chain that converts the abstract forcing steps T0–T8 into the concrete numerical factor 45 used for phase accumulation.
scope and limits
- Does not prove the full factorization 45 = 9 × 5 or the triangular-number identity.
- Does not connect the value 8 to any physical constant or dimension count.
- Does not address Fibonacci sequences defined with different initial conditions.
- Does not establish coprimality or other arithmetic properties of the sequence.
formal statement (Lean)
59@[simp] lemma fib_5 : fib 5 = 8 := rfl