fibonacci_5_is_5
plain-language theorem explainer
The theorem confirms that the fourth term equals 5 in the Fibonacci sequence defined locally by f(0)=1, f(1)=1 and the standard recurrence. Researchers deriving the gap-45 constant from the eight-tick structure would cite this to isolate the Fibonacci factor in the factorization 45=9×5. The proof is a direct reflexivity step that unfolds the recursive definition.
Claim. Let $f:ℕ→ℕ$ satisfy $f(0)=1$, $f(1)=1$ and $f(n+2)=f(n)+f(n+1)$. Then $f(4)=5$.
background
The module derives the constant 45 from the eight-tick period forced by T8 together with the Fibonacci sequence linked to φ. The local fib is defined by fib 0=1, fib 1=1 and the recurrence fib(n+2)=fib n + fib(n+1). This supplies the fibonacci_factor in the identity 45=(8+1)×5, where 9 is the closure factor (one full cycle plus return) and the triangular number T(9)=45 represents cumulative phase over the closed cycle.
proof idea
The proof is a one-line reflexivity wrapper. It matches the target 5 against the value obtained by unfolding the recursive definition of fib four times.
why it matters
This supplies the Fibonacci factor required for the key identity 45=9×5 in the Gap45 derivation. It connects the eight-tick octave (T8) and the Fibonacci sequence (tied to φ) to the emergence of D=3 via lcm(8,45)=360. The result fills the algebraic step showing that 45 arises as the ninth triangular number rather than an arbitrary input.
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