fibonacci_factor_eq
The lemma establishes that the Fibonacci factor equals 5, where this factor is the smallest Fibonacci number greater than 1 coprime to 8. Researchers deriving the 45-gap from the eight-tick cycle and triangular numbers would cite it as the algebraic closure for the factorization 45 = 9 × 5. The proof is a direct reflexivity step on the explicit definition of the factor.
claimLet $f$ be the smallest Fibonacci number greater than 1 that is coprime to 8. Then $f = 5$.
background
The Gap45 module derives the number 45 from the eight-tick structure (T8) combined with the Fibonacci sequence tied to the golden ratio. The closure factor is 9 (one full cycle of 8 plus return), and the Fibonacci factor supplies the multiplier 5 to reach 45, equivalently the ninth triangular number T(9) representing cumulative phase accumulation over the closed cycle. The upstream definition sets this Fibonacci factor explicitly to 5 as the smallest such number coprime to 8; related imports cover collision-free ledger structures and mechanism design but are not invoked in the equality itself.
proof idea
This is a one-line wrapper that applies reflexivity directly to the definition of the Fibonacci factor, which is set to 5.
why it matters in Recognition Science
The lemma supplies the missing algebraic identity in the Gap45 derivation, confirming 45 = (8 + 1) × 5 and thereby linking the eight-tick period (T8) to D = 3 via lcm(8, 45) = 360. It fills the Fibonacci-factor slot referenced in the module's physical-motivation section, where 45 arises as cumulative phase rather than an arbitrary constant. No downstream uses are recorded yet, but the result anchors the claim that 45 is forced by the framework's closure and self-similarity principles.
scope and limits
- Does not derive the Fibonacci factor from the golden-ratio fixed point or Recognition Composition Law.
- Does not prove coprimality of 5 with 8 or uniqueness among Fibonacci numbers.
- Does not address the triangular-number equivalence T(9) = 45 beyond the factorization.
- Does not connect the factor to mass formulas, alpha bounds, or Berry thresholds.
formal statement (Lean)
88@[simp] lemma fibonacci_factor_eq : fibonacci_factor = 5 := rfl
proof body
Term-mode proof.
89
90/-- 5 is a Fibonacci number. -/