Mathematics Fibonacci Sequence From Rs Fib7 Eq 13

The thirteenth Fibonacci number

The Fibonacci sequence begins 1, 1, 2, 3, 5, 8, 13, 21, with each number after the first two being the sum of the two before it. It appears throughout mathematics, from the branching of plants to the Euclidean algorithm for greatest common divisors. The seventh term, counting from zero, is 13: the sequence runs 0, 1, 1, 2, 3, 5, 8, 13. This is not a discovery but a definitional fact, true by the way the sequence is built.

In the Recognition Science framework, the Fibonacci numbers are tied to the golden ratio φ, the number satisfying φ² = φ + 1, approximately 1.618. The framework's machine-checked library of formal theorems records the identity F(n) × φ + F(n-1) = φ^n, which characterizes φ in terms of the Fibonacci numbers. The specific declaration fib7_eq_13 states that the seventh Fibonacci number equals 13, proved by direct computation. The library also records the neighboring values: F(5) = 5, F(6) = 8, and F(8) = 21, the last following from the recurrence F(8) = F(7) + F(6).

These values are not isolated facts in the framework. The library notes that F(4) = 3, which it identifies with the number of spatial dimensions, and F(6) = 8, which it identifies with 2³, the period of a recognition cycle. The seventh term, 13, does not carry such a structural identification; it appears as a step in the sequence and in the recurrence that produces 21. The declaration fib7_eq_13 is a computational theorem, not a claim about physics or geometry.

What the declaration does not claim is more important than what it does. It does not say that 13 has any special role in the framework's derivation of dimensions or cycles. It does not assert that the Fibonacci sequence itself is derived from first principles; the sequence is defined by its recurrence, and the theorem merely evaluates it. The framework's deeper claims about φ and the forcing chain are separate results, not consequences of this single computation.

theorem fib7_eq_13 : Nat.fib 7 = 13 := by decide

theorem fib7_eq_13 : Nat.fib 7 = 13 := by decide

/-- Recurrence: F(8) = F(7) + F(6) = 13 + 8 = 21. -/
theorem fib_recurrence_8 : Nat.fib 8 = Nat.fib 7 + Nat.fib 6 := by decide

What this page does not claim

The declaration does not assign any special structural role to the number 13. It does not derive the Fibonacci sequence from the framework's axioms; the sequence is defined by its recurrence. It does not prove any claim about φ, dimensions, or cycles; those are separate theorems.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/FibonacciSequenceFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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