Mathematics Fibonacci Sequence From Rs Fib8 Eq 21

The eighth Fibonacci number

The Fibonacci sequence begins 1, 1, 2, 3, 5, 8, 13, 21, where each term after the first two is the sum of the two before it. The eighth term, counting from zero, is 21. This is a simple arithmetic fact: 13 plus 8 equals 21. The Recognition Science framework's machine-checked library of formal theorems contains a declaration, fib8_eq_21, that proves exactly this statement, Nat.fib 8 = 21, using a direct computation that requires no additional assumptions.

In the broader mathematical context, the Fibonacci sequence connects to the golden ratio φ, approximately 1.618. The ratio of consecutive Fibonacci numbers approaches φ as the sequence continues. A related identity states that φ raised to the power n equals F(n+1) times φ plus F(n), linking the sequence to the golden ratio's powers. The framework's library also proves several specific values: F(3) = 2, F(4) = 3, F(6) = 8, and F(8) = 21, all by direct computation. It further proves that F(6) equals 2 cubed, which the framework connects to its own derived spatial dimension of three.

In Recognition Science, the framework models certain physical structures as a discrete ledger of recognition events, and the golden ratio emerges as a forced scaling constant from its core cost function. The framework's library shows that specific Fibonacci numbers align with its derived constants: F(3) = 2 and F(4) = 3 relate to its spatial dimension, and F(6) = 8 equals 2 to the third power, matching its ledger period. These are presented as proved identities within the framework's formal system.

The declaration fib8_eq_21 itself, however, establishes only the arithmetic value. It does not claim that 21 has any special role in the framework's physics, nor does it assert anything about the golden ratio, spatial dimensions, or the ledger period. Those connections come from other declarations in the library, such as fib6_eq_2cubeD, which ties F(6) to 2 cubed. The single fact that the eighth Fibonacci number is 21 stands on its own as a verified computation, and any broader significance would require additional theorems to support it.

theorem fib8_eq_21 : Nat.fib 8 = 21 := by decide

/-- F(6) = 8 = 2^D = 2^3. -/
theorem fib6_eq_2cubeD : Nat.fib 6 = 2 ^ 3 := by decide

What this page does not claim

The declaration fib8_eq_21 does not connect the number 21 to the golden ratio or any physical constant. The framework does not claim that every Fibonacci number has a special role in its physics. The proof of fib8_eq_21 by direct computation does not establish any property of the golden ratio.

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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