Mathematics Fibonacci Sequence From Rs Fib6 Eq 2cube D

The sixth Fibonacci number

The Fibonacci sequence begins 1, 1, 2, 3, 5, 8, 13, 21, where each number after the first two is the sum of the two before it. The sixth number in that list, counting from zero, is 8. The number 8 is also two cubed, 2 × 2 × 2 = 8. A machine-checked library of formal theorems contains a proof of this equality, written as fib6_eq_2cubeD, which states that the sixth Fibonacci number equals two raised to the third power.

The proof itself is a direct computation, completed by a method that checks the equality by calculation. The library records this as a theorem with no unproved assumptions and no axioms beyond the standard logical foundations. The declaration is one of a small set of Fibonacci facts the framework stores, including that the third number is 2, the fourth is 3, and the sixth is 8. These values appear in the framework's account of how the golden ratio relates to the Fibonacci sequence, but the declaration itself only establishes the arithmetic identity.

In Recognition Science, the framework models a discrete record of events, called a ledger, and the cost of recognition is forced by a proved theorem. Within that account, the number 3 is identified with the spatial dimension and the number 8 with the period of a recognition cycle. The equality of the sixth Fibonacci number with two cubed links these two values: the framework reads 8 as two raised to the dimension 3. This connection is a definitional choice inside the framework, not a theorem about the physical world.

What the declaration does not claim is any physical content. It does not assert that space has three dimensions or that recognition cycles have eight ticks. Those claims live in other parts of the framework, and the physical bridge from the arithmetic to the geometry is itself an open target. The declaration is a pure statement about the natural numbers, proved by computation, and any larger meaning comes from how the framework chooses to use the numbers.

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

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

theorem fib3_eq_2 : Nat.fib 3 = 2 := by decide

theorem fib4_eq_3 : Nat.fib 4 = 3 := by decide

theorem fib6_eq_8 : Nat.fib 6 = 8 := by decide

What this page does not claim

The declaration does not assert that space has three dimensions. The declaration does not claim that recognition cycles have eight ticks. The equality does not by itself force any physical structure.

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)

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