Mathematics Fibonacci Sequence From Rs Fib Recurrence 8

The eighth Fibonacci number

The Fibonacci sequence begins 0, 1, 1, 2, 3, 5, 8, 13, 21, where each term after the first two is the sum of the two preceding terms. The declaration fib_recurrence_8 establishes the specific case for the eighth term: F(8) = F(7) + F(6), which means 21 = 13 + 8. This is a single instance of the defining recurrence F(n) = F(n-1) + F(n-2), applied at n = 8. The proof is a direct computation, marked in the machine-checked library of formal theorems as a theorem with no unproved assumptions.

The statement belongs to a small family of Fibonacci facts that the framework's library proves by direct calculation. These include F(3) = 2, F(4) = 3, F(6) = 8, and the identity F(8) = F(7) + F(6). Each is a concrete numerical equality, not a general theorem about all Fibonacci numbers. The library also records that F(6) = 8 equals 2³, and that F(4) = 3, numbers that appear elsewhere in the framework's structure. But the declaration itself only certifies the arithmetic of the eighth term.

In Recognition Science, the framework models a discrete record of events, called a ledger, and derives constants from a forced cost function. The Fibonacci numbers appear in this account because the golden ratio φ, which satisfies φ² = φ + 1, connects to the sequence through identities like φ^n = F(n+1)φ + F(n). The specific value F(8) = 21 is not itself a deep structural result; it is a stepping stone in a collection of computed facts. What the declaration establishes is precise and narrow: the recurrence holds at n = 8, and the arithmetic is correct.

What the declaration does not claim is broader. It does not prove the general recurrence for all n, nor does it show that Fibonacci numbers are intrinsic to the framework's physics. The docstring asserts such connections, but the theorem itself is only the numerical equality. The library's other theorems prove F(4) = 3 and F(6) = 8, which the framework links to spatial dimension and ledger period, but fib_recurrence_8 makes no such link. It certifies one line of arithmetic, nothing more.

/-- 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

/-- 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 prove the general Fibonacci recurrence for all n. The declaration does not establish that Fibonacci numbers are intrinsic to the framework's physics. The declaration does not link F(8) to any spatial dimension or ledger period.

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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