Golden Fibonacci Modulus Period
Abstract
Odd Fibonacci values carry an exact fourfold return of the Fibonacci matrix.
Definition 1.1 (Multiplication in the golden residue algebra).
Lean statement: D5/S3/Arith/GoldenFibonacciModulusPeriod.goldenMatrixHom
Formalization. D5/S3/Arith/GoldenFibonacciModulusPeriod.goldenMatrixHom (✓ std3).
Source. Repository-derived.
Commentary.
On the ordered basis consisting of the golden generator and one, the residue a+b*phi acts by the matrix with rows (a+b,b) and (b,a). This assignment preserves zero, one, addition and multiplication over any modulus.
Theorem 1.2 (Exact period at an odd Fibonacci modulus).
Lean statement: D5/S3/Arith/GoldenFibonacciModulusPeriod.golden_fibonacci_modulus_period
Proof. Machine-checked in Lean as D5/S3/Arith/GoldenFibonacciModulusPeriod.golden_fibonacci_modulus_period (✓ std3). ∎
Source. Repository-derived.
Commentary.
For every odd index n at least five, the Fibonacci matrix modulo F_n has multiplicative order exactly 4n. A matrix return forces F_n to divide the Fibonacci number at the return index. Strong divisibility and strict growth then force n to divide that index. At the nth power the matrix is scalar, and Cassini’s identity makes that scalar’s square equal to minus one. Since F_n is greater than two, its scalar has order four, excluding every shorter return.