Golden Cubic Block Prime Periods
Abstract
Prime factors of the two golden cubic blocks have exact Fibonacci matrix periods.
Theorem 1.1 (The Lucas block).
Lean statement: D5/S3/Arith/Primes/GoldenCubicBlockPrimePeriods.cubic_block_b_prime_period
Proof. Machine-checked in Lean as D5/S3/Arith/Primes/GoldenCubicBlockPrimePeriods.cubic_block_b_prime_period (✓ std3). ∎
Source. Repository-derived.
Commentary.
For j at least one, every prime factor p of L_(3^j)^2 + 3 has Fibonacci matrix period exactly 2 times 3^(j+1). The Lucas value at 3^(j+1) vanishes modulo p. The quadratic trace and norm identity then makes the golden generator return at twice that index; its first Fibonacci zero rules out an earlier return.
Theorem 1.2 (The Fibonacci block).
Lean statement: D5/S3/Arith/Primes/GoldenCubicBlockPrimePeriods.cubic_block_c_prime_period
Proof. Machine-checked in Lean as D5/S3/Arith/Primes/GoldenCubicBlockPrimePeriods.cubic_block_c_prime_period (✓ std3). ∎
Source. Repository-derived.
Commentary.
For j at least one, every prime factor p of L_(3^j)^2 + 1 has Fibonacci matrix period exactly 4 times 3^(j+1). At the first Fibonacci zero, the golden generator squares to minus one by Cassini’s identity. The block is odd, so minus one is not one modulo p; its residual order is four.