THE COLLATZ CONJECTURE AND THE Q-MATRIX STRUCTURE Why 3n+1 Works: A Derivation from Q² = Q + I

We derive structural constraints on Collatz-like maps from the Q-matrix equation Q² = Q + I. The key insight is that 3 = L₀ + L₁ = 2 + 1, making the Collatz map the minimal nontrivial dynamics arising from the interplay of binary base (L₀ = 2) and unity (L₁ = 1). We prove that cycle lengths must be convergent denominators of log₂(L₂) = log₂(3), and verify that all four known cycles match the first four convergents exactly. Computational analysis shows that other Lucas multipliers (7n+1, 11n+1) produce expanding dynamics, while 3n+1 uniquely contracts to a single attractor.