A Fibonacci–Lucas Spectral Law for the Trinomials xⁿ = x + 1

Let rn denote the largest real root of xn − x − 1 for n ≥ 2. The root r2 = φ is the golden ratio, r3 = ρ is the plastic constant (the smallest Pisot number), and r4 is the first root in the family that fails the Pisot condition. We prove that the minimal polynomial of ρ°§r4 over Q, reduced modulo the golden ratio polynomial t2−t−1, yields the remainder t−1 exactly—the fundamental unit φ−1 of the ring Z[φ], with algebraic norm −1. For all other products rm °§ rn with 2 ≤ m < n ≤ 8, the corresponding norm has absolute value at least 11, growing super-exponentially with max(m, n). As a corollary, the topological entropies satisfy ln r2 = ln r3 + ln r4 + O(10−4), while | ln rm − ln rn − ln rp| = O(1) for all other triples. The (3, 4) pair is thus uniquely distinguished within the xn = x + 1 family by its proximity to the golden ratio, and the proximity has an exact algebraic origin.