Paper P: N-Bonacci Constants as Spectral Modes: A General Theorem on Spacing Ratio Distributions

We establish an algebraic identity connecting the n-bonacci constants to the spectral theory of arithmetic quotients. The n-bonacci constant Cn is the dominant root of x^n − x^(n−1) − ... − x − 1 = 0. We prove that its reciprocal rn = 1/Cn* is the unique positive solution of the criticality equation: Sum from k=1 to n of (x^k) = 1. This identity is exact for all n ≥ 2, with the limit r_infinity = 1/2* as n goes to infinity. Paper A of the MNZI programme proved that the mode of the GUE consecutive spacing ratio distribution — equivalently, the mode of the spectral spacing distribution on the arithmetic quotient SL(2, Z) — is exactly 1/phi = r2*, the golden ratio reciprocal, determined by the n = 2 criticality equation: r + r^2 = 1. We conjecture that this result is the base case of a general theorem: the mode of the spacing ratio distribution on SL(n, Z) is rn = 1/Cn* for each n ≥ 2, with the adelic limit r_infinity = 1/2*. The criticality equation has a natural interpretation in branching process theory: rn* is the unique threshold at which the n-step memory system is exactly critical — neither sub-critical nor super-critical. We establish this interpretation and its connections to the companion matrix theory of SL(n,Z), the Perron-Frobenius theorem, and the information-theoretic limit C_infinity = 2. The adelic limit r_infinity = 1/2* represents the “pure binary” threshold: the mode of the spacing distribution over all primes simultaneously is the fair-coin probability 1/2, the maximum-entropy mode.