The λ = 2 Uniqueness Theorem and Exoplanetary Orbital Architecture

We present a zero-parameter framework explaining the observed clustering of exoplanetary period ratios at specic values. The framework rests on a fundamental mathematical result:among all positive integers c, only c = 2 satises the condition that the dominant eigenvalue of the recurrence xn = xn−1 + c · xn−2 equals c itself. This uniqueness connects to the Hamiltonian stability boundary |Tr(M)| = 2, yielding predicted period ratios of 11/5 = 2.200 and 43/11 = 3.909 from the resulting Jacobsthal sequence. These predictions match TESS and Kepler observations (2.19 and 3.87 respectively) within 1%. We validate the framework using Kolmogorov-Smirnov tests on Royer et al. (2025) data (D = 0.073, p = 0.142) and preliminary TESS non-adjacent analysis (D = 0.091, p = 0.087). We derive mass-dependent coupling ce = 2 + β(m/3M⋆) 1/3 with β = 0.44 calibrated explicitly to the Jupiter-Saturn ratio. The secular shift prefactor is derived from Laplace-Lagrange theory. We discuss implications for planet formation, particularly how the framework's predictions dier froms to chastic migration and pebble accretion scenarios. The Beta Pictoris prediction (3036AU) awaits direct JWST conrmation. With Special thanks to Dr Shallit, for his criticism and lack of vision.