A Jacobsthal Window in Exoplanet Period Ratios: Derivation of the 71/35 Offset from Symplectic Depletion

The distribution of period ratios in multi-planet systems shows a systematic displacement from the exact 2:1 mean-motion resonance. Using 1038 confirmed multi-planet systems from the NASA Exo-planet Archive (1555 adjacent pairs), we identify 56 pairs within the interval [2.000, 43/21] and measure their mean period ratio as ⟨R⟩ = 2.027±0.004 (boot- strap 95% CI), significantly above exact commensurability (p < 10−5 , Wilcoxon) and statistically consistent with our predicted value (p = 0.67, Wilcoxon). We derive a quantitative model for this offset. The symplectic transfer matrix T(k) governs near-resonant dynamics; at the stability boundary k = 2, it generates the Hamiltonian M¨obius map g(r) = 2 − 1/r, which drives ratios away from 2. We show that adding the unique linear state-dependent dissipation γ(r) = 3−r- stronger near resonance, vanishing far from it - converts g exactly into the Jacobsthal map f(r) = 1+2/r, whose iterates are the Jacobsthal ratios J(n+1)/J(n) converging to 2. The deviation identity J(n + 1)/J(n) − 2 = (−1)n/J(n) places the dominant low-order attractor above 2:1 at 43/21 = 2 + 1/21, defining a resonance window of width ∆ = 1/21. Spectral-edge depletion near the boundary distributes systems within this window as ρ(x) ∝ √ x, yielding ⟨R⟩ − 2 = 3 5 ∆ = 1 35 ≈ 0.02857, predicting ⟨R⟩ = 71/35 ≈ 2.0286. A KS test on the wider interval [1.95, 2.10] gives p = 0.80. Four independent lines of evidence support the framework: (i) REBOUND N-body migration simulations cluster at 43/21 = 2.048, not 2.000; (ii) TRAPPIST-1 non-adjacent period ratios align with Jacobsthal values (permutation p = 0.0004); (iii) the Beta Pictoris system architecture at 30–36 AU matches the Jacobsthal framework, independently confirmed at 30–35 AU; (iv) Monte Carlo experiments confirm the √ ϵ edge exponent under migration-weighted ensembles. The model has no free parameters