Independent Researcher, Belfast, Northern Ireland ORCID: 0009-0009-9192-4797∗ (Dated: March 24th 2026) Updated major revision This is starting to feel like homework, it's not fun anymore. Near the 2:1 mean-motion resonance, the local dynamics of a planetary pair is governed by a transfer matrix T(k) ∈ SL(2,R) with trace k. At the stability boundary k = 2, the conservative M¨obius map g(r) = 2−1/r induces slow parabolic drift away from exact commensurability. A statedependent dissipation satisfying physically motivated boundary conditions—maximum damping at exact resonance, vanishing at the adjacent 3:1 zone boundary—converts g into the JacobsthalLichtenberg map f(r) = 1+2/r exactly. The iterates of f are the Jacobsthal-Lichtenberg ratios, and the dominant above-boundary attractor is 43/21 ≈ 2.048, defining the Jacobsthal window ∆ = 1/21. We derive the spectral-edge density ρ ∝ √ϵ from a Fokker-Planck equation on the SL(2,R) group manifold with an absorbing boundary at exact commensurability, yielding the zero-fitted-parameter prediction ⟨R⟩ = 71/35 ≈ 2.0286. The observed population mean from the NASA Exoplanet Archive (N = 56 pairs in the Jacobsthal window, drawn from 47 independent systems) is 2.027 ± 0.004, consistent with the prediction (p = 0.67 after accounting for within-system correlations). A maximum-likelihood shape fit gives ˆα = 0.47 ± 0.15, consistent with the predicted exponent 1/2. Exploratory tests provide preliminary support: a binary shape test rejects the linear alternative (p = 0.02) but does not discriminate √x from uniform, and a tidal dissipation test (p = 0.029, one-sided) shows the predicted direction of effect. Neither test survives a Bonferroni correction for multiple comparisons, and both should be regarded as hypothesis-generating until confirmed with PLATO-era samples. An inverse Hamiltonian analysis shows that the dissipation profile γ(r) = 3−r is not only an input to the forward derivation but also a prediction of the inverse: it is uniquely determined by the observed density and the first-principles conservative map. REBOUND N-body simulations support the 43/21 attractor across a 25-fold range of migration timescales. Full shape discrimination requires N ≈ 150 (PLATO). We caution that multiple tests are performed; the results should be regarded as hypothesis-generating until confirmed with independent data.
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