Five Domains, One Eigenvalue: Dimensionless Evidence for λ = 2 in Planetary Resonance Structure

Includes a nice surprise 🫢 The Jacobsthal characteristic polynomial r2−r −2 = 0 is derived from Hamiltonian mechanics: it is identically the fixed-point equation of the symplectic trace map Tn+1 = T 2n −2, which follows from the CayleyHamilton theorem for any M ∈ SL(2, R) with det(M) = 1. Its roots λ = 2 and λ− = −1 are therefore physical quantities, not assumed. The Diamond Lattice of Jacobsthal pairslying exactly on y = λx + det(M) = 2x + 1geometrically encodes this structure, with slope equal to the eigenvalue and intercept equal to the symplectic determinant. These eigenvalues generate a thread T = λ − 1/λ = 3/2 and are bounded from below by φ = (1+√5)/2 (the KAM stability ceiling). All empirical results are dimensionless period ratios. In five domains.... (I) pure algebra, (II) the solar system and EarthMoon system, (III) KAM dynamics, (IV) TRAPPIST-1, (V) the exoplanet population, ....the same three numbers emerge. The solar system spans J(11) = 683 Neptune-to-Mercury periods (0.17%). The lunar synodic month satises φn∗ + 1/λ = 29.534 (0.013%), where n∗ = ⌊12 · log2 (3/2)⌋ = 7 is the comma index with zero free parameters. TRAPPIST-1 synodic ratios alternate T, λ, T, λ to four decimal places. The exoplanet mean ratio ⟨R⟩ = λ + 1/(pivot · n∗ ) = 71/35 matches NASA data to four decimal places (FSRE prediction, CANLE validation). A Monte Carlo null test (491 candidate values, three null models) gives p < 3.1 × 10−4 . Falsiable predictions include a hard bound of n∗ = 7 on compact resonant chain length. Keywords: eigenvalue, symplectic trace map, Jacobsthal sequence, Diamond Lattice, Pythagorean comma, planetary resonance, dimensionless ratios Introduction A mathematical constant becomes a physical law when it is both derived from first principles and conrmed by independent measurements. The Jacobsthal characteristic polynomial r2 − r − 2 = 0 is derived from Hamiltonian mechanics: it is the xed-point equation of the symplectic trace map, which follows from the CayleyHamilton theorem for any matrix in SL(2, R). Its rootsλ = 2 and λ− = −1generate a thread T = λ−1/λ = 3/2 and are bounded from below by φ = (1+√ 5)/2, the KAM stability ceiling. The Diamond Lattice of Jacobsthal pairs encodes this structure geometrically: slope = λ, intercept = det(M). All empirical results in this paper are dimensionless period ratios, independent of any unit system. The same three constants λ, T, φemerge from ve domains: algebra, the solar system, KAM dynamics, TRAPPIST-1, and the exoplanet population. This paper derives the eigenvalues from physics, identies the algebraic structure connecting the three constants, validates the predictions against NASA data, and states falsiable tests. Two distinct tools are used. The FSRE (xed-step recurrence equation) operates at the sequence level: it predicts convergent ratios from integer recurrences (J(n+1)/J(n) → λ, Fn+1/Fn → φ). The CANLE (cumulative adjacent normalised log error) operates at the data level: it measures how observed planet period ratios deviate from a reference value across a population. The FSRE generates predictions; the CANLE validates them against NASA data.