Six Domains, One Eigenvalue and Two Stubborn Lambdas

Updated to reflect non matching pairs, Trappist uncertainties to 4 decimal places and monodromy trace clarified to negate confusion. Updated to six domains, I forgot about my kerr paper Abstract. The Jacobsthal characteristic polynomial r2 − r − 2 = 0 is derived from Hamiltonian mechanics: it is identically the xed-point equation of the symplectic trace map Tn+1 = T2n−2, which follows from the CayleyHamilton theorem for any M ∈ SL(2,R) with det(M) = 1. The period-2 orbits of the same map yield the Fibonacci polynomial r2+r−1 = 0. Together, the combined quartic (T2 −T −2)(T2 +T −1) = 0 produces four roots λ = 2, λ− = −1, 1/φ, −φ from which every quantity in this paper descends. In the circular restricted three-body problem, the Poincaré return map has trace Tr(M) = 2cosh(σphys) with σphys ∝ µ1/2; the eigenvalue λ = 2 is the exact zero-mass limit. The Diamond Lattice of Jacobsthal pairs lying on y = λx+det(M) = 2x+1 encodes this structure geometrically. These eigenvalues generate a thread τ = λ − 1/λ = 3/2 selected both algebraically (the eigenvalue-reciprocal gap) and physically (the unique nontrivial Fibonacci convergent resolvable at observed coherence Q ∼ 1030) and are bounded from below by φ = (1+√5)/2 (the KAM stability ceiling). All empirical results are dimensionless. In six domains (I) pure algebra, (II) the solar system and EarthMoon system, (III) KAM dynamics, (IV) TRAPPIST-1, (V) the exoplanet population, (VI) Kerr black holes the same eigenvalue emerges. The exoplanet window mean ⟨R⟩window = 71/35 is validated by an o set-speci c test (p = 0.0015), CANLE cluster bootstrap (p = 0.67 vs 71/35, N = 56), and REBOUND N-body simulations clustering at 43/21. In Domain VI, the Möbius xed-point structure at the Jacobsthal trace k = 5/2 selects the Kerr spin aK/M = ρ = 1/λ. The Smarr decomposition at this spin gives angular fraction f+ = (2 − √3)/2, and the two-horizon angular sum f++f− = 2 = λis proved for all spins. The product Ωdm = f+×λ = 2−√3 ≈ 0.268 lies within 0.3σ of the Planck 2018 measurement. Every factor in this product is derived from Hamiltonian mechanics; the physical identi cation with dark matter density remains a conjecture. A Monte Carlo null test gives p < 3.1 × 10−4. Falsi able 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, Kerr black hole, dark matter density, dimensionless ratios Introduction A mathematical constant becomes a physical law when it is both derived from rst principles and con rmed 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). The period-2 orbits of the same map yield the Fibonacci polynomial r2 + r − 1 = 0. Together, the quartic (T2 − T − 2)(T2 + T − 1) = 0 produces four roots λ = 2, λ− = −1, 1/φ, −φ from which every quantity in this paper descends. In the circular restricted three-body problem, the physical Poincaré trace satis es Tr(M) = 2cosh(σphys) with σphys ∝ µ1/2; the eigenvalue λ = 2 is the exact zero-mass limit. The Diamond Lattice of Jacobsthal pairs encodes this structure geometrically: slope = λ, intercept = det(M). All empirical results in this paper are dimensionless, independent of any unit system. The same eigenvalue λ = 2 emerges from six domains: algebra, the solar system, KAM dynamics, TRAPPIST1, the exoplanet population, and Kerr black hole thermodynamics. This paper derives the eigenvalue from physics, identi es the algebraic structure connecting λ, τ, and φ, validates the predictions against NASA and Planck data, and states falsi able 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.