Seven Domains, One Eigenvalue & Two Stubborn Lambdas

ZERO PARAMETERS: NO FITTING Contains blind envelope predictions 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 seven 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, (VII) photonic quasicrystals ... the same eigenvalue emerges. The exoplanet window mean ⟨R⟩window = 71/35 is validated by an offset-specific 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 fixed-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 = λ is 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 identification with dark matter density remains a conjecture. A Monte Carlo null test gives p < 3.1 × 10−4. Falsiable predictions include a hard bound of n∗ = 7 on compact resonant chain length. 1 Introduction A mathematical constant becomes a physical law when it is both derived from first principles and confirmed by independent measurements. The Jacobsthal characteristic polynomial r2 − r −2 = 0 is derived from Hamiltonian mechanics: it is the fixed-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 seven domains: algebra, the solar system, KAM dynamics, TRAPPIST-1, the exoplanet population, Kerr black hole thermodynamics, and photonic quasicrystals. 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 proposes a laboratory experiment. Two distinct tools are used. The FSRE ( fixed-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: All my own work, the only open question is ..is it worth a pint of Guinness to tell ya the story ?