Seven Domains, One Eigenvalue & Two Stubborn Lambdas and a strange observation

ZERO PARAMETERS: NO FITTING Contains blind envelope predictions & future predictions and lab testable Update : it wasn't really a blind envelope, I just wanted to be honest, it was more like a confirmation my math worked when they matched...so I updated the paper to reflect this, but I can't be bothered explaining it in this abstract window, although it's just occured to me I could have done and saved myself a lot of time...major discovery on that one...anyway I just wanted to make that point clear, other than that this is an honest paper that simply tries to make sense of a pattern that all shared the same eigenvalue, this work has been building in my various other papers, you see the building blocks, the stumbled and the refinement, I am not an academic or anything just a regular dude with a mundane job that allows me time to think. What you see here is a combination of those and older papers, like the Diamond lattice, fsre, canle etc, are works I did a few months ago and the terminology of those foreign terms doesn't take into consideration that everyone (except a few hundred of you ) will never have heard of them, so I wanted the reader to know those terms are other papers, I built the model in my head, dropped a few pieces, found others etc until the structure in my mind seemed stable, in closing judge me and not the math...no wait judge the math and not me... , see just a regular dude who may or may not have got the visual structure in my head wrong, actually there's a lot in my head wrong 😂 Bye. DC Updated with a very humble proposition... hands in the air, just what dropped out, might be evidence of something??? It's in the paper I can't be bothered fixing the text to format it correctly t..ahh I forgot my original discovery was not to make life or math too serious, it is what it is and whatever it is, it's just is. :) if it's too much, well.. it just dropped out of math, head titling brain tilting and ..huh. now Im from Belfast so I saw nothing, wasn't me, don't know anyone who could etc, but you might wanna look at the math and see yourselves, I'm just doing the building here guys, hope some of it helps someone but remember man, peace ffs✌️ I think I'm done with this Bye DC 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 = 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 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 ?