The Coates Quartic: Seven Domains, Two Stubborn Lambdas, and a Seventh-Century Cow

No paramaters are fitted. May I be so bold as to ask you to check out my other paper here: https://doi.org/10.5281/zenodo.19420699 for focused subject matter foundation. Updated yet again, math error and latest results update, Thank you for your patience in watching this paper evolve. Please note this is not for journals, nor will it ever be...if this hit a table it would soon find itself into an editors bin, but what this is...is real, like anything in life everything is about perception, your perception may differ but I've grown attached to the math and its the work I'm most proud of, which is why it's here free for anyone to read, btw..this paper is so honest it would stay out of politics, it's either right or wrong, no win for me if it's right (dog groomer remember) and no loss for me if it's wrong..enjoy what it is. Disclaimer. This paper has not been peer-reviewed. It is offered as an open preprint. Readers are encouraged to check the mathematics against the data sources cited (NASA Exoplanet Archive, Agol et al. 2021). Abstract The combined fixed-point and period-2 equation of the symplectic trace map Tn+1 = T2n − 2 is (T2 − T − 2)(T2 + T − 1) = 0, producing four roots—λ = 2, λ− = −1, 1/φ, −φ—from which every quantity in this paper descends. Part I develops the mathematics: the eigenvalue Riemann surface Σ, the thread map Θ(λ) = λ − 1/λ, the inversion duality network of seven classical integer recurrences (proved bipartite), and the Narayana ladder that probes the full surface. Part II derives the quartic from physics: Hamiltonian mechanics→ symplecticity → Cayley–Hamilton → trace map, with λ = 2 as the exact zero- mass limit of the Poincar´e return map. Part III tests the predictions across seven domains—algebra, solar system, KAM dynamics, TRAPPIST-1, the exoplanet population (⟨R⟩window = 71/35, 95% CI contains the prediction), Kerr black holes (f+ + f− = λ for all spins; QPO frequency ratios at τ = 3/2 in five X-ray binaries; quasi-normal mode monodromy constants ln 3 and ln 5 identified as the Jacobsthal root separation and the pivot), and photonic quasicrystals (log Tn/ log Tn−1 → 2, testable on a laboratory bench). All results are dimensionless. No parameters are fitted.