The Universal Boundary Principle: KAM Stability, Hopf Topology, and WheelerDeWitt Cosmology

The entire framework of this paper flows from a single algebraic object: the characteristic polynomial λ2−kλ+1 = 0 of the symplectic transfer matrix T(k) ∈ SL(2,R). At each Arnold tongue of a parametric resonance, this polynomial assigns a distinguished eigenvalue λn. We prove that the sequence of tongue eigenvalues passes through exactly two algebraically distinguished values: λ2 = φ (the golden ratio, governing damped oscillatory dynamics via the Tau Initiative) and λ3 = 2 (the Jacobsthal eigenvalue, governing tunnelling and instability via the Universal Boundary Principle). No other tongue has an eigenvalue that is a positive integer. The paper develops both frameworks as adjacent cases of a uni ed tongue structure across ve domains: KAM stability, Kerr spacetime QPOs, WheelerDeWitt cosmology, Hopf topology, and exoplanet orbital mechanics. It contains ve proved theorems, three lemmas, two corollaries, and zero free parameters. The central results are: (1) the Tongue Uniqueness Theorem, proving n = 3 is the unique tongue with integer eigenvalue; (2) the Floquet Jacobsthal Theorem, proving µ = ln2 at the Jacobsthal point; (3) the Boundary-Topology Theorem, proving the eigenvector bundle is the Hopf bundle (c1 = 1); (4) the Spatial Dimension Corollary, deriving d = c1 + dimR(Σ) = 1 + 2 = 3 from the bundle structure with no dimensional input; and (5) the λ-decomposition showing the swap identity I0 × Iswap 0 =1−1/λ4 =15/16. Empirical con rmations include: TRAPPIST-1 (p = 3.21 × 10−5); Beta Pictoris (30 36AU consistent with 3035AU, post-hoc); REBOUND clustering at 43/21 = λ+1/J6; and power-law exponent α = −2/φ = −1.236 matching −1.26±0.05 at 0.48σ

Keywords

Universal Boundary Principle, Jacobsthal sequence, SL(2,ℝ) transfer matrix, parabolic boundary, Floquet theory, parametric resonance, eigenvalue Riemann surface, Hopf fibration, Euler class, Wheeler–DeWitt equation, minisuperspace tunnelling, Kerr spacetime, high-frequency quasi-periodic oscillations, KAM stability, symplectic dynamics, mean-motion resonance, TRAPPIST-1, Beta Pictoris, Jacobsthal ratio identity, discrete–continuous duality, Universal Boundary Principle, Tau Initiative, Jacobsthal sequence, Fibonacci sequence, golden ratio, Arnold tongue, tongue uniqueness, SL(2,ℝ), transfer matrix, parabolic boundary, Floquet theory, parametric resonance, eigenvalue Riemann surface, Joukowski map, characteristic polynomial, Hopf fibration, Euler class, first Chern class, Heegaard splitting, spatial dimension, Wheeler–DeWitt equation, minisuperspace, Hartle–Hawking instanton, tunnelling action, Kerr spacetime, high-frequency quasi-periodic oscillations, KAM stability, symplectic dynamics, Abel's theorem, swap identity, discrete–continuous duality, mean-motion resonance, TRAPPIST-1, Beta Pictoris, REBOUND N-body, Jacobsthal ratio identity, Diamond Lattice, Lichtenberg sequence, Mersenne numbers, power-law exponent, Universal Boundary Principle, Tau Initiative, Jacobsthal sequence, Fibonacci sequence, golden ratio, Arnold tongue, tongue uniqueness, SL(2,ℝ) transfer matrix, parabolic boundary, Floquet theory, parametric resonance, eigenvalue Riemann surface, Joukowski map, Hopf fibration, Euler class, Heegaard splitting, Wheeler–DeWitt equation, minisuperspace tunnelling, Hartle–Hawking instanton, Kerr spacetime, high-frequency quasi-periodic oscillations, KAM stability, symplectic dynamics, mean-motion resonance, TRAPPIST-1, Beta Pictoris, REBOUND N-body, Jacobsthal ratio identity, discrete–continuous duality

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