What question did this study set out to answer?

To establish λ = 2 as a universal eigenvalue that organizes structures in various physical domains.

April 4, 2026Open Access

Four Domains, One Eigenvalue: Independent Evidence for λ = 2 as a Physical Law of Resonance Structure

Key Points

To establish λ = 2 as a universal eigenvalue that organizes structures in various physical domains.
Examining the eigenvalue λ = 2 at the boundary of SL(2,R) and its implications across several fields.
Analyzing integer sequences: Jacobsthal, Lichtenberg, and Mersenne to explore their relationships and structures.
Comparing findings in pure algebra, the solar system dynamics, TRAPPIST-1 ratios, and quantum mechanics.
Identified a consistent separation described by the Pythagorean comma across several domains.
Showed that λ = 2 is realized physically in structures from solar system harmonics to quantum mechanical principles.
Confirmed that the integer sequences demonstrate a permanent separation based on even and odd properties.

Abstract

The eigenvalue λ = 2 at the parabolic boundary of SL(2,R) generates a thread T = λ−1/λ = 3/2 and is approached from below by φ = (1+√5)/2. Three integer sequences Jacobsthal (2n side), Lichtenberg (3m side), and Mersenne (λn − 1, linking the two) form a pipeline: λn → Mn →J(n). The Pythagorean comma T12/λ7 = 312/219 is their permanent separation: 2n is even, 3m is odd, they never meet. We show this separation is physically realised in four independent domains: (I) pure algebra; (II) the solar system, where 432 = λ7×T3 is the unique 3-smooth concert pitch and Neptune maps to φ Hz; (III) TRAPPIST-1, where synodic ratios alternate T,λ,T,λ to four decimal places; (IV) quantum mechanics and the periodic table, where the Pauli exclusion principle enforces λ = 2 via the spin degeneracy 2s+1 = 2, and atomic shell capacities 2n2 alternate between powers of λ and powers of 3. From planets to electrons, the eigenvalue and its thread organise structure through the same incommensurable pair of sequences. La valeur propre λ = 2 à la frontière parabolique de SL(2,R) engendre un l T = λ − 1/λ = 3/2 et est approchée par φ = (1+√5)/2. Trois suites d'entiers Jacobsthal (2n), Lichtenberg (3m) et Mersenne (λn − 1) forment un pipeline : λn → Mn → J(n). Le comma pythagoricien T12/λ7 = 312/219 est leur séparation permanente : 2n est pair, 3m est impair. Cette séparation est physiquement réalisée dans quatre domaines indépendants : (I) l'algèbre pure; (II) le système solaire, où 432 = λ7 × T3 et Neptune = φ Hz; (III) TRAPPIST-1, où les rapports synodiques alternent T,λ,T,λ; (IV) la mécanique quantique et le tableau périodique, où le principe d'exclusion de Pauli impose λ = 2 via la dégénérescence de spin 2s + 1 = 2, et les capacités des couches 2n2 alternent entre puissances de λ et puissances de 3. Des planètes aux électrons, la valeur propre et son l organisent la structure par la même paire incommensurable de suites. Posting this to show why the scientific community needs a kick up the ass from an outsider, abandon the aristocratic corporate journals that have stifled progress and to the desk ethic jockey at APS journal, bite me .

Four Domains, One Eigenvalue: Independent Evidence for λ = 2 as a Physical Law of Resonance Structure

Key Points

Abstract

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