Zero free parameters Update includes, Null simulation for statistical variance. 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. These three numbers are carried by three integer sequences: Jacobsthal J(n) = (2n−(−1)n)/3 (the λ sequence), Fibonacci Lucas Fn, Ln (the φ sequences), and Lichtenberg 3m (the T sequence). The Pythagorean comma T12/λ7 = 312/219 is the permanent separation of the first and third: 2n is even, 3m is odd, they never meet. All results reported here are dimensionless period ratios, independent of any unit system. In three domains (I) pure algebra, (II) the solar system and EarthMoon system, (III) the TRAPPIST-1 exoplanet systemthe same three numbers emerge: the solar system spans J(11) = 683 Neptune-to-Mercury periods (0.17%); TRAPPIST-1 synodic ratios alternate T, λ, T, λ to four decimal places, with TRAPPIST-1 synodic periods landing on Lucas numbers L3 = 4 and L6 = 18; the lunar synodic month satises φn∗+1/λ ≈ L7+1/λ = 29.534 (0.013%), where the exponent n∗ = ⌊12·log2 (3/2)⌋ = 7 is the comma index the unique integer at which twelve threads overshoot n∗ eigenvalues and carries zero free parameters; and the Saros/apsidal precession ratio falls within 0.5% of the Jacobsthal convergent 43/21. A falsiable prediction: compact resonant chains discovered by the PLATO mission with mean adjacent ratio below 1.7 will show T-λ alternation in consecutive synodic period ratios. French version to follow Posting this to show why the scientific community needs a kick up the ass from an outsider, from a nobody like me. Science should be inclusive, not a pompous right.
David Coates (Thu,) studied this question.