| _version_ | 1866901656827854848 |
|---|---|
| author | Coates, David |
| author_facet | Coates, David |
| contents | <p>The order-2 linear recurrence an = an−1 + can−2 has dominant eigenvalue λ+(c) = (1 + √1 + 4c)/2. We prove that c= 2 is the unique positive real at which λ+(c) = c (equivalently, |λ−(c)|= 1). At this value the recurrence is Jacobsthal, and the cross-sector product Π(n) = Jn·φ−n rst exceeds λ= 2 at n= 9 = ∆J , yielding a resolution floor σ= φ−9 ≈0.01316.</p> <p>We test whether exoplanetary period ratios near 2:1 exhibit sub-structure at the Jacobsthal convergents Jn+1/Jn, using 1,564 consecutive period ratios from the NASA Exoplanet Archive. Working in log-ratio space, where the analysis is invariant under inversion R↔1/R, four results survive scrutiny. <br>(i) Kernel density estimation, stable across all nine bandwidths tested and under automatic Silverman-rule selection, finds peaks matching the convergents 21/11 and 43/21 to within 0.01.</p> <p>(ii) A focused likelihood ratio test in log-ratio space gives LR = 60.1, ∆BIC = 44.0, p < 10−12, stable across four of five analysis windows.</p> <p>(iii) The scatter of the 21/11 Voronoi cell equals φ−9 to 0.8% (bootstrap 95% CI [0.88, 1.11]).</p> <p>(iv) The thread transform R →(R2−1)/R equals τ = 3/2 to 0.1% at the 2:1 resonance.</p> <p>This prediction extends across seven resonances (3:2 through 5:1, 1,032 planet pairs total): at each integer or simple-fraction resonance R0, the observed mean thread matches (R2 0−1)/R0 to within 2.2%, and these predicted values are expressible as ratios of the quartic's constants λ, D, and ∆F at everyresonance tested.</p> <p>No free constants were fitted. All predictions are derived from (T2−T−2)(T2 + T−1) = 0 before comparison with data.</p> <p> </p> <p>celestial mechanics, planets and satellites: dynamical evolution and stability, methods: statistical, methods: analytical</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20261842 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Jacobsthal Sub-Structure in Exoplanetary Period Ratios and the Discriminant Resolution Floor Coates, David celestial mechanics, planets and satellites: dynamical evolution and stability, methods: statistical, methods: analytical <p>The order-2 linear recurrence an = an−1 + can−2 has dominant eigenvalue λ+(c) = (1 + √1 + 4c)/2. We prove that c= 2 is the unique positive real at which λ+(c) = c (equivalently, |λ−(c)|= 1). At this value the recurrence is Jacobsthal, and the cross-sector product Π(n) = Jn·φ−n rst exceeds λ= 2 at n= 9 = ∆J , yielding a resolution floor σ= φ−9 ≈0.01316.</p> <p>We test whether exoplanetary period ratios near 2:1 exhibit sub-structure at the Jacobsthal convergents Jn+1/Jn, using 1,564 consecutive period ratios from the NASA Exoplanet Archive. Working in log-ratio space, where the analysis is invariant under inversion R↔1/R, four results survive scrutiny. <br>(i) Kernel density estimation, stable across all nine bandwidths tested and under automatic Silverman-rule selection, finds peaks matching the convergents 21/11 and 43/21 to within 0.01.</p> <p>(ii) A focused likelihood ratio test in log-ratio space gives LR = 60.1, ∆BIC = 44.0, p < 10−12, stable across four of five analysis windows.</p> <p>(iii) The scatter of the 21/11 Voronoi cell equals φ−9 to 0.8% (bootstrap 95% CI [0.88, 1.11]).</p> <p>(iv) The thread transform R →(R2−1)/R equals τ = 3/2 to 0.1% at the 2:1 resonance.</p> <p>This prediction extends across seven resonances (3:2 through 5:1, 1,032 planet pairs total): at each integer or simple-fraction resonance R0, the observed mean thread matches (R2 0−1)/R0 to within 2.2%, and these predicted values are expressible as ratios of the quartic's constants λ, D, and ∆F at everyresonance tested.</p> <p>No free constants were fitted. All predictions are derived from (T2−T−2)(T2 + T−1) = 0 before comparison with data.</p> <p> </p> <p>celestial mechanics, planets and satellites: dynamical evolution and stability, methods: statistical, methods: analytical</p> |
| title | Jacobsthal Sub-Structure in Exoplanetary Period Ratios and the Discriminant Resolution Floor |
| topic | celestial mechanics, planets and satellites: dynamical evolution and stability, methods: statistical, methods: analytical |
| url | https://doi.org/10.5281/zenodo.20261842 |