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Zenodo
2026
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| Online Access: | https://doi.org/10.5281/zenodo.20150933 |
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Table of Contents:
- <p>In memory of Shirley.</p> <p>The eigenvalue equation λ2−kλ+ 1 = 0 of SL(2,R) classies dynamics by the trace k: elliptic (|k|<2), parabolic (|k|= 2), hyperbolic (|k|>2). We show that the parabolic boundary λ= 2 is uniquely a two-dimensional phenomenon: among all n-nacci companion matrices (n≥2), the dominant eigenvalue approaches 2 from below as n→∞but reaches it only in the 2 ×2 Jacobsthal case. Forcing a 3 ×3 SL(3,R) matrix to contain eigenvalue m 2 requires the remaining eigenvalues to satisfy ab= 1/2, with real solutions existing only when the trace exceeds 2+√2the silver ratio µ2 plus 1, which maps back to trace index 2 by the Metallic Thread Theorem Θ(µn) = n. The algebraic core includes proved identities: the Coates Quartic (T2−T−2)(T2 + T−1) = T4−4T2−T+ 2 = 0 linking Jacobsthal and Fibonacci recurrences; the regime-alternation theorem showing Jacobsthal ratios strictly alternate across the parabolic boundary; and the normalised period hierarchy, in which exact elliptic rotation periods divided by the base period at k= 0 produce the framework's eigenvalues and Fibonacci numbers (λ= 2 as period 8 in quartic units, 3/2 = Θ(λ) as period 6, F(5) = 5 as period 20).</p> <p>The empirical programme applies transfer-matrix observables to protein sequences (where the Perez Code encoding, derived independently from atomic masses in 1999, arrives pre-calibrated to the parabolic boundary by a forced +2 shift) and to exoplanetary period ratios (where the zero-parameter prediction ⟨R⟩= 71/35 ≈2.0286 is conrmed to within 0.0013 by 57 pairs from the NASA Exoplanet Archive). This document is a map, not a proof. It catalogues what is proved, what is observed, what is conjectured, and what has failed.</p>