Twisted Units, Celestial Symmetries and Entropic Motion: Adelic computations, dilogarithmic identities and hyperbolic dynamics in Q(ζ5)

TWIST constructs an explicit bridge between cyclotomic units in K = \(\mathbb{Q}(\zeta_5)\) and hyperbolic toral automorphisms generated by those units. We compute adelic and archimedean invariants, derive dilogarithmic identities linking the golden ratio \(\varphi\) and \(\pi\), and show how these arithmetic data determine dynamical invariants such as topological entropy and Lyapunov exponents. The deposit contains the main note, explicit formulas, and numerical verifications.Keywords: cyclotomic units; golden ratio; dilogarithm; toral automorphism; entropy; Lyapunov exponents; \(\mathbb{Q}(\zeta_5)\); ideles; adelic invariants Short description: This work makes concrete how a single cyclotomic unit in the 5th cyclotomic field produces an integer matrix acting on \(\mathbb{Z}^4\), how that matrix defines a hyperbolic automorphism of the 4‑torus, and how arithmetic invariants (archimedean norms, idelic data, and polylogarithmic identities) control dynamical quantities such as entropy. The note emphasizes explicit computations and checks, aiming to be accessible to researchers in number theory and dynamical systems.Introduction: Let \(K=\mathbb{Q}(\zeta_5)\) be the 5th cyclotomic field with \(\zeta_5=e^{2\pi i/5}\) and ring of integers \(\mathcal{O}_K=\mathbb{Z}[\zeta_5]\). By Dirichlet's unit theorem the unit rank is one, so a single non‑torsion cyclotomic unit already determines the archimedean logarithmic embedding of \(\mathcal{O}_K^\times\). TWIST focuses on the explicit arithmetic→dynamics map \[ \text{cyclotomic unit } u\in\mathcal{O}_K^\times \quad\Longrightarrow\quad \text{integer matrix }M_u\in\mathrm{GL}(4,\mathbb{Z}) \quad\Longrightarrow\quad \text{hyperbolic toral automorphism }T_u\text{ on }\mathbb{T}^4. \] We work concretely with the cyclotomic unit \[\huge J := 1+\zeta_5^2 \in \mathcal{O}_K^\times \] and present closed‑form computations of its conjugates, archimedean absolute values, the integer matrix representing multiplication by \(J\), and the resulting dynamical invariants.Main results (concise): - Explicit expressions for the Galois conjugates of \(J\) and their moduli; verification that \(J\) is a fundamental cyclotomic unit. - Construction of the integer matrix \(M_J\in\mathrm{SL}(4,\mathbb{Z})\) representing multiplication by \(J\) on \(\mathcal{O}_K\) with respect to the standard \(\mathbb{Z}\)-basis; computation of its characteristic polynomial and spectral data. - Proof that the induced toral automorphism \(T_J\) is hyperbolic, with stable/unstable splitting and Lyapunov exponents \(\pm\ln\varphi\) (each with multiplicity two), and topological entropy \(h(T_J)=2\ln\varphi\). - Derivation and numerical verification of dilogarithmic identities linking \(\varphi\) and \(\pi\), notably \[ \frac{\pi^2}{6}=\operatorname{Li}_2(\varphi^{-2})+\operatorname{Li}_2(\varphi^{-1})+2\ln^2\varphi. \] - Discussion of the idelic/product‑formula perspective tying local factors (archimedean and \(p\)-adic) to global dynamical behavior.The deposit contains: - A short, self‑contained note with proofs and explicit algebraic computations. - A section constructing \(M_J\) from multiplication by \(J\) and analyzing its integer matrix representation. - A dynamics section proving hyperbolicity, computing entropy and Lyapunov exponents, and describing the stable/unstable foliations. - An analytic section deriving the dilogarithmic identities and providing high‑precision numerical checks. - Appendices with numerical data, short proofs of classical lemmas used, and references to standard texts for background.Suggested citation: Please cite this Zenodo record and the accompanying PDF when using the computations or results contained herein.