Quantum Angle: Ideles, Adeles, and Canonical Decomposition

This paper presents a formulation of Quantum Angle entirely in the language of adeles and ideles over the cyclotomic number field $$ K = \mathbb{Q}(\zeta_5), $$ where $$ \zeta_5 = e^{2\pi i/5}. $$ No physical postulates are introduced. All structure follows from the canonical decomposition of the archimedean idelic component and from the global product formula. Canonical Idelic Decomposition For an idele x, define the archimedean logarithmic map $$\ell_\infty(x) = (\log |x|_{v_1}, \log |x|_{v_2}).$$ This admits a canonical sum–difference decomposition $$\Sigma(x) = \log |x|_{v_1} + \log |x|_{v_2}.$$ $$\Delta(x) = \log |x|_{v_1} - \log |x|_{v_2}.$$ The coordinate Σ represents global scale, while Δ represents relative imbalance between the two complex embeddings. Product Formula and Structural Asymmetry For any global element ($$ u \in K^\times $$), the product formula holds: $$ \sum_v \log |u|_v = 0. $$ For ( S )-units, this reduces to the exact constraint $$ \Sigma(u) + \log |u|_{\mathfrak{p}} = 0, $$ where ( \mathfrak{p} ) is the unique prime above ( 5 ). As a consequence: the coordinate ( $$ \Sigma $$) is globally constrained by the ( \mathfrak{p} )-adic valuation, the coordinate ( $$ \Delta $$) is unconstrained by the product formula. This asymmetry is intrinsic and arithmetic. Distinguished Unit and Regulator Direction Define the cyclotomic unit $$ J = 1 + \zeta_5^2, $$ with archimedean absolute values $$|J|_{v_1} = \varphi^{-2}, \qquad |J|_{v_2} = \varphi^{2}.$$ where $$ \varphi = \frac{1+\sqrt{5}}{2}. $$ Thus $$ \Sigma(J) = 0, \qquad \Delta(J) = -4 \log \varphi. $$ The quantity |Delta(J)| defines the fundamental regulator step in the difference coordinate. Emergence of the Cartan Action For ( u \in K^\times ), write $$ \sigma_i(u) = r_i e^{i\theta_i}, \qquad r_i > 0. $$ Then $$ \Delta(u) = 2(\log r_1 - \log r_2). $$ Define the rapidity parameter $$ \eta = \frac{1}{4}\Delta(u). $$ The map $$ A(\eta) = \begin{pmatrix} e^\eta & 0 \ 0 & e^{-\eta} \end{pmatrix} $$ defines the Cartan one‑parameter subgroup of ( \mathrm{SL}(2,\mathbb{R}) ). This defines the Cartan one‑parameter subgroup of SL(2,R). Thus the hyperbolic (Cartan) direction of SL(2,R) emerges canonically from the idelic coordinate $$\Delta$$ without additional assumptions. Quantum Angle The Quantum Angle is defined as the equivalence class $$ \theta_Q := \Delta ;; \mathrm{mod};; \Delta(J)\mathbb{Z}, $$ representing angular position on the circle $$ \mathbb{R} / (4\log\varphi)\mathbb{Z}. $$ Discreteness arises from the ( $$ p $$ )-adic valuation, while continuity arises from the regulator of the unit group. Realizations Two appendices are provided: Appendix A specifies a concrete geometric realization (TWIST‑6D). Appendix B formulates a realization‑dependent selection principle yielding high‑precision numerical constants. These realizations are not required for the validity of the canonical idelic structure. Subject Classification (MSC 2020) 11R42 — Zeta functions and ( L )-functions of number fields11F85 — ( p )-adic theory11R27 — Units and factorization81R05 — Finite‑dimensional groups and algebras No physical assumptions are required. All structure arises from arithmetic.