Global Riemannian, Lie Group and Spectral Geometry of the Three-Sphere: Hopf Fibration, SU(2)×SU(2) Isometry Decomposition and Complete Hyperspherical Harmonic Analysis

This paper systematically fills four persistent gaps in the existing literature on the three-sphere \(S^3\): the absence of a globally non-singular coordinate system; the restriction of Hopf fibration studies to purely topological treatments, without full Riemannian submersion decomposition; the lack of a rigorous proof for the free proper \(\mathbb{Z}_2\) quotient action within the \(SO(4)\) isometry decomposition; the missing unified geometric derivation of multi-scale spectral thresholds. We establish a complete, self-consistent closed-loop theoretical framework for \(S^3\), covering its topology, quaternion Lie group structure, Hopf Riemannian submersion, full hyperspherical harmonic analysis, conformal multi-scale geometry, and global spectral invariant uniqueness. Five logically independent, rigorously proven core theorems are presented: (1) Quaternion isometry between \(\mathrm{Sp}(1)\) and \(S^3\), with a full construction of the global \(SO(4)\) isometric action. (2) The Hopf fibration as a Riemannian submersion; explicit horizontal/vertical tangent space splitting, together with complete connection 1-form and horizontal lift equations. (3) The isomorphism \(SO(4)\cong(\mathrm{SU}(2)\times\mathrm{SU}(2))/\mathbb{Z}_2\), accompanied by full Palais proper action verification and orbit limit analysis. (4) A complete orthonormal basis of hyperspherical harmonics, equipped with an \(O(k^{-2})\) truncation error bound and comprehensive Sobolev regularity classification table. (5) Global uniqueness of the scale-invariant spectral ratio \(\tau^S=3/8\) under all isometric, conformal and symmetric metric deformations, with a self-contained uniqueness proof. We explicitly demonstrate the mutual logical independence of these five results: none of the five theorems relies on the statements or proofs of the remaining four. Additional novel geometric contributions include: A direct Hopf-fibration-based derivation of the multi-scale thresholds \(N_1=\lceil 1/a\rceil\), \(N_2=\lceil 1/a^2\rceil\); A rigorous uniqueness argument showing \(g_a=a^2g_0\) forms the sole family of \(SO(4)\)-invariant conformally flat metrics on \(S^3\); A spectral gap optimality corollary: among all compact 3-manifolds satisfying \(\mathrm{Ric}\ge 2\), the round \(S^3\) uniquely attains the maximal first eigenvalue \(\lambda_1=3\). We further outline direct connections to three longstanding open problems within compact rank-one symmetric space (CROSS) spectral geometry. Crucially, this manuscript is fully self-contained and entirely independent of all companion T/S/T3 preprint series. No theorem, lemma or proposition draws on external unpublished results from companion works. All derivations and proofs are closed within the present text; the manuscript can be fully read, refereed and published without referencing any other paper in the research suite.