This paper systematically fills four persistent gaps in the existing literature on the three-sphere \ (S³\): 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 \ (Z₂\) 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 \ (S³\), 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 \ (Sp (1) \) and \ (S³\), 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) (SU (2) (2) ) /Z₂\), 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 \ (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 \ (N₁= 1/a\), \ (N₂= 1/a²\) ; A rigorous uniqueness argument showing \ (gₐ=a²g₀\) forms the sole family of \ (SO (4) \) -invariant conformally flat metrics on \ (S³\) ; A spectral gap optimality corollary: among all compact 3-manifolds satisfying \ (Ric 2\), the round \ (S³\) uniquely attains the maximal first eigenvalue \ (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.
Q Zhao (Fri,) studied this question.