Universal Power-Law Spectral Hierarchy on Compact Rank-One Symmetric Spaces Under Conformal Scaling: Critical Symmetry Breaking, Multi-Scale Error Control, and Spectral Invariant Coupling

This paper develops a complete analytical theory of the spectral geometry of all compact rank-one symmetric spaces (CROSS) — including spheres \(S^n\) (\(n≥2\)), real projective spaces \(RP^n\), complex projective spaces \(CP^n\), quaternionic projective spaces \(HP^n\), and the Cayley projective plane \(CaP^2\) — under global conformal scaling deformations \(g_a=a^2 g_0\), where \(a∈(0,+∞)\) is a scale parameter. For each space in this class, we derive the complete conformal-scaling spectral hierarchy with explicit coefficients determined by the space's dimension and algebraic type. The results are formulated in terms of the spectral parameters \((α,β)\) that uniformly characterize the CROSS Laplacian spectrum [1, 2, 3]. We establish four principal mathematical results. First, we derive the complete high-order power-law asymptotic hierarchy for the eigenvalues \(λ_k(a)\): \[ λ_k(α,β)(a)=k(k+α+β+1)/a^2+{γ_1(α,β)}{a^2}1/k^2+{γ_2(α,β)}{a^2}1/k^4+O(1/k^6), \] with explicit coefficients \(γ_1(α,β)\) and \(γ_2(α,β)\), and rigorous remainder bounds. For \(S^3\) (i.e., \(α=β=1/2\)), this reduces to the exact expansion with \(γ_1=1/4\), \(γ_2=-1/32\). Second, we identify the universal critical scale \[ acrit(α,β)(k)=√k+α+β+1/(α+β+1)k \] at which the scaling symmetry of the \(k\)-th eigenvalue breaks down, prove its monotonicity and limiting behavior, and construct the complete two-dimensional \((a,ε)\) phase diagram. For \(S^3\), this gives \(acrit(k)=√(k+2)/(2k)\). Third, we develop a three-layer multi-scale stratification — macroscopic, resonant, and microscopic — with layer-specific rigorous error bounds and the closed-form formula \[ N_0(α,β)(a,δ_0)= Γ({K_2(α,β)(a)}{δ_0})1/6 \] for the minimal truncation order required to achieve a prescribed uniform error tolerance. Fourth, we derive the complete system of coupled differential equations relating the scale parameter to the compactness invariant \(τ(a)=λ_1(a)/λ_2(a)\), the scale-dependent critical threshold \(εcrit(a)\), and the spectral entropy functional \(Sspec(a)\). The analytical formulas provide exact benchmarks for recent numerical studies [39,40,41,42] of conformally deformed spheres and other CROSS, correcting quantitative estimates, identifying exact symmetry-breaking thresholds with precision unattainable numerically, and offering rigorous error control with significant computational savings. The paper is purely mathematical in scope, with brief cross-disciplinary illustrations provided only as examples of the geometric framework’s applicability. The results fill a fundamental gap in the spectral geometry of compact curved manifolds under metric deformations and provide the analytical foundation for the companion papers in the T2 series [8, 9, 10].