Observability Spectrum of Hopf Bifurcation: Mode Visibility Order, Defect Stratification, Observation Calculus, and Statistical Transfer

Problem. A nondegenerate Hopf bifurcation produces a periodic orbit with square-root amplitude scaling, but a general smooth observable does not, in general, inherit a unique statistical critical exponent. The first order at which phase modes appear, cancellations induced by symmetry, and the choice of a reference value along the equilibrium branch can all alter the critical behavior of an observable in distributional, fluctuation, and frequency-domain statistics. This paper studies this observation-layer structure and organizes it into a mode-resolved theory of critical transfer. Method. For a local family of Hopf periodic orbits Γ_ρ(θ), we define the observation excess Y_h relative to the moving equilibrium-branch value h(x_s(η)) and subject it to a finite-order weighted Fourier–Taylor expansion. Let q̲_m = 2 for m = 0 and q̲_m = |m| for m ≠ 0, and define the visibility defect order d_m(h) := (q_m(h) − q̲_m)/2, where q_m(h) is the first nonzero critical order of the m-th dynamical Fourier mode. We prove q_m = q̲_m + 2d_m and strictly distinguish q̲_m, determined by the Hopf weight lattice, from d_m, determined by finite-jet cancellations of the observation. Main results. First, we prove that the periodic invariant measure satisfies W_p(ν_ρ, δ_{x_c}) = C_{p,ν} ρ + O(ρ²) and has a first-order blow-up limit in the fixed critical tangent space. Second, we establish the factorization a_m(ρ) = ρ^{|m|} φ_m(ρ²), which yields on the supercritical parameter side a_m(η) = C_m(h) η^{q_m(h)/2}[1 + O(η)]. We then prove a generic visibility baseline in the finite-jet sense, discrete-symmetry selection rules, a finite-codimension stratification of visibility defects, and stability of visible order under small observational perturbations. We also establish composition rules under addition, multiplication, and smooth scalar post-processing, including the equilibrium-reference term. At the statistical level, we unify the distribution, W_p distance, periodic mean, variance, general absolute moments, power spectrum, and autocorrelation of the leading visible waveform through the same first-order visibility data. We prove a critical limit for the renormalized spectral measure and a uniform sharp convergence result for autocorrelation relative to the critical phase variable. We further show that a nondegenerate scalar post-processing can preserve the overall leading order while generally changing the modal spectrum. Subcritical periodic branches, the strictly degenerate Bautin path, and a general geometric exponent are discussed as theoretical interfaces. Theoretical positioning. We do not present either the harmonic order of classical Hopf periodic solutions or the recent singular behavior of the time-averaged zero mode as the main original contribution. Rather, building on these results, we study the full modal visibility of general smooth observables, visibility-defect stratification, composition calculus, and transfer to probability measures and frequency-domain statistics. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently determined by the author. Multiple AI tools assisted with code implementation, data presentation, and text drafting. The author bears full academic responsibility for all research content.