## 6 Conclusions
* We proved that transitions to the infinite indigation theory of \(k=\{1,2,3,\ldots\}\), which is already at a main frequency of \(\Omega_{1}=\{\omega\}\), result in nonlinearities that are connected with other nonlinearities (to numbers of \(n_{1},n_{2},\ldots n_{\rm k}\)) and relaxation modes \(\rho_{k}^{(\omega)}(\omega)=\sum_{n=1}^{\infty}\Re_{k}^{(\omega)}(n_{i}\tau) \mathrm{cos}\left(\frac{\tau\mathrm{m}\chi}{\mathrm{d}}\right)\) (expression (A10a)); moreover, they are connected to fundamentally new dependencies \(\mathrm{Re}\left[\hat{\varepsilon}^{(\Omega_{1}=\{\omega\})}(\omega;T)\right]\) and \(\mathrm{Im}\left[\hat{\varepsilon}^{(\Omega_{1}=\{\omega\})}(\omega;T)\right]\) over a wide range of temperatures and field strengths (expression (106)). Ion tunneling has a significant effect on the frequency laws (106) in the field of diffusion relaxation. Moreover, the results of the linear theory of dielectric losses in the HBC [1] are a more specific case of more general Formulas (106). * Transcendental Equations (137) that were independent of each other were constructed to calculate critical temperature \(T_{cr,relax}\), separating the temperature regions (zones) into diffusion (\(T<T_{cr,relax}:T_{n,D}=\frac{T_{D}}{n^{2}}<T_{M};\frac{T_{D}}{T_{M}}<n^{2}\)) and Maxwell (\(T>T_{cr,relax}\)\(T_{M}<T_{n,D}=\frac{T_{D}}{n^{2}};n^{2}\frac{T_{M}}{T_{D}}<1\)) dielectric relaxation types. In the quasiclassical kinetic theory [2; 3; 4], the relaxation time for the n-oh relaxation mode \(T_{n}=\frac{T_{n,D}T_{M}}{T_{nD}+T_{M}}\) is revealed via the diffusive relaxation times of \(T_{n,D}(T)=\frac{T_{D}(T)}{n^{2}}\), \(T_{D}(T)\); the same is the case for the 1-oh relaxation mode and Maxwell relaxation time \(T_{M}(T)\). * Based on the quasi-classical expressions generalized at the fundamental frequency of the alternating field \((\omega)\) for the real and imaginary components of complex dielectric permittivity (CDP) (\(\mathrm{Re}\left[\hat{\varepsilon}(\omega;T)\right]\),\(\mathrm{Im}\left[\hat{\varepsilon}(\omega;T)\right]\)--see (106)), an expression was constructed to calculate the tangent of the dielectric loss angle (150) \(\mathrm{tg}^{(\omega)}(\omega;T)=\frac{\Gamma_{2}^{(\omega)}(T)}{1-\Gamma_{1}^{( \omega)}(T)}\), in combination with quasi-classical relaxation parameters \(\Gamma_{1}^{(\omega)}(T)\) and \(\Gamma_{2}^{(\omega)}(T)\) (see (119a) and (119b) in analytical representations (120), (121), and (122)). * The quasiclassical formulas generalized at the main frequency of variation field \((\omega)\) are constructed and analyzed for the CDP component \(\left[\hat{\varepsilon}^{(\omega)}\right]^{\prime}_{M}\), \(\left[\hat{\varepsilon}^{(\omega)}\right]^{\prime}_{M}\), and \(\left[\mathrm{tg}^{(\omega)}\left(\frac{T_{D}}{T_{M}};\omega T_{M}\right) \right]_{M}\) in the Maxwell relaxation field (\(T>T_{cr,relax}:n^{2}\frac{T_{M}}{T_{D}}<1\)) (see (157)-(159)). On this basis, theoretical graphs of the studied functional dependencies \(\left[\mathrm{tg}^{(\omega)}\left(\frac{T_{D}}{T_{M}};\omega T_{M}\right) \right]_{M}\) and \(\left[\hat{\varepsilon}^{(\omega)}\left(\frac{T_{D}}{T_{M}};\omega T_{M}\right) \right]_{M}\) were obtained, reflecting the nonlinear effects associated with the influence of dimensionless parameters \(\alpha_{1}=\frac{T_{\rm D}}{T_{\rm M}}\) and \(\alpha_{2}=\omega T_{\rm M}\) on the dielectric loss mechanism (Figures A1 and A2). Theoretical spectra analyses were performed for \(\left[\varepsilon^{(\omega)}\right]^{\prime}_{\rm M}\)\(\left[\varepsilon^{(\omega)}\right]^{\prime}_{\rm M}\), and \(\left[\text{tg}\delta^{(\omega)}\left(\frac{T_{\rm D}}{T_{\rm M}};\omega T_{ \rm M}\right)\right]_{\rm M}\) in the \"deep\" Maxwell relaxation area (T \(\gg\text{T}_{\text{cr, }relax},n^{2}\frac{T_{\rm M}}{T_{\rm M}}\ \ll\ 1\)) (see (161)-(163)).