Context: Beginning with the formal definition of transfinite spaces together with the Picker-Hansel theorem, we obviously get a relation $$ \bigcap\xi_{|\sigma|\mapsto \theta^*} \oplus_\psi \left(\frac{1}{1+\sigma^{(i+j+k)_{\tilde{\mathbb{P}}}}}\right) \sim \ell^{|\alpha|+1}+\frac{t^5}{\varphi(\mathbf{\Omega})}. $$ Thus, a transfinite space with quintic characteristic is not Hershel in the sense of $\epsilon$-Gram isometric transformation, but, is instead homomorphic to its $\psi^k$ Frobisher category. That is, given a form $\beta(a_\nu, \{\wp\mapsto \mathbb{P}\})$ of any partial order, we obtain $$\prod_{|\tau|\sim 1/\delta^2}\frac{{\mathbf{Y}}d\gamma}{\Delta\cup_{i\in\Sigma}\eta_k}=\left[\beta:\mu_+\right]$$ for (almost) all the continuous connections relating $\eta_{\tilde{k}}$ to its representation in Norman form. Hence the projection $$T_{\epsilon^{+}}(\kappa;\aleph^{*})\longrightarrow_{\nu_0} \mathbb{C}_{*}$$ is simply the Hodgkin map of even permuting scales of $\beta$. So, we get that all forms of $\psi_{x}$ are ascending to a non-trivial sequence solution of $$\Psi_x+\Omega_y=\sum_{\sigma\in \wp}\mathbf{Hodj}(\sigma)$$ but neither are connected to any of the $\langle\phi^i|\psi^j\rangle$ transpositions of $\mathbb{N}$ into $\chi(U)$. For example, if the operator $\mathcal{P}_\ast$ is simply connected (in the $\hat{\delta}$-Freighter sense), then its $\Gamma$ behaviour gives the non-vanishing decomposition $$\mathbb{T}_\mu=\bigoplus_{\ell\in\{\varphi\sim 1/\mathbf{G}\}} \mathbf{A}_\ell(\mathbb{N};(i,j,k)_{\omega}).$$ which is easily seen to be of quintic characteristic (by the Field theorem on transfinite proportions).
Question: Using these results, we get that $\chi_\theta(\xi;x^\alpha)$ of $\mathbb{T}_\mu$ is in any case a non-Hodgkin principal field transform. So, are all $\kappa$-subsets of the spectrum $\mathcal{B}_{\tau}$ equivalent to a system $$\mathcal{Y+I}=\omega_5(1,t)\quad(\text{mod}\ \overline{\mathcal{Y}}=0)$$ of Hall-Categories that are in overdiagonal expansions?
Partial Result: Computing the $\Pi$ map of a transfinite open spaces with quintic characteristics of second kind, I obtained that they transform according to $$ \mathfrak{R}(\mathbf{\Psi})=\int_{\gamma(t_1)}^{\gamma(t_5)} \mathbf{Kor} \begin{bmatrix} \alpha_0&\alpha_1&\alpha_2&\cdots&\alpha_n \\ \alpha_1&\alpha_2&\alpha_3&\cdots&\alpha_{n+1} \\ \alpha_2&\alpha_3&\alpha_4&\cdots&\alpha_{n+2} \\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \alpha_n&\alpha_{n+1}&\alpha_{n+2}&\cdots&\alpha_{2n} \end{bmatrix} d\mathfrak{H}. $$ It is then clear that from the Lie spectrum of $\mathfrak{F}$ we get that $\mathbf{\Psi}^{\overline{i\cdot j\cdot k}}$ is given by $$ \begin{array}{cccc} \mathfrak{H}: & \bigsqcup_{\alpha^{\mu_i}} \oplus \bigsqcup_{\beta^{\mu_j}}\oplus \bigsqcup_{\gamma^{\mu_k}}& \longrightarrow & \mathbb{C} \odot\mathbb{C}. \end{array} $$ But this is of course not sufficient to prove the general case.