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The aim of the present manuscript is to derive an expression for the lowerbound of the modulus of the Dirichlet eta function on vertical lines$\Re(s)=\alpha$. The approach is based on a two-dimensional principal componentanalysis built on a parametric ellipse, to match the dimensionality of thecomplex plane. The lower bound $\forall s \in \mathbb{C}$ s.t. $\Re(s) \in\mathcal{P}$, $| \eta(s) | \geq \left| 1 - \frac{\sqrt{2}}{2^\alpha} \right|$,where $\eta$ is the Dirichlet eta function is related with the Riemannhypothesis as $|\eta(s)| > 0$ for any $s \in \mathbb{C}$ s.t. $\Re(s) \in\mathcal{P}$, where $\mathcal{P}$ is a partition spanning one half of thecritical strip on either side of the critical line $\Re(s)=1/2$ depending upona variable. We propose the composite lower bound $\forall s \in \, \mathbb{C}$s.t. $\Re(s) \in \,]1/2,1[$, $|\eta(s)| \geq \text{Min}\left(1-\frac{\sqrt{2}}{2^{\alpha}},\frac{\sqrt{2}}{2^\alpha}-\frac{\sqrt{2}}{2}\right)$,resulting from transitivity of $\eta: \mathbb{C} \to \mathbb{C}$ in $\eta(s) =\left(1-\frac{2}{2^s} \right) \zeta(s)$. The solution space of the set ofsolutions referring to such $\mathcal{L}^2$-problem is a representation of thespace spanned by explanatory variables satisfying its algebraic form, as afoundation of a Harrison Ford proof (see the little proof).

A lower bound for the modulus of the Dirichlet eta function on a partition $\mathcal{P}$ from 2-D principal component analysis