A Truncated Quasiconformal Energy for the Riemann ξ-Function and Sharp Extremal–Length/Teichmüller Lower Bounds

We define a concrete “truncated quasiconformal energy” $E_xi(T)$ associated to the Riemann $xi$–function on a height window $|t|<T$. The definition is geometric: one selects a canonical family of disjoint level corridors, anchored to representative off–critical zeros (one per level, if any exist), and considers the least possible quasiconformal dilatation needed to move those symmetric puncture pairs toward the critical line subject to a corridor–control constraint. We then prove sharp extremal–length lower bounds of the form [ E_xi(T) ge log!left(frac{Mod(Gamma^{mathrm{src}}_xi(T))}{Mod(Gamma^{mathrm{tgt}}_xi(T))}right), qquad d_xi(T):=tfrac12 E_xi(T) ge tfrac12log!left(frac{Mod(Gamma^{mathrm{src}}_xi(T))}{Mod(Gamma^{mathrm{tgt}}_xi(T))}right), ] and we compute the moduli explicitly in terms of corridor widths in a uniform level decomposition. These inequalities are unconditional consequences of extremal length and do not prove the Riemann Hypothesis. Their role is to produce a mathematically precise “energy ladder” $Tmapsto E_xi(T)$: each finite window yields a finite-stage energy optimization problem, while any divergence $E_xi(T)toinfty$ as $Ttoinfty$ is an infinite-energy obstruction to a global bounded-distortion axis-landing deformation in the chosen corridor-controlled class.