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 \[ E_\xi(T)\ \ge\ \log\!\left(\frac{\Mod(\Gamma^{\mathrm{src}}_\xi(T))}{\Mod(\Gamma^{\mathrm{tgt}}_\xi(T))}\right), \qquad d_\xi(T):=\tfrac12 E_\xi(T)\ \ge\ \tfrac12\log\!\left(\frac{\Mod(\Gamma^{\mathrm{src}}_\xi(T))}{\Mod(\Gamma^{\mathrm{tgt}}_\xi(T))}\right), \] 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'' $T\mapsto E_\xi(T)$: each finite window yields a finite-stage energy optimization problem, while any divergence $E_\xi(T)\to\infty$ as $T\to\infty$ is an infinite-energy obstruction to a global bounded-distortion axis-landing deformation in the chosen corridor-controlled class.