TEBAC HP IV: Complex-Time Wedge Continuation and Rigidity (E3 Closure)

This module develops the complex-time (CT) Laplace representation and wedge bounds for the HP--III discrepancy transform\[E_{\mathrm{TP}}(s)\;:=\;\partial_s \log\!\Bigg(\frac{D_{\mathrm{comp}}(s)}{\xi_R(s)}\Bigg),\qquadz := s-\tfrac12.\]We express the normalized transform\[G(z)\;=\;\frac{E_{\mathrm{TP}}\!\left(\tfrac12+z\right)}{2z}\]as a Laplace transform of an $L^1$ kernel and derive boundary/sector control on CT rays. We then formulate a resolvent/Laplace forcing functional (Route~B) whose vanishing implies $E_{\mathrm{TP}}\equiv 0$ on the wedge, hence $D_{\mathrm{comp}}/\xi_R$ is constant and fixed by the $E2N$ normalization. The Route~B forcing step is discharged in HP--III by the tight-frame/Parseval package: combining the exact bookkeeping identity with the seed-cancellation proposition yields $\Phi_N(z)\to 0$ on an interval $z>0$ and hence identically by the identity theorem; see HP--III (Prop.~III-seed-cancel, Thm.~III-parseval-force). This Zenodo record contains the LaTeX source and compiled PDF of the HP-IV module. Cross-module references to HP-III are resolved in the master build (shared bibliography/aux files).