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\[ETP(s)\;:=\;∂_s log\!({Dcomp(s)}{ξ_R(s)}), := s-12.\]We express the normalized transform\[G(z)\;=\;{ETP\!(12+z)}{2z}\]as a Laplace transform of an L¹ kernel and derive boundary/sector control on CT rays. We then formulate a resolvent/Laplace forcing functional (Route~B) whose vanishing implies ETP≡ 0 on the wedge, hence Dcomp/ξR 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 ΦN(z)→ 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).