We close the E3 uniqueness step in the TEBAC Hilbert--Pólya program for $\mathrm{GL}(1)$ by deriving a full $\mathrm{Wedge}(\mathrm{GL}_1)$ package for the reference-subtracted remainder kernel $R(t)$ from complex-time heat kernel bounds of Davies type. These sectorial complex-time estimates on the remainder channel imply that the associated odd remainder transform $H(s)$ (in the centred variable $s=\tfrac12+z$) extends to an entire function of order $\le 1$ with uniform strip and half-plane control, and a concrete Phragmén--Lindelöf uniqueness argument then forces $H\equiv 0$. An interface lemma identifies $\partial_s\log Q(s)$ in terms of $H(s)/z$, so that the determinant quotient $Q(s)=D_{\mathrm{GL}(1)}(s)/\xi(s)$ is forced to be constant; the canonical normalization finally fixes $Q\equiv 1$, and hence $D_{\mathrm{GL}(1)}(s)\equiv \xi(s)$. Build: pdflatex (run twice for cross-references).\\ Companion baseline: ''TEBAC Hilbert--Pólya for GL(1): baseline construction (E2/GS5 companion paper)''.