An Explicit Rigid Transfer Sign for the Inner Form of GSp4 and Two-Member Packet

Let $L/\mathbb{Q}_p$ be a finite extension, let $G = \mathrm{GSp}_4/L$, and let $J = \mathrm{GU}_2(D)$ be its unique non-split inner form, where $D/L$ is the quaternion division algebra. In the two-member endoscopic packet regime, fix the refined endoscopic datum used in the $\mathrm{GSp}_4/\mathrm{GU}_2(D)$ comparison, a Whittaker datum, and the corresponding rigid-inner-form normalization. The comparison of the inner-form transfer normalization with the quasi-split normalization reduces on the Siegel Levi $M = \mathrm{GL}_2 \times \mathbb{G}_m$ to a Kottwitz pairing, which is computed explicitly. The unique basic lift $b_{1,M}$ has Kottwitz invariant $\kappa_M(b_{1,M}) = (1,1)$. After fixing the exceptional $B_2 = C_2$ derived-root identification, a full lattice calculation shows that exactly two compatible unimodular extensions exist in the central direction; in the two extensions the class $(1,1)$ is represented on the dual Levi center by $E_2 - E_0$ and $E_2$, and both evaluate to $-1$ on the refined endoscopic element. Consequently, $$\Delta_{\mathrm{rig}}^M / \Delta_{\mathrm{qs}}^M = \langle \kappa_M(b_{1,M}), s_M \rangle = -1.$$ The assertion is relative: no absolute normalization $\Delta_{\mathrm{qs}}^M = 1$ is used. The two Levi identifications in the Chan–Gan endoscopic comparison differ by sign. Hence the computed rigid correction determines the member of the corresponding two-element orientation torsor compatible with the fixed modern rigid/refined datum. After removing the common group-level Kottwitz sign $e(J)$, the stable and non-trivial $\mathbb{Z}/2$ Fourier modes recover the two Harish-Chandra characters by rank-two Fourier inversion. The intrinsic sign computation uses no unramified or $p > 2$ hypothesis and therefore applies over every finite extension $L/\mathbb{Q}_p$ in the stated packet regime. This does not extend the stronger Fargues–Scholze/local-Shimura compatibility statements whose known proofs retain unramified $p > 2$ hypotheses.