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

Let L/Qₚ be a finite extension, let G = GSp₄/L, and let J = GU₂(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 GSp₄/GU₂(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 = GL₂ × Gₘ to a Kottwitz pairing, which is computed explicitly. The unique basic lift b1,M has Kottwitz invariant κM(b1,M) = (1,1). After fixing the exceptional B₂ = C₂ 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 E₂ - E₀ and E₂, and both evaluate to $-1$ on the refined endoscopic element. Consequently, ΔrigM / ΔqsM = κM(b1,M), sM = -1. The assertion is relative: no absolute normalization ΔqsM = 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 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/Qₚ 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.