Ricci-flow instability on $G_{2,4}\mathbb{R}$

We study the equal-radius product Einstein metric on$\mathbb{S}^2(a)\times\mathbb{S}^2(a)$ and its quotient by the diagonal antipodalinvolution. The geometric setting is motivated by the identifications\[G_2^+(\mathbb{R}^4)\simeq \mathbb{S}^2\times\mathbb{S}^2,\qquadG_2(\mathbb{R}^4)\simeq(\mathbb{S}^2\times\mathbb{S}^2)/\mathbb{Z}_{2}.\]The product metric has Einstein constant $\Lambda=a^{-2}$. We giveexplicit curvature and normalized Ricci-flow calculations, and weidentify the parallel transverse-traceless tensor $h=g_1-g_2$ satisfying$\Delta_Eh=-2\Lambda h$. This negative direction is invariant underthe diagonal antipodal involution and therefore descends to theunoriented quotient. The main spectral result is a dimension-independent parity theorem forproducts of round spheres: it gives the complete scalar spectrum, anexplicit formula for the first positive eigenvalue, and an exactheat-trace projection identity. The four-dimensional Grassmannian isthen treated as a distinguished specialization. We also derive aparity-selection rule for the one-form spectrum on that quotient. Scalar product harmonicsof bidegree $(\ell,m)$ descend precisely when $\ell+m$ is even. Itfollows that the first positive scalar eigenvalue is $4a^{-2}=4\Lambda$on the quotient, whereas it is $2a^{-2}=2\Lambda$ on the orienteddouble cover. We also derive the corresponding parity rule for exactand coexact one-form harmonics. Thus the quotient changes the scalarspectral threshold but does not remove the negative TT direction.By the instability theorem of Kr"oncke, both metrics are dynamicallyunstable under volume-normalized Ricci flow. The product instability itself is known in the general theory ofpositive Einstein products. Accordingly, we do not claim thatinstability as a new theorem. The contribution developed here is anexplicit, convention-controlled comparison of the oriented cover andthe diagonal antipodal quotient, including the resulting spectralselection rules and their interaction with the TT instability mechanism.We also prove that the zero eigenspace of the TT Lichnerowiczoperator is exactly one-dimensional on both the cover and quotient.A complete decomposition of its positive TT spectrum is not claimed.