A Covariant Action for the Elastic Dark-Energy Medium: the Verlinde-class anharmonic solid with the a0 = c^2 sqrt(Lambda/32pi) displacement law and the derived y_c = Z/2 budget cutoff (Branch B written, not adopted)

v3 update (the two 'defined computations' carried out): v1/v2 promised that the medium's shear fraction w and Cassini quadrupole Q2 are computations of the written theory, not posits. v3 carries them out with two independent solvers (an l=2 Navier boundary-value problem and a 2D variational energy minimization), both gated on reproducing the single-invariant scalar quadrupole in the mu_s->0 limit. They reduce to the identical P-wave form Q2 = Q2_scalar/(1 + 4 beta/kappa_t), with kappa_t = V''(1)/K_eff = 0.5 PINNED (footing-independent) and beta = mu_s/3K_eff the shear share. Shear rigidity genuinely suppresses the quadrupole (w beta_crit ~ 0.40. But the action fixes only the combination K_F + mu_s/6 = K_eff (the l=0 monopole), leaving beta -- the shear POISSON RATIO -- a genuinely free material parameter (bounded only to (0,2) by ghost-freedom and causality). This is a real two-moduli feature of isotropic elasticity, not an omission (the three-scalar solid EFT has shear and bulk as independent Lagrangian data). Neither microphysics nor cosmology closes it: Verlinde's emergent elasticity pins a framework-inadmissible Poisson ratio (nu=+1, negative bulk, excluded by K_F>0), and cosmological rigidity bounds on a w=-1 solid (induced gravitational slip ~ Omega_DE*beta/67 ~ 0.4-2%, below ~10% sensitivity; the customary c_v^2=mu/(rho+P) diverges at w=-1 and must be replaced by the finite mu_s/(rho_L c^2)=beta/67) do NOT cap beta below beta_crit. VERDICT: Cassini is decided by one free scalar straddling beta_crit; every natural elasticity value (Poisson-1/4 -> beta=2/7; the Verlinde shear magnitude -> 1/3; the natural-elasticity cluster [0.18,0.33]) lands below and marginally FAILS (x1.1-1.8), while a pass needs a medium shear-stiffer than any natural elasticity motivates. The branch is FORMALLY OPEN but the weight of evidence tilts to FAIL -- not the clean pass v1 hoped for, and not a clean kill. Honest floor of the Branch-B quadrupole question. v2 had corrected v1's minimal-sourcing overclaim to the nonminimal [1+h(J)]R gravitational sector. Branch B written, not adopted; no theory-of-everything claim.