Elastic Completion of the Transient Generalized Chaplygin Gas: A Dimensional Cascade with Evolving Rigidity

We revisit the transient generalized Chaplygin gas (GCG) with negative exponent, a dark energy model first identified by Sen and Scherrer (2005), in light of two recent developments: the DESI DR2 preference for thawing dark energy, and the long standing observation that supernova data mildly favor negative GCG exponents while perturbation theory forbids them through an imaginary adiabatic sound speed. We show that for the specific exponent alpha = -2 the model admits a closed one parameter family, w(a) = -1/(1 + epsilon a^3), whose CPL projection obeys w_a = 3 w_0 (1 + w_0) and remains inside the thawing, non phantom region preferred by current data. Inverting the DESI DR2 pivot measurement w_p = -0.954 +/- 0.024 at z_p = 0.31 yields a data preferred epsilon = 0.108 (range 0.05 to 0.17). The adiabatic instability, c_a^2 = 2w < 0, is cured by promoting the fluid to a relativistic elastic solid in the sense of Bucher and Spergel and of Battye and Moss: a shear modulus mu restores a real longitudinal sound speed. Defining an effective dimension d(a) = -3 w(a), we find that the minimal rigidity required for stability takes the parameter free form mu_min/rho = d(3 - d)/6, a parabola in the effective dimension peaking at 3/8 when d = 3/2, with the causal band 0 <= c_s^2 <= 1 giving mu/rho between d(3 - d)/6 and (3 + 2d)(3 - d)/12. Today this requires a soft network, mu/rho between 0.13 and 0.21. Finally, reading d(a) as the dimensionality of a forming network suggests a quantitative correlation with the measured morphology of the cosmic matter web: if both follow the same one parameter cascade with different clocks, the lag is exactly tau = (1/3) ln(epsilon_web / epsilon_DE). Using published NEXUS+ mass fractions to anchor the matter web at D_web(0) = 1.35 to 1.66 gives tau = 0.67 to 0.81 e-folds of expansion, equivalently the statement that the dark energy medium today occupies the dimensional state the matter web occupied at z of about 1. This and two further consequences are falsifiable with existing simulation outputs and upcoming DESI and Euclid releases.