From Cain to Matignon: Exact Positive-Diagonal Stability in Dimension Three and a Nonlinear Caputo Ecological Realization

Positive-diagonal D-stability asks whether stability persists under every positive diagonal left scaling. We replace the classical Hurwitz half-plane by the Matignon sector for commensurate Caputo systems and solve the resulting real three-dimensional problem on the full-dimensional strict-$P(-A)$ stratum. Four scale-invariant quantities $({\beta }_{12},{\beta }_{13},{\beta }_{23},\kappa )$ reduce the complete positive-diagonal orbit to an open simplex, and for $2/3<\alpha <1$ we prove the exact criterion $\kappa <T_{\alpha }\left(\beta \right)$. At $\alpha =1$, $T_{\alpha }$ collapses to Cain’s classical threshold, while $T_{\alpha }>T_1$ for every $2/3<\alpha <1$, producing an explicit genuinely fractional band. The threshold objective is strictly convex in logit coordinates, has a unique nondegenerate optimizer, and yields the minimum-dimension result: three is the minimum dimension in which the genuinely fractional multiplicative-D-stable class has nonempty full-dimensional interior. We then construct a nonlinear three-species Caputo Kolmogorov realization with double-Allee basal growth and intraguild predation. A natural competing-consumer architecture is proved unable to enter the fractional-only band, whereas the intraguild-predation architecture realizes an open biological subset of it. Varying only the Allee threshold moves the fixed biological system transversally across Cain’s classical D-stability boundary into the fractional-only Matignon region. A reproducible integration of the full nonlinear model at a certified orbit point displays decaying Caputo transients at $\alpha =0.9$, while the same parameters, rate scaling and initial perturbation leave a fixed local neighborhood at $\alpha =1$.