Late-Time Fractional-Order Identification in Caputo Diffusion Equation

We study late-time identification of the Caputo order in a linear diffusion equation generated by a strictly positive self-adjoint operator with compact resolvent. For signed scalar observations $M_\alpha(t)=\sum_n a_nE_{\alpha,1}(-\lambda_nt^\alpha)$ satisfying $\sum_n|a_n|/\lambda_n<\infty$, we show that, after eigenspace grouping, every nontrivial observation has a finite first nonzero resolvent moment $S_m=\sum_n a_n/\lambda_n^m$. A uniform differentiated large-argument expansion of the Mittag-Leffler factor yields eventual strict monotonicity of $\alpha\mapsto M_\alpha(t)$ on admissible intervals avoiding the zeros of $1/\Gamma(1-m\alpha)$, hence uniqueness from one sufficiently late scalar measurement. For two measurements, $M_\alpha(\rho t)/M_\alpha(t)=\rho^{-m\alpha}(1+O(t^{-\alpha_0}))$, giving a log-ratio estimator with asymptotic-bias and relative-noise error bounds. For bounded observations, $S_m=\langle\mathcal A^{-m}\varphi,h\rangle$; for a finite rod, the leading point-sensor condition is $(\mathcal A^{-1}\varphi)(x_*)\ne0$. Counterexamples show the sharpness of the exclusions and noise interpretation.