Coercivity structure of positive-type memory: exact gaps, critical horizons, and singular limits

We study diffusion equations with positive-type memory in the degenerate regime where the instantaneous diffusion may lose coercivity. The basic question is simple: can a completely monotone memory term replace the missing $L^{2}(0,\Tend;V)$ coercivity? The answer is negative in the instantaneous energy space. The obstruction is measured by the memory coercivity symbol $m$, defined through the Bernstein representation of the kernel and equal to $\operatorname{Re}\hat{k}$ whenever $k\in L^{1}(0,\infty)$. For kernels of finite $L^{1}$-mass, an exact frequency identity expresses the gap between the instantaneous energy and the memory dissipation as the spectral weight $1-m(\omega)$; provided that the memory form is non-trivial, the gap is non-negative for all states and all time horizons precisely when $\norm{k}_{L^{1}(0,\infty)}\leq1$. At a fixed horizon, the threshold is instead the finite-horizon coercivity profile $\Lambda_{k}(\Tend)$, whose unit crossing defines a critical horizon and which applies also to kernels of infinite $L^{1}$-mass, including the fractional kernels. For every locally integrable completely monotone kernel, however, $m(\omega)\to0$ as $|\omega|\to\infty$. Therefore, positive-type memory is dissipative, but it is not frequency-uniformly coercive: no constant $c>0$ makes the memory dissipation dominate $c\int_{0}^{\Tend}a_{1}(u,u)$. This is a no-go theorem, and we make the deficit quantitative through a coercivity-gap index $\rho\in[0,2]$, valid for every non-constant kernel. Finally, the whole coercivity structure is discontinuous under weak-$*$ convergence of the associated time measures. The graph-space well-posedness theory motivated by this no-go result, and the certified stability it targets, are developed in a companion paper.