Domain Number Mathematics III: Finite-Resolution Canonical-Chart Coverage for Second-Order Differential Equations

Domain number mathematics treats requested mathematical resolution as part of the problem data and permits relation-valued or region-valued targets to be exact objects. This paper develops a concrete differential-equation application of that viewpoint. The underlying transformations are classical: second-order linear equations can be reduced to normal form, Liouville--Green transformations isolate oscillatory or exponential behavior, Pr"ufer-type coordinates separate phase and amplitude, and simple turning points are described by Airy functions. The finite-resolution layer studied here is not a rediscovery of these transformations. Instead, a declared class of simple canonical charts is treated as a \emph{certification language}, and the question is changed from ``solve the special-function equation'' to ``at what requested resolution does this chart class cover the target domain?'' For the normal form $\psi''+p(x)\psi=0,$ we derive the exact Liouville--Green defect $D_{LG}(x)=\frac{|5p'(x)^2-4p(x)p''(x)|}{16|p(x)|^3},$ which measures the residual after converting $p>0$ to unit-circle dynamics and $p<0$ to unit-hyperbolic dynamics. The same defect yields explicit phase and amplitude certificates. Near a simple turning point $p(x_0)=0$, $p'(x_0)\neq0$, a minimal tangent-Airy chart has defect $D_A(x)=|p'(x_0)|^{-2/3}|p(x)-p'(x_0)(x-x_0)|.$ Locally, $D_{LG}\sim d^{-3}$ while $D_A\sim d^2$, giving a generic leading crossover law. We then define chart-class-relative certification sets and the minimax coverage resolution $\delta_{cover}^{\mathcal{K}}(P;I)=\sup_{x\in I}\inf_{\chi\in\mathcal{K}} d_\chi(x).$ For Hermite normal form, the full-line coverage threshold is determined exactly by a unique algebraic crossover and satisfies $\delta_{cover}^{\mathcal{K}}\sim(5/(64A))^{2/5}$. For Bessel normal form, the three parameter regimes $\lambda=\nu^2-1/4<0$, $=0$, and $>0$ are resolved separately. The no-turning-point branch has the exact threshold $8/(25|\lambda|)$; the exceptional orders $|\nu|=1/2$ give exact unit-circle dynamics and zero threshold; and the turning-point branch has an exact half-line threshold equal to the maximum of a singular-endpoint contribution and two unique Airy--Liouville crossover values. The resulting full-half-line threshold is highly nonuniform at $|\nu|=1/2$, while the nonuniformity disappears on every truncated interval $[a,\infty)$ with $a>0$. Finally, the Hermite and Bessel coverage conditions are lowered to finite rational degree-two relation networks. Thus, in these examples, global coverage at prescribed resolution can be certified without evaluating Hermite or Bessel functions themselves. The results provide an explicit model of resolution-dependent solvability, chart-class dependence, and finite certification in domain number mathematics.