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 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 ψ''+p(x)ψ=0, we derive the exact Liouville--Green defect DLG(x)=|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(x₀)=0, p'(x₀)≠0, a minimal tangent-Airy chart has defect DA(x)=|p'(x₀)|-2/3|p(x)-p'(x₀)(x-x₀)|. Locally, DLG~ d⁻³ while DA~ d², giving a generic leading crossover law. We then define chart-class-relative certification sets and the minimax coverage resolution δcoverK(P;I)=x∈ Iinfχ d_χ(x). For Hermite normal form, the full-line coverage threshold is determined exactly by a unique algebraic crossover and satisfies δcoverK~(5/(64A))2/5. For Bessel normal form, the three parameter regimes λ=ν²-1/4<0, $=0$, and $>0$ are resolved separately. The no-turning-point branch has the exact threshold 8/(25|λ|); the exceptional orders |ν|=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 |ν|=1/2, while the nonuniformity disappears on every truncated interval [a,∞) 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.