Domain Number Mathematics IV: Finite-Resolution Integral Charts, Positive-Density Geometry, and Unit-Circle Weight Certification

Domain number mathematics treats requested mathematical resolution as part of the problem data and permits finite relations or regions to be exact target objects. Earlier work developed this viewpoint for dependency-preserving rational relation networks, finite increment calculus, differential and summation domains, finite gluing, and canonical-chart coverage for second-order differential equations. The present paper develops the integral branch. For positive integral densities we introduce a factored chart representation $W=BR$, where the base weight B is part of a declared certification language and is fixed before residual-chart optimization. Only the positive residual shape R is assigned the chart resolution studied here; any cost of certifying base-cell masses is recorded separately. The symmetric multiplicative density defect d_Σ(R,C)=|R-C|/R+C is equivalent to the log-density metric artanhd_Σ=12|log R-log C|. It is invariant under common positive factors and under common changes of integration variable. Positive linear integral operators do not increase this defect. Classical operations such as linearity, change of variables, integration by parts, symmetry reduction, domain decomposition, algebraic decomposition, recurrence generation, and finite-core/tail splitting remain exact relation transformations; resolution loss occurs only at a later compression or certification step. For a continuous positive residual R on a compact interval and a declared class of M positive constant-residual charts, the exact minimax coverage threshold is δM^*(R)=\!(osc(log R)/4M). This gives an exact chart-budget law. The family R(t)=(1-zt)-γ is solved explicitly, including optimal cell boundaries, chart constants, and finite integral enclosures. On the rational-parameter slice, it also exhibits a certification-class transition at γ=1: finite-resolution chart geometry varies continuously, while the zero-resolution integral changes from algebraic to logarithmic. With a Jacobi base weight, the same theorem yields a unit-circle weight family containing Beta, Gauss-hypergeometric, Appell/Lauricella, and complete elliptic integral examples. For K, E, and Π, the residual coverage problem is explicit and algebraic after removing logarithms. Finally, integral certification is separated into shape resolution and base-mass resolution. Their multiplicative resolutions compose by δₜₒₜ≤ δₛₕₐₚₑ+δₘₐₛₛ1+δₛₕₐₚₑδₘₐₛₛ, or additively in artanh coordinates. For the elliptic Jacobi base, a unit-circle coordinate followed by a rational tangent coordinate reduces every base-cell mass to an integral of 1/(1+u²) on $[0,1]$, producing explicit rational finite-stage certificates. Fixed-budget residual-coverage certificates for the Jacobi--Lauricella family, and finite algebraic enclosing regions for the elliptic integral values K, E, and Π, can therefore be lowered to rational {} networks without evaluating those special functions themselves.