An exact dyonic black hole at the integrable locus of quadratic nonlinear electrodynamics

In the quadratic nonlinear electrodynamics $-F+aF^{2}+bG^{2}$, dyonic black holes are known in closed form only when $a=0$, because the electric field solves a cubic whose coefficients carry the radius. We identify the locus $b=a/2$, where that dependence cancels, the magnetic charge decouples from the constitutive relation, and the field equations integrate exactly. Trading the areal radius for the field strength then gives an exact dyonic solution, the radial quadrature evaluating to Gauss hypergeometric functions, with the cosmological constant retained. It recovers the dyonic Reissner--Nordstr\"om, magnetic and Euler--Heisenberg branches, and at first order resums the perturbative dyonic lapse function, whose symmetry under interchange of the charges breaks at second order. The horizon structure contains a window of three horizons requiring both charges. The same change of variable renders the electric Smarr integral, previously stated to have no closed form, hypergeometric. Closed forms for the density and pressures show the electric charge enlarging the region where the dominant energy condition holds beyond the magnetic bound $\rH^{4}>4ap^{2}$; three curvature laws arise at the centre, one a relegated singularity. The extended thermodynamics closes with the coupling and pressure as variables, and the critical ratio falls below the Maxwell value $3/8$. The two photon cones split the shadow into a doublet reaching half a percent, while the geodesic light ring and scalar ringdown stay almost blind to the coupling.