Finite mass two throat wormholes: global light rings, branch resolved strong lensing, and scalar transmission

Within general relativity, we study a static, horizonless, asymptotically flat two throat geometry constructed as an inverse problem: the metric is prescribed and the anisotropic stress tensor required by the Einstein equations is reconstructed, without assuming a specific matter Lagrangian. A finite mass deformation of an earlier quartic embedding profile produces two symmetric throats and an intermediate equator. The deformation fixes the same finite geometrized mass at both spatial infinities, while the lapse uses the same mass parameter as an asymptotic consistency condition. The complete radial averaged null functional is related explicitly to the affinely parametrized ANEC integral, obtained exactly, and shown to be strictly negative. We classify the global null potential for all positive masses and nonnegative redshift deformation. Depending on the parameters, the unstable set consists of throat rings, four off throat rings, or exterior rings accompanied by a subcritical equatorial ring. In the four ring phase, all unstable rings share one critical impact scale but the inner and outer branches have different Lyapunov exponents. For nondegenerate critical rings, we derive the logarithmic strong deflection coefficients analytically; the cross throat coefficient is the sum of the local contributions from every global maximum traversed by the ray, as verified by direct numerical integration. The minimally coupled massless scalar exhibits phase dependent barriers and resonant transmission as a test field propagation diagnostic. The model therefore isolates finite mass effects on averaged energy conditions, global light rings, branch resolved strong lensing, and scalar transmission within a single prescribed geometry.