Strong field response maps and cross-sector consistency relations for an inverse cubic Schwarzschild exterior

We construct a first-order response map for null and timelike strong field observables in a generic static spherical deformation of Schwarzschild spacetime written in areal radius gauge. The map separates lapse and radial metric contributions and gives closed expressions for the photon critical scale, photon orbit instability, circular orbit energy and angular momentum, innermost stable circular orbit frequency, and radial epicyclic dynamics. We then specialize the result to a Schwarzschild-completed inverse cubic exterior motivated by nonlocal gravitational effective actions. Eliminating the single deformation parameter yields parameter independent closure relations among asymptotic observables and a frequency resolved relation between the critical curve shift and the radial to azimuthal epicyclic frequency ratio. For an idealized axisymmetric polarized brightness ring tied to the critical angular scale, the total intensity and complex linear polarization visibilities follow zeroth and second-order Bessel responses; their nulls share one geometric dilation. Numerical solutions of the unexpanded truncated metric verify the linear response over a positive coupling interval in which the metric corrections remain below 1% in the photon region. Current horizon-scale measurements do not yet reach this calibration interval, so the formulation is an overconstrained theoretical consistency benchmark rather than a completed observational likelihood.

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