Bootstrapping black holes at low impact parameter
We use the stringy dispersion relation (SDR) to ask the following question about gravitational effective field theories: once the universal large-impact-parameter eikonal carrier is supplied, where does the remaining positive spectrum go? Working in six dimensions for concreteness, we include the complete high-spin continuum tail of the carrier and first work at weak coupling, where $M_{\rm Pl}>M_{\rm EFT}$. The extremal spectra contain a saturated low-impact band whose outer edge stays at roughly five to six times the inverse EFT scale even as the gravitational radius shrinks. The same edge is reproduced by a strict $G_N=0$ capped-SDR problem on the present grids. Thus the weak-coupling band approaches an intrinsic non-gravitational baseline of the capped extremal problem. On a common high-energy grid, a coupling ladder crossing $M_{\rm Pl}=M_{\rm EFT}$ resolves the cap-saturated support as a wedge: its outer envelope has the rotating black-hole homogeneity $G_N^{1/3}E^ {4/3}$, while its approximately linear lower envelope flattens as $G_N$ grows. We then use the deliberately reversed hierarchy $M_{\rm Pl}<M_ {\rm EFT}$ as a microscope for strong-gravity structure. In this regime a cap-saturated low-impact band follows an order-one Giddings-Porto rotating black-hole scale, a separate Regge-like ridge appears at high spin and low energies, the broad available region between these structures and the eikonal layer remains mostly empty, while at the far-tail of energy, series of Regge trajectories emerge.