  We present a construction employing a type IIA supergravity and 3-form flux
background together with an NS5-brane that localises massless gravity near the
5-brane worldvolume. The nonsingular underlying type IIA solution is a lift to
10D of the vacuum solution of the 6D Salam-Sezgin model and has a hyperbolic
${\cal H}^{(2,2)}\times S^1$ structure in the lifting dimensions. A fully
back-reacted solution including the NS5-brane is constructed by recognising the
10D Salam-Sezgin vacuum solution as a "brane resolved through transgression."
The background hyperbolic structure plays a key r\^ole in generating a mass gap
in the spectrum of the transverse-space wave operator, which gives rise to the
localisation of gravity on the 6D NS5-brane worldvolume, or, equally, in a
further compactification to 4D. Also key to the successful localisation of
gravity is the specific form of the corresponding transverse wavefunction
Schr\"odinger problem, which asymptotically involves a $V=-1/(4\rho^2)$
potential, where $\rho$ is the transverse-space radius, and for which the
NS5-brane source gives rise to a specific choice of self-adjoint extension for
the transverse wave operator. The corresponding boundary condition as
$\rho\to0$ ensures the masslessness of gravity in the effective braneworld
theory. Above the mass gap, there is a continuum of massive states which give
rise to small corrections to Newton's law.
