What does this equation mean?

The bill here is tidal, not electromagnetic or mechanical, and it is the one place this construction is bounded directly by what has actually been observed rather than by an idealization. A rigid processor of size $\ell$ falling radially experiences a stretching tidal acceleration between its two ends of approximately $a_{\rm tidal}(r) \approx \frac{2GM}{r^3}\,\ell$. which at the horizon itself scales as $\ell$ $c^6$/(4$G^2$$M^2$) — falling as the inverse square of the mass. For a processor of laboratory size $\ell$=1\,$\mathrm{cm}$ at the horizon of a black hole with the mass measured for the progenitors of the first LIGO gravitational-wave detection, around 30\,$M_\odot$ , this tidal stress is already about 1.1$\times$10^5\,$\mathrm{m\,s^{-2}}$ , on the order of ten thousand Earth gravities across a single centimeter — enough to disassemble ordinary hardware well before the horizon is reached [ 16 ] . For the Sagittarius A* mass used above, the same formula gives roughly 6$\times$10^{-6}\,$\mathrm{m\,s^{-2}}$ across the same centimeter: negligible, and consistent with the free-fall approximation actually holding at the horizon of a sufficiently large hole [ 14 ] . “Free fall is locally ordinary” is therefore not a universal statement about black holes; it is a statement whose validity is bounded, by real astrophysical data, to holes massive enough that the tidal term stays small over the processor’s own size — a condition satisfied at Sagittarius A*'s horizon and badly violated at a stellar-mass horizon of the kind LIGO has actually detected.