Reasoning:
- Define inelasticity y ≡ MBH/√ŝ. The energy not trapped behind the apparent horizon is radiated gravitationally in the collision; classical results (from Penrose / D’eath-Payne in 4D, extended to higher D and nonzero impact parameter by Eardley-Giddings and refined by Yoshino et al.) give y ≲ 0.6 at b=0 for both the 10D (ADD) and 5D (RS) cases, falling with b - to y ≈ 0 (ADD) and y ≈ 0.2 (RS) at the largest horizon-forming impact parameter.
- To make a black hole of mass MBH one therefore needs parton energies larger by 1/y; because parton luminosities fall steeply, the impact-parameter-averaged cross section (eq. 2.21, weighting by 2z dz with z = b/bmax and threshold (xmin MD)²/(y(z)² s)) drops by orders of magnitude versus the geometric estimate (Figure 4), more strongly for ADD than RS.
- These classical y values are lower bounds computed near the Planck scale where quantum-gravity corrections are unknown - one more way the naive production picture is unreliable near threshold.
- Together with the thermality thresholds, this pushes genuinely semiclassical production even further out of the LHC’s effective reach.
Validity verdict (step 6)
Reconstruction: premises = classical trapped-surface inelasticity values (y <~ 0.6 at b=0, falling with b) and steeply falling parton luminosities; conclusion = impact-parameter-weighted cross section falls orders of magnitude below the geometric estimate. Traced the load-bearing step: making mass MBH requires parton CM energy MBH/y, and a steeply falling luminosity evaluated at higher invariant mass suppresses the rate by orders of magnitude; averaging over b with y(b) decreasing only strengthens this. Valid conditional on the y values; their classical near-Planck provenance is flagged in the body and priced downstream.