Attachment to the letter on arXiv:2609.14218 (Yoo, Escrivà, Harada & Kohri), collisionless-particle runs at e = 0.2

Files
  ltb_check_2609_14218.py            self-contained script (python3 + numpy + scipy); reproduces both tables in 1-3 min
  table1_collapse_times.csv          exact growing-mode LTB collapse times vs. the alpha_0 drop times read from Fig. 7
  table2_nakedness_causal_bound.csv  global-nakedness boundary (limit null ray from the central singularity) and
                                     causal-bound quantities (trapping radius r_1, enclosed mass, trapping time)

Conventions (paper's): k = 1, a_i = 1, H_i = 5k, entry at k/(aH) = sqrt6, r_m = sqrt6/k, t_H = 100 sqrt6/k (eq. 4.16).
The LTB solution is the constant-bang-time solution of eqs. (4.10) and (4.12) written with the Misner-Sharp mass
m(r) = (H_i^2/2)(Psi^2 r)^3 (= m_paper r^3/6) and E(r) = -k~ r^2/2.

Table 1 columns
  t_C(0)/t_H          central shell-focusing time, = 0.589 e^{3 mu} mu^{-3/2} t_H (the paper's t_s(0))
  t_C(r_m)/t_H        collapse time of the shell at r_m
  t_AH(r_m)/t_H       time at which the r_m shell reaches R = 2m in the collapsing phase
  linear deadline     1.686 / (0.883 mu) to the power 3/2 (top-hat with the linear mean density inside r_m)
  alpha0 drop Fig.7   midpoint of the alpha_0 drop in Fig. 7, read by eye (about +-10%); please replace by your numbers
  ratio               alpha_0 drop / t_C(r_m): 0.71-0.77 for mu = 0.05-0.30
  0.735 x t_C(r_m)    horizon time predicted if that ratio persisted (219, 260, 314, 392 t_H for mu = 0.045, 0.04, 0.035, 0.03)
  Fig.10 onset        time at which the max-norm Hamiltonian violation in Fig. 10 rises to O(1), read by eye; compare with t_C(0)

Table 2
  The limiting outgoing radial null ray from the regular centre launched at t -> t_C(0)^- (Cauchy horizon of the
  central singularity) is integrated with dt/dr = R'/sqrt(1+2E).  TRAPPED = the ray enters a region with 2m/R > 1 and
  dR/dt < 0 before reaching the underdense region (r = 3 r_m); ESCAPED = globally naked.  Boundary between mu = 0.325
  and 0.330, i.e. the paper's "globally naked at 0.32, PBH at 0.33".  For trapped cases, all shells beyond r_1 are
  inside their own apparent horizon before the ray reaches them, so those trapped spheres lie outside the causal future
  of the central singularity; m(r_1) is the mass they enclose, in units of M_H(t_k) = 1/(2 H_k).

The Hellaby-Lake block of the script checks that no shell crossing precedes the central singularity for this profile
(M' > 0, t_B' = 0, t_C'(r) >= 0 for every mu up to 1), which the causal argument needs.
