docs: correct sigma_LT/sigma_0 attribution in dPIE docstrings

autogalaxy/profiles/mass/total/dual_pseudo_isothermal_mass.py

@@ -25,8 +25,9 @@ def _b0_from_lenstool_sigma(
         b0 [arcsec] = 6 * 648000 * (sigma_LT / c)^2 * (D_LS / D_S)
 
     This is identical to ``4 * pi * (sigma_0 / c)^2 * D_LS / D_S`` (in radians) for the
-    central velocity dispersion ``sigma_0 = sqrt(3/2) * sigma_LT`` — the fiducial-vs-central
-    distinction of Eliasdottir et al. (2007) App. A / Bergamini et al. (2019).
+    central velocity dispersion ``sigma_0 = sqrt(3/2) * sigma_LT`` — a Lenstool parameter
+    convention (an E07-style b0 coefficient paired with the K93/L05 deflection amplitude),
+    as quoted by Bergamini et al. (2019); see the :class:`dPIEMass` docstring.
     """
     c_km_s = 299792.458
 
@@ -571,9 +572,10 @@ def from_lenstool(
 
         - ``sigma`` is Lenstool's **fiducial** velocity dispersion ``v_disp`` (sigma_LT),
           *not* the central velocity dispersion sigma_0 of the dPIE profile. They differ
-          by sigma_0 = sqrt(3/2) * sigma_LT (Eliasdottir et al. 2007, App. A; Bergamini
-          et al. 2019, Eq. 5) — quoting a measured central/aperture dispersion here
-          overestimates the mass by 50%. The lens strength is
+          by sigma_0 = sqrt(3/2) * sigma_LT — a Lenstool parameter convention (E07-style
+          b0 coefficient paired with the K93/L05 deflection amplitude; see the dPIEMass
+          class docstring), quoted by Bergamini et al. 2019, Eq. 5 — so a measured
+          central/aperture dispersion entered here overestimates the mass by 50%. The lens strength is
           b0 = 6 * 648000 * (sigma_LT / c)^2 * (D_LS / D_S) arcsec, where Lenstool's
           stored ``b0 = 6 * pia_c2 * sigma^2`` (``set_potfile.c``) carries no distance
           ratio — Lenstool applies D_LS / D_S separately at deflection time
@@ -1095,18 +1097,27 @@ class dPIEMass(dPIEMassB0):
       must be handled when ingesting real-data catalogues).
     - ``sigma`` — Lenstool's **fiducial** velocity dispersion ``v_disp``
       (sigma_LT, km/s), *not* the central dispersion:
-      sigma_0 = sqrt(3/2) * sigma_LT (Elíasdóttir et al. 2007, App. A;
-      Bergamini et al. 2019, Eq. 5). Quoting a measured stellar dispersion here
-      overestimates the mass by 50%.
+      sigma_0 = sqrt(3/2) * sigma_LT (the relation quoted by Bergamini et al.
+      2019, Eq. 5). The sqrt(3/2) is Lenstool's parameter convention, not a
+      physical definition from Elíasdóttir et al. (2007): it arises because
+      Lenstool computes ``b0 = 6 * pia_c2 * sigma^2`` (an E07-style
+      coefficient, ``set_potfile.c``) but evaluates deflections with the
+      Kassiola & Kovner (1993) / Limousin et al. (2005) amplitude, so the
+      input sigma must be read through the L05 parameterization (see the
+      contributed derivation note "On the definitions of b0 and velocity
+      dispersion in Lenstool / dPIE", H. Ding 2026). Quoting a measured
+      stellar dispersion here overestimates the mass by 50%.
     - ``r_core`` / ``r_cut`` — Lenstool ``core_radius`` / ``cut_radius`` in
       arcseconds (the internal ``ra`` / ``rs``). For ``.par`` files using the kpc
       variants, pre-convert with
       ``r_core = r_core_kpc / cosmology.kpc_per_arcsec_from(redshift=redshift_object)``.
 
     The lens strength is fully normalized internally:
     b0 = 6 * 648000 * (sigma_LT / c)^2 * (D_LS / D_S) arcsec — equivalently
-    E_0 = 6 pi (D_LS / D_S) (sigma_LT / c)^2 in radians (Elíasdóttir et al. 2007,
-    Eq. A24) with the E_0-to-b0 prefactor folded in. Lenstool stores its ``b0``
+    E_0 = 6 pi (D_LS / D_S) (sigma_LT / c)^2 in radians (the coefficient of
+    Elíasdóttir et al. 2007, Eq. A24) with the E_0-to-b0 prefactor folded in;
+    written in the central dispersion this is
+    b0 = 4 * 648000 * (sigma_0 / c)^2 * (D_LS / D_S). Lenstool stores its ``b0``
     without the distance ratio and applies D_LS / D_S at deflection time
     (``e_grad.c``); the two conventions are verified equivalent against the
     Lenstool C source and reference deflections
@@ -1149,6 +1160,8 @@ class dPIEMass(dPIEMassB0):
     Elíasdóttir et al. (2007), arXiv:0710.5636 (App. A).
     Limousin et al. (2005), MNRAS, 356, 309.
     Bergamini et al. (2019), A&A, 631, A130.
+    Ding, H. (2026), "On the definitions of b0 and velocity dispersion in Lenstool / dPIE",
+    contributed derivation note (PyAutoLens Slack, 2026-07).
     """
 
     def __init__(