Two models compared per (m, k, chord-set):
  STEINER-RICH: G triangulates each O-face using internal Steiner vertices,
    so each B_in edge dualizes to its own (degree-1) inside-O triangle.
    Chord set has NO effect on T'_{f'}; equivalent to step-1 baseline.
  STEINER-POOR: G does NOT further sub-triangulate O-faces beyond O itself.
    Each O-face becomes one G-face with degree = (B_in-edge count).
    Edge-3-colorable only if every O-face has ≤ 3 B_in edges.

case                                          n   P_e SR     D SR     U SR  |  P_e SP     D SP     U SP  | faces (cycle-position groups)
-------------------------------------------------------------------------------------------------------------------------------------------------
(4,4) no chord  [baseline]                    8      258   81/81    81/81   |       0    0/81     0/81   | {0,2,4,6}
(4,4) chord (0,2)  faces 2+2                  8      258   81/81    81/81   |     144   36/81    54/81   | {0,2} | {4,6}
(5,5) no chord  [baseline]                   10     1026  243/243  243/243  |       0    0/243    0/243  | {0,2,4,6,8}
(5,5) chord (0,2)  faces 2+3                 10     1026  243/243  243/243  |     240   36/243   90/243  | {0,2} | {4,6,8}
(5,5) chord (0,3)  faces 3+2                 10     1026  243/243  243/243  |     240   36/243   90/243  | {0,2,4} | {6,8}
(6,6) no chord  [baseline]                   12     4098  729/729  729/729  |       0    0/729    0/729  | {0,2,4,6,8,10}
(6,6) chord (0,3)  antipodal  faces 3+3      12     4098  729/729  729/729  |     408   36/729   90/729  | {0,2,4} | {6,8,10}
(6,6) chords (0,2)(3,5)  faces 2+2+2         12     4098  729/729  729/729  |    1536  216/729  456/729  | {0,2} | {4,10} | {6,8}
(3,4) no chord                                7      126   78/81    27/27   |       0    0/81     0/27   | {0,2,4,5}
(3,4) chord (0,2)                             7      126   78/81    27/27   |      48   36/81    24/27   | {0,2} | {4,5}
(6,4) no chord                               10     1026   81/81   549/729  |       0    0/81     0/729  | {0,2,5,8}
(6,4) chord (0,2)                            10     1026   81/81   549/729  |     480   36/81   342/729  | {0,2} | {5,8}
