General validity of the extended Euler-type identity V − E + F = 2(T − N_i + S_i)

Prove that the extended Euler-type identity V − E + F = 2(T − N_i + S_i) holds in full generality for genus‑0 decompositions described by the paper’s internal variables—T tetrahedra, N_i internal gluing triangles, and S_i internal triangulation segments—beyond the specific construction sequences for which it has been verified.

Background

The author introduces an extended Euler-type relation linking boundary counts (V, E, F) to internal decomposition counts (T, N_i, S_i). While the exact topological identity T − N_i + E_i − V_i = 1 is proved (and reduces to T − N_i + S_i = 1 in normal form), the explicit boundary-incidence formulation V − E + F = 2(T − N_i + S_i) is only established for certain construction sequences and presented as conjectural more broadly.

A general proof would formalize the incidence bridge between internal tetrahedralization parameters and boundary combinatorics for all admissible genus‑0 decompositions in the paper’s setting, removing reliance on construction‑specific arguments.