Realizability of all admissible flatness values S

Establish that for every integer V ≥ 4 and every integer flatness value S in the admissible parity-sensitive range (0 ≤ S ≤ 3V/2 − 6 for even V and 0 ≤ S ≤ 3V/2 − 13/2 for odd V), there exists a genus‑0 polyhedron in normal form (equivalently, a simple 3‑connected planar graph) realizing E = 3V − 6 − S and F = 2V − 4 − S, i.e., that every admissible S is realized by some such polyhedron.

Background

The paper defines the flatness parameter S := ∑f(deg f − 3), which measures deviation from a fully triangulated boundary. Using proved identities E = 3V − 6 − S and F = 2V − 4 − S together with vertex-degree constraints, the author derives tight parity-sensitive ranges for S. The claim that every integer S within these ranges is realizable is treated as a heuristic/conjectural assertion supported by constructive families and exhaustive checks for small V, but without a general proof.

Formally confirming realizability of all admissible S would complete the external combinatorial classification at this level, showing that the derived ranges are not only necessary but also sufficient. This would bridge the gap between the proved bounds and the existence of corresponding genus‑0 normal‑form polyhedra for each S.