Prove
Lemma 10. Let T be a six-label triangle system on C . Every vertex link is a…Lemma 11. Suppose f:RP^3→Δ^9 is admissible, let K be its support complex, and…Proposition
First observe that if two tetrahedra differ by exchanging labels a,b , the same exchange carries their complementary triangle systems to one another. Their complements share five labels, on which both systems are the triangles of K . After the exchange these triangles remain fixed, so the systems agree everywhere by Lemma 10(4). Suppose three tetrahedra T∪a , T∪b , T∪c contain the same triangle T . Put R=[10](T∪a,b,c) , which has four elements. Transporting the complementary systems successively by R∪b,c (a b)R∪a,c (b c)R∪a,b (c a)R∪b,c preserves the initial system. The composite fixes R and exchanges b and c , contradicting Lemma 10(3). Thus at most two tetrahedra contain any triangle. Positive even multiplicity gives exactly two.