O18 established the minimal fibre condition: Born-Infeld parity is the only global symmetry of S₁₈, ensuring that each fibre of the non-injective projection is exactly \, -\. O22 proved the shell-alignment conjecture of O21, and O23 derived the threshold condition c (n₃) = 3 from the quaternionic maximality of the admissible neutral traceless sector. Together, these results show that the integer 3 in c (n₃) = 3 is a property of Im Nₓₑ₋, not of the cardinality of the fibres. The condition of O18 is therefore stronger than necessary: the programme requires only that any symmetry of S₁₈ acts vertically with respect to, i. e. does not enlarge the real rank of the admissible neutral traceless sector. The present paper proves this verticality condition directly from Born-Infeld admissibility and the quaternionic maximality established in O23, without requiring a classification of C₄₅₅. As a corollary, the transfer chain_ \;\; ₀₈ₑ \;\; ^* = 1₀₈ₑ + 1{2} 0. 126 established unconditionally with respect to the fibre structure of. Non-injectivity may increase microscopic multiplicity, but cannot enlarge the admissible observable structure.
Jérôme Beau (Thu,) studied this question.