Three Stable Directions from Quaternionic Minimality: Derivation of Σc(n3) = 3 from Born–Infeld Fibre Admissibility

O21 defined the observable shell $n_{3}$ via the threshold condition $\Sigma_{c}(n_{3}) = 3$ and used it as structural input for the fibre-level transfer ${\delta_{\mathrm{pair}}} \to {\beta^{*}}$. O22 established that Born–Infeld projection locking forces saturation to occur on a BFS shell, closing the question ``why a shell?'' The present paper closes the complementary question left open by O21: why is the relevant saturation threshold three-dimensional, i.e.\ why $\Sigma_{c}(n_{3}) = 3$ rather than $2$ or $4$? We show that this value is not an arithmetic input but a structural consequence of Born–Infeld fibre admissibility: the value three is not a spectral artefact but the unique dimension of the imaginary subspace of the minimal admissible associative non-commutative algebra. The argument proceeds in three steps. First, an exclusion lemma rules out abelian realizations of the admissible neutral sector and limits the candidate structures to finite subgroups of ${\mathrm{SU}}(2)$. Second, a quaternionic minimality theorem — anchored in the Hurwitz-to-${\mathfrak{su}}(2)$ bridge established in SpectralCapacity — identifies $\mathbb{H}$ as the minimal associative non-commutative structure compatible with Born–Infeld parity, and shows that the admissible neutral directions correspond to the imaginary subspace ${\mathrm{Im}\,\mathbb{H}}$, which carries exactly three independent axes. Third, a spectral realization section shows that the ADE binary graphs physically relevant to the programme realize this abstract ``$3$'' as exactly three distinct non-trivial spectral levels, confirming the structural derivation at the observable level. The result is conditional on the absence of additional effective symmetries beyond Born–Infeld parity, in accordance with the explicit scope of O18.