Three Neutral Directions from a Spinor Carrier: Conditional Status of the Threefold Admissibility Threshold

Abstract

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 ${\delta_{\mathrm{pair}}} \to {\beta^{*}}$. The present paper determines the exact status of the value three in that threshold. The positive result is conditional: if the admissible neutral sector is carried by an irreducible two-dimensional ${\mathrm{SU}}(2)$-valued representation $V_{\rho} \cong \mathbb{C}^{2}$, then its traceless neutral sector is the real Lie algebra ${\mathfrak{su}}(2) \cong {\mathrm{Im}\,\mathbb{H}}$, of real dimension exactly three. This is a representation-theoretic theorem about the supplied carrier, and it explains why a spinorial neutral sector carries three independent directions. Two bridges required to convert this theorem into a derivation of $\Sigma_{c}(n_{3}) = 3$ are open and are stated here explicitly. First, the carrier selection: no result of the programme derives $V_{\rho} \cong \mathbb{C}^{2}$ from Born–Infeld parity; the parity input of O18 is a covariance of the response family and does not force the fibre structure, so the spinor carrier is a supplied input. Second, the observable identification: no theorem identifies the cumulative Gram–Schmidt span $\Sigma_{c}$ of O21 with the dimension of the neutral traceless module; since $n_{3}$ is defined as the shell where $\Sigma_{c}$ reaches three, the threshold is a supplied selection rule, not a derived quantity. A structural caution accompanies the positive result: a symmetric neutral generating set need not span three axes. In the faithful two-dimensional representation of $Q_{8}$, the set $\{\pm\mathbf{i}, \pm\mathbf{j}\}$ generates the group with only two generator axes; the third direction arises only under commutator closure, an operation not supplied by admissibility. The ADE spectral observation of three non-trivial eigenvalue classes is retained as a consistency check compatible with, but not probative of, the threefold structure.