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

Abstract

O21 defined the observable shell $n_{3}$ by the threshold $\Sigma_{c}(n_{3}) = 3$ and used it as structural input. This paper determines the exact status of the value three. The positive result is conditional: if the admissible neutral sector is carried by an irreducible two-dimensional ${\mathrm{SU}}(2)$-valued $V_{\rho} \cong \mathbb{C}^{2}$, its traceless neutral sector is ${\mathfrak{su}}(2) \cong {\mathrm{Im}\,\mathbb{H}}$, of real dimension exactly three. Two bridges separate this theorem from a derivation of the threshold, and both are open. On the first, carrier selection, the paper determines what one candidate source supplies. The fingerprint vectors of the deposited O12 filtration are pure Fourier modes, so every proper level is a coordinate subspace with single-line Weyl support, and the exact stabiliser of that level in the Weil image of ${\mathrm{SL}}(2,\mathbb{Z}/q\mathbb{Z})$ is $U \rtimes M_{n}$, inside a Borel subgroup. This excludes the exceptional binary polyhedral groups $2T$, $2O$ and $2I$ at every proper level, including at primes where they exist in the ambient group. It does not exclude a spinor carrier: at explicit generic levels at $q = 53$ and $q = 101$ the stabiliser is $\mathrm{Dic}_{q}$, whose restricted Weil module is multiplicity-free, one character plus four inequivalent two-dimensional irreducibles, exactly two of them faithful and ${\mathrm{SU}}(2)$-valued. The source therefore supplies genuine spinorial content, and the difficulty is selection, not existence. It splits in two: selecting which prime, block and depth the projection singles out within the generic filtration, which nothing here addresses; and, inside a level already known to be dicyclic, choosing between the two admissible carriers, for which a structurally motivated candidate — the unique odd member of $\{c_{\Sigma}, q - c_{\Sigma}\}$ — succeeds on all $17$ audited instances over $13$ distinct subspaces, with one step observed and not proved. The second bridge is observable identification: no theorem identifies the cumulative Gram–Schmidt span $\Sigma_{c}$ with the dimension of the neutral traceless module, so the threshold remains a supplied selection rule. A caution accompanies the positive result: 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 ADE observation of three eigenvalue classes is retained as a consistency check, not as evidence.