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

Core contributions

• Conditional adjoint-dimension theorem (Theorem 3.1): if the neutral sector is carried by an irreducible two-dimensional unitary representation \[ V_\rho \cong \mathbb{C}^2, \] then its traceless sector is \[ \mathfrak{su}(2)\cong \mathrm{Im}\,\mathbb{H}, \] of real dimension 3. The carrier is a supplied input, not a derived one.
• Failure of generator counting: the neutral generating set \(\{\pm i,\pm j\}\) of \(Q_8\) spans only two axes of \(\mathrm{Im}\,\mathbb{H}\); counting neutral generators cannot yield the value 3.
• Exact stabiliser of a filtration level (Theorem 4.2, v2.1): the fingerprint vectors of the deposited O12 shell-span filtration are pure Fourier modes, so every proper level is a coordinate subspace whose Weyl support is a single line \(L\). Its exact stabiliser in the Weil image of \(\mathrm{SL}(2,\mathbb{Z}/q\mathbb{Z})\) is \[ \mathrm{Stab}(W_{<n}) = U \rtimes M_n \subseteq B(L), \qquad M_n=\{s\in\mathbb{F}_q^{\times} : sF_n=F_n\}, \] for every odd prime, every generic block and every proper nonzero level.
• Exceptional binary polyhedral groups excluded (Corollary 4.3, v2.1): \(B(L)\) is metacyclic, so none of \(2T\), \(2O\), \(2I\) is selected — including at the primes where they genuinely exist inside the ambient group, such as \(2I\) for \(q\equiv\pm1\pmod 5\).
• But a spinor carrier is not excluded (Proposition 4.5, v2.1): dicyclic groups are metacyclic too. At explicit generic levels at \(q=53\) and \(q=101\) the stabiliser is \(\mathrm{Dic}_q\), and the level decomposes multiplicity-freely as one character plus four inequivalent two-dimensional irreducibles, of which exactly two are faithful and \(\mathrm{SU}(2)\)-valued. The filtration therefore supplies genuine spinorial content.
• The obstruction is selection, not existence (v2.1): the carrier bridge splits into two open steps. Selecting which prime, block and depth the projection singles out within the generic filtration is untouched by any result. Inside a level already known to be dicyclic, choosing between the two admissible carriers has a structurally motivated candidate — the unique odd member of \(\{c_\Sigma, q-c_\Sigma\}\) — which succeeds on all 17 audited instances over 13 distinct subspaces, with one step observed and not proved.
• Open carrier selection: no result in the programme currently derives the choice of the irreducible two-dimensional unitary carrier from admissibility constraints alone.
• Open identification: the identification of the observable threshold \(\Sigma_c\) with the dimension of the traceless sector of the carrier is not proved; it is an open bridge.
• Status of the threshold: \(\Sigma_c(n_3)=3\) is a supplied selection rule in the spectral admissibility pipeline, consistent with — but not derived from — the conditional adjoint-dimension theorem.

Interpretation

O23 clarifies the logical status of the number 3 within the programme.

• O21: the threshold \(\Sigma_c(n_3)=3\) is used
• O23: the threshold \(\Sigma_c(n_3)=3\) is given a conditional algebraic reading

Under the supplied spinor-carrier hypothesis, the number 3 is the dimension of the traceless sector \(\mathfrak{su}(2)\cong\mathrm{Im}\,\mathbb{H}\) of the carrier. This is a genuine theorem — but it is conditional on an input that the framework does not currently select, and the identification of the observable threshold with that dimension is itself an open bridge.

In other words, the programme holds:

• a proved conditional theorem (carrier \(\Rightarrow\) three neutral directions)
• a proved no-go (generator counting cannot give 3)
• a supplied selection rule (\(\Sigma_c(n_3)=3\)) awaiting carrier selection and identification

Relation to the Cosmochrony program

O23 occupies a clarifying position in the O-series. After the intrinsic construction of the saturation rank in O21 and the derivation of shell locking in O22, O23 delimits exactly what is proved and what is supplied about the three-dimensionality of the stable sector.

The programme now reads: O16 (pair observable), O17 (pair dynamics), O18 (fibre structure), O19 (canonical normalisation), O20 (persistence criterion), O21 (intrinsic saturation rank), O22 (projection locking and shell condition), O23 (conditional status of the threshold dimension).

After O23, the observable threshold is intrinsic and shell-realised, and its value 3 is understood conditionally: it matches the adjoint dimension of a supplied spinor carrier, while carrier selection and the identification of \(\Sigma_c\) with that dimension remain open.

Current outcome and open directions

O23 establishes that, on a supplied irreducible two-dimensional unitary carrier, the neutral traceless sector is \[ \mathrm{Im}\,\mathbb{H}, \] of dimension 3. The threshold value \[ \Sigma_c(n_3)=3 \] remains a supplied selection rule: the threshold problem is conditionally understood, not structurally closed.

Remaining directions include:

• Carrier selection: derive from admissibility constraints why the neutral sector should be carried by an irreducible two-dimensional unitary representation.
• Threshold identification: prove (or refute) the identification of \(\Sigma_c\) with the dimension of the traceless sector of the carrier.
• Effective shell selection: determine the numerical value of \(n_3\), not only the value of the threshold.
• Large-\(q\) asymptotics: study the behaviour of the mechanism in the large-graph limit.
• Universality: test extension beyond SU(2)-type structures and Heisenberg graphs.
• Transfer closure (settled negatively): the continuation \[ \delta_{\mathrm{pair}} \to \beta^* \] is not an open deliverable: the span-growth no-go proves it has no derived carrier on the Heisenberg measurement substrate, so the numerical agreement with the charged-lepton window stands only as a cross-substrate phenomenological check.

References

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