Status
The notice withdraws eight result families: individual-profile colour co-admissibility; the claimed arithmetic criterion for colour triplets; the metaplectic colour step; the proposition concerning $\Delta(27)$; the rank-eight test; uniqueness of $\mathrm{SU}(3)$; irreducibility as a consequence of the stated conditions; and every derivation of the Standard Model gauge group.
No weakened gauge-group theorem replaces them. The colour factor, determinant-one condition, commutation of factors, direct-product structure, and complete gauge group are open.
Explicit counterexamples
- Uniqueness: the proper subgroup $\mathrm{SO}(3)\subset\mathrm{SU}(3)$ acts irreducibly on $\mathbb{C}^3$ after complexification of its defining representation. The published uniqueness lemma is therefore false.
- Irreducibility: the diagonal torus $T^2=\{\operatorname{diag}(e^{i\alpha},e^{i\beta},e^{-i(\alpha+\beta)}):\alpha,\beta\in\mathbb{R}\}\subset\mathrm{SU}(3)$ satisfies the stated sector-preservation conditions while acting reducibly. Those conditions do not imply irreducibility.
- Arithmetic criterion: for every prime $q\geq7$, $\{1,2,q-3\}$ is an admissible triplet under the published definition. Hence existence is not equivalent to $q\equiv1\pmod3$.
Material under re-audit
Scalar observations, arithmetic statements, discrete-group claims and generator-twist isospectrality are not affirmed by the withdrawal notice. They require independent proofs before any part can be republished. The numerical values reported in O32 remain measurements, but their pointwise co-admissibility and gauge-group interpretation is withdrawn.
Programme consequence
O31 no longer supplies an edge from the spectral admissibility programme to a colour gauge factor. Any downstream Yang–Mills construction is conditional on a separately supplied compact gauge group. Neither $\mathrm{SU}(3)$ nor $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ is derived by this chain.
Reference
Jérôme Beau. Withdrawal Notice for O31: Unsupported Colour-Triplet Co-admissibility and Gauge-Group Claims. Version 2.0, 2026.