Withdrawal Notice for O31: Unsupported Colour-Triplet Co-admissibility and Gauge-Group Claims

Abstract

This version replaces the O31 paper in the same documentary chain. It is a withdrawal notice, not a corrected paper. Eight results of the previous version are withdrawn: the uniqueness of ${\mathrm{SU}}(3)$ among connected compact subgroups acting irreducibly on ${\mathbb{C}}^3$; the irreducibility of the group of Born–Infeld-compatible fibre-preserving transformations; the co-admissibility of the individual profiles $\sigma_{c_i}$; the arithmetic criterion that colour triplets exist if and only if $q \equiv 1 \pmod 3$; the metaplectic identity as used, which changes the central character while the automorphism invoked fixes the centre; the $\Delta(27)$ spectral proposition; the mistyped rank-eight test in $\mathrm{End}({\mathbb{C}}^3)$; and every derivation of a gauge factor or of the Standard Model gauge group ${\mathrm{SU}}(3) \times {\mathrm{SU}}(2) \times \mathrm{U}(1)$. Three counterexamples are given in full. ${\mathrm{SO}}(3) \subset {\mathrm{SU}}(3)$ is a proper connected compact subgroup acting irreducibly on ${\mathbb{C}}^3$, so irreducibility with determinant one never singles out ${\mathrm{SU}}(3)$. The diagonal torus $T^2 \subset {\mathrm{SU}}(3)$ satisfies every constraint that was imposed while acting reducibly, and requiring each sector to be mapped to itself in fact forces reducibility. For every prime $q \geq 7$ the set $\{1, 2, q-3\}$ is an admissible colour triplet under the published definition, so the arithmetic criterion is false. The remaining scalar, arithmetic and discrete-group material is under re-audit and is not asserted. No new positive conclusion about groups, gauge factors, or the Standard Model gauge group is asserted here, and none should be inferred. The previous version remains available and citable in this record's version history, for traceability.