A Clifford-Orbit Assembly of the Binary-Tetrahedral Fourier Transform

The group Fourier transform of the binary tetrahedral group 2T = SL(2,3) = Q8 : C3 resists a naive tensorfactorisation F3 ⊗ FQ8 because the group is a genuine semidirect (crossed) product. We reconstruct the correctassembly explicitly. Writing the transform as W = U2 · (I3 ⊗ FQ8)†, the residual operator U2 is not a globalproduct but a direct sum over the C3-orbits of the Q8 dual: a trivial-character sector, a sign-character orbitsector, and a two-dimensional-irrep sector, of dimensions 3, 9 and 12. We give the representation identitybehind the shorthand 24 = 3 ⊕ 9 ⊕ (2×6), verified to machine precision by exact block-diagonalisation of theregular representation (maximum off-block 1.35×10−15). Two structural facts organise the synthesis. First, thesign sector maps a monomial action to a monomial target, so its intertwiner can be a pure basis permutation;the two-dimensional-irrep sector maps a monomial action to a non-monomial target, so its intertwiner provablycannot be permutation-diagonal. Second, the twelve-dimensional sector factors as ρ0 ⊗ D6 with D6 = D3 ⊗ I2,isolating a single common two-dimensional representation factor. We stress the distinction betweenrepresentation structure and circuit cost: the factorisation is a statement about the assembly, not yet a verifiedentangling-gate count. A self-audit table records which invariant refuted each earlier, incorrect decomposition. ※ The license for codes is AGPL3.0-or -later