The Remaining Construction, Constructed: A Closed-Form Twiddle for the 24-Cell Quantum Reader and a Semidirect-Product Fourier Obstruction

The parent paper [1] reduced the circuit synthesis of the optimal 24-outcome reader for the binary tetrahedralgroup 2T to the synthesis of a single fixed operator: the 2T group Fourier transform F, whichblock-diagonalizes the regular representation into irreducible blocks [1,1,1, 2,2,2, 2,2,2, 3,3,3]. That paperreported, as an honest negative (its Result D), that the naive tensor product FC3 ⊗ FQ8 fails to realize F,because 2T = Q8 : C3 is a crossed (semidirect) product, and it named the missing piece — the twiddle fromthe Clifford theory of Q8 : C3 — as a remaining construction. Here we construct it. We show (i) the twiddle hasan exact closed form: in coset-times-fiber coordinates, each two-dimensional-sector column of F equals (ωmM0)t applied to the Q8-Fourier column, where M0 is the intertwiner of the two-dimensional irrep ρ with itsC3-conjugate (rebuild residual 1.7 × 10-16); (ii) a semidirect-product Fourier obstruction: nodirect-product-form two-factor split of F exists, because the C3-action moves the Fourier dual of Q8 nontrivially(M0 ≠ I) — stated and verified as a direct-versus-semidirect contrast, not as an observed residual; and (iii) adirect measurement of the binary-embedding tax: the abstract group-FFT gate count is O(101), but ageneric-Givens emitter on the 24-into-32 five-qubit embedding cannot realize it, so the O(102) target isreachable in principle yet unreached with a structure-blind emitter. The line's firmly established compressionremains the parent paper's Lever-1 (25,210 → 5,980 CNOT, 4.2×). No quantum-advantage claim is madeanywhere. All results are exact statevector-level numpy computations at machine precision. ※The license for codes is AGPL3.0-or-later