FQ8 × WCM Quantum Reader

The parent paper [1] reduced the circuit synthesis of the optimal 24-outcome reader for the binarytetrahedral group 2T to a single fixed operator, the 2T group Fourier transform, and identified its dominantcost as FQ8, the 8×8 group Fourier transform of the quaternion group Q8. The companion addendum [2]reported, as an honest negative, that on the parent paper's generic Givens-based emitter astructurally-built FQ8 still costs 218-234 CNOT -- far above the O(101) structural estimate -- and framed thegap as a binary-embedding and generic-decomposition tax: an emitter artifact rather than arepresentation-theoretic obstruction. This note discharges that claim with verified circuits. Using only thestructure of FQ8 -- the plus/minus pairing induced by the center Z2 of Q8, which lets a single Hadamardseparate the character sector from the two-dimensional-representation sector -- we build an exactfactorization in which the sole entangling object is a single controlled two-qubit gate Cq2[Wr]. We thenshow that Wr has Cartan (KAK) coordinates (π/4, 0, 0), the CNOT class, so its two-qubit nonlocal core isone CNOT. Three constructions of FQ8 are emitted and verified equal to FQ8 up to global phase atmachine precision, giving a monotone progression of 234 → 33 → 23 emitted CNOT. Every number belowis a real emitted, verified circuit, not a cost-model estimate. We are explicit about what is not achieved: thesingle-digit floor implied by "one CNOT core" is not reached, because lifting Wr to its controlled form costsone Toffoli plus controlled-local flanks (14 CNOT as emitted); reducing this further is a compiler task, not anew representation-theoretic result. The result establishes that the 200-CNOT wall was largely an emitterartifact. ※ The license for codes is AGPL3.0-or-later