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
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
Prior work by the authors [1] showed that the 24 vertices of the 24-cell, as unit quaternions, form the binarytetrahedral group 2T, that a controlled probe lifts the phase-blind projective group 2T/{±1} to all 24 gates, and thatthe optimal single-shot measurement discriminating them compiles, gate by gate, to an explicit RY/RZ/X/CNOTcircuit. That compilation was symmetry-blind: a textbook Givens synthesis of a generic 32×32 unitary, emitting25,210 CNOTs. Here we ask what the group structure of the input buys back. We report four findings. (A) The 24controlled-probe states span only a rank-3 subspace, so a reader need be correct only there; restricting thesynthesis to that support cuts the emitted CNOT count from 25,210 to 5,980 (4.2×) with the same single-shotarchitecture and the identical optimum. (B) The optimal POVM is exactly 2T-covariant: its 24 elements are onegroup orbit of a single seed, and the analysis map is a genuine tight-frame isometry that intertwines the supportrepresentation with the regular representation of 2T. (C) Consequently the reader factors as a data-independent2T group-Fourier transform followed by a placement whose data-dependent part is tiny (72 CNOT); the rank-3signal occupies just 5 of the 24 Fourier lines, which is the structural reason (A) works. (D, honest negative) Acheap synthesis of that group-Fourier transform is not achieved: the naive tensor product of the C3 and Q8transforms fails to block-diagonalize because 2T = Q8 : C3 is a crossed product, and the missing twiddle absorbsall the cost. We make no quantum-advantage claim; every number is a real emitted gate count or a verified matrixidentity from exact, numpy-only statevector simulation. ※ The license for codes is AGPL3.0-or-later
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
The Watabe-Claude Method reads a signal's structure by assigning 4D-embeddedwindows to the 24 vertices of the 24-cell and decomposing the occupancy with thegroup B4. This note takes the 24-cell the other way -- into a qubit. Its 24vertices, as unit quaternions, are exactly the binary tetrahedral group 2T, afinite nonabelian subgroup of SU(2); under the quaternion-to-SU(2) map they are24 distinct single-qubit gates. Three exact facts, all verified by classical simulation of one or two qubits(no external quantum or group-theory libraries): 1. Foundation. The 24 vertices form 2T (order 24): closed Cayley table, Latin square, 24-cell inner-product signature {-1,-1/2,0,1/2,1}. As SU(2) matrices they are 24 distinct, nonabelian gates containing the Pauli set. 2. An honest negative. Encoding the vertex assignment for a quantum optimiser is a dead end: the 5-qubit binary QUBO has 31 terms (vs 300 one-hot) but up to five-body couplings, and a shallow QAOA recovers the right vertex only 3-10% of the time -- the assignment is a classical argmax with no role for a quantum optimiser. 3. The finding. Applied to a single probe qubit, the 24 gates collapse to 12 distinct outputs, because 2T's center {+-1} is exactly the global phase a lone qubit cannot observe (U(-q)=-U(q) verified for all 24); the observable group is the projective image 2T/{+-1}, the order-12 tetrahedral rotation group. A controlled application makes the center's phase relative and separates all 24. Optimal single-shot (square-root) discrimination succeeds 16.7% over the 12 classes and 12.1% over all 24 -- 2.0x and 2.9x the random baselines, ratios fixed by the 24-cell geometry. The through-line with the classical WCM series: there, B4 reads which symmetrychannel carries a signal's structure; here, the same polytope's group structuregoverns which distinctions a quantum measurement can make. No quantum speedup oradvantage of any kind is claimed; the contribution is an exact correspondencebetween a polytope's group structure and the observability limits of a qubit.Self-audit: 12 findings, all NUMERICAL / NEGATIVE / STRUCTURAL, none overclaimed. v1.0.1 (2026-08-27): Adds an explicit, end-to-end-verified measurement circuit. The optimal square-root discrimination is certified minimum-error (Yuen-Kennedy-Lax conditions plus an independent iteration from random measurements), realised as an explicit Naimark unitary, embedded in a power-of-two qubit register, and compiled to an atomic gate list of RY, RZ, X, and CNOT only -- verified end to end (every gate one of those four types, the product reproducing the target measurement within floating-point accumulation; ~2e-11 over the ~84,000-gate 24-way circuit). The lone-qubit collapse is sharpened to an orbit hierarchy (24 -> 12 generic -> 6 symmetry-axis), naming the measure-zero exceptional probes (including the computational basis). Terminology and scope clarifications throughout (B4 vs 2T; central quotient; coherent-control assumption; honest symmetry-blind CNOT upper bounds 25,210 / 2,022). No quantum speedup is claimed. Status: preprint, not peer-reviewed. This is an independent research note; its revisions were shaped by critique from an AI collaborator, which is not a substitute for formal peer review. Text CC-BY-4.0; code AGPL-3.0-or-later.
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
An important result from the mid nineties shows that any unitary evolution may be realized as a sequence of controlled-not and one-qubit gates. This work surveys especially efficient circuits in this library, in the special case of evolutions on two-quantum bits. In particular, we show that to construct an arbitrary two-qubit state from |00>, one CNOT gate suffices. To simulate an arbitrary two-qubit operator up to relative phases, two CNOTs suffice. To simulate an arbitrary two-qubit operator up to global phase, three CNOTs suffice. In each case, we construct an explicit circuit and prove optimality in the generic case. We also contribute a procedure to determine the minimal number of CNOT gates necessary to simulate a given two-qubit operator up to global phase. We use this procedure to discuss timing a given Hamiltonian to simulate the CNOT and to determine an optimal circuit for the two-qubit Quantum Fourier Transform. Our constructive proofs amount to circuit synthesis algorithms and have been coded in C++.
A three-state object is not yet a qutrit. The claim requires a three-dimensional complex Hilbert sector, coherent state transformations, sufficient local control, a two-qutrit entangler, and an experimental route that distinguishes quantum coherence from classical population switching. The exact covering arithmetic of the pentachoron K5 fixes three as the minimal equal support size for a two-set cover of five vertices and gives the partition (2, 1, 2) on covering triangular faces; identification of such a face with a physical interface remains an explicit model bridge. On that declared support, the fixed-one sector and the antisymmetric fixed-two sector of three fermionic modes in one resolved internal channel are both 3-dimensional qutrit sectors and are related by particle–hole duality. Hard-core exclusion alone is not sufficient for the exterior-square gate lift. After removal of the uniform offset, the most general real, site-permutation-invariant effective Hamiltonian has the form H0 = cL(K3) and spectrum {0, 3c, 3c}; symmetry fixes neither the sign nor the scale of c. On the associated singlet–doublet decomposition, U△ = P0 + ω²PE, ω² + ω + 1 = 0, ω ≠ 1, is an exact balanced tritter: every transition probability equals 1/3. Diagonal Clifford factors make U△ projectively equivalent to the qutrit Fourier gate F3. The ideal logical generators AK3 = X3 + X3† and D01 = diag(1, −1, 0) generate a real control Lie algebra of dimension 8, isomorphic to su(3). A programmable rank-one phase compiles into CZ3 using 4 commuting factors, and CZ3 compiles into SUM3. Under the stated control assumptions this gives an ideal universal qutrit architecture. Exact counterexamples delimit every promotion: three sites do not force an oriented cycle, symmetry fixes neither the sign nor the scale of c, connected hopping plus a non-degenerate diagonal need not generate su(3), localization is not coherence, and a population-sensitive observable is not tomography. Oxo-centred mixed-valence Fe3O complexes therefore enter only as a falsifiable implementation route. Published structural, Mössbauer, IVCT, and polarization data establish relevant populations and observables, not coherent logical gates. No logical gate, effective-Hamiltonian parameter, or laboratory pulse duration is identified in this paper with a K5 tick, local generation, or commit.