Five GAP witnesses turn the QR-137 code from a one-block curiosity into an exact encoded exceptional-symmetry tower:
[ [[137m,m,21]],\qquad \overline C_{\mathbb R}(m)=O^+(2m,2),\qquad [Sp(2m,2):O^+(2m,2)]=2^{m-1}(2^m+1). ]
At four blocks, the logical Pauli-label space is explicitly isometric to the Pass-124 model of (E_8/2E_8), so the direct-sum code realizes the complete graph tower
[ \operatorname{SRG}(255,126,61,63) \supset \operatorname{SRG}(135,70,37,35) \sqcup \operatorname{SRG}(120,63,30,36). ]
At three blocks, changing the quadratic refinement gives (O^-(6,2)\cong W(E_6)) on 36 nonsingular vectors. GAP constructs an explicit (PSp(4,3))-equivariant bijection from those vectors to the 36 W33 spreads and supplies a named stabilizer-normalizing encoded phase lift.
The packet also closes two tempting overreads. The three order-1,152 character kernels inside the real two-qubit Clifford group are pairwise nonisomorphic, so the formal (S_3=\operatorname{Aut}(C_2^2)) does not lift. And four recurring exchange parities form one synchronized (C_2)-graded fiber product, but the QR affine extension is nonsplit, so no common involutory (C_2) action exists.
Let (C_{\mathbb R}(2)) be the order-2,304 real two-qubit Clifford matrix group generated by (H_1,H_2,X_i,Z_i), and CNOT. Its abelianization is (C_2^2), hence it has exactly three nonzero characters. GAP computes their kernels:
| character | kernel | classes | center | derived order |
|---|---|---|---|---|
| total Hadamard parity | (W(F_4)) | 25 | 2 | 288 |
| determinant/CNOT parity | ((2O\times2O)/\langle(-1,-1)\rangle) | 34 | 2 | 288 |
| mixed parity | (2_+^{1+4}:((C_3^2):C_4)) | 19 | 2 | 288 |
Every pair intersects in (C_{\mathbb R}(2)'), every pair generates the ambient group, and the derived tower has orders
[ 2304,576,288,32,2,1 ]
with successive quotients (C_2^2,C_2,C_3^2,C_2^4,C_2). The different class counts and element-order profiles prove that the kernels are pairwise nonisomorphic and therefore characteristic. There is no lifted kernel triality.
Projectively, the three kernels become the three index-two subgroups (T_0,S_4\times S_4,T_1) of (\operatorname{Aut}(K_{4,4})=(S_4\times S_4):C_2). Radially normalizing the 48 roots of (F_4) gives one shell with support distribution (8+24+16). The determinant kernel is transitive on this shell, while (W(F_4)) retains its (24+24) short/long split and alone preserves the two-length metric root system.
This calculation also corrects a corpus error:
[ GL(2,3)=\operatorname{SmallGroup}(48,29)\not\cong 2O=\operatorname{SmallGroup}(48,28). ]
The former has 13 involutions and the latter one; both have central quotient (S_4), which is why order/quotient shorthand obscured the distinction.
Four independent QR blocks give a direct-sum code
[ [[548,4,21]],\qquad \operatorname{rank}S=544. ]
GAP acts on the actual 1,096-coordinate binary Pauli-label stabilizer. All four nonresidue-permuted encoded Hadamards and all twelve directed transversal CNOTs preserve its row space. On ((x_1,x_2,x_3,x_4,z_1,z_2,z_3,z_4)), they generate the full group
[ O^+(8,2),\qquad |O^+(8,2)|=348{,}364{,}800,\qquad [Sp(8,2):O^+(8,2)]=136. ]
The coordinate map
[ (x_1,x_2,x_3,x_4,z_1,z_2,z_3,z_4) \longmapsto (x_1,z_1,x_2,z_2,x_3,z_3,x_4,z_4) ]
preserves both the quadratic and polar forms on every vector and every ordered vector pair. Direct common-neighbor enumeration then recovers the Pass-124 graph and both induced quadratic fibers, including their exact spectra. The abstract graph tower already belonged to Pass 124; the new result is its physical four-block QR realization and this chosen isometry.
Use interleaved coordinates ((x_1,z_1,x_2,z_2,x_3,z_3)) and
[ q_+(v)=x_1z_1+x_2z_2+x_3z_3,\qquad a=(0,0,0,0,1,1). ]
Because (q_+(a)=1),
[ q_-(v)=q_+(v)+B(a,v) =x_1z_1+x_2z_2+x_3z_3+x_3+z_3 ]
has minus type. Its 36 nonsingular vectors form (NO^-(6,2)=\operatorname{SRG}(36,15,6,6)), and their transvections generate
[ O^-(6,2)\cong W(E_6),\qquad |O^-(6,2)|=51{,}840, ]
with derived subgroup (PSp(4,3)) of order 25,920. GAP conjugates this derived degree-36 action to the repo's native action on the 36 W33 spreads. The resulting spread-index to nonsingular-vector-index bijection is
[1,12,2,14,5,4,29,31,34,20,25,8,18,33,32,9,24,27,
36,10,22,16,3,13,21,30,19,11,23,7,17,26,35,28,6,15]
and GAP checks all 630 pairs:
[ |S_i\cap S_j|=4 \quad\Longleftrightarrow\quad B(v_{\phi(i)},v_{\phi(j)})=0. ]
Relative to (q_+), the 36 directions split as (16+20): 16 already have (q_+=1), while 20 require the added phase refinement. On the physical ([[411,3,21]]) code, the named lift
[ \widehat R_{z_3}=\exp(-\pi i\bar Z_3/4), \qquad \bar Z_3=Z^{\otimes137}, ]
fixes every stabilizer row and sends (\bar X_3\mapsto\bar X_3\bar Z_3). Together with the encoded Hadamards and transversal CNOTs, the three such rotations generate the full (Sp(6,2)), so the displayed (O^-(6,2)) has an encoded Clifford lift. This exact lift is a weight-137 Pauli rotation; no locality, transversality, or fault-tolerance claim is made for it.
For every positive integer (m), direct sum gives ([[137m,m,21]]): the stabilizer rank is (136m), and the distance is the minimum distance 21 of one component. The block Hadamards and directed CNOTs preserve
[ q_m(x,z)=\sum_{i=1}^m x_i z_i ]
and generate the standard real-Clifford quotient (O^+(2m,2)). GAP checks the physical ranks and complete logical groups through the exceptional range:
| (m) | code | (|O^+(2m,2)|) | nonzero orbits | index in (Sp) | |---:|---|---:|---:|---:| | 1 | ([[137,1,21]]) | 2 | (2+1) | 3 | | 2 | ([[274,2,21]]) | 72 | (9+6) | 10 | | 3 | ([[411,3,21]]) | 40,320 | (35+28) | 36 | | 4 | ([[548,4,21]]) | 348,364,800 | (135+120) | 136 |
For (q_a(v)=q_m(v)+B(a,v)), exhaustive GAP counts are
[ (3,1),(10,6),(36,28),(136,120) ]
for (plus, minus) refinements at (m=1,2,3,4). In general,
[ q_a\text{ is plus}\Longleftrightarrow q_m(a)=0, ]
and the counts are (2^{2m-1}\pm2^{m-1}). GAP constructs the (Sp(8,2)) action on all 256 refinements, proves transitivity on the 136 plus and 120 minus fibers, and finds an order-348,364,800 stabilizer of a plus refinement. Thus
[ [Sp(2m,2):O^+(2m,2)] =2^{m-1}(2^m+1) =#{\text{plus refinements}}. ]
The exceptional boundary is exact: (O^+(6,2)\cong S_8), not (W(E_6)); the minus refinement in Pass 365 gives (W(E_6)). At four blocks, (O^+(8,2)\cong W(E_8)/{\pm1}).
Four exact index-two extensions recur in the current atlas:
| parent | even kernel | split? |
|---|---|---|
| (W(E_6)), order 51,840 | (PSp(4,3)), order 25,920 | yes |
| (2.U_4(2).2), order 103,680 | (2.U_4(2)), order 51,840 | yes |
| (C_{137}:C_{136}), order 18,632 | (C_{137}:C_{68}), order 9,316 | no |
| (C_{\mathbb R}(2)), order 2,304 | (W(F_4)), order 1,152 | yes |
Synchronizing their four quotient maps produces the fiber product
[ F=G_1\times_{C_2}G_2\times_{C_2}G_3\times_{C_2}G_4 ]
and exact sequence
[ 1\to PSp(4,3)\times2.U_4(2)\times(C_{137}:C_{68})\times W(F_4) \to F\to C_2\to1. ]
Its even kernel and total orders are
[ 14{,}420{,}554{,}127{,}769{,}600,\qquad 28{,}841{,}108{,}255{,}539{,}200. ]
But the odd coset of (C_{137}:C_{68}) inside (C_{137}:C_{136}) contains elements only of orders 8 and 136. Therefore an odd tuple in (F) can never be an involution. GAP constructs an odd tuple of order 8, proving that 8 is sharp. The universal object is a common grading, not a semidirect product by one exchange gate.
The real-Clifford quotient, its orthogonal structure, and Clifford generation are standard; see Real Randomized Benchmarking and Circuit Relations for Real Stabilizers. The two-qubit ambient subgroup landscape is independently catalogued in Classification of the Subgroups of the Two-Qubit Clifford Group. The length-137 QR weight distribution comes from Tjhai–Tomlinson–Ambroze–Ahmed.
The repo contribution is the exact QR-137 physical realization, the explicit maps to its existing W33/E8 objects, the normalized-shell and character-kernel separations, and the synchronized-extension no-go. None of these finite theorems derives a measured mass, coupling, chirality choice, or continuum dynamics.
gap -q analysis/w33_pass363_real_clifford_character_diamond.g
gap -q analysis/w33_pass364_qr548_e8_phase_space.g
gap -q analysis/w33_pass365_qr411_e6_minus_polar_lift.g
gap -q analysis/w33_pass366_qr137m_real_clifford_refinement_tower.g
gap -q analysis/w33_pass367_universal_c2_exchange_gate_pullback.g
python3 -m pytest tests/test_pass363_367_gap_qr_clifford_refinement.py -qThe generated certificates live under data/ with matching basenames.