The finite two qubit stabilizer identification

E8.CLIFFORD2Q.01

Original ledger classification: Exact [E] finite group/lattice identification (H1-H3) with the H4 layer honestly PARTIAL (the index bit identified as the sqrt2/Hadamard Galois bit, not J <-> -J); no marker move (Python-only per GATE.WOLFRAM.02).

Original claim text

The two-qubit stabilizer identification of the compiler carrier (32/32 checks, ~4 s; discovery probe two_qubit_clifford_probe.py CLIFFORD-PARTIAL(H4), 2026-08-05). H1 EXACT: under the single predeclared chart U (the coordinate-class frame v_k = 2 e_{2k} -> the computational basis; z_k = r_2k + i r_2k+1, J -> multiplication by i) the 60 Gaussian E8 lines ARE the 60 pure two-qubit stabilizer rays, entry-exactly over Z[i] -- 60/60 projective matches with support census 4 + 24 + 32 on both sides; no fallback frame needed. H2 EXACT: U G31 U^-1 is an INDEX-2 subgroup of the two-qubit Clifford matrix group C2 = <H(x)I, I(x)H, S(x)I, I(x)S, CNOT> -- |C2| = 92160 = 2 x 46080 with coset zeta8 G31; projectively |C2/U(1)| = 11520 = |G31|/|mu4| (equal). H3 EXACT (the semantics carry): 46080 = 4 x 16 x 720 = mu4 x Pauli16 x Sp(4,2), with ker(G31 -> Sp(4,2)) = mu4 * Pauli16 and the 16 Paulis mod phase acting trivially on the 16 code states of V = L/(1+i)L; the 15 Gaussian classes ARE the 15 maximal commuting Pauli contexts (the W(3,2) = GQ(2,2) polar-space identification: hbar(x,y) = 0 iff the contexts share a Pauli, all 105 pairs; v752's B transported); the 6 spreads are the MUB pentads (cross-context |<psi|phi>|^2 = 1/4 EXACT) pulling back to ovoids (the self-dual GQ(2,2) point-line duality). H4 FAILS AS PREDECLARED (the honest partial): the index-2 character chi conjugates NEITHER J nor the mu4 phases (zeta8 is central) -- chi is the GALOIS sqrt2/HADAMARD bit Gal(Q(zeta8)/Q(i)), and it exchanges the two mu8-lift classes {mu4 R, zeta8 mu4 R} of the 240 roots; the J <-> -J orientation bit is the separate ANTIUNITARY outer Z2 (which fixes chi). All four controls fire (Haar-random 0/60; the T-gate residual-torus twist gives EXACTLY the predeclared 16/60; the non-context controlled-Hadamard frame 12/60 < 60; seeded random rays 0/60) -- the frame alone does not force the identification, the lattice does. Stabilizer-formalism mathematics about the carrier: the fired ROOTCLASS-MIXED kill (v775 / ARF.ROOTCLASS.01) is UNAFFECTED, no matter reading is made. No marker move, NO RH claim. NOTE (2026-08-06, v798 -- marker UNCHANGED): the missing coset zeta8 G31 is IDENTIFIED -- the per-binary-factor Hadamard is the metaplectic/Weil lift of modular S at tau = i (exact fixed-point relations; Weil constant zeta8), the total Fourier H (x) H lies inside G31 with anomaly (T3 K3)^3 = iI = J, so this row's sqrt2/Hadamard Galois bit is the tau = i metaplectic bit; the strict q*-fixing census over the coset is empty (the Arf defect unavoidable) with exactly one G31-conjugacy class of Theta-real metaplectic survivors. See SEAM.CLIFFORD.MODULAR_S.01.

Original classification detail

CLIFFORD-PARTIAL(H4): the 60 Gaussian lines = the 60 two-qubit stabilizer rays entry-exactly over Z[i] (60/60, census 4+24+32); G31 = index-2 subgroup of the Clifford matrix group (coset zeta8 G31, projectively equal); 46080 = mu4 x Pauli16 x Sp(4,2) with classes = contexts and spreads = MUB pentads; the index bit = the sqrt2/Hadamard Galois bit (H4 as predeclared); controls fire; no matter reading; no RH statement.

Dependencies as recorded

v689; v752; v634; E8.GAUSSIAN.CODE.01; ARF.ROOTCLASS.01