Original ledger status
E8.CLIFFORD2Q.01
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.
Recorded status: 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).
Dependencies, status detail and source
Canonical status: 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: v689; v752; v634; E8.GAUSSIAN.CODE.01; ARF.ROOTCLASS.01
Scripts: v783_two_qubit_clifford.py
Location: tfpt_1;tfpt_2;tfpt_3
Original CSV, line 686Original ledger status
SEAM.CLIFFORD.MODULAR_S.01
the missing Clifford index-2 bit IS the metaplectic lift of the modular S transformation at its fixed point tau = i -- coset decision positive, deck placed, anchor named; the frozen uniqueness decider fails sharply (27/27 checks, ~50 s; discovery probe seam_clifford_modular_s_probe.py, 2026-08-06, verdict MODULAR-S-PARTIAL). COSET DECISION POSITIVE (exact cyclotomic arithmetic): the per-binary-factor Weil/metaplectic lift K1 = H (x) I (and K2 = I (x) H) lands EXACTLY in the missing coset zeta8 G31 of v783 (BFS membership certificates for zeta8^{-1} K_i in G31), with the exact tau = i fixed-point relations S^2 = +I, T^2 = the Pauli parity element, and (T K)^3 = zeta8 I -- the Weil constant gamma = zeta8, c = 1 per binary factor. R2 SOLVED (deck placement): the TOTAL symplectic Fourier K3 = H (x) H (the toric-code modular S) lies INSIDE G31 (chi = 0) and its metaplectic anomaly is (T3 K3)^3 = i I = J -- THE DECK IS THE METAPLECTIC ANOMALY OF THE TOTAL MODULAR LIFT (c = 2); zeta8 = (1 + J)/sqrt2 is the real orthogonal HALF-DECK ((1+J)^2 = 2J exact) swapping the seam pair {L, zeta8 L}: lift = half-deck turn x lattice automorphism. The Z4 deck-clock Fourier F4 is NOT a carrier symmetry (fails the Clifford gate; typed out). OS reflection gate exact: Theta K1 Theta = K1 and Theta T1 Theta = T1^{-1} -- the antiunitary reality involution acts as the GL(2,Z) orientation flip tau -> -tau-bar, which FIXES tau = i. THE FROZEN UNIQUENESS DECIDER FAILS SHARPLY (the PARTIAL): the strict census over all 46080 coset elements (order gate 480 -> Theta-reality 96 -> q*-fixed + metaplectic pairing) is EMPTY (0/46080) -- NO Theta-real metaplectic involution in the missing coset fixes the v774 Arf refinement q*: the Arf defect q* -> q* + hbar(., c) is UNAVOIDABLE; the retyped census (defect recorded instead of forbidden) has 32 survivors = 16 (+-1)-pairs = EXACTLY ONE G31-conjugacy class (w1 ~ w2; tier-A 4 = the Pauli/phase gauge fiber of H (x) I). R3 NAMED, NOT FIXED: the sigma-orbit of lifts {w1, sigma w1 sigma^{-1}, w2} has Arf defects EXACTLY {A + F_Sigma + F_i : i = 1, 2, 3} (one per family) whose XOR is THE ANCHOR A itself; the total Fourier's defect is A + F_Sigma; every retyped survivor's defect carries the anchor flag chi_NSR(c) = 1 -- the lift does not fix the anchor, it NAMES it through the Arf defects. CONTROLS: CZ (real, order 2, Clifford, non-metaplectic) lies INSIDE G31 (chi = 0); the wrong phase convention zeta8 K1 breaks S^2 in {+-I} ((zeta8 K1)^2 = iI); census selectivity 480 -> 96 -> 32 of 46080. ROOTCLASS-MIXED (v775) unaffected -- finite stabilizer/Weil mathematics about the carrier, no physics claim. No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02. CONTRACT (registered, partial state): coset decision POSITIVE and R2 solved (the deck = the metaplectic anomaly of the total Fourier); R3 NAMED through the sigma-orbit Arf defects (XOR = the anchor A); remaining: derive the physical selection of the defect class (the anchor naming from seam data) or type the two-variant/defect residue as intrinsic gauge; kill: a Theta-real metaplectic coset element fixing q* (excluded here by the exhaustive 0/46080 census) would falsify the defect law.
Recorded status: Exact [E] cyclotomic coset/metaplectic identities (BFS certificates; exact tau = i relations) with the strict uniqueness census EMPTY by its own frozen gate -- the Arf defect unavoidable, a sharp typed fact; R2 placed, R3 named; no marker move (Python-only per GATE.WOLFRAM.02).
Dependencies, status detail and source
Canonical status: MODULAR-S-PARTIAL: the per-factor Hadamard IS the metaplectic S-lift at tau = i, landing exactly in the missing coset zeta8 G31 (S^2 = I, T^2 = Pauli parity, (TK)^3 = zeta8 I); the deck J = the metaplectic anomaly of the total Fourier (R2 solved); zeta8 = (1+J)/sqrt2 the half-deck swapping the seam pair; strict census EMPTY (0/46080 fix q*), one G31-conjugacy class of retyped survivors, sigma-orbit Arf defects XOR to the anchor A (R3 named, not fixed); no RH statement.
Dependencies: E8.CLIFFORD2Q.01; ARF.BOUNDARY.CODE.01; ARF.SPINORCOMPILER.01; v783; v774; v752; v689; ARF.ROOTCLASS.01
Scripts: v798_seam_clifford_modular_s.py
Location: tfpt_1;tfpt_3;tfpt_research_contracts
Original CSV, line 690