A registered result in its source context

SEAM.CLIFFORD.MODULAR_S.01

Original ledger classification: 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).

Original claim text

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.

Original classification detail

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 as recorded

E8.CLIFFORD2Q.01; ARF.BOUNDARY.CODE.01; ARF.SPINORCOMPILER.01; v783; v774; v752; v689; ARF.ROOTCLASS.01