A registered result in its source context

GNET.PPINDEX.01

Original ledger classification: Exact [E] index results (PP = 1/4 optimal, Watatani = 4 with forced quasi-basis weights; mu4 derived, not inserted) + [E] state-preserving Ramond healing + [M] outerness witnesses (Python-only per GATE.WOLFRAM.02).

Original claim text

PHYS T2 main slice -- Pimsner-Popa / Watatani index 4 on the CAR ladder, with the mu4 action DERIVED from the clock and the Ramond zero-mode test (one module, 57 checks, ~7 s; discovery probe phys_car_pp_index_probe.py 57/57, verdict T2-SLICE-GO (PP-INDEX-4-ON-LADDER)). THE DERIVATION (circularity rule): mu4 is NOT inserted -- the clock L = S^{N/12} of the v679/v716 frame gives the half-turn H = L^6 whose order 4 is FORCED by the NS spin structure (H^2 = S^N = -1, H^4 = 1); the deck Dk = L^4 commutes and is unused in E. THE INDEX: on windows W_k of the N = 48/96/192/384 ladder the mu4-average conditional expectation E = (1/4) sum_j U^j . U^-j (U = Gamma(H_W), vacuum-fixing phase, no convention freedom) has optimal Pimsner-Popa constant EXACTLY 1/4 (explicit rank-one minimizer + pinching theorem; HONEST SPEC CORRECTION documented: charge-homogeneous elements give lambda = 1 identically -- the bound lives on maximally sector-mixing elements) and Watatani index EXACTLY 4 via an explicit quasi-basis (sum v v* == 4 * 1; weights FORCED by the reconstruction identity -- uniform weights BREAK scalarness (kill K4), the Z2-only average gives lambda = 1/2, index 2: mu4 distinguished from mu2). THE RAMOND HEALING: undressed, H_R = S^{N/2} has order 2 and the census DEGENERATES (m = 2, lambda = 1/2, index 2 -- measured); the v679 Klein zero-mode dressing U' = Gamma(H_R) e^{i pi/2 n_0} restores order 4 and index 4 STATE-PRESERVINGLY ([rho, U'] = 0 exactly; the zero mode maximally mixed, c_0 = 1/2) -- kill K2 does not fire. THE (1+i) DICTIONARY (with v722): NS momenta half-integer <-> every window mode carries H-eigenvalue +-i (chi = 1); R momenta integer <-> chi = 0 with the dressed zero mode the unique chi = 1 carrier; the sector-dim anomaly N_0 - N_2 = 2^k sits ONLY in ker(chi) -- the ramified defect lives in ker(chi) on BOTH sides. Outerness witnesses w_q -> 1/4 and |omega(U^j)| -> 0 typed as witnesses, not proof (kill K3 stays a witness question). GNET.RAMIFIED.01 contract candidate printed with kills. No marker move (GATE.METRIC.08/10 stay as typed). [2026-08-03 LIII, Lean round 5: the exact algebraic index-4 core is now FORMALLY VERIFIED in Lean 4 (WatataniIndexFour.lean, lake build green, kernel-checked, no sorry): the clock-order shadow (S^12 = -1 => H^4 = 1, mu4 order forced by the NS spin structure), the explicit 13-element quasi-basis with sum v v* = 4*1 EXACTLY = N(1+i)^2, the sharp Pimsner-Popa constant lambda* = 1/4 via an SOS certificate at the mixing minimizer, and the mu2/three-sector controls (index 2/3). Honest scope: the smallest faithful matrix model M4(Z[i]) -- the CAR/Fock layer, Gaussian states, the k-window sector dimensions, the Ramond dressing and the ladder witnesses stay numeric (v726).] [2026-08-05 LXII (v779): the l = 1 three-sector index-3 anomaly is now EXPLAINED by the exact dim law d_c(l) = 4^(l-1) + 2^(l-1) cos(pi c/2) (d_2(1) = 0); the C[Z4] 4-element quasi-basis is obstructed at every finite l with relative gap exactly 2^-l (asymptotically restored); the index-4 structure transports literally along the Haar tower with uniform sector-resolved split bounds -- the measured finite precursor of index continuity in the GNS limit.]

Original classification detail

T2-SLICE-GO (PP-INDEX-4-ON-LADDER): optimal Pimsner-Popa constant exactly 1/4 and Watatani index exactly 4 on the CAR ladder with mu4 DERIVED from the clock; Ramond healed state-preservingly by the Klein dressing; (1+i) dictionary closes with v722; Q-system identification = the open contract half; no status move.

Dependencies as recorded

GATE.METRIC.08; GATE.METRIC.10; GNET.RAMIFIED.01; v679