A registered result in its source context

GNET.GNSLIMIT.01

Original ledger classification: Exact [E] finite coherence data + exact [E] dim law + measured [MEASURED] precursors and the honest negative, all typed finite evidence for the named continuum theorems -- never the theorems (Python-only per GATE.WOLFRAM.02).

Original claim text

The G_net continuum strand -- finite-level precursors of the GNS limit net COHERENT, the exact C[Z4] obstruction dim law, and the honest Arf negative; one module from two probes (27 + 27 checks, ~50 s; discovery probes gnet_gns_limit_probe.py 27/27 GNS-PRECURSORS-COHERENT and gnet_arf_qsystem_probe.py 27/27 GNET-ARF-NO-CORRESPONDENCE, both 2026-08-05). THE PRECURSORS (typed finite evidence FOR the five v746 continuum theorems, never a continuum proof): Haar scaling isometry tower intertwines EXACTLY (V isometric, V H_N = H_2N V, V L_N = L_2N V <= 2.2e-16; window data literally level-independent; Fock coherence <= 1.5e-14); the index-4 expectation squares COMMUTE exactly (restriction coherence, [E_I, E_J] = 0, joint idempotency -- the finite half of the Longo Q-system identification); the canonical chiral-sea state is Cauchy with geometric contraction ~1/4 per doubling (d5 = 2.4e-4; parity-odd exactly 0); the index census is machine-exact at EVERY level with zero degradation (lambda* = 1/4, index 4*1, l = 1 anomaly stable at 3, PP margins clean); the split witness stabilizes and decreases with separation; the finite Haag-duality census gives twisted-duality DEFECT = 0 (dim A' = 4^{n_c} exhausted by the twisted complement) and dim B'/dim A' = 4 = the local Watatani index (l = 1: 3); a 10-section continuum-paper skeleton is the work product. THE EXACT DIM LAW: d_c(l) = 4^(l-1) + 2^(l-1) cos(pi c/2) for l = 1..6 -- the C[Z4]/Longo 4-element quasi-basis is OBSTRUCTED at every finite l with relative gap exactly 2^-l -> 0 (the crossed product is exactly asymptotic; the finite precursor of the GATE.METRIC.08/11 identification), and d_2(1) = 0 IS the l = 1 three-sector index-3 anomaly of v746; the explicit 185-element Watatani quasi-basis at l = 2 has sum v v* = 4*1 exact; sector-resolved split bounds are uniform along the tower; the sigma register exists internally on the Z6-quotient window W6 (deck + half-turn joint, dims (20,16,12,16), lambda* = 1/4) -- but dim Fock(W6) = 64 != 16. THE HONEST NEGATIVE, EXHAUSTIVE: the 16 = 1 + 5 + 10 counting echo is REAL and exact but NOT geometric -- F2-rank certificate 3 vs 4 (linear bijection impossible), GL(4,2) census 0/20160 for EVERY Arf-1 form's 5-set, no nontrivial internal order-3 symmetry (6144-element census); K_matter does NOT govern the tower weight transport (x5 = (0.414, 0.586) vs pi = (1/3, 2/3); the degenerate-decimation caution preregistered and confirmed by control C4). BINDING HONESTY RULE: index 4 = |mu4| and the channel eigenvalue 4/49 = (-2/7)^2 are different numbers from different registers -- nothing equates them. All controls fire (bond scramble, Z2-only, scrambled sea, decimation, wrong Arf form). Spec corrections carried verbatim (G4b spanning basis; S4 bar anchor). No marker move, NO RH claim.

Original classification detail

GNS-PRECURSORS-COHERENT + GNET-ARF-NO-CORRESPONDENCE: exact Haar tower intertwining, commuting index-4 expectation squares, geometric state Cauchy ~1/4, machine-exact index census with zero degradation, twisted-duality defect 0 with ratio 4; dim law d_c = 4^(l-1) + 2^(l-1) cos(pi c/2) => C[Z4] obstruction exactly 2^-l (explains the l = 1 anomaly); the 1+5+10 echo exact but non-geometric (rank 3 vs 4, GL census 0/20160); K_matter does not govern transport; the two fours stay in their registers; no RH statement.

Dependencies as recorded

GNET.LOCALNET.01; GNET.PPINDEX.01; GATE.METRIC.08; GATE.METRIC.11; ARF.SPINORCOMPILER.01; v746; v726; v754