A registered result in its source context

SEAM.EQUIV.CROSSEDPRODUCT.01

Original ledger classification: Net-level crossed-product certification + R1 invariant reduction [E]/[C]/[O]: h_s = 16/16 = 1 in Z => the LR locality criterion holds, the Z2 crossed product SO(16)_1 -> (E8)_1 is a LOCAL index-2 extension by 1995-2001 PEER-REVIEWED subfactor theory (LR/Boeckenhauer/BE/KLM), AGT/AMT demoted to second witness; mu4 glue h(J^k)={1,1,1} parallel; R1 reduced to computed invariants (FHS |C|=1, nu=16, c_-=8, the Kitaev-E8 16-fold-way class); SEAM.EQUIV.01 stays [O]

Original claim text

The NET-LEVEL CROSSED-PRODUCT certification of the 128-spinor extension + the R1 -> R1' invariant-level reduction (2026-07-02, closure-route round) -- re-founds BOTH halves of the SEAM.EQUIV.01 residual on harder-to-reject ground; does NOT close it. [E] LR LOCALITY INTEGER (index-2): the SO(16)_1 spinor has h_s = N/16 = 16/16 = 1 in Z => statistics phase e^{2 pi i h_s} = +1 -- EXACTLY the Longo-Rehren locality criterion, so the Z2 simple-current crossed product SO(16)_1 x| {1,s} is a LOCAL index-2 extension (Longo-Rehren RMP 7 (1995); the level-1 so(N) net WITH its DHR sectors from even-CAR: Boeckenhauer RMP 8 (1996); simple-current extensions of nets: Boeckenhauer-Evans CMP 197 (1998)); KLM gives mu(B) = mu(SO16)/[B:A]^2 = 4/4 = 1 => B HOLOMORPHIC => B = (E8)_1 by the v463 classification pin (E4/eta^8 q^1 = 248 re-verified). THE 2024/2025-PREPRINT LEG (AGT arXiv:2506.01008 + AMT arXiv:2407.18222, v459) IS DEMOTED TO AN INDEPENDENT SECOND WITNESS -- the extension certification now rests on 1995-2001 peer-reviewed journals. [E] INDEX-4 PARALLEL: the mu4 glue J = (s, Lambda1) of (D5)_1 x (A3)_1 has h(J^k) = {1,1,1} for k=1,2,3 -- ALL integer (5/8+3/8=1; 1/2+1/2=1), the v125 isotropy q(k(1,1))=k^2 restated at net level; mu = 16/4^2 = 1, the SAME holomorphic endpoint. [E] R1' INVARIANTS COMPUTED: per-copy FHS Chern C(M=1)=1 vs C(M=3)=0 (control), nu = 16 x |C| = 16, c_- = nu/2 = 8 = g_car + N_fam = rank E8; nu mod 16 = 0 is EXACTLY the class-D phase whose edge admits a purely bosonic description = the (E8)_1 level-1 state (Kitaev Ann. Phys. 321 (2006), sixteen-fold way) -- the identification endpoint pinned from published condensed-matter theory. [E] THE CHAIN COMPOSES (v297 typing): T1 invariants -> nu=16 phase (Kitaev 2006/2009; c_- a PHASE invariant: Kapustin-Spodyneiko PRB 101 (2020), Kim et al. PRL 129 (2022)) -> T2 SO(16)_1 net (MMST CMP 397 (2023) + Boeckenhauer 1996) -> T3 local holomorphic extension (LR/BE + KLM CMP 219 (2001)) -> T4 (E8)_1 (Dong-Mason + v463). [C] R1 -> R1': the Lean axiom collar_realizes (model-level fiat 'the seam IS this lattice model') is replaceable by collar_invariants (invariant-level: quasi-free [C v155/v160] + gap [E v302] + class D + c_- = 8 [E v456, from P1]) -- any realisation with these invariants is phase-equivalent to the v367 stack, hence has the SAME edge scaling limit; quasi-freeness stays the ONE [C] hypothesis. [O] NOT CLOSED: the cited theorems stay cited (certification, not re-proof); the abstract continuum existence remains; SEAM.EQUIV.01 stays [O].

Original classification detail

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Dependencies as recorded

SEAM.EQUIV.01; SEAM.EQUIV.MMST.AUDIT.01; SEAM.EQUIV.LATTICEVOA.01; SEAM.EQUIV.UNIQ.01; SEAM.EQUIV.ONEPARTICLE.01; v125; v154; v297; v336; v367; v456; v458; v459; v463; v464